text: # Copyright (c) 1999-2007 Gary Strangman; All Rights Reserved.
  #
  # Permission is hereby granted, free of charge, to any person obtaining a copy
  # of this software and associated documentation files (the "Software"), to deal
  # in the Software without restriction, including without limitation the rights
  # to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
  # copies of the Software, and to permit persons to whom the Software is
  # furnished to do so, subject to the following conditions:
  # 
  # The above copyright notice and this permission notice shall be included in
  # all copies or substantial portions of the Software.
  # 
  # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
  # IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
  # FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
  # AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
  # LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
  # OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
  # THE SOFTWARE.
  #
  # Comments and/or additions are welcome (send e-mail to:
  # strang@nmr.mgh.harvard.edu).
  # 
  """
  stats.py module
  
  (Requires pstat.py module.)
  
  #################################################
  #######  Written by:  Gary Strangman  ###########
  #######  Last modified:  Dec 18, 2007 ###########
  #################################################
  
  A collection of basic statistical functions for python.  The function
  names appear below.
  
  IMPORTANT:  There are really *3* sets of functions.  The first set has an 'l'
  prefix, which can be used with list or tuple arguments.  The second set has
  an 'a' prefix, which can accept NumPy array arguments.  These latter
  functions are defined only when NumPy is available on the system.  The third
  type has NO prefix (i.e., has the name that appears below).  Functions of
  this set are members of a "Dispatch" class, c/o David Ascher.  This class
  allows different functions to be called depending on the type of the passed
  arguments.  Thus, stats.mean is a member of the Dispatch class and
  stats.mean(range(20)) will call stats.lmean(range(20)) while
  stats.mean(Numeric.arange(20)) will call stats.amean(Numeric.arange(20)).
  This is a handy way to keep consistent function names when different
  argument types require different functions to be called.  Having
  implementated the Dispatch class, however, means that to get info on
  a given function, you must use the REAL function name ... that is
  "print(stats.lmean.__doc__)" or "print(stats.amean.__doc__)" work fine,
  while "print(stats.mean.__doc__)" will print the doc for the Dispatch
  class.  NUMPY FUNCTIONS ('a' prefix) generally have more argument options
  but should otherwise be consistent with the corresponding list functions.
  
  Disclaimers:  The function list is obviously incomplete and, worse, the
  functions are not optimized.  All functions have been tested (some more
  so than others), but they are far from bulletproof.  Thus, as with any
  free software, no warranty or guarantee is expressed or implied. :-)  A
  few extra functions that don't appear in the list below can be found by
  interested treasure-hunters.  These functions don't necessarily have
  both list and array versions but were deemed useful
  
  CENTRAL TENDENCY:  geometricmean
                     harmonicmean
                     mean
                     median
                     medianscore
                     mode
  
  MOMENTS:  moment
            variation
            skew
            kurtosis
            skewtest   (for Numpy arrays only)
            kurtosistest (for Numpy arrays only)
            normaltest (for Numpy arrays only)
  
  ALTERED VERSIONS:  tmean  (for Numpy arrays only)
                     tvar   (for Numpy arrays only)
                     tmin   (for Numpy arrays only)
                     tmax   (for Numpy arrays only)
                     tstdev (for Numpy arrays only)
                     tsem   (for Numpy arrays only)
                     describe
  
  FREQUENCY STATS:  itemfreq
                    scoreatpercentile
                    percentileofscore
                    histogram
                    cumfreq
                    relfreq
  
  VARIABILITY:  obrientransform
                samplevar
                samplestdev
                signaltonoise (for Numpy arrays only)
                var
                stdev
                sterr
                sem
                z
                zs
                zmap (for Numpy arrays only)
  
  TRIMMING FCNS:  threshold (for Numpy arrays only)
                  trimboth
                  trim1
                  round (round all vals to 'n' decimals; Numpy only)
  
  CORRELATION FCNS:  covariance  (for Numpy arrays only)
                     correlation (for Numpy arrays only)
                     paired
                     pearsonr
                     spearmanr
                     pointbiserialr
                     kendalltau
                     linregress
  
  INFERENTIAL STATS:  ttest_1samp
                      ttest_ind
                      ttest_rel
                      chisquare
                      ks_2samp
                      mannwhitneyu
                      ranksums
                      wilcoxont
                      kruskalwallish
                      friedmanchisquare
  
  PROBABILITY CALCS:  chisqprob
                      erfcc
                      zprob
                      ksprob
                      fprob
                      betacf
                      gammln 
                      betai
  
  ANOVA FUNCTIONS:  F_oneway
                    F_value
  
  SUPPORT FUNCTIONS:  writecc
                      incr
                      sign  (for Numpy arrays only)
                      sum
                      cumsum
                      ss
                      summult
                      sumdiffsquared
                      square_of_sums
                      shellsort
                      rankdata
                      outputpairedstats
                      findwithin
  """
  ## CHANGE LOG:
  ## ===========
  ## 07-11.26 ... conversion for numpy started
  ## 07-05-16 ... added Lin's Concordance Correlation Coefficient (alincc) and acov
  ## 05-08-21 ... added "Dice's coefficient"
  ## 04-10-26 ... added ap2t(), an ugly fcn for converting p-vals to T-vals
  ## 04-04-03 ... added amasslinregress() function to do regression on N-D arrays
  ## 03-01-03 ... CHANGED VERSION TO 0.6
  ##              fixed atsem() to properly handle limits=None case
  ##              improved histogram and median functions (estbinwidth) and
  ##                   fixed atvar() function (wrong answers for neg numbers?!?)
  ## 02-11-19 ... fixed attest_ind and attest_rel for div-by-zero Overflows
  ## 02-05-10 ... fixed lchisqprob indentation (failed when df=even)
  ## 00-12-28 ... removed aanova() to separate module, fixed licensing to
  ##                   match Python License, fixed doc string & imports
  ## 00-04-13 ... pulled all "global" statements, except from aanova()
  ##              added/fixed lots of documentation, removed io.py dependency
  ##              changed to version 0.5
  ## 99-11-13 ... added asign() function
  ## 99-11-01 ... changed version to 0.4 ... enough incremental changes now
  ## 99-10-25 ... added acovariance and acorrelation functions
  ## 99-10-10 ... fixed askew/akurtosis to avoid divide-by-zero errors
  ##              added aglm function (crude, but will be improved)
  ## 99-10-04 ... upgraded acumsum, ass, asummult, asamplevar, avar, etc. to
  ##                   all handle lists of 'dimension's and keepdims
  ##              REMOVED ar0, ar2, ar3, ar4 and replaced them with around
  ##              reinserted fixes for abetai to avoid math overflows
  ## 99-09-05 ... rewrote achisqprob/aerfcc/aksprob/afprob/abetacf/abetai to
  ##                   handle multi-dimensional arrays (whew!)
  ## 99-08-30 ... fixed l/amoment, l/askew, l/akurtosis per D'Agostino (1990)
  ##              added anormaltest per same reference
  ##              re-wrote azprob to calc arrays of probs all at once
  ## 99-08-22 ... edited attest_ind printing section so arrays could be rounded
  ## 99-08-19 ... fixed amean and aharmonicmean for non-error(!) overflow on
  ##                   short/byte arrays (mean of #s btw 100-300 = -150??)
  ## 99-08-09 ... fixed asum so that the None case works for Byte arrays
  ## 99-08-08 ... fixed 7/3 'improvement' to handle t-calcs on N-D arrays
  ## 99-07-03 ... improved attest_ind, attest_rel (zero-division errortrap)
  ## 99-06-24 ... fixed bug(?) in attest_ind (n1=a.shape[0])
  ## 04/11/99 ... added asignaltonoise, athreshold functions, changed all
  ##                   max/min in array section to N.maximum/N.minimum,
  ##                   fixed square_of_sums to prevent integer overflow
  ## 04/10/99 ... !!! Changed function name ... sumsquared ==> square_of_sums
  ## 03/18/99 ... Added ar0, ar2, ar3 and ar4 rounding functions
  ## 02/28/99 ... Fixed aobrientransform to return an array rather than a list
  ## 01/15/99 ... Essentially ceased updating list-versions of functions (!!!)
  ## 01/13/99 ... CHANGED TO VERSION 0.3
  ##              fixed bug in a/lmannwhitneyu p-value calculation
  ## 12/31/98 ... fixed variable-name bug in ldescribe
  ## 12/19/98 ... fixed bug in findwithin (fcns needed pstat. prefix)
  ## 12/16/98 ... changed amedianscore to return float (not array) for 1 score
  ## 12/14/98 ... added atmin and atmax functions
  ##              removed umath from import line (not needed)
  ##              l/ageometricmean modified to reduce chance of overflows (take
  ##                   nth root first, then multiply)
  ## 12/07/98 ... added __version__variable (now 0.2)
  ##              removed all 'stats.' from anova() fcn
  ## 12/06/98 ... changed those functions (except shellsort) that altered
  ##                   arguments in-place ... cumsum, ranksort, ...
  ##              updated (and fixed some) doc-strings
  ## 12/01/98 ... added anova() function (requires NumPy)
  ##              incorporated Dispatch class
  ## 11/12/98 ... added functionality to amean, aharmonicmean, ageometricmean
  ##              added 'asum' function (added functionality to N.add.reduce)
  ##              fixed both moment and amoment (two errors)
  ##              changed name of skewness and askewness to skew and askew
  ##              fixed (a)histogram (which sometimes counted points <lowerlimit)
  
  from __future__ import absolute_import
  from __future__ import print_function
  from builtins import input
  from . import pstat        # required 3rd party module
  import math, string, copy  # required python modules
  from types import *
  
  __version__ = 0.6
  
  ############# DISPATCH CODE ##############
  
  
  class Dispatch:
      """
  The Dispatch class, care of David Ascher, allows different functions to
  be called depending on the argument types.  This way, there can be one
  function name regardless of the argument type.  To access function doc
  in stats.py module, prefix the function with an 'l' or 'a' for list or
  array arguments, respectively.  That is, print stats.lmean.__doc__ or
  print stats.amean.__doc__ or whatever.
  """
  
      def __init__(self, *tuples):
          self._dispatch = {}
          for func, types in tuples:
              for t in types:
                  if t in self._dispatch:
                      raise ValueError("can't have two dispatches on " + str(t))
                  self._dispatch[t] = func
  
      def __call__(self, arg1, *args, **kw):
          if type(arg1) not in self._dispatch:
              raise TypeError("don't know how to dispatch %s arguments" % type(arg1))
          return self._dispatch[type(arg1)](*((arg1,) + args), **kw)
  
  
  ##########################################################################
  ########################   LIST-BASED FUNCTIONS   ########################
  ##########################################################################
  
  ### Define these regardless
  
  ####################################
  #######  CENTRAL TENDENCY  #########
  ####################################
  
  def lgeometricmean (inlist):
      """
  Calculates the geometric mean of the values in the passed list.
  That is:  n-th root of (x1 * x2 * ... * xn).  Assumes a '1D' list.
  
  Usage:   lgeometricmean(inlist)
  """
      mult = 1.0
      one_over_n = 1.0/len(inlist)
      for item in inlist:
          mult = mult * pow(item, one_over_n)
      return mult
  
  
  def lharmonicmean (inlist):
      """
  Calculates the harmonic mean of the values in the passed list.
  That is:  n / (1/x1 + 1/x2 + ... + 1/xn).  Assumes a '1D' list.
  
  Usage:   lharmonicmean(inlist)
  """
      sum = 0
      for item in inlist:
          sum = sum + 1.0/item
      return len(inlist) / sum
  
  
  def lmean (inlist):
      """
  Returns the arithematic mean of the values in the passed list.
  Assumes a '1D' list, but will function on the 1st dim of an array(!).
  
  Usage:   lmean(inlist)
  """
      sum = 0
      for item in inlist:
          sum = sum + item
      return sum/float(len(inlist))
  
  
  def lmedian (inlist,numbins=1000):
      """
  Returns the computed median value of a list of numbers, given the
  number of bins to use for the histogram (more bins brings the computed value
  closer to the median score, default number of bins = 1000).  See G.W.
  Heiman's Basic Stats (1st Edition), or CRC Probability & Statistics.
  
  Usage:   lmedian (inlist, numbins=1000)
  """
      (hist, smallest, binsize, extras) = histogram(inlist, numbins, [min(inlist), max(inlist)]) # make histog
      cumhist = cumsum(hist)              # make cumulative histogram
      for i in range(len(cumhist)):        # get 1st(!) index holding 50%ile score
          if cumhist[i]>=len(inlist)/2.0:
              cfbin = i
              break
      LRL = smallest + binsize*cfbin        # get lower read limit of that bin
      cfbelow = cumhist[cfbin-1]
      freq = float(hist[cfbin])                # frequency IN the 50%ile bin
      median = LRL + ((len(inlist)/2.0 - cfbelow)/float(freq))*binsize  # median formula
      return median
  
  
  def lmedianscore (inlist):
      """
  Returns the 'middle' score of the passed list.  If there is an even
  number of scores, the mean of the 2 middle scores is returned.
  
  Usage:   lmedianscore(inlist)
  """
  
      newlist = sorted(copy.deepcopy(inlist))
      if len(newlist) % 2 == 0:   # if even number of scores, average middle 2
          index = len(newlist) // 2  # integer division correct
          median = float(newlist[index] + newlist[index-1]) /2
      else:
          index = len(newlist) // 2  # int divsion gives mid value when count from 0
          median = newlist[index]
      return median
  
  
  def lmode(inlist):
      """
  Returns a list of the modal (most common) score(s) in the passed
  list.  If there is more than one such score, all are returned.  The
  bin-count for the mode(s) is also returned.
  
  Usage:   lmode(inlist)
  Returns: bin-count for mode(s), a list of modal value(s)
  """
  
      scores = sorted(pstat.unique(inlist))
      freq = []
      for item in scores:
          freq.append(inlist.count(item))
      maxfreq = max(freq)
      mode = []
      stillmore = 1
      while stillmore:
          try:
              indx = freq.index(maxfreq)
              mode.append(scores[indx])
              del freq[indx]
              del scores[indx]
          except ValueError:
              stillmore=0
      return maxfreq, mode
  
  
  ####################################
  ############  MOMENTS  #############
  ####################################
  
  def lmoment(inlist,moment=1):
      """
  Calculates the nth moment about the mean for a sample (defaults to
  the 1st moment).  Used to calculate coefficients of skewness and kurtosis.
  
  Usage:   lmoment(inlist,moment=1)
  Returns: appropriate moment (r) from ... 1/n * SUM((inlist(i)-mean)**r)
  """
      if moment == 1:
          return 0.0
      else:
          mn = mean(inlist)
          n = len(inlist)
          s = 0
          for x in inlist:
              s = s + (x-mn)**moment
          return s/float(n)
  
  
  def lvariation(inlist):
      """
  Returns the coefficient of variation, as defined in CRC Standard
  Probability and Statistics, p.6.
  
  Usage:   lvariation(inlist)
  """
      return 100.0*samplestdev(inlist)/float(mean(inlist))
  
  
  def lskew(inlist):
      """
  Returns the skewness of a distribution, as defined in Numerical
  Recipies (alternate defn in CRC Standard Probability and Statistics, p.6.)
  
  Usage:   lskew(inlist)
  """
      return moment(inlist, 3)/pow(moment(inlist, 2), 1.5)
  
  
  def lkurtosis(inlist):
      """
  Returns the kurtosis of a distribution, as defined in Numerical
  Recipies (alternate defn in CRC Standard Probability and Statistics, p.6.)
  
  Usage:   lkurtosis(inlist)
  """
      return moment(inlist, 4)/pow(moment(inlist, 2), 2.0)
  
  
  def ldescribe(inlist):
      """
  Returns some descriptive statistics of the passed list (assumed to be 1D).
  
  Usage:   ldescribe(inlist)
  Returns: n, mean, standard deviation, skew, kurtosis
  """
      n = len(inlist)
      mm = (min(inlist), max(inlist))
      m = mean(inlist)
      sd = stdev(inlist)
      sk = skew(inlist)
      kurt = kurtosis(inlist)
      return n, mm, m, sd, sk, kurt
  
  
  ####################################
  #######  FREQUENCY STATS  ##########
  ####################################
  
  def litemfreq(inlist):
      """
  Returns a list of pairs.  Each pair consists of one of the scores in inlist
  and it's frequency count.  Assumes a 1D list is passed.
  
  Usage:   litemfreq(inlist)
  Returns: a 2D frequency table (col [0:n-1]=scores, col n=frequencies)
  """
      scores = sorted(pstat.unique(inlist))
      freq = []
      for item in scores:
          freq.append(inlist.count(item))
      return pstat.abut(scores, freq)
  
  
  def lscoreatpercentile (inlist, percent):
      """
  Returns the score at a given percentile relative to the distribution
  given by inlist.
  
  Usage:   lscoreatpercentile(inlist,percent)
  """
      if percent > 1:
          print("\nDividing percent>1 by 100 in lscoreatpercentile().\n")
          percent = percent / 100.0
      targetcf = percent*len(inlist)
      h, lrl, binsize, extras = histogram(inlist)
      cumhist = cumsum(copy.deepcopy(h))
      for i in range(len(cumhist)):
          if cumhist[i] >= targetcf:
              break
      score = binsize * ((targetcf - cumhist[i-1]) / float(h[i])) + (lrl+binsize*i)
      return score
  
  
  def lpercentileofscore (inlist, score,histbins=10,defaultlimits=None):
      """
  Returns the percentile value of a score relative to the distribution
  given by inlist.  Formula depends on the values used to histogram the data(!).
  
  Usage:   lpercentileofscore(inlist,score,histbins=10,defaultlimits=None)
  """
  
      h, lrl, binsize, extras = histogram(inlist, histbins, defaultlimits)
      cumhist = cumsum(copy.deepcopy(h))
      i = int((score - lrl)/float(binsize))
      pct = (cumhist[i-1]+((score-(lrl+binsize*i))/float(binsize))*h[i])/float(len(inlist)) * 100
      return pct
  
  
  def lhistogram (inlist,numbins=10,defaultreallimits=None,printextras=0):
      """
  Returns (i) a list of histogram bin counts, (ii) the smallest value
  of the histogram binning, and (iii) the bin width (the last 2 are not
  necessarily integers).  Default number of bins is 10.  If no sequence object
  is given for defaultreallimits, the routine picks (usually non-pretty) bins
  spanning all the numbers in the inlist.
  
  Usage:   lhistogram (inlist, numbins=10, defaultreallimits=None,suppressoutput=0)
  Returns: list of bin values, lowerreallimit, binsize, extrapoints
  """
      if (defaultreallimits is not None):
          if not isinstance(defaultreallimits, (list, tuple)) or len(defaultreallimits) == 1: # only one limit given, assumed to be lower one & upper is calc'd
              lowerreallimit = defaultreallimits
              upperreallimit = 1.000001 * max(inlist)
          else: # assume both limits given
              lowerreallimit = defaultreallimits[0]
              upperreallimit = defaultreallimits[1]
          binsize = (upperreallimit-lowerreallimit)/float(numbins)
      else:     # no limits given for histogram, both must be calc'd
          estbinwidth=(max(inlist)-min(inlist))/float(numbins) +1e-6 #1=>cover all
          binsize = ((max(inlist)-min(inlist)+estbinwidth))/float(numbins)
          lowerreallimit = min(inlist) - binsize/2 #lower real limit,1st bin
      bins = [0]*(numbins)
      extrapoints = 0
      for num in inlist:
          try:
              if (num-lowerreallimit) < 0:
                  extrapoints = extrapoints + 1
              else:
                  bintoincrement = int((num-lowerreallimit)/float(binsize))
                  bins[bintoincrement] = bins[bintoincrement] + 1
          except:
              extrapoints = extrapoints + 1
      if (extrapoints > 0 and printextras == 1):
          print('\nPoints outside given histogram range =', extrapoints)
      return (bins, lowerreallimit, binsize, extrapoints)
  
  
  def lcumfreq(inlist,numbins=10,defaultreallimits=None):
      """
  Returns a cumulative frequency histogram, using the histogram function.
  
  Usage:   lcumfreq(inlist,numbins=10,defaultreallimits=None)
  Returns: list of cumfreq bin values, lowerreallimit, binsize, extrapoints
  """
      h, l, b, e = histogram(inlist, numbins, defaultreallimits)
      cumhist = cumsum(copy.deepcopy(h))
      return cumhist, l, b, e
  
  
  def lrelfreq(inlist,numbins=10,defaultreallimits=None):
      """
  Returns a relative frequency histogram, using the histogram function.
  
  Usage:   lrelfreq(inlist,numbins=10,defaultreallimits=None)
  Returns: list of cumfreq bin values, lowerreallimit, binsize, extrapoints
  """
      h, l, b, e = histogram(inlist, numbins, defaultreallimits)
      for i in range(len(h)):
          h[i] = h[i]/float(len(inlist))
      return h, l, b, e
  
  
  ####################################
  #####  VARIABILITY FUNCTIONS  ######
  ####################################
  
  def lobrientransform(*args):
      """
  Computes a transform on input data (any number of columns).  Used to
  test for homogeneity of variance prior to running one-way stats.  From
  Maxwell and Delaney, p.112.
  
  Usage:   lobrientransform(*args)
  Returns: transformed data for use in an ANOVA
  """
      TINY = 1e-10
      k = len(args)
      n = [0.0]*k
      v = [0.0]*k
      m = [0.0]*k
      nargs = []
      for i in range(k):
          nargs.append(copy.deepcopy(args[i]))
          n[i] = float(len(nargs[i]))
          v[i] = var(nargs[i])
          m[i] = mean(nargs[i])
      for j in range(k):
          for i in range(n[j]):
              t1 = (n[j]-1.5)*n[j]*(nargs[j][i]-m[j])**2
              t2 = 0.5*v[j]*(n[j]-1.0)
              t3 = (n[j]-1.0)*(n[j]-2.0)
              nargs[j][i] = (t1-t2) / float(t3)
      check = 1
      for j in range(k):
          if v[j] - mean(nargs[j]) > TINY:
              check = 0
      if check != 1:
          raise ValueError('Problem in obrientransform.')
      else:
          return nargs
  
  
  def lsamplevar (inlist):
      """
  Returns the variance of the values in the passed list using
  N for the denominator (i.e., DESCRIBES the sample variance only).
  
  Usage:   lsamplevar(inlist)
  """
      n = len(inlist)
      mn = mean(inlist)
      deviations = []
      for item in inlist:
          deviations.append(item-mn)
      return ss(deviations)/float(n)
  
  
  def lsamplestdev (inlist):
      """
  Returns the standard deviation of the values in the passed list using
  N for the denominator (i.e., DESCRIBES the sample stdev only).
  
  Usage:   lsamplestdev(inlist)
  """
      return math.sqrt(samplevar(inlist))
  
  
  def lcov (x,y, keepdims=0):
      """
  Returns the estimated covariance of the values in the passed
  array (i.e., N-1).  Dimension can equal None (ravel array first), an
  integer (the dimension over which to operate), or a sequence (operate
  over multiple dimensions).  Set keepdims=1 to return an array with the
  same number of dimensions as inarray.
  
  Usage:   lcov(x,y,keepdims=0)
  """
  
      n = len(x)
      xmn = mean(x)
      ymn = mean(y)
      xdeviations = [0]*len(x)
      ydeviations = [0]*len(y)
      for i in range(len(x)):
          xdeviations[i] = x[i] - xmn
          ydeviations[i] = y[i] - ymn
      ss = 0.0
      for i in range(len(xdeviations)):
          ss = ss + xdeviations[i]*ydeviations[i]
      return ss/float(n-1)
  
  
  def lvar (inlist):
      """
  Returns the variance of the values in the passed list using N-1
  for the denominator (i.e., for estimating population variance).
  
  Usage:   lvar(inlist)
  """
      n = len(inlist)
      mn = mean(inlist)
      deviations = [0]*len(inlist)
      for i in range(len(inlist)):
          deviations[i] = inlist[i] - mn
      return ss(deviations)/float(n-1)
  
  
  def lstdev (inlist):
      """
  Returns the standard deviation of the values in the passed list
  using N-1 in the denominator (i.e., to estimate population stdev).
  
  Usage:   lstdev(inlist)
  """
      return math.sqrt(var(inlist))
  
  
  def lsterr(inlist):
      """
  Returns the standard error of the values in the passed list using N-1
  in the denominator (i.e., to estimate population standard error).
  
  Usage:   lsterr(inlist)
  """
      return stdev(inlist) / float(math.sqrt(len(inlist)))
  
  
  def lsem (inlist):
      """
  Returns the estimated standard error of the mean (sx-bar) of the
  values in the passed list.  sem = stdev / sqrt(n)
  
  Usage:   lsem(inlist)
  """
      sd = stdev(inlist)
      n = len(inlist)
      return sd/math.sqrt(n)
  
  
  def lz (inlist, score):
      """
  Returns the z-score for a given input score, given that score and the
  list from which that score came.  Not appropriate for population calculations.
  
  Usage:   lz(inlist, score)
  """
      z = (score-mean(inlist))/samplestdev(inlist)
      return z
  
  
  def lzs (inlist):
      """
  Returns a list of z-scores, one for each score in the passed list.
  
  Usage:   lzs(inlist)
  """
      zscores = []
      for item in inlist:
          zscores.append(z(inlist, item))
      return zscores
  
  
  ####################################
  #######  TRIMMING FUNCTIONS  #######
  ####################################
  
  def ltrimboth (l, proportiontocut):
      """
  Slices off the passed proportion of items from BOTH ends of the passed
  list (i.e., with proportiontocut=0.1, slices 'leftmost' 10% AND 'rightmost'
  10% of scores.  Assumes list is sorted by magnitude.  Slices off LESS if
  proportion results in a non-integer slice index (i.e., conservatively
  slices off proportiontocut).
  
  Usage:   ltrimboth (l,proportiontocut)
  Returns: trimmed version of list l
  """
      lowercut = int(proportiontocut*len(l))
      uppercut = len(l) - lowercut
      return l[lowercut:uppercut]
  
  
  def ltrim1 (l,proportiontocut,tail='right'):
      """
  Slices off the passed proportion of items from ONE end of the passed
  list (i.e., if proportiontocut=0.1, slices off 'leftmost' or 'rightmost'
  10% of scores).  Slices off LESS if proportion results in a non-integer
  slice index (i.e., conservatively slices off proportiontocut).
  
  Usage:   ltrim1 (l,proportiontocut,tail='right')  or set tail='left'
  Returns: trimmed version of list l
  """
      if tail == 'right':
          lowercut = 0
          uppercut = len(l) - int(proportiontocut*len(l))
      elif tail == 'left':
          lowercut = int(proportiontocut*len(l))
          uppercut = len(l)
      return l[lowercut:uppercut]
  
  
  ####################################
  #####  CORRELATION FUNCTIONS  ######
  ####################################
  
  def lpaired(x, y):
      """
  Interactively determines the type of data and then runs the
  appropriated statistic for paired group data.
  
  Usage:   lpaired(x,y)
  Returns: appropriate statistic name, value, and probability
  """
      samples = ''
      while samples not in ['i', 'r', 'I', 'R', 'c', 'C']:
          print('\nIndependent or related samples, or correlation (i,r,c): ', end=' ')
          samples = input()
  
      if samples in ['i', 'I', 'r', 'R']:
          print('\nComparing variances ...', end=' ')
  # USE O'BRIEN'S TEST FOR HOMOGENEITY OF VARIANCE, Maxwell & delaney, p.112
          r = obrientransform(x, y)
          f, p = F_oneway(pstat.colex(r, 0), pstat.colex(r, 1))
          if p<0.05:
              vartype='unequal, p='+str(round(p, 4))
          else:
              vartype='equal'
          print(vartype)
          if samples in ['i', 'I']:
              if vartype[0]=='e':
                  t, p = ttest_ind(x, y, 0)
                  print('\nIndependent samples t-test:  ', round(t, 4), round(p, 4))
              else:
                  if len(x)>20 or len(y)>20:
                      z, p = ranksums(x, y)
                      print('\nRank Sums test (NONparametric, n>20):  ', round(z, 4), round(p, 4))
                  else:
                      u, p = mannwhitneyu(x, y)
                      print('\nMann-Whitney U-test (NONparametric, ns<20):  ', round(u, 4), round(p, 4))
  
          else:  # RELATED SAMPLES
              if vartype[0]=='e':
                  t, p = ttest_rel(x, y, 0)
                  print('\nRelated samples t-test:  ', round(t, 4), round(p, 4))
              else:
                  t, p = ranksums(x, y)
                  print('\nWilcoxon T-test (NONparametric):  ', round(t, 4), round(p, 4))
      else:  # CORRELATION ANALYSIS
          corrtype = ''
          while corrtype not in ['c', 'C', 'r', 'R', 'd', 'D']:
              print('\nIs the data Continuous, Ranked, or Dichotomous (c,r,d): ', end=' ')
              corrtype = input()
          if corrtype in ['c', 'C']:
              m, b, r, p, see = linregress(x, y)
              print('\nLinear regression for continuous variables ...')
              lol = [['Slope', 'Intercept', 'r', 'Prob', 'SEestimate'], [round(m, 4), round(b, 4), round(r, 4), round(p, 4), round(see, 4)]]
              pstat.printcc(lol)
          elif corrtype in ['r', 'R']:
              r, p = spearmanr(x, y)
              print('\nCorrelation for ranked variables ...')
              print("Spearman's r: ", round(r, 4), round(p, 4))
          else: # DICHOTOMOUS
              r, p = pointbiserialr(x, y)
              print('\nAssuming x contains a dichotomous variable ...')
              print('Point Biserial r: ', round(r, 4), round(p, 4))
      print('\n\n')
      return None
  
  
  def lpearsonr(x, y):
      """
  Calculates a Pearson correlation coefficient and the associated
  probability value.  Taken from Heiman's Basic Statistics for the Behav.
  Sci (2nd), p.195.
  
  Usage:   lpearsonr(x,y)      where x and y are equal-length lists
  Returns: Pearson's r value, two-tailed p-value
  """
      TINY = 1.0e-30
      if len(x) != len(y):
          raise ValueError('Input values not paired in pearsonr.  Aborting.')
      n = len(x)
      x = list(map(float, x))
      y = list(map(float, y))
      xmean = mean(x)
      ymean = mean(y)
      r_num = n*(summult(x, y)) - sum(x)*sum(y)
      r_den = math.sqrt((n*ss(x) - square_of_sums(x))*(n*ss(y)-square_of_sums(y)))
      r = (r_num / r_den)  # denominator already a float
      df = n-2
      t = r*math.sqrt(df/((1.0-r+TINY)*(1.0+r+TINY)))
      prob = betai(0.5*df, 0.5, df/float(df+t*t))
      return r, prob
  
  
  def llincc(x, y):
      """
  Calculates Lin's concordance correlation coefficient.
  
  Usage:   alincc(x,y)    where x, y are equal-length arrays
  Returns: Lin's CC
  """
      covar = lcov(x, y)*(len(x)-1)/float(len(x))  # correct denom to n
      xvar = lvar(x)*(len(x)-1)/float(len(x))  # correct denom to n
      yvar = lvar(y)*(len(y)-1)/float(len(y))  # correct denom to n
      lincc = (2 * covar) / ((xvar+yvar) +((amean(x)-amean(y))**2))
      return lincc
  
  
  def lspearmanr(x, y):
      """
  Calculates a Spearman rank-order correlation coefficient.  Taken
  from Heiman's Basic Statistics for the Behav. Sci (1st), p.192.
  
  Usage:   lspearmanr(x,y)      where x and y are equal-length lists
  Returns: Spearman's r, two-tailed p-value
  """
      TINY = 1e-30
      if len(x) != len(y):
          raise ValueError('Input values not paired in spearmanr.  Aborting.')
      n = len(x)
      rankx = rankdata(x)
      ranky = rankdata(y)
      dsq = sumdiffsquared(rankx, ranky)
      rs = 1 - 6*dsq / float(n*(n**2-1))
      t = rs * math.sqrt((n-2) / ((rs+1.0)*(1.0-rs)))
      df = n-2
      probrs = betai(0.5*df, 0.5, df/(df+t*t))  # t already a float
  # probability values for rs are from part 2 of the spearman function in
  # Numerical Recipies, p.510.  They are close to tables, but not exact. (?)
      return rs, probrs
  
  
  def lpointbiserialr(x, y):
      """
  Calculates a point-biserial correlation coefficient and the associated
  probability value.  Taken from Heiman's Basic Statistics for the Behav.
  Sci (1st), p.194.
  
  Usage:   lpointbiserialr(x,y)      where x,y are equal-length lists
  Returns: Point-biserial r, two-tailed p-value
  """
      TINY = 1e-30
      if len(x) != len(y):
          raise ValueError('INPUT VALUES NOT PAIRED IN pointbiserialr.  ABORTING.')
      data = pstat.abut(x, y)
      categories = pstat.unique(x)
      if len(categories) != 2:
          raise ValueError("Exactly 2 categories required for pointbiserialr().")
      else:   # there are 2 categories, continue
          codemap = pstat.abut(categories, list(range(2)))
          recoded = pstat.recode(data, codemap, 0)
          x = pstat.linexand(data, 0, categories[0])
          y = pstat.linexand(data, 0, categories[1])
          xmean = mean(pstat.colex(x, 1))
          ymean = mean(pstat.colex(y, 1))
          n = len(data)
          adjust = math.sqrt((len(x)/float(n))*(len(y)/float(n)))
          rpb = (ymean - xmean)/samplestdev(pstat.colex(data, 1))*adjust
          df = n-2
          t = rpb*math.sqrt(df/((1.0-rpb+TINY)*(1.0+rpb+TINY)))
          prob = betai(0.5*df, 0.5, df/(df+t*t))  # t already a float
          return rpb, prob
  
  
  def lkendalltau(x, y):
      """
  Calculates Kendall's tau ... correlation of ordinal data.  Adapted
  from function kendl1 in Numerical Recipies.  Needs good test-routine.@@@
  
  Usage:   lkendalltau(x,y)
  Returns: Kendall's tau, two-tailed p-value
  """
      n1 = 0
      n2 = 0
      iss = 0
      for j in range(len(x)-1):
          for k in range(j, len(y)):
              a1 = x[j] - x[k]
              a2 = y[j] - y[k]
              aa = a1 * a2
              if (aa):             # neither list has a tie
                  n1 = n1 + 1
                  n2 = n2 + 1
                  if aa > 0:
                      iss = iss + 1
                  else:
                      iss = iss -1
              else:
                  if (a1):
                      n1 = n1 + 1
                  else:
                      n2 = n2 + 1
      tau = iss / math.sqrt(n1*n2)
      svar = (4.0*len(x)+10.0) / (9.0*len(x)*(len(x)-1))
      z = tau / math.sqrt(svar)
      prob = erfcc(abs(z)/1.4142136)
      return tau, prob
  
  
  def llinregress(x, y):
      """
  Calculates a regression line on x,y pairs.  
  
  Usage:   llinregress(x,y)      x,y are equal-length lists of x-y coordinates
  Returns: slope, intercept, r, two-tailed prob, sterr-of-estimate
  """
      TINY = 1.0e-20
      if len(x) != len(y):
          raise ValueError('Input values not paired in linregress.  Aborting.')
      n = len(x)
      x = list(map(float, x))
      y = list(map(float, y))
      xmean = mean(x)
      ymean = mean(y)
      r_num = float(n*(summult(x, y)) - sum(x)*sum(y))
      r_den = math.sqrt((n*ss(x) - square_of_sums(x))*(n*ss(y)-square_of_sums(y)))
      r = r_num / r_den
      z = 0.5*math.log((1.0+r+TINY)/(1.0-r+TINY))
      df = n-2
      t = r*math.sqrt(df/((1.0-r+TINY)*(1.0+r+TINY)))
      prob = betai(0.5*df, 0.5, df/(df+t*t))
      slope = r_num / float(n*ss(x) - square_of_sums(x))
      intercept = ymean - slope*xmean
      sterrest = math.sqrt(1-r*r)*samplestdev(y)
      return slope, intercept, r, prob, sterrest
  
  
  ####################################
  #####  INFERENTIAL STATISTICS  #####
  ####################################
  
  def lttest_1samp(a,popmean,printit=0,name='Sample',writemode='a'):
      """
  Calculates the t-obtained for the independent samples T-test on ONE group
  of scores a, given a population mean.  If printit=1, results are printed
  to the screen.  If printit='filename', the results are output to 'filename'
  using the given writemode (default=append).  Returns t-value, and prob.
  
  Usage:   lttest_1samp(a,popmean,Name='Sample',printit=0,writemode='a')
  Returns: t-value, two-tailed prob
  """
      x = mean(a)
      v = var(a)
      n = len(a)
      df = n-1
      svar = ((n-1)*v)/float(df)
      t = (x-popmean)/math.sqrt(svar*(1.0/n))
      prob = betai(0.5*df, 0.5, float(df)/(df+t*t))
  
      if printit != 0:
          statname = 'Single-sample T-test.'
          outputpairedstats(printit, writemode,
                            'Population', '--', popmean, 0, 0, 0,
                            name, n, x, v, min(a), max(a),
                            statname, t, prob)
      return t, prob
  
  
  def lttest_ind (a, b, printit=0, name1='Samp1', name2='Samp2', writemode='a'):
      """
  Calculates the t-obtained T-test on TWO INDEPENDENT samples of
  scores a, and b.  From Numerical Recipies, p.483.  If printit=1, results
  are printed to the screen.  If printit='filename', the results are output
  to 'filename' using the given writemode (default=append).  Returns t-value,
  and prob.
  
  Usage:   lttest_ind(a,b,printit=0,name1='Samp1',name2='Samp2',writemode='a')
  Returns: t-value, two-tailed prob
  """
      x1 = mean(a)
      x2 = mean(b)
      v1 = stdev(a)**2
      v2 = stdev(b)**2
      n1 = len(a)
      n2 = len(b)
      df = n1+n2-2
      svar = ((n1-1)*v1+(n2-1)*v2)/float(df)
      t = (x1-x2)/math.sqrt(svar*(1.0/n1 + 1.0/n2))
      prob = betai(0.5*df, 0.5, df/(df+t*t))
  
      if printit != 0:
          statname = 'Independent samples T-test.'
          outputpairedstats(printit, writemode,
                            name1, n1, x1, v1, min(a), max(a),
                            name2, n2, x2, v2, min(b), max(b),
                            statname, t, prob)
      return t, prob
  
  
  def lttest_rel (a,b,printit=0,name1='Sample1',name2='Sample2',writemode='a'):
      """
  Calculates the t-obtained T-test on TWO RELATED samples of scores,
  a and b.  From Numerical Recipies, p.483.  If printit=1, results are
  printed to the screen.  If printit='filename', the results are output to
  'filename' using the given writemode (default=append).  Returns t-value,
  and prob.
  
  Usage:   lttest_rel(a,b,printit=0,name1='Sample1',name2='Sample2',writemode='a')
  Returns: t-value, two-tailed prob
  """
      if len(a) != len(b):
          raise ValueError('Unequal length lists in ttest_rel.')
      x1 = mean(a)
      x2 = mean(b)
      v1 = var(a)
      v2 = var(b)
      n = len(a)
      cov = 0
      for i in range(len(a)):
          cov = cov + (a[i]-x1) * (b[i]-x2)
      df = n-1
      cov = cov / float(df)
      sd = math.sqrt((v1+v2 - 2.0*cov)/float(n))
      t = (x1-x2)/sd
      prob = betai(0.5*df, 0.5, df/(df+t*t))
  
      if printit != 0:
          statname = 'Related samples T-test.'
          outputpairedstats(printit, writemode,
                            name1, n, x1, v1, min(a), max(a),
                            name2, n, x2, v2, min(b), max(b),
                            statname, t, prob)
      return t, prob
  
  
  def lchisquare(f_obs,f_exp=None):
      """
  Calculates a one-way chi square for list of observed frequencies and returns
  the result.  If no expected frequencies are given, the total N is assumed to
  be equally distributed across all groups.
  
  Usage:   lchisquare(f_obs, f_exp=None)   f_obs = list of observed cell freq.
  Returns: chisquare-statistic, associated p-value
  """
      k = len(f_obs)                 # number of groups
      if f_exp is None:
          f_exp = [sum(f_obs)/float(k)] * len(f_obs) # create k bins with = freq.
      chisq = 0
      for i in range(len(f_obs)):
          chisq = chisq + (f_obs[i]-f_exp[i])**2 / float(f_exp[i])
      return chisq, chisqprob(chisq, k-1)
  
  
  def lks_2samp (data1, data2):
      """
  Computes the Kolmogorov-Smirnof statistic on 2 samples.  From
  Numerical Recipies in C, page 493.
  
  Usage:   lks_2samp(data1,data2)   data1&2 are lists of values for 2 conditions
  Returns: KS D-value, associated p-value
  """
      j1 = 0
      j2 = 0
      fn1 = 0.0
      fn2 = 0.0
      n1 = len(data1)
      n2 = len(data2)
      en1 = n1
      en2 = n2
      d = 0.0
      data1.sort()
      data2.sort()
      while j1 < n1 and j2 < n2:
          d1=data1[j1]
          d2=data2[j2]
          if d1 <= d2:
              fn1 = (j1)/float(en1)
              j1 = j1 + 1
          if d2 <= d1:
              fn2 = (j2)/float(en2)
              j2 = j2 + 1
          dt = (fn2-fn1)
          if math.fabs(dt) > math.fabs(d):
              d = dt
      try:
          en = math.sqrt(en1*en2/float(en1+en2))
          prob = ksprob((en+0.12+0.11/en)*abs(d))
      except:
          prob = 1.0
      return d, prob
  
  
  def lmannwhitneyu(x, y):
      """
  Calculates a Mann-Whitney U statistic on the provided scores and
  returns the result.  Use only when the n in each condition is < 20 and
  you have 2 independent samples of ranks.  NOTE: Mann-Whitney U is
  significant if the u-obtained is LESS THAN or equal to the critical
  value of U found in the tables.  Equivalent to Kruskal-Wallis H with
  just 2 groups.
  
  Usage:   lmannwhitneyu(data)
  Returns: u-statistic, one-tailed p-value (i.e., p(z(U)))
  """
      n1 = len(x)
      n2 = len(y)
      ranked = rankdata(x+y)
      rankx = ranked[0:n1]       # get the x-ranks
      ranky = ranked[n1:]        # the rest are y-ranks
      u1 = n1*n2 + (n1*(n1+1))/2.0 - sum(rankx)  # calc U for x
      u2 = n1*n2 - u1                            # remainder is U for y
      bigu = max(u1, u2)
      smallu = min(u1, u2)
      T = math.sqrt(tiecorrect(ranked))  # correction factor for tied scores
      if T == 0:
          raise ValueError('All numbers are identical in lmannwhitneyu')
      sd = math.sqrt(T*n1*n2*(n1+n2+1)/12.0)
      z = abs((bigu-n1*n2/2.0) / sd)  # normal approximation for prob calc
      return smallu, 1.0 - zprob(z)
  
  
  def ltiecorrect(rankvals):
      """
  Corrects for ties in Mann Whitney U and Kruskal Wallis H tests.  See
  Siegel, S. (1956) Nonparametric Statistics for the Behavioral Sciences.
  New York: McGraw-Hill.  Code adapted from |Stat rankind.c code.
  
  Usage:   ltiecorrect(rankvals)
  Returns: T correction factor for U or H
  """
      sorted, posn = shellsort(rankvals)
      n = len(sorted)
      T = 0.0
      i = 0
      while (i<n-1):
          if sorted[i] == sorted[i+1]:
              nties = 1
              while (i<n-1) and (sorted[i] == sorted[i+1]):
                  nties = nties +1
                  i = i +1
              T = T + nties**3 - nties
          i = i+1
      T = T / float(n**3-n)
      return 1.0 - T
  
  
  def lranksums(x, y):
      """
  Calculates the rank sums statistic on the provided scores and
  returns the result.  Use only when the n in each condition is > 20 and you
  have 2 independent samples of ranks.
  
  Usage:   lranksums(x,y)
  Returns: a z-statistic, two-tailed p-value
  """
      n1 = len(x)
      n2 = len(y)
      alldata = x+y
      ranked = rankdata(alldata)
      x = ranked[:n1]
      y = ranked[n1:]
      s = sum(x)
      expected = n1*(n1+n2+1) / 2.0
      z = (s - expected) / math.sqrt(n1*n2*(n1+n2+1)/12.0)
      prob = 2*(1.0 -zprob(abs(z)))
      return z, prob
  
  
  def lwilcoxont(x, y):
      """
  Calculates the Wilcoxon T-test for related samples and returns the
  result.  A non-parametric T-test.
  
  Usage:   lwilcoxont(x,y)
  Returns: a t-statistic, two-tail probability estimate
  """
      if len(x) != len(y):
          raise ValueError('Unequal N in wilcoxont.  Aborting.')
      d=[]
      for i in range(len(x)):
          diff = x[i] - y[i]
          if diff != 0:
              d.append(diff)
      count = len(d)
      absd = list(map(abs, d))
      absranked = rankdata(absd)
      r_plus = 0.0
      r_minus = 0.0
      for i in range(len(absd)):
          if d[i] < 0:
              r_minus = r_minus + absranked[i]
          else:
              r_plus = r_plus + absranked[i]
      wt = min(r_plus, r_minus)
      mn = count * (count+1) * 0.25
      se =  math.sqrt(count*(count+1)*(2.0*count+1.0)/24.0)
      z = math.fabs(wt-mn) / se
      prob = 2*(1.0 -zprob(abs(z)))
      return wt, prob
  
  
  def lkruskalwallish(*args):
      """
  The Kruskal-Wallis H-test is a non-parametric ANOVA for 3 or more
  groups, requiring at least 5 subjects in each group.  This function
  calculates the Kruskal-Wallis H-test for 3 or more independent samples
  and returns the result.  
  
  Usage:   lkruskalwallish(*args)
  Returns: H-statistic (corrected for ties), associated p-value
  """
      args = list(args)
      n = [0]*len(args)
      all = []
      n = list(map(len, args))
      for i in range(len(args)):
          all = all + args[i]
      ranked = rankdata(all)
      T = tiecorrect(ranked)
      for i in range(len(args)):
          args[i] = ranked[0:n[i]]
          del ranked[0:n[i]]
      rsums = []
      for i in range(len(args)):
          rsums.append(sum(args[i])**2)
          rsums[i] = rsums[i] / float(n[i])
      ssbn = sum(rsums)
      totaln = sum(n)
      h = 12.0 / (totaln*(totaln+1)) * ssbn - 3*(totaln+1)
      df = len(args) - 1
      if T == 0:
          raise ValueError('All numbers are identical in lkruskalwallish')
      h = h / float(T)
      return h, chisqprob(h, df)
  
  
  def lfriedmanchisquare(*args):
      """
  Friedman Chi-Square is a non-parametric, one-way within-subjects
  ANOVA.  This function calculates the Friedman Chi-square test for repeated
  measures and returns the result, along with the associated probability
  value.  It assumes 3 or more repeated measures.  Only 3 levels requires a
  minimum of 10 subjects in the study.  Four levels requires 5 subjects per
  level(??).
  
  Usage:   lfriedmanchisquare(*args)
  Returns: chi-square statistic, associated p-value
  """
      k = len(args)
      if k < 3:
          raise ValueError('Less than 3 levels.  Friedman test not appropriate.')
      n = len(args[0])
      data = pstat.abut(*tuple(args))
      for i in range(len(data)):
          data[i] = rankdata(data[i])
      ssbn = 0
      for i in range(k):
          ssbn = ssbn + sum(args[i])**2
      chisq = 12.0 / (k*n*(k+1)) * ssbn - 3*n*(k+1)
      return chisq, chisqprob(chisq, k-1)
  
  
  ####################################
  ####  PROBABILITY CALCULATIONS  ####
  ####################################
  
  def lchisqprob(chisq, df):
      """
  Returns the (1-tailed) probability value associated with the provided
  chi-square value and df.  Adapted from chisq.c in Gary Perlman's |Stat.
  
  Usage:   lchisqprob(chisq,df)
  """
      BIG = 20.0
      def ex(x):
          BIG = 20.0
          if x < -BIG:
              return 0.0
          else:
              return math.exp(x)
  
      if chisq <=0 or df < 1:
          return 1.0
      a = 0.5 * chisq
      if df%2 == 0:
          even = 1
      else:
          even = 0
      if df > 1:
          y = ex(-a)
      if even:
          s = y
      else:
          s = 2.0 * zprob(-math.sqrt(chisq))
      if (df > 2):
          chisq = 0.5 * (df - 1.0)
          if even:
              z = 1.0
          else:
              z = 0.5
          if a > BIG:
              if even:
                  e = 0.0
              else:
                  e = math.log(math.sqrt(math.pi))
              c = math.log(a)
              while (z <= chisq):
                  e = math.log(z) + e
                  s = s + ex(c*z-a-e)
                  z = z + 1.0
              return s
          else:
              if even:
                  e = 1.0
              else:
                  e = 1.0 / math.sqrt(math.pi) / math.sqrt(a)
              c = 0.0
              while (z <= chisq):
                  e = e * (a/float(z))
                  c = c + e
                  z = z + 1.0
              return (c*y+s)
      else:
          return s
  
  
  def lerfcc(x):
      """
  Returns the complementary error function erfc(x) with fractional
  error everywhere less than 1.2e-7.  Adapted from Numerical Recipies.
  
  Usage:   lerfcc(x)
  """
      z = abs(x)
      t = 1.0 / (1.0+0.5*z)
      ans = t * math.exp(-z*z-1.26551223 + t*(1.00002368+t*(0.37409196+t*(0.09678418+t*(-0.18628806+t*(0.27886807+t*(-1.13520398+t*(1.48851587+t*(-0.82215223+t*0.17087277)))))))))
      if x >= 0:
          return ans
      else:
          return 2.0 - ans
  
  
  def lzprob(z):
      """
  Returns the area under the normal curve 'to the left of' the given z value.
  Thus, 
      for z<0, zprob(z) = 1-tail probability
      for z>0, 1.0-zprob(z) = 1-tail probability
      for any z, 2.0*(1.0-zprob(abs(z))) = 2-tail probability
  Adapted from z.c in Gary Perlman's |Stat.
  
  Usage:   lzprob(z)
  """
      Z_MAX = 6.0    # maximum meaningful z-value
      if z == 0.0:
          x = 0.0
      else:
          y = 0.5 * math.fabs(z)
          if y >= (Z_MAX*0.5):
              x = 1.0
          elif (y < 1.0):
              w = y*y
              x = ((((((((0.000124818987 * w
                          -0.001075204047) * w +0.005198775019) * w
                        -0.019198292004) * w +0.059054035642) * w
                      -0.151968751364) * w +0.319152932694) * w
                    -0.531923007300) * w +0.797884560593) * y * 2.0
          else:
              y = y - 2.0
              x = (((((((((((((-0.000045255659 * y
                               +0.000152529290) * y -0.000019538132) * y
                             -0.000676904986) * y +0.001390604284) * y
                           -0.000794620820) * y -0.002034254874) * y
                         +0.006549791214) * y -0.010557625006) * y
                       +0.011630447319) * y -0.009279453341) * y
                     +0.005353579108) * y -0.002141268741) * y
                   +0.000535310849) * y +0.999936657524
      if z > 0.0:
          prob = ((x+1.0)*0.5)
      else:
          prob = ((1.0-x)*0.5)
      return prob
  
  
  def lksprob(alam):
      """
  Computes a Kolmolgorov-Smirnov t-test significance level.  Adapted from
  Numerical Recipies.
  
  Usage:   lksprob(alam)
  """
      fac = 2.0
      sum = 0.0
      termbf = 0.0
      a2 = -2.0*alam*alam
      for j in range(1, 201):
          term = fac*math.exp(a2*j*j)
          sum = sum + term
          if math.fabs(term) <= (0.001*termbf) or math.fabs(term) < (1.0e-8*sum):
              return sum
          fac = -fac
          termbf = math.fabs(term)
      return 1.0             # Get here only if fails to converge; was 0.0!!
  
  
  def lfprob (dfnum, dfden, F):
      """
  Returns the (1-tailed) significance level (p-value) of an F
  statistic given the degrees of freedom for the numerator (dfR-dfF) and
  the degrees of freedom for the denominator (dfF).
  
  Usage:   lfprob(dfnum, dfden, F)   where usually dfnum=dfbn, dfden=dfwn
  """
      p = betai(0.5*dfden, 0.5*dfnum, dfden/float(dfden+dfnum*F))
      return p
  
  
  def lbetacf(a, b, x):
      """
  This function evaluates the continued fraction form of the incomplete
  Beta function, betai.  (Adapted from: Numerical Recipies in C.)
  
  Usage:   lbetacf(a,b,x)
  """
      ITMAX = 200
      EPS = 3.0e-7
  
      bm = az = am = 1.0
      qab = a+b
      qap = a+1.0
      qam = a-1.0
      bz = 1.0-qab*x/qap
      for i in range(ITMAX+1):
          em = float(i+1)
          tem = em + em
          d = em*(b-em)*x/((qam+tem)*(a+tem))
          ap = az + d*am
          bp = bz+d*bm
          d = -(a+em)*(qab+em)*x/((qap+tem)*(a+tem))
          app = ap+d*az
          bpp = bp+d*bz
          aold = az
          am = ap/bpp
          bm = bp/bpp
          az = app/bpp
          bz = 1.0
          if (abs(az-aold)<(EPS*abs(az))):
              return az
      print('a or b too big, or ITMAX too small in Betacf.')
  
  
  def lgammln(xx):
      """
  Returns the gamma function of xx.
      Gamma(z) = Integral(0,infinity) of t^(z-1)exp(-t) dt.
  (Adapted from: Numerical Recipies in C.)
  
  Usage:   lgammln(xx)
  """
  
      coeff = [76.18009173, -86.50532033, 24.01409822, -1.231739516,
               0.120858003e-2, -0.536382e-5]
      x = xx - 1.0
      tmp = x + 5.5
      tmp = tmp - (x+0.5)*math.log(tmp)
      ser = 1.0
      for j in range(len(coeff)):
          x = x + 1
          ser = ser + coeff[j]/x
      return -tmp + math.log(2.50662827465*ser)
  
  
  def lbetai(a, b, x):
      """
  Returns the incomplete beta function:
  
      I-sub-x(a,b) = 1/B(a,b)*(Integral(0,x) of t^(a-1)(1-t)^(b-1) dt)
  
  where a,b>0 and B(a,b) = G(a)*G(b)/(G(a+b)) where G(a) is the gamma
  function of a.  The continued fraction formulation is implemented here,
  using the betacf function.  (Adapted from: Numerical Recipies in C.)
  
  Usage:   lbetai(a,b,x)
  """
      if (x<0.0 or x>1.0):
          raise ValueError('Bad x in lbetai')
      if (x==0.0 or x==1.0):
          bt = 0.0
      else:
          bt = math.exp(gammln(a+b)-gammln(a)-gammln(b)+a*math.log(x)+b*
                        math.log(1.0-x))
      if (x<(a+1.0)/(a+b+2.0)):
          return bt*betacf(a, b, x)/float(a)
      else:
          return 1.0-bt*betacf(b, a, 1.0-x)/float(b)
  
  
  ####################################
  #######  ANOVA CALCULATIONS  #######
  ####################################
  
  def lF_oneway(*lists):
      """
  Performs a 1-way ANOVA, returning an F-value and probability given
  any number of groups.  From Heiman, pp.394-7.
  
  Usage:   F_oneway(*lists)    where *lists is any number of lists, one per
                                    treatment group
  Returns: F value, one-tailed p-value
  """
      a = len(lists)           # ANOVA on 'a' groups, each in it's own list
      alldata = []
      for l in lists:
          alldata.extend(l)
      alldata = N.array(alldata)
      bign = len(alldata)
      sstot = ass(alldata)-(asquare_of_sums(alldata)/float(bign))
      ssbn = 0
      for list in lists:
          ssbn = ssbn + asquare_of_sums(N.array(list))/float(len(list))
      ssbn = ssbn - (asquare_of_sums(alldata)/float(bign))
      sswn = sstot-ssbn
      dfbn = a-1
      dfwn = bign - a
      msb = ssbn/float(dfbn)
      msw = sswn/float(dfwn)
      f = msb/msw
      prob = fprob(dfbn, dfwn, f)
      return f, prob
  
  
  def lF_value (ER, EF, dfnum, dfden):
      """
  Returns an F-statistic given the following:
          ER  = error associated with the null hypothesis (the Restricted model)
          EF  = error associated with the alternate hypothesis (the Full model)
          dfR-dfF = degrees of freedom of the numerator
          dfF = degrees of freedom associated with the denominator/Full model
  
  Usage:   lF_value(ER,EF,dfnum,dfden)
  """
      return ((ER-EF)/float(dfnum) / (EF/float(dfden)))
  
  
  ####################################
  ########  SUPPORT FUNCTIONS  #######
  ####################################
  
  def writecc (listoflists,file,writetype='w',extra=2):
      """
  Writes a list of lists to a file in columns, customized by the max
  size of items within the columns (max size of items in col, +2 characters)
  to specified file.  File-overwrite is the default.
  
  Usage:   writecc (listoflists,file,writetype='w',extra=2)
  Returns: None
  """
      if not isinstance(listoflists[0], (list, tuple)):
          listoflists = [listoflists]
      outfile = open(file, writetype)
      rowstokill = []
      list2print = copy.deepcopy(listoflists)
      for i in range(len(listoflists)):
          if listoflists[i] == ['\n'] or listoflists[i]=='\n' or listoflists[i]=='dashes':
              rowstokill = rowstokill + [i]
      rowstokill.reverse()
      for row in rowstokill:
          del list2print[row]
      maxsize = [0]*len(list2print[0])
      for col in range(len(list2print[0])):
          items = pstat.colex(list2print, col)
          maxsize[col] = (max(map(lambda item: len(pstat.makestr(item)), items))
                          + extra)
      for row in listoflists:
          if row == ['\n'] or row == '\n':
              outfile.write('\n')
          elif row == ['dashes'] or row == 'dashes':
              dashes = [0]*len(maxsize)
              for j in range(len(maxsize)):
                  dashes[j] = '-'*(maxsize[j]-2)
              outfile.write(pstat.lineincustcols(dashes, maxsize))
          else:
              outfile.write(pstat.lineincustcols(row, maxsize))
          outfile.write('\n')
      outfile.close()
      return None
  
  
  def lincr(l, cap):        # to increment a list up to a max-list of 'cap'
      """
  Simulate a counting system from an n-dimensional list.
  
  Usage:   lincr(l,cap)   l=list to increment, cap=max values for each list pos'n
  Returns: next set of values for list l, OR -1 (if overflow)
  """
      l[0] = l[0] + 1     # e.g., [0,0,0] --> [2,4,3] (=cap)
      for i in range(len(l)):
          if l[i] > cap[i] and i < len(l)-1: # if carryover AND not done
              l[i] = 0
              l[i+1] = l[i+1] + 1
          elif l[i] > cap[i] and i == len(l)-1: # overflow past last column, must be finished
              l = -1
      return l
  
  
  def lsum (inlist):
      """
  Returns the sum of the items in the passed list.
  
  Usage:   lsum(inlist)
  """
      s = 0
      for item in inlist:
          s = s + item
      return s
  
  
  def lcumsum (inlist):
      """
  Returns a list consisting of the cumulative sum of the items in the
  passed list.
  
  Usage:   lcumsum(inlist)
  """
      newlist = copy.deepcopy(inlist)
      for i in range(1, len(newlist)):
          newlist[i] = newlist[i] + newlist[i-1]
      return newlist
  
  
  def lss(inlist):
      """
  Squares each value in the passed list, adds up these squares and
  returns the result.
  
  Usage:   lss(inlist)
  """
      ss = 0
      for item in inlist:
          ss = ss + item*item
      return ss
  
  
  def lsummult (list1, list2):
      """
  Multiplies elements in list1 and list2, element by element, and
  returns the sum of all resulting multiplications.  Must provide equal
  length lists.
  
  Usage:   lsummult(list1,list2)
  """
      if len(list1) != len(list2):
          raise ValueError("Lists not equal length in summult.")
      s = 0
      for item1, item2 in pstat.abut(list1, list2):
          s = s + item1*item2
      return s
  
  
  def lsumdiffsquared(x, y):
      """
  Takes pairwise differences of the values in lists x and y, squares
  these differences, and returns the sum of these squares.
  
  Usage:   lsumdiffsquared(x,y)
  Returns: sum[(x[i]-y[i])**2]
  """
      sds = 0
      for i in range(len(x)):
          sds = sds + (x[i]-y[i])**2
      return sds
  
  
  def lsquare_of_sums(inlist):
      """
  Adds the values in the passed list, squares the sum, and returns
  the result.
  
  Usage:   lsquare_of_sums(inlist)
  Returns: sum(inlist[i])**2
  """
      s = sum(inlist)
      return float(s)*s
  
  
  def lshellsort(inlist):
      """
  Shellsort algorithm.  Sorts a 1D-list.
  
  Usage:   lshellsort(inlist)
  Returns: sorted-inlist, sorting-index-vector (for original list)
  """
      n = len(inlist)
      svec = copy.deepcopy(inlist)
      ivec = list(range(n))
      gap = n // 2   # integer division needed
      while gap >0:
          for i in range(gap, n):
              for j in range(i-gap, -1, -gap):
                  while j>=0 and svec[j]>svec[j+gap]:
                      temp        = svec[j]
                      svec[j]     = svec[j+gap]
                      svec[j+gap] = temp
                      itemp       = ivec[j]
                      ivec[j]     = ivec[j+gap]
                      ivec[j+gap] = itemp
          gap = gap // 2  # integer division needed
  # svec is now sorted inlist, and ivec has the order svec[i] = vec[ivec[i]]
      return svec, ivec
  
  
  def lrankdata(inlist):
      """
  Ranks the data in inlist, dealing with ties appropritely.  Assumes
  a 1D inlist.  Adapted from Gary Perlman's |Stat ranksort.
  
  Usage:   lrankdata(inlist)
  Returns: a list of length equal to inlist, containing rank scores
  """
      n = len(inlist)
      svec, ivec = shellsort(inlist)
      sumranks = 0
      dupcount = 0
      newlist = [0]*n
      for i in range(n):
          sumranks = sumranks + i
          dupcount = dupcount + 1
          if i == n - 1 or svec[i] != svec[i + 1]:
              averank = sumranks / float(dupcount) + 1
              for j in range(i-dupcount+1, i+1):
                  newlist[ivec[j]] = averank
              sumranks = 0
              dupcount = 0
      return newlist
  
  
  def outputpairedstats(fname, writemode, name1, n1, m1, se1, min1, max1, name2, n2, m2, se2, min2, max2, statname, stat, prob):
      """
  Prints or write to a file stats for two groups, using the name, n,
  mean, sterr, min and max for each group, as well as the statistic name,
  its value, and the associated p-value.
  
  Usage:   outputpairedstats(fname,writemode,
                             name1,n1,mean1,stderr1,min1,max1,
                             name2,n2,mean2,stderr2,min2,max2,
                             statname,stat,prob)
  Returns: None
  """
      suffix = ''                       # for *s after the p-value
      try:
          x = prob.shape
          prob = prob[0]
      except:
          pass
      if  prob < 0.001:  suffix = '  ***'
      elif prob < 0.01:  suffix = '  **'
      elif prob < 0.05:  suffix = '  *'
      title = [['Name', 'N', 'Mean', 'SD', 'Min', 'Max']]
      lofl = title+[[name1, n1, round(m1, 3), round(math.sqrt(se1), 3), min1, max1],
                    [name2, n2, round(m2, 3), round(math.sqrt(se2), 3), min2, max2]]
      if not isinstance(fname, str) or len(fname) == 0:
          print()
          print(statname)
          print()
          pstat.printcc(lofl)
          print()
          try:
              if stat.shape == ():
                  stat = stat[0]
              if prob.shape == ():
                  prob = prob[0]
          except:
              pass
          print('Test statistic = ', round(stat, 3), '   p = ', round(prob, 3), suffix)
          print()
      else:
          file = open(fname, writemode)
          file.write('\n'+statname+'\n\n')
          file.close()
          writecc(lofl, fname, 'a')
          file = open(fname, 'a')
          try:
              if stat.shape == ():
                  stat = stat[0]
              if prob.shape == ():
                  prob = prob[0]
          except:
              pass
          file.write(pstat.list2string(['\nTest statistic = ', round(stat, 4), '   p = ', round(prob, 4), suffix, '\n\n']))
          file.close()
      return None
  
  
  def lfindwithin (data):
      """
  Returns an integer representing a binary vector, where 1=within-
  subject factor, 0=between.  Input equals the entire data 2D list (i.e.,
  column 0=random factor, column -1=measured values (those two are skipped).
  Note: input data is in |Stat format ... a list of lists ("2D list") with 
  one row per measured value, first column=subject identifier, last column=
  score, one in-between column per factor (these columns contain level
  designations on each factor).  See also stats.anova.__doc__.
  
  Usage:   lfindwithin(data)     data in |Stat format
  """
  
      numfact = len(data[0])-1
      withinvec = 0
      for col in range(1, numfact):
          examplelevel = pstat.unique(pstat.colex(data, col))[0]
          rows = pstat.linexand(data, col, examplelevel)  # get 1 level of this factor
          factsubjs = pstat.unique(pstat.colex(rows, 0))
          allsubjs = pstat.unique(pstat.colex(data, 0))
          if len(factsubjs) == len(allsubjs):  # fewer Ss than scores on this factor?
              withinvec = withinvec + (1 << col)
      return withinvec
  
  
  #########################################################
  #########################################################
  ####### DISPATCH LISTS AND TUPLES TO ABOVE FCNS #########
  #########################################################
  #########################################################
  
  ## CENTRAL TENDENCY:
  geometricmean = Dispatch ( (lgeometricmean, (list, tuple)), )
  harmonicmean = Dispatch ( (lharmonicmean, (list, tuple)), )
  mean = Dispatch ( (lmean, (list, tuple)), )
  median = Dispatch ( (lmedian, (list, tuple)), )
  medianscore = Dispatch ( (lmedianscore, (list, tuple)), )
  mode = Dispatch ( (lmode, (list, tuple)), )
  
  ## MOMENTS:
  moment = Dispatch ( (lmoment, (list, tuple)), )
  variation = Dispatch ( (lvariation, (list, tuple)), )
  skew = Dispatch ( (lskew, (list, tuple)), )
  kurtosis = Dispatch ( (lkurtosis, (list, tuple)), )
  describe = Dispatch ( (ldescribe, (list, tuple)), )
  
  ## FREQUENCY STATISTICS:
  itemfreq = Dispatch ( (litemfreq, (list, tuple)), )
  scoreatpercentile = Dispatch ( (lscoreatpercentile, (list, tuple)), )
  percentileofscore = Dispatch ( (lpercentileofscore, (list, tuple)), )
  histogram = Dispatch ( (lhistogram, (list, tuple)), )
  cumfreq = Dispatch ( (lcumfreq, (list, tuple)), )
  relfreq = Dispatch ( (lrelfreq, (list, tuple)), )
  
  ## VARIABILITY:
  obrientransform = Dispatch ( (lobrientransform, (list, tuple)), )
  samplevar = Dispatch ( (lsamplevar, (list, tuple)), )
  samplestdev = Dispatch ( (lsamplestdev, (list, tuple)), )
  var = Dispatch ( (lvar, (list, tuple)), )
  stdev = Dispatch ( (lstdev, (list, tuple)), )
  sterr = Dispatch ( (lsterr, (list, tuple)), )
  sem = Dispatch ( (lsem, (list, tuple)), )
  z = Dispatch ( (lz, (list, tuple)), )
  zs = Dispatch ( (lzs, (list, tuple)), )
  
  ## TRIMMING FCNS:
  trimboth = Dispatch ( (ltrimboth, (list, tuple)), )
  trim1 = Dispatch ( (ltrim1, (list, tuple)), )
  
  ## CORRELATION FCNS:
  paired = Dispatch ( (lpaired, (list, tuple)), )
  pearsonr = Dispatch ( (lpearsonr, (list, tuple)), )
  spearmanr = Dispatch ( (lspearmanr, (list, tuple)), )
  pointbiserialr = Dispatch ( (lpointbiserialr, (list, tuple)), )
  kendalltau = Dispatch ( (lkendalltau, (list, tuple)), )
  linregress = Dispatch ( (llinregress, (list, tuple)), )
  
  ## INFERENTIAL STATS:
  ttest_1samp = Dispatch ( (lttest_1samp, (list, tuple)), )
  ttest_ind = Dispatch ( (lttest_ind, (list, tuple)), )
  ttest_rel = Dispatch ( (lttest_rel, (list, tuple)), )
  chisquare = Dispatch ( (lchisquare, (list, tuple)), )
  ks_2samp = Dispatch ( (lks_2samp, (list, tuple)), )
  mannwhitneyu = Dispatch ( (lmannwhitneyu, (list, tuple)), )
  ranksums = Dispatch ( (lranksums, (list, tuple)), )
  tiecorrect = Dispatch ( (ltiecorrect, (list, tuple)), )
  wilcoxont = Dispatch ( (lwilcoxont, (list, tuple)), )
  kruskalwallish = Dispatch ( (lkruskalwallish, (list, tuple)), )
  friedmanchisquare = Dispatch ( (lfriedmanchisquare, (list, tuple)), )
  
  ## PROBABILITY CALCS:
  chisqprob = Dispatch ( (lchisqprob, (int, float)), )
  zprob = Dispatch ( (lzprob, (int, float)), )
  ksprob = Dispatch ( (lksprob, (int, float)), )
  fprob = Dispatch ( (lfprob, (int, float)), )
  betacf = Dispatch ( (lbetacf, (int, float)), )
  betai = Dispatch ( (lbetai, (int, float)), )
  erfcc = Dispatch ( (lerfcc, (int, float)), )
  gammln = Dispatch ( (lgammln, (int, float)), )
  
  ## ANOVA FUNCTIONS:
  F_oneway = Dispatch ( (lF_oneway, (list, tuple)), )
  F_value = Dispatch ( (lF_value, (list, tuple)), )
  
  ## SUPPORT FUNCTIONS:
  incr = Dispatch ( (lincr, (list, tuple)), )
  sum = Dispatch ( (lsum, (list, tuple)), )
  cumsum = Dispatch ( (lcumsum, (list, tuple)), )
  ss = Dispatch ( (lss, (list, tuple)), )
  summult = Dispatch ( (lsummult, (list, tuple)), )
  square_of_sums = Dispatch ( (lsquare_of_sums, (list, tuple)), )
  sumdiffsquared = Dispatch ( (lsumdiffsquared, (list, tuple)), )
  shellsort = Dispatch ( (lshellsort, (list, tuple)), )
  rankdata = Dispatch ( (lrankdata, (list, tuple)), )
  findwithin = Dispatch ( (lfindwithin, (list, tuple)), )
  
  
  #=============  THE ARRAY-VERSION OF THE STATS FUNCTIONS  ===============
  #=============  THE ARRAY-VERSION OF THE STATS FUNCTIONS  ===============
  #=============  THE ARRAY-VERSION OF THE STATS FUNCTIONS  ===============
  #=============  THE ARRAY-VERSION OF THE STATS FUNCTIONS  ===============
  #=============  THE ARRAY-VERSION OF THE STATS FUNCTIONS  ===============
  #=============  THE ARRAY-VERSION OF THE STATS FUNCTIONS  ===============
  #=============  THE ARRAY-VERSION OF THE STATS FUNCTIONS  ===============
  #=============  THE ARRAY-VERSION OF THE STATS FUNCTIONS  ===============
  #=============  THE ARRAY-VERSION OF THE STATS FUNCTIONS  ===============
  #=============  THE ARRAY-VERSION OF THE STATS FUNCTIONS  ===============
  #=============  THE ARRAY-VERSION OF THE STATS FUNCTIONS  ===============
  #=============  THE ARRAY-VERSION OF THE STATS FUNCTIONS  ===============
  #=============  THE ARRAY-VERSION OF THE STATS FUNCTIONS  ===============
  #=============  THE ARRAY-VERSION OF THE STATS FUNCTIONS  ===============
  #=============  THE ARRAY-VERSION OF THE STATS FUNCTIONS  ===============
  #=============  THE ARRAY-VERSION OF THE STATS FUNCTIONS  ===============
  #=============  THE ARRAY-VERSION OF THE STATS FUNCTIONS  ===============
  #=============  THE ARRAY-VERSION OF THE STATS FUNCTIONS  ===============
  #=============  THE ARRAY-VERSION OF THE STATS FUNCTIONS  ===============
  
  try:                         # DEFINE THESE *ONLY* IF NUMERIC IS AVAILABLE
   import numpy as N
   import numpy.linalg as LA
  
  
  #####################################
  ########  ACENTRAL TENDENCY  ########
  #####################################
  
   def ageometricmean (inarray,dimension=None,keepdims=0):
      """
  Calculates the geometric mean of the values in the passed array.
  That is:  n-th root of (x1 * x2 * ... * xn).  Defaults to ALL values in
  the passed array.  Use dimension=None to flatten array first.  REMEMBER: if
  dimension=0, it collapses over dimension 0 ('rows' in a 2D array) only, and
  if dimension is a sequence, it collapses over all specified dimensions.  If
  keepdims is set to 1, the resulting array will have as many dimensions as
  inarray, with only 1 'level' per dim that was collapsed over.
  
  Usage:   ageometricmean(inarray,dimension=None,keepdims=0)
  Returns: geometric mean computed over dim(s) listed in dimension
  """
      inarray = N.array(inarray, N.float_)
      if dimension is None:
          inarray = N.ravel(inarray)
          size = len(inarray)
          mult = N.power(inarray, 1.0/size)
          mult = N.multiply.reduce(mult)
      elif isinstance(dimension, (int, float)):
          size = inarray.shape[dimension]
          mult = N.power(inarray, 1.0/size)
          mult = N.multiply.reduce(mult, dimension)
          if keepdims == 1:
              shp = list(inarray.shape)
              shp[dimension] = 1
              sum = N.reshape(sum, shp)
      else: # must be a SEQUENCE of dims to average over
          dims = sorted(dimension, reverse=True)
          size = N.array(N.multiply.reduce(N.take(inarray.shape, dims)), N.float_)
          mult = N.power(inarray, 1.0/size)
          for dim in dims:
              mult = N.multiply.reduce(mult, dim)
          if keepdims == 1:
              shp = list(inarray.shape)
              for dim in dims:
                  shp[dim] = 1
              mult = N.reshape(mult, shp)
      return mult
  
  
   def aharmonicmean (inarray,dimension=None,keepdims=0):
      """
  Calculates the harmonic mean of the values in the passed array.
  That is:  n / (1/x1 + 1/x2 + ... + 1/xn).  Defaults to ALL values in
  the passed array.  Use dimension=None to flatten array first.  REMEMBER: if
  dimension=0, it collapses over dimension 0 ('rows' in a 2D array) only, and
  if dimension is a sequence, it collapses over all specified dimensions.  If
  keepdims is set to 1, the resulting array will have as many dimensions as
  inarray, with only 1 'level' per dim that was collapsed over.
  
  Usage:   aharmonicmean(inarray,dimension=None,keepdims=0)
  Returns: harmonic mean computed over dim(s) in dimension
  """
      inarray = inarray.astype(N.float_)
      if dimension is None:
          inarray = N.ravel(inarray)
          size = len(inarray)
          s = N.add.reduce(1.0 / inarray)
      elif isinstance(dimension, (int, float)):
          size = float(inarray.shape[dimension])
          s = N.add.reduce(1.0/inarray, dimension)
          if keepdims == 1:
              shp = list(inarray.shape)
              shp[dimension] = 1
              s = N.reshape(s, shp)
      else: # must be a SEQUENCE of dims to average over
          dims = sorted(dimension)
          nondims = []
          for i in range(len(inarray.shape)):
              if i not in dims:
                  nondims.append(i)
          tinarray = N.transpose(inarray, nondims+dims) # put keep-dims first
          idx = [0] *len(nondims)
          if idx == []:
              size = len(N.ravel(inarray))
              s = asum(1.0 / inarray)
              if keepdims == 1:
                  s = N.reshape([s], N.ones(len(inarray.shape)))
          else:
              idx[0] = -1
              loopcap = N.array(tinarray.shape[0:len(nondims)]) -1
              s = N.zeros(loopcap+1, N.float_)
              while incr(idx, loopcap) != -1:
                  s[idx] = asum(1.0/tinarray[idx])
              size = N.multiply.reduce(N.take(inarray.shape, dims))
              if keepdims == 1:
                  shp = list(inarray.shape)
                  for dim in dims:
                      shp[dim] = 1
                  s = N.reshape(s, shp)
      return size / s
  
  
   def amean (inarray,dimension=None,keepdims=0):
      """
  Calculates the arithmatic mean of the values in the passed array.
  That is:  1/n * (x1 + x2 + ... + xn).  Defaults to ALL values in the
  passed array.  Use dimension=None to flatten array first.  REMEMBER: if
  dimension=0, it collapses over dimension 0 ('rows' in a 2D array) only, and
  if dimension is a sequence, it collapses over all specified dimensions.  If
  keepdims is set to 1, the resulting array will have as many dimensions as
  inarray, with only 1 'level' per dim that was collapsed over.
  
  Usage:   amean(inarray,dimension=None,keepdims=0)
  Returns: arithematic mean calculated over dim(s) in dimension
  """
      if inarray.dtype in [N.int_, N.short, N.ubyte]:
          inarray = inarray.astype(N.float_)
      if dimension is None:
          inarray = N.ravel(inarray)
          sum = N.add.reduce(inarray)
          denom = float(len(inarray))
      elif isinstance(dimension, (int, float)):
          sum = asum(inarray, dimension)
          denom = float(inarray.shape[dimension])
          if keepdims == 1:
              shp = list(inarray.shape)
              shp[dimension] = 1
              sum = N.reshape(sum, shp)
      else: # must be a TUPLE of dims to average over
          dims = sorted(dimension, reverse=True)
          sum = inarray *1.0
          for dim in dims:
              sum = N.add.reduce(sum, dim)
          denom = N.array(N.multiply.reduce(N.take(inarray.shape, dims)), N.float_)
          if keepdims == 1:
              shp = list(inarray.shape)
              for dim in dims:
                  shp[dim] = 1
              sum = N.reshape(sum, shp)
      return sum/denom
  
  
   def amedian (inarray,numbins=1000):
      """
  Calculates the COMPUTED median value of an array of numbers, given the
  number of bins to use for the histogram (more bins approaches finding the
  precise median value of the array; default number of bins = 1000).  From
  G.W. Heiman's Basic Stats, or CRC Probability & Statistics.
  NOTE:  THIS ROUTINE ALWAYS uses the entire passed array (flattens it first).
  
  Usage:   amedian(inarray,numbins=1000)
  Returns: median calculated over ALL values in inarray
  """
      inarray = N.ravel(inarray)
      (hist, smallest, binsize, extras) = ahistogram(inarray, numbins, [min(inarray), max(inarray)])
      cumhist = N.cumsum(hist)            # make cumulative histogram
      otherbins = N.greater_equal(cumhist, len(inarray)/2.0)
      otherbins = list(otherbins)         # list of 0/1s, 1s start at median bin
      cfbin = otherbins.index(1)                # get 1st(!) index holding 50%ile score
      LRL = smallest + binsize*cfbin        # get lower read limit of that bin
      cfbelow = N.add.reduce(hist[0:cfbin])        # cum. freq. below bin
      freq = hist[cfbin]                        # frequency IN the 50%ile bin
      median = LRL + ((len(inarray)/2.0-cfbelow)/float(freq))*binsize # MEDIAN
      return median
  
  
   def amedianscore (inarray,dimension=None):
      """
  Returns the 'middle' score of the passed array.  If there is an even
  number of scores, the mean of the 2 middle scores is returned.  Can function
  with 1D arrays, or on the FIRST dimension of 2D arrays (i.e., dimension can
  be None, to pre-flatten the array, or else dimension must equal 0).
  
  Usage:   amedianscore(inarray,dimension=None)
  Returns: 'middle' score of the array, or the mean of the 2 middle scores
  """
      if dimension is None:
          inarray = N.ravel(inarray)
          dimension = 0
      inarray = N.sort(inarray, dimension)
      if inarray.shape[dimension] % 2 == 0:   # if even number of elements
          indx = inarray.shape[dimension]/2   # integer division correct
          median = N.asarray(inarray[indx]+inarray[indx-1]) / 2.0
      else:
          indx = inarray.shape[dimension] / 2 # integer division correct
          median = N.take(inarray, [indx], dimension)
          if median.shape == (1,):
              median = median[0]
      return median
  
  
   def amode(a, dimension=None):
      """
  Returns an array of the modal (most common) score in the passed array.
  If there is more than one such score, ONLY THE FIRST is returned.
  The bin-count for the modal values is also returned.  Operates on whole
  array (dimension=None), or on a given dimension.
  
  Usage:   amode(a, dimension=None)
  Returns: array of bin-counts for mode(s), array of corresponding modal values
  """
  
      if dimension is None:
          a = N.ravel(a)
          dimension = 0
      scores = pstat.aunique(N.ravel(a))       # get ALL unique values
      testshape = list(a.shape)
      testshape[dimension] = 1
      oldmostfreq = N.zeros(testshape)
      oldcounts = N.zeros(testshape)
      for score in scores:
          template = N.equal(a, score)
          counts = asum(template, dimension, 1)
          mostfrequent = N.where(counts>oldcounts, score, oldmostfreq)
          oldcounts = N.where(counts>oldcounts, counts, oldcounts)
          oldmostfreq = mostfrequent
      return oldcounts, mostfrequent
  
  
   def atmean(a,limits=None,inclusive=(1, 1)):
       """
  Returns the arithmetic mean of all values in an array, ignoring values
  strictly outside the sequence passed to 'limits'.   Note: either limit
  in the sequence, or the value of limits itself, can be set to None.  The
  inclusive list/tuple determines whether the lower and upper limiting bounds
  (respectively) are open/exclusive (0) or closed/inclusive (1).
  
  Usage:   atmean(a,limits=None,inclusive=(1,1))
  """
       if a.dtype in [N.int_, N.short, N.ubyte]:
           a = a.astype(N.float_)
       if limits is None:
           return mean(a)
       assert isinstance(limits, (list, tuple, N.ndarray)), "Wrong type for limits in atmean"
       if inclusive[0]:         lowerfcn = N.greater_equal
       else:               lowerfcn = N.greater
       if inclusive[1]:         upperfcn = N.less_equal
       else:               upperfcn = N.less
       if limits[0] > N.maximum.reduce(N.ravel(a)) or limits[1] < N.minimum.reduce(N.ravel(a)):
           raise ValueError("No array values within given limits (atmean).")
       elif limits[0] is None and limits[1] is not None:
           mask = upperfcn(a, limits[1])
       elif limits[0] is not None and limits[1] is None:
           mask = lowerfcn(a, limits[0])
       elif limits[0] is not None and limits[1] is not None:
           mask = lowerfcn(a, limits[0])*upperfcn(a, limits[1])
       s = float(N.add.reduce(N.ravel(a*mask)))
       n = float(N.add.reduce(N.ravel(mask)))
       return s/n
  
  
   def atvar(a,limits=None,inclusive=(1, 1)):
       """
  Returns the sample variance of values in an array, (i.e., using N-1),
  ignoring values strictly outside the sequence passed to 'limits'.  
  Note: either limit in the sequence, or the value of limits itself,
  can be set to None.  The inclusive list/tuple determines whether the lower
  and upper limiting bounds (respectively) are open/exclusive (0) or
  closed/inclusive (1). ASSUMES A FLAT ARRAY (OR ELSE PREFLATTENS).
  
  Usage:   atvar(a,limits=None,inclusive=(1,1))
  """
       a = a.astype(N.float_)
       if limits is None or limits == [None, None]:
           return avar(a)
       assert isinstance(limits, (list, tuple, N.ndarray)), "Wrong type for limits in atvar"
       if inclusive[0]:    lowerfcn = N.greater_equal
       else:               lowerfcn = N.greater
       if inclusive[1]:    upperfcn = N.less_equal
       else:               upperfcn = N.less
       if limits[0] > N.maximum.reduce(N.ravel(a)) or limits[1] < N.minimum.reduce(N.ravel(a)):
           raise ValueError("No array values within given limits (atvar).")
       elif limits[0] is None and limits[1] is not None:
           mask = upperfcn(a, limits[1])
       elif limits[0] is not None and limits[1] is None:
           mask = lowerfcn(a, limits[0])
       elif limits[0] is not None and limits[1] is not None:
           mask = lowerfcn(a, limits[0])*upperfcn(a, limits[1])
  
       a = N.compress(mask, a)  # squish out excluded values
       return avar(a)
  
  
   def atmin(a,lowerlimit=None,dimension=None,inclusive=1):
       """
  Returns the minimum value of a, along dimension, including only values less
  than (or equal to, if inclusive=1) lowerlimit.  If the limit is set to None,
  all values in the array are used.
  
  Usage:   atmin(a,lowerlimit=None,dimension=None,inclusive=1)
  """
       if inclusive:         lowerfcn = N.greater
       else:               lowerfcn = N.greater_equal
       if dimension is None:
           a = N.ravel(a)
           dimension = 0
       if lowerlimit is None:
           lowerlimit = N.minimum.reduce(N.ravel(a))-11
       biggest = N.maximum.reduce(N.ravel(a))
       ta = N.where(lowerfcn(a, lowerlimit), a, biggest)
       return N.minimum.reduce(ta, dimension)
  
  
   def atmax(a,upperlimit,dimension=None,inclusive=1):
       """
  Returns the maximum value of a, along dimension, including only values greater
  than (or equal to, if inclusive=1) upperlimit.  If the limit is set to None,
  a limit larger than the max value in the array is used.
  
  Usage:   atmax(a,upperlimit,dimension=None,inclusive=1)
  """
       if inclusive:         upperfcn = N.less
       else:               upperfcn = N.less_equal
       if dimension is None:
           a = N.ravel(a)
           dimension = 0
       if upperlimit is None:
           upperlimit = N.maximum.reduce(N.ravel(a))+1
       smallest = N.minimum.reduce(N.ravel(a))
       ta = N.where(upperfcn(a, upperlimit), a, smallest)
       return N.maximum.reduce(ta, dimension)
  
  
   def atstdev(a,limits=None,inclusive=(1, 1)):
       """
  Returns the standard deviation of all values in an array, ignoring values
  strictly outside the sequence passed to 'limits'.   Note: either limit
  in the sequence, or the value of limits itself, can be set to None.  The
  inclusive list/tuple determines whether the lower and upper limiting bounds
  (respectively) are open/exclusive (0) or closed/inclusive (1).
  
  Usage:   atstdev(a,limits=None,inclusive=(1,1))
  """
       return N.sqrt(tvar(a, limits, inclusive))
  
  
   def atsem(a,limits=None,inclusive=(1, 1)):
       """
  Returns the standard error of the mean for the values in an array,
  (i.e., using N for the denominator), ignoring values strictly outside
  the sequence passed to 'limits'.   Note: either limit in the sequence,
  or the value of limits itself, can be set to None.  The inclusive list/tuple
  determines whether the lower and upper limiting bounds (respectively) are
  open/exclusive (0) or closed/inclusive (1).
  
  Usage:   atsem(a,limits=None,inclusive=(1,1))
  """
       sd = tstdev(a, limits, inclusive)
       if limits is None or limits == [None, None]:
           n = float(len(N.ravel(a)))
           limits = [min(a)-1, max(a)+1]
       assert isinstance(limits, (list, tuple, N.ndarray)), "Wrong type for limits in atsem"
       if inclusive[0]:         lowerfcn = N.greater_equal
       else:               lowerfcn = N.greater
       if inclusive[1]:         upperfcn = N.less_equal
       else:               upperfcn = N.less
       if limits[0] > N.maximum.reduce(N.ravel(a)) or limits[1] < N.minimum.reduce(N.ravel(a)):
           raise ValueError("No array values within given limits (atsem).")
       elif limits[0] is None and limits[1] is not None:
           mask = upperfcn(a, limits[1])
       elif limits[0] is not None and limits[1] is None:
           mask = lowerfcn(a, limits[0])
       elif limits[0] is not None and limits[1] is not None:
           mask = lowerfcn(a, limits[0])*upperfcn(a, limits[1])
       term1 = N.add.reduce(N.ravel(a*a*mask))
       n = float(N.add.reduce(N.ravel(mask)))
       return sd/math.sqrt(n)
  
  
  #####################################
  ############  AMOMENTS  #############
  #####################################
  
   def amoment(a,moment=1,dimension=None):
      """
  Calculates the nth moment about the mean for a sample (defaults to the
  1st moment).  Generally used to calculate coefficients of skewness and
  kurtosis.  Dimension can equal None (ravel array first), an integer
  (the dimension over which to operate), or a sequence (operate over
  multiple dimensions).
  
  Usage:   amoment(a,moment=1,dimension=None)
  Returns: appropriate moment along given dimension
  """
      if dimension is None:
          a = N.ravel(a)
          dimension = 0
      if moment == 1:
          return 0.0
      else:
          mn = amean(a, dimension, 1)  # 1=keepdims
          s = N.power((a-mn), moment)
          return amean(s, dimension)
  
  
   def avariation(a,dimension=None):
      """
  Returns the coefficient of variation, as defined in CRC Standard
  Probability and Statistics, p.6. Dimension can equal None (ravel array
  first), an integer (the dimension over which to operate), or a
  sequence (operate over multiple dimensions).
  
  Usage:   avariation(a,dimension=None)
  """
      return 100.0*asamplestdev(a, dimension)/amean(a, dimension)
  
  
   def askew(a,dimension=None): 
      """ 
  Returns the skewness of a distribution (normal ==> 0.0; >0 means extra
  weight in left tail).  Use askewtest() to see if it's close enough.
  Dimension can equal None (ravel array first), an integer (the
  dimension over which to operate), or a sequence (operate over multiple
  dimensions).
  
  Usage:   askew(a, dimension=None)
  Returns: skew of vals in a along dimension, returning ZERO where all vals equal
  """
      denom = N.power(amoment(a, 2, dimension), 1.5)
      zero = N.equal(denom, 0)
      if isinstance(denom, N.ndarray) and asum(zero) != 0:
          print("Number of zeros in askew: ", asum(zero))
      denom = denom + zero  # prevent divide-by-zero
      return N.where(zero, 0, amoment(a, 3, dimension)/denom)
  
  
   def akurtosis(a,dimension=None):
      """
  Returns the kurtosis of a distribution (normal ==> 3.0; >3 means
  heavier in the tails, and usually more peaked).  Use akurtosistest()
  to see if it's close enough.  Dimension can equal None (ravel array
  first), an integer (the dimension over which to operate), or a
  sequence (operate over multiple dimensions).
  
  Usage:   akurtosis(a,dimension=None)
  Returns: kurtosis of values in a along dimension, and ZERO where all vals equal
  """
      denom = N.power(amoment(a, 2, dimension), 2)
      zero = N.equal(denom, 0)
      if isinstance(denom, N.ndarray) and asum(zero) != 0:
          print("Number of zeros in akurtosis: ", asum(zero))
      denom = denom + zero  # prevent divide-by-zero
      return N.where(zero, 0, amoment(a, 4, dimension)/denom)
  
  
   def adescribe(inarray,dimension=None):
       """
  Returns several descriptive statistics of the passed array.  Dimension
  can equal None (ravel array first), an integer (the dimension over
  which to operate), or a sequence (operate over multiple dimensions).
  
  Usage:   adescribe(inarray,dimension=None)
  Returns: n, (min,max), mean, standard deviation, skew, kurtosis
  """
       if dimension is None:
           inarray = N.ravel(inarray)
           dimension = 0
       n = inarray.shape[dimension]
       mm = (N.minimum.reduce(inarray), N.maximum.reduce(inarray))
       m = amean(inarray, dimension)
       sd = astdev(inarray, dimension)
       skew = askew(inarray, dimension)
       kurt = akurtosis(inarray, dimension)
       return n, mm, m, sd, skew, kurt
  
  
  #####################################
  ########  NORMALITY TESTS  ##########
  #####################################
  
   def askewtest(a,dimension=None):
      """
  Tests whether the skew is significantly different from a normal
  distribution.  Dimension can equal None (ravel array first), an
  integer (the dimension over which to operate), or a sequence (operate
  over multiple dimensions).
  
  Usage:   askewtest(a,dimension=None)
  Returns: z-score and 2-tail z-probability
  """
      if dimension is None:
          a = N.ravel(a)
          dimension = 0
      b2 = askew(a, dimension)
      n = float(a.shape[dimension])
      y = b2 * N.sqrt(((n+1)*(n+3)) / (6.0*(n-2)) )
      beta2 = ( 3.0*(n*n+27*n-70)*(n+1)*(n+3) ) / ( (n-2.0)*(n+5)*(n+7)*(n+9) )
      W2 = -1 + N.sqrt(2*(beta2-1))
      delta = 1/N.sqrt(N.log(N.sqrt(W2)))
      alpha = N.sqrt(2/(W2-1))
      y = N.where(y==0, 1, y)
      Z = delta*N.log(y/alpha + N.sqrt((y/alpha)**2+1))
      return Z, (1.0-zprob(Z))*2
  
  
   def akurtosistest(a,dimension=None):
      """
  Tests whether a dataset has normal kurtosis (i.e.,
  kurtosis=3(n-1)/(n+1)) Valid only for n>20.  Dimension can equal None
  (ravel array first), an integer (the dimension over which to operate),
  or a sequence (operate over multiple dimensions).
  
  Usage:   akurtosistest(a,dimension=None)
  Returns: z-score and 2-tail z-probability, returns 0 for bad pixels
  """
      if dimension is None:
          a = N.ravel(a)
          dimension = 0
      n = float(a.shape[dimension])
      if n<20:
          print("akurtosistest only valid for n>=20 ... continuing anyway, n=", n)
      b2 = akurtosis(a, dimension)
      E = 3.0*(n-1) /(n+1)
      varb2 = 24.0*n*(n-2)*(n-3) / ((n+1)*(n+1)*(n+3)*(n+5))
      x = (b2-E)/N.sqrt(varb2)
      sqrtbeta1 = 6.0*(n*n-5*n+2)/((n+7)*(n+9)) * N.sqrt((6.0*(n+3)*(n+5))/
                                                         (n*(n-2)*(n-3)))
      A = 6.0 + 8.0/sqrtbeta1 *(2.0/sqrtbeta1 + N.sqrt(1+4.0/(sqrtbeta1**2)))
      term1 = 1 -2/(9.0*A)
      denom = 1 +x*N.sqrt(2/(A-4.0))
      denom = N.where(N.less(denom, 0), 99, denom)
      term2 = N.where(N.equal(denom, 0), term1, N.power((1-2.0/A)/denom, 1/3.0))
      Z = ( term1 - term2 ) / N.sqrt(2/(9.0*A))
      Z = N.where(N.equal(denom, 99), 0, Z)
      return Z, (1.0-zprob(Z))*2
  
  
   def anormaltest(a,dimension=None):
      """
  Tests whether skew and/OR kurtosis of dataset differs from normal
  curve.  Can operate over multiple dimensions.  Dimension can equal
  None (ravel array first), an integer (the dimension over which to
  operate), or a sequence (operate over multiple dimensions).
  
  Usage:   anormaltest(a,dimension=None)
  Returns: z-score and 2-tail probability
  """
      if dimension is None:
          a = N.ravel(a)
          dimension = 0
      s, p = askewtest(a, dimension)
      k, p = akurtosistest(a, dimension)
      k2 = N.power(s, 2) + N.power(k, 2)
      return k2, achisqprob(k2, 2)
  
  
  #####################################
  ######  AFREQUENCY FUNCTIONS  #######
  #####################################
  
   def aitemfreq(a):
      """
  Returns a 2D array of item frequencies.  Column 1 contains item values,
  column 2 contains their respective counts.  Assumes a 1D array is passed.
  @@@sorting OK?
  
  Usage:   aitemfreq(a)
  Returns: a 2D frequency table (col [0:n-1]=scores, col n=frequencies)
  """
      scores = pstat.aunique(a)
      scores = N.sort(scores)
      freq = N.zeros(len(scores))
      for i in range(len(scores)):
          freq[i] = N.add.reduce(N.equal(a, scores[i]))
      return N.array(pstat.aabut(scores, freq))
  
  
   def ascoreatpercentile (inarray, percent):
      """
  Usage:   ascoreatpercentile(inarray,percent)   0<percent<100
  Returns: score at given percentile, relative to inarray distribution
  """
      percent = percent / 100.0
      targetcf = percent*len(inarray)
      h, lrl, binsize, extras = histogram(inarray)
      cumhist = cumsum(h*1)
      for i in range(len(cumhist)):
          if cumhist[i] >= targetcf:
              break
      score = binsize * ((targetcf - cumhist[i-1]) / float(h[i])) + (lrl+binsize*i)
      return score
  
  
   def apercentileofscore (inarray,score,histbins=10,defaultlimits=None):
      """
  Note: result of this function depends on the values used to histogram
  the data(!).
  
  Usage:   apercentileofscore(inarray,score,histbins=10,defaultlimits=None)
  Returns: percentile-position of score (0-100) relative to inarray
  """
      h, lrl, binsize, extras = histogram(inarray, histbins, defaultlimits)
      cumhist = cumsum(h*1)
      i = int((score - lrl)/float(binsize))
      pct = (cumhist[i-1]+((score-(lrl+binsize*i))/float(binsize))*h[i])/float(len(inarray)) * 100
      return pct
  
  
   def ahistogram (inarray,numbins=10,defaultlimits=None,printextras=1):
      """
  Returns (i) an array of histogram bin counts, (ii) the smallest value
  of the histogram binning, and (iii) the bin width (the last 2 are not
  necessarily integers).  Default number of bins is 10.  Defaultlimits
  can be None (the routine picks bins spanning all the numbers in the
  inarray) or a 2-sequence (lowerlimit, upperlimit).  Returns all of the
  following: array of bin values, lowerreallimit, binsize, extrapoints.
  
  Usage:   ahistogram(inarray,numbins=10,defaultlimits=None,printextras=1)
  Returns: (array of bin counts, bin-minimum, min-width, #-points-outside-range)
  """
      inarray = N.ravel(inarray)               # flatten any >1D arrays
      if (defaultlimits is not None):
          lowerreallimit = defaultlimits[0]
          upperreallimit = defaultlimits[1]
          binsize = (upperreallimit-lowerreallimit) / float(numbins)
      else:
          Min = N.minimum.reduce(inarray)
          Max = N.maximum.reduce(inarray)
          estbinwidth = float(Max - Min)/float(numbins) + 1e-6
          binsize = (Max-Min+estbinwidth)/float(numbins)
          lowerreallimit = Min - binsize/2.0  #lower real limit,1st bin
      bins = N.zeros(numbins)
      extrapoints = 0
      for num in inarray:
          try:
              if (num-lowerreallimit) < 0:
                  extrapoints = extrapoints + 1
              else:
                  bintoincrement = int((num-lowerreallimit) / float(binsize))
                  bins[bintoincrement] = bins[bintoincrement] + 1
          except:                           # point outside lower/upper limits
              extrapoints = extrapoints + 1
      if (extrapoints > 0 and printextras == 1):
          print('\nPoints outside given histogram range =', extrapoints)
      return (bins, lowerreallimit, binsize, extrapoints)
  
  
   def acumfreq(a,numbins=10,defaultreallimits=None):
      """
  Returns a cumulative frequency histogram, using the histogram function.
  Defaultreallimits can be None (use all data), or a 2-sequence containing
  lower and upper limits on values to include.
  
  Usage:   acumfreq(a,numbins=10,defaultreallimits=None)
  Returns: array of cumfreq bin values, lowerreallimit, binsize, extrapoints
  """
      h, l, b, e = histogram(a, numbins, defaultreallimits)
      cumhist = cumsum(h*1)
      return cumhist, l, b, e
  
  
   def arelfreq(a,numbins=10,defaultreallimits=None):
      """
  Returns a relative frequency histogram, using the histogram function.
  Defaultreallimits can be None (use all data), or a 2-sequence containing
  lower and upper limits on values to include.
  
  Usage:   arelfreq(a,numbins=10,defaultreallimits=None)
  Returns: array of cumfreq bin values, lowerreallimit, binsize, extrapoints
  """
      h, l, b, e = histogram(a, numbins, defaultreallimits)
      h = N.array(h/float(a.shape[0]))
      return h, l, b, e
  
  
  #####################################
  ######  AVARIABILITY FUNCTIONS  #####
  #####################################
  
   def aobrientransform(*args):
      """
  Computes a transform on input data (any number of columns).  Used to
  test for homogeneity of variance prior to running one-way stats.  Each
  array in *args is one level of a factor.  If an F_oneway() run on the
  transformed data and found significant, variances are unequal.   From
  Maxwell and Delaney, p.112.
  
  Usage:   aobrientransform(*args)    *args = 1D arrays, one per level of factor
  Returns: transformed data for use in an ANOVA
  """
      TINY = 1e-10
      k = len(args)
      n = N.zeros(k, N.float_)
      v = N.zeros(k, N.float_)
      m = N.zeros(k, N.float_)
      nargs = []
      for i in range(k):
          nargs.append(args[i].astype(N.float_))
          n[i] = float(len(nargs[i]))
          v[i] = var(nargs[i])
          m[i] = mean(nargs[i])
      for j in range(k):
          for i in range(n[j]):
              t1 = (n[j]-1.5)*n[j]*(nargs[j][i]-m[j])**2
              t2 = 0.5*v[j]*(n[j]-1.0)
              t3 = (n[j]-1.0)*(n[j]-2.0)
              nargs[j][i] = (t1-t2) / float(t3)
      check = 1
      for j in range(k):
          if v[j] - mean(nargs[j]) > TINY:
              check = 0
      if check != 1:
          raise ValueError('Lack of convergence in obrientransform.')
      else:
          return N.array(nargs)
  
  
   def asamplevar (inarray,dimension=None,keepdims=0):
      """
  Returns the sample standard deviation of the values in the passed
  array (i.e., using N).  Dimension can equal None (ravel array first),
  an integer (the dimension over which to operate), or a sequence
  (operate over multiple dimensions).  Set keepdims=1 to return an array
  with the same number of dimensions as inarray.
  
  Usage:   asamplevar(inarray,dimension=None,keepdims=0)
  """
      if dimension is None:
          inarray = N.ravel(inarray)
          dimension = 0
      if dimension == 1:
          mn = amean(inarray, dimension)[:, N.NewAxis]
      else:
          mn = amean(inarray, dimension, keepdims=1)
      deviations = inarray - mn 
      if isinstance(dimension, list):
          n = 1
          for d in dimension:
              n = n*inarray.shape[d]
      else:
          n = inarray.shape[dimension]
      svar = ass(deviations, dimension, keepdims) / float(n)
      return svar
  
  
   def asamplestdev (inarray, dimension=None, keepdims=0):
      """
  Returns the sample standard deviation of the values in the passed
  array (i.e., using N).  Dimension can equal None (ravel array first),
  an integer (the dimension over which to operate), or a sequence
  (operate over multiple dimensions).  Set keepdims=1 to return an array
  with the same number of dimensions as inarray.
  
  Usage:   asamplestdev(inarray,dimension=None,keepdims=0)
  """
      return N.sqrt(asamplevar(inarray, dimension, keepdims))
  
  
   def asignaltonoise(instack,dimension=0):
      """
  Calculates signal-to-noise.  Dimension can equal None (ravel array
  first), an integer (the dimension over which to operate), or a
  sequence (operate over multiple dimensions).
  
  Usage:   asignaltonoise(instack,dimension=0):
  Returns: array containing the value of (mean/stdev) along dimension,
           or 0 when stdev=0
  """
      m = mean(instack, dimension)
      sd = stdev(instack, dimension)
      return N.where(sd==0, 0, m/sd)
  
  
   def acov (x,y, dimension=None,keepdims=0):
      """
  Returns the estimated covariance of the values in the passed
  array (i.e., N-1).  Dimension can equal None (ravel array first), an
  integer (the dimension over which to operate), or a sequence (operate
  over multiple dimensions).  Set keepdims=1 to return an array with the
  same number of dimensions as inarray.
  
  Usage:   acov(x,y,dimension=None,keepdims=0)
  """
      if dimension is None:
          x = N.ravel(x)
          y = N.ravel(y)
          dimension = 0
      xmn = amean(x, dimension, 1)  # keepdims
      xdeviations = x - xmn
      ymn = amean(y, dimension, 1)  # keepdims
      ydeviations = y - ymn
      if isinstance(dimension, list):
          n = 1
          for d in dimension:
              n = n*x.shape[d]
      else:
          n = x.shape[dimension]
      covar = N.sum(xdeviations*ydeviations)/float(n-1)
      return covar
  
  
   def avar (inarray, dimension=None,keepdims=0):
      """
  Returns the estimated population variance of the values in the passed
  array (i.e., N-1).  Dimension can equal None (ravel array first), an
  integer (the dimension over which to operate), or a sequence (operate
  over multiple dimensions).  Set keepdims=1 to return an array with the
  same number of dimensions as inarray.
  
  Usage:   avar(inarray,dimension=None,keepdims=0)
  """
      if dimension is None:
          inarray = N.ravel(inarray)
          dimension = 0
      mn = amean(inarray, dimension, 1)
      deviations = inarray - mn
      if isinstance(dimension, list):
          n = 1
          for d in dimension:
              n = n*inarray.shape[d]
      else:
          n = inarray.shape[dimension]
      var = ass(deviations, dimension, keepdims)/float(n-1)
      return var
  
  
   def astdev (inarray, dimension=None, keepdims=0):
      """
  Returns the estimated population standard deviation of the values in
  the passed array (i.e., N-1).  Dimension can equal None (ravel array
  first), an integer (the dimension over which to operate), or a
  sequence (operate over multiple dimensions).  Set keepdims=1 to return
  an array with the same number of dimensions as inarray.
  
  Usage:   astdev(inarray,dimension=None,keepdims=0)
  """
      return N.sqrt(avar(inarray, dimension, keepdims))
  
  
   def asterr (inarray, dimension=None, keepdims=0):
      """
  Returns the estimated population standard error of the values in the
  passed array (i.e., N-1).  Dimension can equal None (ravel array
  first), an integer (the dimension over which to operate), or a
  sequence (operate over multiple dimensions).  Set keepdims=1 to return
  an array with the same number of dimensions as inarray.
  
  Usage:   asterr(inarray,dimension=None,keepdims=0)
  """
      if dimension is None:
          inarray = N.ravel(inarray)
          dimension = 0
      return astdev(inarray, dimension, keepdims) / float(N.sqrt(inarray.shape[dimension]))
  
  
   def asem (inarray, dimension=None, keepdims=0):
      """
  Returns the standard error of the mean (i.e., using N) of the values
  in the passed array.  Dimension can equal None (ravel array first), an
  integer (the dimension over which to operate), or a sequence (operate
  over multiple dimensions).  Set keepdims=1 to return an array with the
  same number of dimensions as inarray.
  
  Usage:   asem(inarray,dimension=None, keepdims=0)
  """
      if dimension is None:
          inarray = N.ravel(inarray)
          dimension = 0
      if isinstance(dimension, list):
          n = 1
          for d in dimension:
              n = n*inarray.shape[d]
      else:
          n = inarray.shape[dimension]
      s = asamplestdev(inarray, dimension, keepdims) / N.sqrt(n-1)
      return s
  
  
   def az (a, score):
      """
  Returns the z-score of a given input score, given thearray from which
  that score came.  Not appropriate for population calculations, nor for
  arrays > 1D.
  
  Usage:   az(a, score)
  """
      z = (score-amean(a)) / asamplestdev(a)
      return z
  
  
   def azs (a):
      """
  Returns a 1D array of z-scores, one for each score in the passed array,
  computed relative to the passed array.
  
  Usage:   azs(a)
  """
      zscores = []
      for item in a:
          zscores.append(z(a, item))
      return N.array(zscores)
  
  
   def azmap (scores, compare, dimension=0):
      """
  Returns an array of z-scores the shape of scores (e.g., [x,y]), compared to
  array passed to compare (e.g., [time,x,y]).  Assumes collapsing over dim 0
  of the compare array.
  
  Usage:   azs(scores, compare, dimension=0)
  """
      mns = amean(compare, dimension)
      sstd = asamplestdev(compare, 0)
      return (scores - mns) / sstd
  
  
  #####################################
  #######  ATRIMMING FUNCTIONS  #######
  #####################################
  
  ## deleted around() as it's in numpy now
  
   def athreshold(a,threshmin=None,threshmax=None,newval=0):
      """
  Like Numeric.clip() except that values <threshmid or >threshmax are replaced
  by newval instead of by threshmin/threshmax (respectively).
  
  Usage:   athreshold(a,threshmin=None,threshmax=None,newval=0)
  Returns: a, with values <threshmin or >threshmax replaced with newval
  """
      mask = N.zeros(a.shape)
      if threshmin is not None:
          mask = mask + N.where(a<threshmin, 1, 0)
      if threshmax is not None:
          mask = mask + N.where(a>threshmax, 1, 0)
      mask = N.clip(mask, 0, 1)
      return N.where(mask, newval, a)
  
  
   def atrimboth (a, proportiontocut):
      """
  Slices off the passed proportion of items from BOTH ends of the passed
  array (i.e., with proportiontocut=0.1, slices 'leftmost' 10% AND
  'rightmost' 10% of scores.  You must pre-sort the array if you want
  "proper" trimming.  Slices off LESS if proportion results in a
  non-integer slice index (i.e., conservatively slices off
  proportiontocut).
  
  Usage:   atrimboth (a,proportiontocut)
  Returns: trimmed version of array a
  """
      lowercut = int(proportiontocut*len(a))
      uppercut = len(a) - lowercut
      return a[lowercut:uppercut]
  
  
   def atrim1 (a,proportiontocut,tail='right'):
      """
  Slices off the passed proportion of items from ONE end of the passed
  array (i.e., if proportiontocut=0.1, slices off 'leftmost' or 'rightmost'
  10% of scores).  Slices off LESS if proportion results in a non-integer
  slice index (i.e., conservatively slices off proportiontocut).
  
  Usage:   atrim1(a,proportiontocut,tail='right')  or set tail='left'
  Returns: trimmed version of array a
  """
      if string.lower(tail) == 'right':
          lowercut = 0
          uppercut = len(a) - int(proportiontocut*len(a))
      elif string.lower(tail) == 'left':
          lowercut = int(proportiontocut*len(a))
          uppercut = len(a)
      return a[lowercut:uppercut]
  
  
  #####################################
  #####  ACORRELATION FUNCTIONS  ######
  #####################################
  
   def acovariance(X):
      """
  Computes the covariance matrix of a matrix X.  Requires a 2D matrix input.
  
  Usage:   acovariance(X)
  Returns: covariance matrix of X
  """
      if len(X.shape) != 2:
          raise TypeError("acovariance requires 2D matrices")
      n = X.shape[0]
      mX = amean(X, 0)
      return N.dot(N.transpose(X), X) / float(n) - N.multiply.outer(mX, mX)
  
  
   def acorrelation(X):
      """
  Computes the correlation matrix of a matrix X.  Requires a 2D matrix input.
  
  Usage:   acorrelation(X)
  Returns: correlation matrix of X
  """
      C = acovariance(X)
      V = N.diagonal(C)
      return C / N.sqrt(N.multiply.outer(V, V))
  
  
   def apaired(x, y):
      """
  Interactively determines the type of data in x and y, and then runs the
  appropriated statistic for paired group data.
  
  Usage:   apaired(x,y)     x,y = the two arrays of values to be compared
  Returns: appropriate statistic name, value, and probability
  """
      samples = ''
      while samples not in ['i', 'r', 'I', 'R', 'c', 'C']:
          print('\nIndependent or related samples, or correlation (i,r,c): ', end=' ')
          samples = input()
  
      if samples in ['i', 'I', 'r', 'R']:
          print('\nComparing variances ...', end=' ')
  # USE O'BRIEN'S TEST FOR HOMOGENEITY OF VARIANCE, Maxwell & delaney, p.112
          r = obrientransform(x, y)
          f, p = F_oneway(pstat.colex(r, 0), pstat.colex(r, 1))
          if p<0.05:
              vartype='unequal, p='+str(round(p, 4))
          else:
              vartype='equal'
          print(vartype)
          if samples in ['i', 'I']:
              if vartype[0]=='e':
                  t, p = ttest_ind(x, y, None, 0)
                  print('\nIndependent samples t-test:  ', round(t, 4), round(p, 4))
              else:
                  if len(x)>20 or len(y)>20:
                      z, p = ranksums(x, y)
                      print('\nRank Sums test (NONparametric, n>20):  ', round(z, 4), round(p, 4))
                  else:
                      u, p = mannwhitneyu(x, y)
                      print('\nMann-Whitney U-test (NONparametric, ns<20):  ', round(u, 4), round(p, 4))
  
          else:  # RELATED SAMPLES
              if vartype[0]=='e':
                  t, p = ttest_rel(x, y, 0)
                  print('\nRelated samples t-test:  ', round(t, 4), round(p, 4))
              else:
                  t, p = ranksums(x, y)
                  print('\nWilcoxon T-test (NONparametric):  ', round(t, 4), round(p, 4))
      else:  # CORRELATION ANALYSIS
          corrtype = ''
          while corrtype not in ['c', 'C', 'r', 'R', 'd', 'D']:
              print('\nIs the data Continuous, Ranked, or Dichotomous (c,r,d): ', end=' ')
              corrtype = input()
          if corrtype in ['c', 'C']:
              m, b, r, p, see = linregress(x, y)
              print('\nLinear regression for continuous variables ...')
              lol = [['Slope', 'Intercept', 'r', 'Prob', 'SEestimate'], [round(m, 4), round(b, 4), round(r, 4), round(p, 4), round(see, 4)]]
              pstat.printcc(lol)
          elif corrtype in ['r', 'R']:
              r, p = spearmanr(x, y)
              print('\nCorrelation for ranked variables ...')
              print("Spearman's r: ", round(r, 4), round(p, 4))
          else: # DICHOTOMOUS
              r, p = pointbiserialr(x, y)
              print('\nAssuming x contains a dichotomous variable ...')
              print('Point Biserial r: ', round(r, 4), round(p, 4))
      print('\n\n')
      return None
  
  
   def dices(x, y):
      """
  Calculates Dice's coefficient ... (2*number of common terms)/(number of terms in x +
  number of terms in y). Returns a value between 0 (orthogonal) and 1.
  
  Usage:  dices(x,y)
  """
      import sets
      x = sets.Set(x)
      y = sets.Set(y)
      common = len(x.intersection(y))
      total = float(len(x) + len(y))
      return 2*common/total
  
  
   def icc(x,y=None,verbose=0):
      """
  Calculates intraclass correlation coefficients using simple, Type I sums of squares.
  If only one variable is passed, assumed it's an Nx2 matrix
  
  Usage:   icc(x,y=None,verbose=0)
  Returns: icc rho, prob ####PROB IS A GUESS BASED ON PEARSON
  """
      TINY = 1.0e-20
      if y:
          all = N.concatenate([x, y], 0)
      else:
          all = x+0
          x = all[:, 0]
          y = all[:, 1]
      totalss = ass(all-mean(all))
      pairmeans = (x+y)/2.
      withinss = ass(x-pairmeans) + ass(y-pairmeans)
      withindf = float(len(x))
      betwdf = float(len(x)-1)
      withinms = withinss / withindf
      betweenms = (totalss-withinss) / betwdf
      rho = (betweenms-withinms)/(withinms+betweenms)
      t = rho*math.sqrt(betwdf/((1.0-rho+TINY)*(1.0+rho+TINY)))
      prob = abetai(0.5*betwdf, 0.5, betwdf/(betwdf+t*t), verbose)
      return rho, prob
  
  
   def alincc(x, y):
      """
  Calculates Lin's concordance correlation coefficient.
  
  Usage:   alincc(x,y)    where x, y are equal-length arrays
  Returns: Lin's CC
  """
      x = N.ravel(x)
      y = N.ravel(y)
      covar = acov(x, y)*(len(x)-1)/float(len(x))  # correct denom to n
      xvar = avar(x)*(len(x)-1)/float(len(x))  # correct denom to n
      yvar = avar(y)*(len(y)-1)/float(len(y))  # correct denom to n
      lincc = (2 * covar) / ((xvar+yvar) +((amean(x)-amean(y))**2))
      return lincc
  
  
   def apearsonr(x,y,verbose=1):
      """
  Calculates a Pearson correlation coefficient and returns p.  Taken
  from Heiman's Basic Statistics for the Behav. Sci (2nd), p.195.
  
  Usage:   apearsonr(x,y,verbose=1)      where x,y are equal length arrays
  Returns: Pearson's r, two-tailed p-value
  """
      TINY = 1.0e-20
      n = len(x)
      xmean = amean(x)
      ymean = amean(y)
      r_num = n*(N.add.reduce(x*y)) - N.add.reduce(x)*N.add.reduce(y)
      r_den = math.sqrt((n*ass(x) - asquare_of_sums(x))*(n*ass(y)-asquare_of_sums(y)))
      r = (r_num / r_den)
      df = n-2
      t = r*math.sqrt(df/((1.0-r+TINY)*(1.0+r+TINY)))
      prob = abetai(0.5*df, 0.5, df/(df+t*t), verbose)
      return r, prob
  
  
   def aspearmanr(x, y):
      """
  Calculates a Spearman rank-order correlation coefficient.  Taken
  from Heiman's Basic Statistics for the Behav. Sci (1st), p.192.
  
  Usage:   aspearmanr(x,y)      where x,y are equal-length arrays
  Returns: Spearman's r, two-tailed p-value
  """
      TINY = 1e-30
      n = len(x)
      rankx = rankdata(x)
      ranky = rankdata(y)
      dsq = N.add.reduce((rankx-ranky)**2)
      rs = 1 - 6*dsq / float(n*(n**2-1))
      t = rs * math.sqrt((n-2) / ((rs+1.0)*(1.0-rs)))
      df = n-2
      probrs = abetai(0.5*df, 0.5, df/(df+t*t))
  # probability values for rs are from part 2 of the spearman function in
  # Numerical Recipies, p.510.  They close to tables, but not exact.(?)
      return rs, probrs
  
  
   def apointbiserialr(x, y):
      """
  Calculates a point-biserial correlation coefficient and the associated
  probability value.  Taken from Heiman's Basic Statistics for the Behav.
  Sci (1st), p.194.
  
  Usage:   apointbiserialr(x,y)      where x,y are equal length arrays
  Returns: Point-biserial r, two-tailed p-value
  """
      TINY = 1e-30
      categories = pstat.aunique(x)
      data = pstat.aabut(x, y)
      if len(categories) != 2:
          raise ValueError("Exactly 2 categories required (in x) for pointbiserialr().")
      else:   # there are 2 categories, continue
          codemap = pstat.aabut(categories, N.arange(2))
          recoded = pstat.arecode(data, codemap, 0)
          x = pstat.alinexand(data, 0, categories[0])
          y = pstat.alinexand(data, 0, categories[1])
          xmean = amean(pstat.acolex(x, 1))
          ymean = amean(pstat.acolex(y, 1))
          n = len(data)
          adjust = math.sqrt((len(x)/float(n))*(len(y)/float(n)))
          rpb = (ymean - xmean)/asamplestdev(pstat.acolex(data, 1))*adjust
          df = n-2
          t = rpb*math.sqrt(df/((1.0-rpb+TINY)*(1.0+rpb+TINY)))
          prob = abetai(0.5*df, 0.5, df/(df+t*t))
          return rpb, prob
  
  
   def akendalltau(x, y):
      """
  Calculates Kendall's tau ... correlation of ordinal data.  Adapted
  from function kendl1 in Numerical Recipies.  Needs good test-cases.@@@
  
  Usage:   akendalltau(x,y)
  Returns: Kendall's tau, two-tailed p-value
  """
      n1 = 0
      n2 = 0
      iss = 0
      for j in range(len(x)-1):
          for k in range(j, len(y)):
              a1 = x[j] - x[k]
              a2 = y[j] - y[k]
              aa = a1 * a2
              if (aa):             # neither array has a tie
                  n1 = n1 + 1
                  n2 = n2 + 1
                  if aa > 0:
                      iss = iss + 1
                  else:
                      iss = iss -1
              else:
                  if (a1):
                      n1 = n1 + 1
                  else:
                      n2 = n2 + 1
      tau = iss / math.sqrt(n1*n2)
      svar = (4.0*len(x)+10.0) / (9.0*len(x)*(len(x)-1))
      z = tau / math.sqrt(svar)
      prob = erfcc(abs(z)/1.4142136)
      return tau, prob
  
  
   def alinregress(*args):
      """
  Calculates a regression line on two arrays, x and y, corresponding to x,y
  pairs.  If a single 2D array is passed, alinregress finds dim with 2 levels
  and splits data into x,y pairs along that dim.
  
  Usage:   alinregress(*args)    args=2 equal-length arrays, or one 2D array
  Returns: slope, intercept, r, two-tailed prob, sterr-of-the-estimate, n
  """
      TINY = 1.0e-20
      if len(args) == 1:  # more than 1D array?
          args = args[0]
          if len(args) == 2:
              x = args[0]
              y = args[1]
          else:
              x = args[:, 0]
              y = args[:, 1]
      else:
          x = args[0]
          y = args[1]
      n = len(x)
      xmean = amean(x)
      ymean = amean(y)
      r_num = n*(N.add.reduce(x*y)) - N.add.reduce(x)*N.add.reduce(y)
      r_den = math.sqrt((n*ass(x) - asquare_of_sums(x))*(n*ass(y)-asquare_of_sums(y)))
      r = r_num / r_den
      z = 0.5*math.log((1.0+r+TINY)/(1.0-r+TINY))
      df = n-2
      t = r*math.sqrt(df/((1.0-r+TINY)*(1.0+r+TINY)))
      prob = abetai(0.5*df, 0.5, df/(df+t*t))
      slope = r_num / (float(n)*ass(x) - asquare_of_sums(x))
      intercept = ymean - slope*xmean
      sterrest = math.sqrt(1-r*r)*asamplestdev(y)
      return slope, intercept, r, prob, sterrest, n
  
   def amasslinregress(*args):
      """
  Calculates a regression line on one 1D array (x) and one N-D array (y).
  
  Returns: slope, intercept, r, two-tailed prob, sterr-of-the-estimate, n
  """
      TINY = 1.0e-20
      if len(args) == 1:  # more than 1D array?
          args = args[0]
          if len(args) == 2:
              x = N.ravel(args[0])
              y = args[1]
          else:
              x = N.ravel(args[:, 0])
              y = args[:, 1]
      else:
          x = args[0]
          y = args[1]
      x = x.astype(N.float_)
      y = y.astype(N.float_)
      n = len(x)
      xmean = amean(x)
      ymean = amean(y, 0)
      shp = N.ones(len(y.shape))
      shp[0] = len(x)
      x.shape = shp
      print(x.shape, y.shape)
      r_num = n*(N.add.reduce(x*y, 0)) - N.add.reduce(x)*N.add.reduce(y, 0)
      r_den = N.sqrt((n*ass(x) - asquare_of_sums(x))*(n*ass(y, 0)-asquare_of_sums(y, 0)))
      zerodivproblem = N.equal(r_den, 0)
      r_den = N.where(zerodivproblem, 1, r_den)  # avoid zero-division in 1st place
      r = r_num / r_den  # need to do this nicely for matrix division
      r = N.where(zerodivproblem, 0.0, r)
      z = 0.5*N.log((1.0+r+TINY)/(1.0-r+TINY))
      df = n-2
      t = r*N.sqrt(df/((1.0-r+TINY)*(1.0+r+TINY)))
      prob = abetai(0.5*df, 0.5, df/(df+t*t))
  
      ss = float(n)*ass(x)-asquare_of_sums(x)
      s_den = N.where(ss==0, 1, ss)  # avoid zero-division in 1st place
      slope = r_num / s_den
      intercept = ymean - slope*xmean
      sterrest = N.sqrt(1-r*r)*asamplestdev(y, 0)
      return slope, intercept, r, prob, sterrest, n
  
  
  #####################################
  #####  AINFERENTIAL STATISTICS  #####
  #####################################
  
   def attest_1samp(a,popmean,printit=0,name='Sample',writemode='a'):
      """
  Calculates the t-obtained for the independent samples T-test on ONE group
  of scores a, given a population mean.  If printit=1, results are printed
  to the screen.  If printit='filename', the results are output to 'filename'
  using the given writemode (default=append).  Returns t-value, and prob.
  
  Usage:   attest_1samp(a,popmean,Name='Sample',printit=0,writemode='a')
  Returns: t-value, two-tailed prob
  """
      if not isinstance(a, N.ndarray):
          a = N.array(a)
      x = amean(a)
      v = avar(a)
      n = len(a)
      df = n-1
      svar = ((n-1)*v) / float(df)
      t = (x-popmean)/math.sqrt(svar*(1.0/n))
      prob = abetai(0.5*df, 0.5, df/(df+t*t))
  
      if printit != 0:
          statname = 'Single-sample T-test.'
          outputpairedstats(printit, writemode,
                            'Population', '--', popmean, 0, 0, 0,
                            name, n, x, v, N.minimum.reduce(N.ravel(a)),
                            N.maximum.reduce(N.ravel(a)),
                            statname, t, prob)
      return t, prob
  
  
   def attest_ind (a, b, dimension=None, printit=0, name1='Samp1', name2='Samp2',writemode='a'):
      """
  Calculates the t-obtained T-test on TWO INDEPENDENT samples of scores
  a, and b.  From Numerical Recipies, p.483.  If printit=1, results are
  printed to the screen.  If printit='filename', the results are output
  to 'filename' using the given writemode (default=append).  Dimension
  can equal None (ravel array first), or an integer (the dimension over
  which to operate on a and b).
  
  Usage:   attest_ind (a,b,dimension=None,printit=0,
                       Name1='Samp1',Name2='Samp2',writemode='a')
  Returns: t-value, two-tailed p-value
  """
      if dimension is None:
          a = N.ravel(a)
          b = N.ravel(b)
          dimension = 0
      x1 = amean(a, dimension)
      x2 = amean(b, dimension)
      v1 = avar(a, dimension)
      v2 = avar(b, dimension)
      n1 = a.shape[dimension]
      n2 = b.shape[dimension]
      df = n1+n2-2
      svar = ((n1-1)*v1+(n2-1)*v2) / float(df)
      zerodivproblem = N.equal(svar, 0)
      svar = N.where(zerodivproblem, 1, svar)  # avoid zero-division in 1st place
      t = (x1-x2)/N.sqrt(svar*(1.0/n1 + 1.0/n2))  # N-D COMPUTATION HERE!!!!!!
      t = N.where(zerodivproblem, 1.0, t)     # replace NaN/wrong t-values with 1.0
      probs = abetai(0.5*df, 0.5, float(df)/(df+t*t))
  
      if isinstance(t, N.ndarray):
          probs = N.reshape(probs, t.shape)
      if probs.shape == (1,):
          probs = probs[0]
          
      if printit != 0:
          if isinstance(t, N.ndarray):
              t = t[0]
          if isinstance(probs, N.ndarray):
              probs = probs[0]
          statname = 'Independent samples T-test.'
          outputpairedstats(printit, writemode,
                            name1, n1, x1, v1, N.minimum.reduce(N.ravel(a)),
                            N.maximum.reduce(N.ravel(a)),
                            name2, n2, x2, v2, N.minimum.reduce(N.ravel(b)),
                            N.maximum.reduce(N.ravel(b)),
                            statname, t, probs)
          return
      return t, probs
  
   def ap2t(pval, df):
      """
  Tries to compute a t-value from a p-value (or pval array) and associated df.
  SLOW for large numbers of elements(!) as it re-computes p-values 20 times
  (smaller step-sizes) at which point it decides it's done. Keeps the signs
  of the input array. Returns 1000 (or -1000) if t>100.
  
  Usage:  ap2t(pval,df)
  Returns: an array of t-values with the shape of pval
      """
      pval = N.array(pval)
      signs = N.sign(pval)
      pval = abs(pval)
      t = N.ones(pval.shape, N.float_)*50
      step = N.ones(pval.shape, N.float_)*25
      print("Initial ap2t() prob calc")
      prob = abetai(0.5*df, 0.5, float(df)/(df+t*t))
      print('ap2t() iter: ', end=' ')
      for i in range(10):
          print(i, ' ', end=' ')
          t = N.where(pval<prob, t+step, t-step)
          prob = abetai(0.5*df, 0.5, float(df)/(df+t*t))
          step = step/2
      print()
      # since this is an ugly hack, we get ugly boundaries
      t = N.where(t>99.9, 1000, t)      # hit upper-boundary
      t = t+signs
      return t #, prob, pval
  
  
   def attest_rel (a,b,dimension=None,printit=0,name1='Samp1',name2='Samp2',writemode='a'):
      """
  Calculates the t-obtained T-test on TWO RELATED samples of scores, a
  and b.  From Numerical Recipies, p.483.  If printit=1, results are
  printed to the screen.  If printit='filename', the results are output
  to 'filename' using the given writemode (default=append).  Dimension
  can equal None (ravel array first), or an integer (the dimension over
  which to operate on a and b).
  
  Usage:   attest_rel(a,b,dimension=None,printit=0,
                      name1='Samp1',name2='Samp2',writemode='a')
  Returns: t-value, two-tailed p-value
  """
      if dimension is None:
          a = N.ravel(a)
          b = N.ravel(b)
          dimension = 0
      if len(a) != len(b):
          raise ValueError('Unequal length arrays.')
      x1 = amean(a, dimension)
      x2 = amean(b, dimension)
      v1 = avar(a, dimension)
      v2 = avar(b, dimension)
      n = a.shape[dimension]
      df = float(n-1)
      d = (a-b).astype('d')
  
      denom = N.sqrt((n*N.add.reduce(d*d, dimension) - N.add.reduce(d, dimension)**2) /df)
      zerodivproblem = N.equal(denom, 0)
      denom = N.where(zerodivproblem, 1, denom)  # avoid zero-division in 1st place
      t = N.add.reduce(d, dimension) / denom      # N-D COMPUTATION HERE!!!!!!
      t = N.where(zerodivproblem, 1.0, t)     # replace NaN/wrong t-values with 1.0
      probs = abetai(0.5*df, 0.5, float(df)/(df+t*t))
      if isinstance(t, N.ndarray):
          probs = N.reshape(probs, t.shape)
      if probs.shape == (1,):
          probs = probs[0]
  
      if printit != 0:
          statname = 'Related samples T-test.'
          outputpairedstats(printit, writemode,
                            name1, n, x1, v1, N.minimum.reduce(N.ravel(a)),
                            N.maximum.reduce(N.ravel(a)),
                            name2, n, x2, v2, N.minimum.reduce(N.ravel(b)),
                            N.maximum.reduce(N.ravel(b)),
                            statname, t, probs)
          return
      return t, probs
  
  
   def achisquare(f_obs,f_exp=None):
      """
  Calculates a one-way chi square for array of observed frequencies and returns
  the result.  If no expected frequencies are given, the total N is assumed to
  be equally distributed across all groups.
  @@@NOT RIGHT??
  
  Usage:   achisquare(f_obs, f_exp=None)   f_obs = array of observed cell freq.
  Returns: chisquare-statistic, associated p-value
  """
  
      k = len(f_obs)
      if f_exp is None:
          f_exp = N.array([sum(f_obs)/float(k)] * len(f_obs), N.float_)
      f_exp = f_exp.astype(N.float_)
      chisq = N.add.reduce((f_obs-f_exp)**2 / f_exp)
      return chisq, achisqprob(chisq, k-1)
  
  
   def aks_2samp (data1, data2):
      """
  Computes the Kolmogorov-Smirnof statistic on 2 samples.  Modified from
  Numerical Recipies in C, page 493.  Returns KS D-value, prob.  Not ufunc-
  like.
  
  Usage:   aks_2samp(data1,data2)  where data1 and data2 are 1D arrays
  Returns: KS D-value, p-value
  """
      j1 = 0    # N.zeros(data1.shape[1:]) TRIED TO MAKE THIS UFUNC-LIKE
      j2 = 0    # N.zeros(data2.shape[1:])
      fn1 = 0.0 # N.zeros(data1.shape[1:],N.float_)
      fn2 = 0.0 # N.zeros(data2.shape[1:],N.float_)
      n1 = data1.shape[0]
      n2 = data2.shape[0]
      en1 = n1*1
      en2 = n2*1
      d = N.zeros(data1.shape[1:], N.float_)
      data1 = N.sort(data1, 0)
      data2 = N.sort(data2, 0)
      while j1 < n1 and j2 < n2:
          d1=data1[j1]
          d2=data2[j2]
          if d1 <= d2:
              fn1 = (j1)/float(en1)
              j1 = j1 + 1
          if d2 <= d1:
              fn2 = (j2)/float(en2)
              j2 = j2 + 1
          dt = (fn2-fn1)
          if abs(dt) > abs(d):
              d = dt
  #    try:
      en = math.sqrt(en1*en2/float(en1+en2))
      prob = aksprob((en+0.12+0.11/en)*N.fabs(d))
  #    except:
  #        prob = 1.0
      return d, prob
  
  
   def amannwhitneyu(x, y):
      """
  Calculates a Mann-Whitney U statistic on the provided scores and
  returns the result.  Use only when the n in each condition is < 20 and
  you have 2 independent samples of ranks.  REMEMBER: Mann-Whitney U is
  significant if the u-obtained is LESS THAN or equal to the critical
  value of U.
  
  Usage:   amannwhitneyu(x,y)     where x,y are arrays of values for 2 conditions
  Returns: u-statistic, one-tailed p-value (i.e., p(z(U)))
  """
      n1 = len(x)
      n2 = len(y)
      ranked = rankdata(N.concatenate((x, y)))
      rankx = ranked[0:n1]       # get the x-ranks
      ranky = ranked[n1:]        # the rest are y-ranks
      u1 = n1*n2 + (n1*(n1+1))/2.0 - sum(rankx)  # calc U for x
      u2 = n1*n2 - u1                            # remainder is U for y
      bigu = max(u1, u2)
      smallu = min(u1, u2)
      T = math.sqrt(tiecorrect(ranked))  # correction factor for tied scores
      if T == 0:
          raise ValueError('All numbers are identical in amannwhitneyu')
      sd = math.sqrt(T*n1*n2*(n1+n2+1)/12.0)
      z = abs((bigu-n1*n2/2.0) / sd)  # normal approximation for prob calc
      return smallu, 1.0 - azprob(z)
  
  
   def atiecorrect(rankvals):
      """
  Tie-corrector for ties in Mann Whitney U and Kruskal Wallis H tests.
  See Siegel, S. (1956) Nonparametric Statistics for the Behavioral
  Sciences.  New York: McGraw-Hill.  Code adapted from |Stat rankind.c
  code.
  
  Usage:   atiecorrect(rankvals)
  Returns: T correction factor for U or H
  """
      sorted, posn = ashellsort(N.array(rankvals))
      n = len(sorted)
      T = 0.0
      i = 0
      while (i<n-1):
          if sorted[i] == sorted[i+1]:
              nties = 1
              while (i<n-1) and (sorted[i] == sorted[i+1]):
                  nties = nties +1
                  i = i +1
              T = T + nties**3 - nties
          i = i+1
      T = T / float(n**3-n)
      return 1.0 - T
  
  
   def aranksums(x, y):
      """
  Calculates the rank sums statistic on the provided scores and returns
  the result.
  
  Usage:   aranksums(x,y)     where x,y are arrays of values for 2 conditions
  Returns: z-statistic, two-tailed p-value
  """
      n1 = len(x)
      n2 = len(y)
      alldata = N.concatenate((x, y))
      ranked = arankdata(alldata)
      x = ranked[:n1]
      y = ranked[n1:]
      s = sum(x)
      expected = n1*(n1+n2+1) / 2.0
      z = (s - expected) / math.sqrt(n1*n2*(n1+n2+1)/12.0)
      prob = 2*(1.0 - azprob(abs(z)))
      return z, prob
  
  
   def awilcoxont(x, y):
      """
  Calculates the Wilcoxon T-test for related samples and returns the
  result.  A non-parametric T-test.
  
  Usage:   awilcoxont(x,y)     where x,y are equal-length arrays for 2 conditions
  Returns: t-statistic, two-tailed p-value
  """
      if len(x) != len(y):
          raise ValueError('Unequal N in awilcoxont.  Aborting.')
      d = x-y
      d = N.compress(N.not_equal(d, 0), d) # Keep all non-zero differences
      count = len(d)
      absd = abs(d)
      absranked = arankdata(absd)
      r_plus = 0.0
      r_minus = 0.0
      for i in range(len(absd)):
          if d[i] < 0:
              r_minus = r_minus + absranked[i]
          else:
              r_plus = r_plus + absranked[i]
      wt = min(r_plus, r_minus)
      mn = count * (count+1) * 0.25
      se =  math.sqrt(count*(count+1)*(2.0*count+1.0)/24.0)
      z = math.fabs(wt-mn) / se
      z = math.fabs(wt-mn) / se
      prob = 2*(1.0 -zprob(abs(z)))
      return wt, prob
  
  
   def akruskalwallish(*args):
      """
  The Kruskal-Wallis H-test is a non-parametric ANOVA for 3 or more
  groups, requiring at least 5 subjects in each group.  This function
  calculates the Kruskal-Wallis H and associated p-value for 3 or more
  independent samples.
  
  Usage:   akruskalwallish(*args)     args are separate arrays for 3+ conditions
  Returns: H-statistic (corrected for ties), associated p-value
  """
      assert len(args) == 3, "Need at least 3 groups in stats.akruskalwallish()"
      args = list(args)
      n = [0]*len(args)
      n = list(map(len, args))
      all = []
      for i in range(len(args)):
          all = all + args[i].tolist()
      ranked = rankdata(all)
      T = tiecorrect(ranked)
      for i in range(len(args)):
          args[i] = ranked[0:n[i]]
          del ranked[0:n[i]]
      rsums = []
      for i in range(len(args)):
          rsums.append(sum(args[i])**2)
          rsums[i] = rsums[i] / float(n[i])
      ssbn = sum(rsums)
      totaln = sum(n)
      h = 12.0 / (totaln*(totaln+1)) * ssbn - 3*(totaln+1)
      df = len(args) - 1
      if T == 0:
          raise ValueError('All numbers are identical in akruskalwallish')
      h = h / float(T)
      return h, chisqprob(h, df)
  
  
   def afriedmanchisquare(*args):
      """
  Friedman Chi-Square is a non-parametric, one-way within-subjects
  ANOVA.  This function calculates the Friedman Chi-square test for
  repeated measures and returns the result, along with the associated
  probability value.  It assumes 3 or more repeated measures.  Only 3
  levels requires a minimum of 10 subjects in the study.  Four levels
  requires 5 subjects per level(??).
  
  Usage:   afriedmanchisquare(*args)   args are separate arrays for 2+ conditions
  Returns: chi-square statistic, associated p-value
  """
      k = len(args)
      if k < 3:
          raise ValueError('\nLess than 3 levels.  Friedman test not appropriate.\n')
      n = len(args[0])
      data = pstat.aabut(*args)
      data = data.astype(N.float_)
      for i in range(len(data)):
          data[i] = arankdata(data[i])
      ssbn = asum(asum(args, 1)**2)
      chisq = 12.0 / (k*n*(k+1)) * ssbn - 3*n*(k+1)
      return chisq, achisqprob(chisq, k-1)
  
  
  #####################################
  ####  APROBABILITY CALCULATIONS  ####
  #####################################
  
   def achisqprob(chisq, df):
      """
  Returns the (1-tail) probability value associated with the provided chi-square
  value and df.  Heavily modified from chisq.c in Gary Perlman's |Stat.  Can
  handle multiple dimensions.
  
  Usage:   achisqprob(chisq,df)    chisq=chisquare stat., df=degrees of freedom
  """
      BIG = 200.0
      def ex(x):
          BIG = 200.0
          exponents = N.where(N.less(x, -BIG), -BIG, x)
          return N.exp(exponents)
  
      if isinstance(chisq, N.ndarray):
          arrayflag = 1
      else:
          arrayflag = 0
          chisq = N.array([chisq])
      if df < 1:
          return N.ones(chisq.shape, N.float)
      probs = N.zeros(chisq.shape, N.float_)
      probs = N.where(N.less_equal(chisq, 0), 1.0, probs)  # set prob=1 for chisq<0
      a = 0.5 * chisq
      if df > 1:
          y = ex(-a)
      if df%2 == 0:
          even = 1
          s = y*1
          s2 = s*1
      else:
          even = 0
          s = 2.0 * azprob(-N.sqrt(chisq))
          s2 = s*1
      if (df > 2):
          chisq = 0.5 * (df - 1.0)
          if even:
              z = N.ones(probs.shape, N.float_)
          else:
              z = 0.5 *N.ones(probs.shape, N.float_)
          if even:
              e = N.zeros(probs.shape, N.float_)
          else:
              e = N.log(N.sqrt(N.pi)) *N.ones(probs.shape, N.float_)
          c = N.log(a)
          mask = N.zeros(probs.shape)
          a_big = N.greater(a, BIG)
          a_big_frozen = -1 *N.ones(probs.shape, N.float_)
          totalelements = N.multiply.reduce(N.array(probs.shape))
          while asum(mask) != totalelements:
              e = N.log(z) + e
              s = s + ex(c*z-a-e)
              z = z + 1.0
  #            print z, e, s
              newmask = N.greater(z, chisq)
              a_big_frozen = N.where(newmask*N.equal(mask, 0)*a_big, s, a_big_frozen)
              mask = N.clip(newmask+mask, 0, 1)
          if even:
              z = N.ones(probs.shape, N.float_)
              e = N.ones(probs.shape, N.float_)
          else:
              z = 0.5 *N.ones(probs.shape, N.float_)
              e = 1.0 / N.sqrt(N.pi) / N.sqrt(a) * N.ones(probs.shape, N.float_)
          c = 0.0
          mask = N.zeros(probs.shape)
          a_notbig_frozen = -1 *N.ones(probs.shape, N.float_)
          while asum(mask) != totalelements:
              e = e * (a/z.astype(N.float_))
              c = c + e
              z = z + 1.0
  #            print '#2', z, e, c, s, c*y+s2
              newmask = N.greater(z, chisq)
              a_notbig_frozen = N.where(newmask*N.equal(mask, 0)*(1-a_big),
                                        c*y+s2, a_notbig_frozen)
              mask = N.clip(newmask+mask, 0, 1)
          probs = N.where(N.equal(probs, 1), 1,
                          N.where(N.greater(a, BIG), a_big_frozen, a_notbig_frozen))
          return probs
      else:
          return s
  
  
   def aerfcc(x):
      """
  Returns the complementary error function erfc(x) with fractional error
  everywhere less than 1.2e-7.  Adapted from Numerical Recipies.  Can
  handle multiple dimensions.
  
  Usage:   aerfcc(x)
  """
      z = abs(x)
      t = 1.0 / (1.0+0.5*z)
      ans = t * N.exp(-z*z-1.26551223 + t*(1.00002368+t*(0.37409196+t*(0.09678418+t*(-0.18628806+t*(0.27886807+t*(-1.13520398+t*(1.48851587+t*(-0.82215223+t*0.17087277)))))))))
      return N.where(N.greater_equal(x, 0), ans, 2.0-ans)
  
  
   def azprob(z):
      """
  Returns the area under the normal curve 'to the left of' the given z value.
  Thus, 
      for z<0, zprob(z) = 1-tail probability
      for z>0, 1.0-zprob(z) = 1-tail probability
      for any z, 2.0*(1.0-zprob(abs(z))) = 2-tail probability
  Adapted from z.c in Gary Perlman's |Stat.  Can handle multiple dimensions.
  
  Usage:   azprob(z)    where z is a z-value
  """
      def yfunc(y):
          x = (((((((((((((-0.000045255659 * y
                           +0.000152529290) * y -0.000019538132) * y
                         -0.000676904986) * y +0.001390604284) * y
                       -0.000794620820) * y -0.002034254874) * y
                     +0.006549791214) * y -0.010557625006) * y
                   +0.011630447319) * y -0.009279453341) * y
                 +0.005353579108) * y -0.002141268741) * y
               +0.000535310849) * y +0.999936657524
          return x
  
      def wfunc(w):
          x = ((((((((0.000124818987 * w
                      -0.001075204047) * w +0.005198775019) * w
                    -0.019198292004) * w +0.059054035642) * w
                  -0.151968751364) * w +0.319152932694) * w
                -0.531923007300) * w +0.797884560593) * N.sqrt(w) * 2.0
          return x
  
      Z_MAX = 6.0    # maximum meaningful z-value
      x = N.zeros(z.shape, N.float_) # initialize
      y = 0.5 * N.fabs(z)
      x = N.where(N.less(y, 1.0), wfunc(y*y), yfunc(y-2.0)) # get x's
      x = N.where(N.greater(y, Z_MAX*0.5), 1.0, x)          # kill those with big Z
      prob = N.where(N.greater(z, 0), (x+1)*0.5, (1-x)*0.5)
      return prob
  
  
   def aksprob(alam):
       """
  Returns the probability value for a K-S statistic computed via ks_2samp.
  Adapted from Numerical Recipies.  Can handle multiple dimensions.
  
  Usage:   aksprob(alam)
  """
       if isinstance(alam, N.ndarray):
           frozen = -1 *N.ones(alam.shape, N.float64)
           alam = alam.astype(N.float64)
           arrayflag = 1
       else:
           frozen = N.array(-1.)
           alam = N.array(alam, N.float64)
           arrayflag = 1
       mask = N.zeros(alam.shape)
       fac = 2.0 *N.ones(alam.shape, N.float_)
       sum = N.zeros(alam.shape, N.float_)
       termbf = N.zeros(alam.shape, N.float_)
       a2 = N.array(-2.0*alam*alam, N.float64)
       totalelements = N.multiply.reduce(N.array(mask.shape))
       for j in range(1, 201):
           if asum(mask) == totalelements:
               break
           exponents = (a2*j*j)
           overflowmask = N.less(exponents, -746)
           frozen = N.where(overflowmask, 0, frozen)
           mask = mask+overflowmask
           term = fac*N.exp(exponents)
           sum = sum + term
           newmask = N.where(N.less_equal(abs(term), (0.001*termbf)) +
                             N.less(abs(term), 1.0e-8*sum), 1, 0)
           frozen = N.where(newmask*N.equal(mask, 0), sum, frozen)
           mask = N.clip(mask+newmask, 0, 1)
           fac = -fac
           termbf = abs(term)
       if arrayflag:
           return N.where(N.equal(frozen, -1), 1.0, frozen)  # 1.0 if doesn't converge
       else:
           return N.where(N.equal(frozen, -1), 1.0, frozen)[0]  # 1.0 if doesn't converge
  
  
   def afprob (dfnum, dfden, F):
      """
  Returns the 1-tailed significance level (p-value) of an F statistic
  given the degrees of freedom for the numerator (dfR-dfF) and the degrees
  of freedom for the denominator (dfF).  Can handle multiple dims for F.
  
  Usage:   afprob(dfnum, dfden, F)   where usually dfnum=dfbn, dfden=dfwn
  """
      if isinstance(F, N.ndarray):
          return abetai(0.5*dfden, 0.5*dfnum, dfden/(1.0*dfden+dfnum*F))
      else:
          return abetai(0.5*dfden, 0.5*dfnum, dfden/float(dfden+dfnum*F))
  
  
   def abetacf(a,b,x,verbose=1):
      """
  Evaluates the continued fraction form of the incomplete Beta function,
  betai.  (Adapted from: Numerical Recipies in C.)  Can handle multiple
  dimensions for x.
  
  Usage:   abetacf(a,b,x,verbose=1)
  """
      ITMAX = 200
      EPS = 3.0e-7
  
      arrayflag = 1
      if isinstance(x, N.ndarray):
          frozen = N.ones(x.shape, N.float_) *-1  #start out w/ -1s, should replace all
      else:
          arrayflag = 0
          frozen = N.array([-1])
          x = N.array([x])
      mask = N.zeros(x.shape)
      bm = az = am = 1.0
      qab = a+b
      qap = a+1.0
      qam = a-1.0
      bz = 1.0-qab*x/qap
      for i in range(ITMAX+1):
          if N.sum(N.ravel(N.equal(frozen, -1)))==0:
              break
          em = float(i+1)
          tem = em + em
          d = em*(b-em)*x/((qam+tem)*(a+tem))
          ap = az + d*am
          bp = bz+d*bm
          d = -(a+em)*(qab+em)*x/((qap+tem)*(a+tem))
          app = ap+d*az
          bpp = bp+d*bz
          aold = az*1
          am = ap/bpp
          bm = bp/bpp
          az = app/bpp
          bz = 1.0
          newmask = N.less(abs(az-aold), EPS*abs(az))
          frozen = N.where(newmask*N.equal(mask, 0), az, frozen)
          mask = N.clip(mask+newmask, 0, 1)
      noconverge = asum(N.equal(frozen, -1))
      if noconverge != 0 and verbose:
          print('a or b too big, or ITMAX too small in Betacf for ', noconverge, ' elements')
      if arrayflag:
          return frozen
      else:
          return frozen[0]
  
  
   def agammln(xx):
      """
  Returns the gamma function of xx.
      Gamma(z) = Integral(0,infinity) of t^(z-1)exp(-t) dt.
  Adapted from: Numerical Recipies in C.  Can handle multiple dims ... but
  probably doesn't normally have to.
  
  Usage:   agammln(xx)
  """
      coeff = [76.18009173, -86.50532033, 24.01409822, -1.231739516,
               0.120858003e-2, -0.536382e-5]
      x = xx - 1.0
      tmp = x + 5.5
      tmp = tmp - (x+0.5)*N.log(tmp)
      ser = 1.0
      for j in range(len(coeff)):
          x = x + 1
          ser = ser + coeff[j]/x
      return -tmp + N.log(2.50662827465*ser)
  
  
   def abetai(a,b,x,verbose=1):
      """
  Returns the incomplete beta function:
  
      I-sub-x(a,b) = 1/B(a,b)*(Integral(0,x) of t^(a-1)(1-t)^(b-1) dt)
  
  where a,b>0 and B(a,b) = G(a)*G(b)/(G(a+b)) where G(a) is the gamma
  function of a.  The continued fraction formulation is implemented
  here, using the betacf function.  (Adapted from: Numerical Recipies in
  C.)  Can handle multiple dimensions.
  
  Usage:   abetai(a,b,x,verbose=1)
  """
      TINY = 1e-15
      if isinstance(a, N.ndarray):
          if asum(N.less(x, 0) + N.greater(x, 1)) != 0:
              raise ValueError('Bad x in abetai')
      x = N.where(N.equal(x, 0), TINY, x)
      x = N.where(N.equal(x, 1.0), 1-TINY, x)
  
      bt = N.where(N.equal(x, 0)+N.equal(x, 1), 0, -1)
      exponents = ( gammln(a+b)-gammln(a)-gammln(b)+a*N.log(x)+b*
                    N.log(1.0-x) )
      # 746 (below) is the MAX POSSIBLE BEFORE OVERFLOW
      exponents = N.where(N.less(exponents, -740), -740, exponents)
      bt = N.exp(exponents)
      if isinstance(x, N.ndarray):
          ans = N.where(N.less(x, (a+1)/(a+b+2.0)),
                        bt*abetacf(a, b, x, verbose)/float(a),
                        1.0-bt*abetacf(b, a, 1.0-x, verbose)/float(b))
      else:
          if x<(a+1)/(a+b+2.0):
              ans = bt*abetacf(a, b, x, verbose)/float(a)
          else:
              ans = 1.0-bt*abetacf(b, a, 1.0-x, verbose)/float(b)
      return ans
  
  
  #####################################
  #######  AANOVA CALCULATIONS  #######
  #####################################
  
   import LinearAlgebra, operator
   LA = LinearAlgebra
  
   def aglm(data, para):
      """
  Calculates a linear model fit ... anova/ancova/lin-regress/t-test/etc. Taken
  from:
      Peterson et al. Statistical limitations in functional neuroimaging
      I. Non-inferential methods and statistical models.  Phil Trans Royal Soc
      Lond B 354: 1239-1260.
  
  Usage:   aglm(data,para)
  Returns: statistic, p-value ???
  """
      if len(para) != len(data):
          print("data and para must be same length in aglm")
          return
      n = len(para)
      p = pstat.aunique(para)
      x = N.zeros((n, len(p)))  # design matrix
      for l in range(len(p)):
          x[:, l] = N.equal(para, p[l])
      b = N.dot(N.dot(LA.inv(N.dot(N.transpose(x), x)),  # i.e., b=inv(X'X)X'Y
                      N.transpose(x)),
                data)
      diffs = (data - N.dot(x, b))
      s_sq = 1./(n-len(p)) * N.dot(N.transpose(diffs), diffs)
  
      if len(p) == 2:  # ttest_ind
          c = N.array([1, -1])
          df = n-2
          fact = asum(1.0/asum(x, 0))  # i.e., 1/n1 + 1/n2 + 1/n3 ...
          t = N.dot(c, b) / N.sqrt(s_sq*fact)
          probs = abetai(0.5*df, 0.5, float(df)/(df+t*t))
          return t, probs
  
  
   def aF_oneway(*args):
      """
  Performs a 1-way ANOVA, returning an F-value and probability given
  any number of groups.  From Heiman, pp.394-7.
  
  Usage:   aF_oneway (*args)    where *args is 2 or more arrays, one per
                                    treatment group
  Returns: f-value, probability
  """
      na = len(args)            # ANOVA on 'na' groups, each in it's own array
      alldata = []
      alldata = N.concatenate(args)
      bign = len(alldata)
      sstot = ass(alldata)-(asquare_of_sums(alldata)/float(bign))
      ssbn = 0
      for a in args:
          ssbn = ssbn + asquare_of_sums(N.array(a))/float(len(a))
      ssbn = ssbn - (asquare_of_sums(alldata)/float(bign))
      sswn = sstot-ssbn
      dfbn = na-1
      dfwn = bign - na
      msb = ssbn/float(dfbn)
      msw = sswn/float(dfwn)
      f = msb/msw
      prob = fprob(dfbn, dfwn, f)
      return f, prob
  
  
   def aF_value (ER, EF, dfR, dfF):
      """
  Returns an F-statistic given the following:
          ER  = error associated with the null hypothesis (the Restricted model)
          EF  = error associated with the alternate hypothesis (the Full model)
          dfR = degrees of freedom the Restricted model
          dfF = degrees of freedom associated with the Restricted model
  """
      return ((ER-EF)/float(dfR-dfF) / (EF/float(dfF)))
  
  
   def outputfstats(Enum, Eden, dfnum, dfden, f, prob):
       Enum = round(Enum, 3)
       Eden = round(Eden, 3)
       dfnum = round(Enum, 3)
       dfden = round(dfden, 3)
       f = round(f, 3)
       prob = round(prob, 3)
       suffix = ''                       # for *s after the p-value
       if  prob < 0.001:  suffix = '  ***'
       elif prob < 0.01:  suffix = '  **'
       elif prob < 0.05:  suffix = '  *'
       title = [['EF/ER', 'DF', 'Mean Square', 'F-value', 'prob', '']]
       lofl = title+[[Enum, dfnum, round(Enum/float(dfnum), 3), f, prob, suffix],
                     [Eden, dfden, round(Eden/float(dfden), 3), '', '', '']]
       pstat.printcc(lofl)
       return
  
  
   def F_value_multivariate(ER, EF, dfnum, dfden):
       """
  Returns an F-statistic given the following:
          ER  = error associated with the null hypothesis (the Restricted model)
          EF  = error associated with the alternate hypothesis (the Full model)
          dfR = degrees of freedom the Restricted model
          dfF = degrees of freedom associated with the Restricted model
  where ER and EF are matrices from a multivariate F calculation.
  """
       if isinstance(ER, (int, float)):
           ER = N.array([[ER]])
       if isinstance(EF, (int, float)):
           EF = N.array([[EF]])
       n_um = (LA.det(ER) - LA.det(EF)) / float(dfnum)
       d_en = LA.det(EF) / float(dfden)
       return n_um / d_en
  
  
  #####################################
  #######  ASUPPORT FUNCTIONS  ########
  #####################################
  
   def asign(a):
      """
  Usage:   asign(a)
  Returns: array shape of a, with -1 where a<0 and +1 where a>=0
  """
      a = N.asarray(a)
      if isinstance(a, (int, float)):
          return a-a-N.less(a, 0)+N.greater(a, 0)
      else:
          return N.zeros(N.shape(a))-N.less(a, 0)+N.greater(a, 0)
  
  
   def asum (a, dimension=None,keepdims=0):
       """
  An alternative to the Numeric.add.reduce function, which allows one to
  (1) collapse over multiple dimensions at once, and/or (2) to retain
  all dimensions in the original array (squashing one down to size.
  Dimension can equal None (ravel array first), an integer (the
  dimension over which to operate), or a sequence (operate over multiple
  dimensions).  If keepdims=1, the resulting array will have as many
  dimensions as the input array.
  
  Usage:   asum(a, dimension=None, keepdims=0)
  Returns: array summed along 'dimension'(s), same _number_ of dims if keepdims=1
  """
       if isinstance(a, N.ndarray) and a.dtype in [N.int_, N.short, N.ubyte]:
           a = a.astype(N.float_)
       if dimension is None:
           s = N.sum(N.ravel(a))
       elif isinstance(dimension, (int, float)):
           s = N.add.reduce(a, dimension)
           if keepdims == 1:
               shp = list(a.shape)
               shp[dimension] = 1
               s = N.reshape(s, shp)
       else: # must be a SEQUENCE of dims to sum over
          dims = sorted(dimension, reverse=True)
          s = a *1.0
          for dim in dims:
              s = N.add.reduce(s, dim)
          if keepdims == 1:
              shp = list(a.shape)
              for dim in dims:
                  shp[dim] = 1
              s = N.reshape(s, shp)
       return s
  
  
   def acumsum (a,dimension=None):
      """
  Returns an array consisting of the cumulative sum of the items in the
  passed array.  Dimension can equal None (ravel array first), an
  integer (the dimension over which to operate), or a sequence (operate
  over multiple dimensions, but this last one just barely makes sense).
  
  Usage:   acumsum(a,dimension=None)
  """
      if dimension is None:
          a = N.ravel(a)
          dimension = 0
      if isinstance(dimension, (list, tuple, N.ndarray)):
          dimension = sorted(dimension, reverse=True)
          for d in dimension:
              a = N.add.accumulate(a, d)
          return a
      else:
          return N.add.accumulate(a, dimension)
  
  
   def ass(inarray, dimension=None, keepdims=0):
      """
  Squares each value in the passed array, adds these squares & returns
  the result.  Unfortunate function name. :-) Defaults to ALL values in
  the array.  Dimension can equal None (ravel array first), an integer
  (the dimension over which to operate), or a sequence (operate over
  multiple dimensions).  Set keepdims=1 to maintain the original number
  of dimensions.
  
  Usage:   ass(inarray, dimension=None, keepdims=0)
  Returns: sum-along-'dimension' for (inarray*inarray)
  """
      if dimension is None:
          inarray = N.ravel(inarray)
          dimension = 0
      return asum(inarray*inarray, dimension, keepdims)
  
  
   def asummult (array1,array2,dimension=None,keepdims=0):
      """
  Multiplies elements in array1 and array2, element by element, and
  returns the sum (along 'dimension') of all resulting multiplications.
  Dimension can equal None (ravel array first), an integer (the
  dimension over which to operate), or a sequence (operate over multiple
  dimensions).  A trivial function, but included for completeness.
  
  Usage:   asummult(array1,array2,dimension=None,keepdims=0)
  """
      if dimension is None:
          array1 = N.ravel(array1)
          array2 = N.ravel(array2)
          dimension = 0
      return asum(array1*array2, dimension, keepdims)
  
  
   def asquare_of_sums(inarray, dimension=None, keepdims=0):
      """
  Adds the values in the passed array, squares that sum, and returns the
  result.  Dimension can equal None (ravel array first), an integer (the
  dimension over which to operate), or a sequence (operate over multiple
  dimensions).  If keepdims=1, the returned array will have the same
  NUMBER of dimensions as the original.
  
  Usage:   asquare_of_sums(inarray, dimension=None, keepdims=0)
  Returns: the square of the sum over dim(s) in dimension
  """
      if dimension is None:
          inarray = N.ravel(inarray)
          dimension = 0
      s = asum(inarray, dimension, keepdims)
      if isinstance(s, N.ndarray):
          return s.astype(N.float_)*s
      else:
          return float(s)*s
  
  
   def asumdiffsquared(a,b, dimension=None, keepdims=0):
      """
  Takes pairwise differences of the values in arrays a and b, squares
  these differences, and returns the sum of these squares.  Dimension
  can equal None (ravel array first), an integer (the dimension over
  which to operate), or a sequence (operate over multiple dimensions).
  keepdims=1 means the return shape = len(a.shape) = len(b.shape)
  
  Usage:   asumdiffsquared(a,b)
  Returns: sum[ravel(a-b)**2]
  """
      if dimension is None:
          inarray = N.ravel(a)
          dimension = 0
      return asum((a-b)**2, dimension, keepdims)
  
  
   def ashellsort(inarray):
      """
  Shellsort algorithm.  Sorts a 1D-array.
  
  Usage:   ashellsort(inarray)
  Returns: sorted-inarray, sorting-index-vector (for original array)
  """
      n = len(inarray)
      svec = inarray *1.0
      ivec = list(range(n))
      gap = n // 2   # integer division needed
      while gap >0:
          for i in range(gap, n):
              for j in range(i-gap, -1, -gap):
                  while j>=0 and svec[j]>svec[j+gap]:
                      temp        = svec[j]
                      svec[j]     = svec[j+gap]
                      svec[j+gap] = temp
                      itemp       = ivec[j]
                      ivec[j]     = ivec[j+gap]
                      ivec[j+gap] = itemp
          gap = gap // 2  # integer division needed
  #    svec is now sorted input vector, ivec has the order svec[i] = vec[ivec[i]]
      return svec, ivec
  
  
   def arankdata(inarray):
      """
  Ranks the data in inarray, dealing with ties appropritely.  Assumes
  a 1D inarray.  Adapted from Gary Perlman's |Stat ranksort.
  
  Usage:   arankdata(inarray)
  Returns: array of length equal to inarray, containing rank scores
  """
      n = len(inarray)
      svec, ivec = ashellsort(inarray)
      sumranks = 0
      dupcount = 0
      newarray = N.zeros(n, N.float_)
      for i in range(n):
          sumranks = sumranks + i
          dupcount = dupcount + 1
          if i == n - 1 or svec[i] != svec[i + 1]:
              averank = sumranks / float(dupcount) + 1
              for j in range(i-dupcount+1, i+1):
                  newarray[ivec[j]] = averank
              sumranks = 0
              dupcount = 0
      return newarray
  
  
   def afindwithin(data):
      """
  Returns a binary vector, 1=within-subject factor, 0=between.  Input
  equals the entire data array (i.e., column 1=random factor, last
  column = measured values.
  
  Usage:   afindwithin(data)     data in |Stat format
  """
      numfact = len(data[0])-2
      withinvec = [0]*numfact
      for col in range(1, numfact+1):
          rows = pstat.linexand(data, col, pstat.unique(pstat.colex(data, 1))[0])  # get 1 level of this factor
          if len(pstat.unique(pstat.colex(rows, 0))) < len(rows):   # if fewer subjects than scores on this factor
              withinvec[col-1] = 1
      return withinvec
  
  
   #########################################################
   #########################################################
   ######  RE-DEFINE DISPATCHES TO INCLUDE ARRAYS  #########
   #########################################################
   #########################################################
  
  ## CENTRAL TENDENCY:
   geometricmean = Dispatch ( (lgeometricmean, (list, tuple)),
                              (ageometricmean, (N.ndarray,)) )
   harmonicmean = Dispatch ( (lharmonicmean, (list, tuple)),
                             (aharmonicmean, (N.ndarray,)) )
   mean = Dispatch ( (lmean, (list, tuple)),
                     (amean, (N.ndarray,)) )
   median = Dispatch ( (lmedian, (list, tuple)),
                       (amedian, (N.ndarray,)) )
   medianscore = Dispatch ( (lmedianscore, (list, tuple)),
                            (amedianscore, (N.ndarray,)) )
   mode = Dispatch ( (lmode, (list, tuple)),
                     (amode, (N.ndarray,)) )
   tmean = Dispatch ( (atmean, (N.ndarray,)) )
   tvar = Dispatch ( (atvar, (N.ndarray,)) )
   tstdev = Dispatch ( (atstdev, (N.ndarray,)) )
   tsem = Dispatch ( (atsem, (N.ndarray,)) )
  
  ## VARIATION:
   moment = Dispatch ( (lmoment, (list, tuple)),
                       (amoment, (N.ndarray,)) )
   variation = Dispatch ( (lvariation, (list, tuple)),
                          (avariation, (N.ndarray,)) )
   skew = Dispatch ( (lskew, (list, tuple)),
                     (askew, (N.ndarray,)) )
   kurtosis = Dispatch ( (lkurtosis, (list, tuple)),
                         (akurtosis, (N.ndarray,)) )
   describe = Dispatch ( (ldescribe, (list, tuple)),
                         (adescribe, (N.ndarray,)) )
  
  ## DISTRIBUTION TESTS
  
   skewtest = Dispatch ( (askewtest, (list, tuple)),
                         (askewtest, (N.ndarray,)) )
   kurtosistest = Dispatch ( (akurtosistest, (list, tuple)),
                             (akurtosistest, (N.ndarray,)) )
   normaltest = Dispatch ( (anormaltest, (list, tuple)),
                           (anormaltest, (N.ndarray,)) )
  
  ## FREQUENCY STATS:
   itemfreq = Dispatch ( (litemfreq, (list, tuple)),
                         (aitemfreq, (N.ndarray,)) )
   scoreatpercentile = Dispatch ( (lscoreatpercentile, (list, tuple)),
                                  (ascoreatpercentile, (N.ndarray,)) )
   percentileofscore = Dispatch ( (lpercentileofscore, (list, tuple)),
                                   (apercentileofscore, (N.ndarray,)) )
   histogram = Dispatch ( (lhistogram, (list, tuple)),
                          (ahistogram, (N.ndarray,)) )
   cumfreq = Dispatch ( (lcumfreq, (list, tuple)),
                        (acumfreq, (N.ndarray,)) )
   relfreq = Dispatch ( (lrelfreq, (list, tuple)),
                        (arelfreq, (N.ndarray,)) )
   
  ## VARIABILITY:
   obrientransform = Dispatch ( (lobrientransform, (list, tuple)),
                                (aobrientransform, (N.ndarray,)) )
   samplevar = Dispatch ( (lsamplevar, (list, tuple)),
                          (asamplevar, (N.ndarray,)) )
   samplestdev = Dispatch ( (lsamplestdev, (list, tuple)),
                            (asamplestdev, (N.ndarray,)) )
   signaltonoise = Dispatch( (asignaltonoise, (N.ndarray,)),)
   var = Dispatch ( (lvar, (list, tuple)),
                    (avar, (N.ndarray,)) )
   stdev = Dispatch ( (lstdev, (list, tuple)),
                      (astdev, (N.ndarray,)) )
   sterr = Dispatch ( (lsterr, (list, tuple)),
                      (asterr, (N.ndarray,)) )
   sem = Dispatch ( (lsem, (list, tuple)),
                    (asem, (N.ndarray,)) )
   z = Dispatch ( (lz, (list, tuple)),
                  (az, (N.ndarray,)) )
   zs = Dispatch ( (lzs, (list, tuple)),
                   (azs, (N.ndarray,)) )
   
  ## TRIMMING FCNS:
   threshold = Dispatch( (athreshold, (N.ndarray,)),)
   trimboth = Dispatch ( (ltrimboth, (list, tuple)),
                         (atrimboth, (N.ndarray,)) )
   trim1 = Dispatch ( (ltrim1, (list, tuple)),
                      (atrim1, (N.ndarray,)) )
   
  ## CORRELATION FCNS:
   paired = Dispatch ( (lpaired, (list, tuple)),
                       (apaired, (N.ndarray,)) )
   lincc = Dispatch ( (llincc, (list, tuple)),
                         (alincc, (N.ndarray,)) )
   pearsonr = Dispatch ( (lpearsonr, (list, tuple)),
                         (apearsonr, (N.ndarray,)) )
   spearmanr = Dispatch ( (lspearmanr, (list, tuple)),
                          (aspearmanr, (N.ndarray,)) )
   pointbiserialr = Dispatch ( (lpointbiserialr, (list, tuple)),
                               (apointbiserialr, (N.ndarray,)) )
   kendalltau = Dispatch ( (lkendalltau, (list, tuple)),
                           (akendalltau, (N.ndarray,)) )
   linregress = Dispatch ( (llinregress, (list, tuple)),
                           (alinregress, (N.ndarray,)) )
   
  ## INFERENTIAL STATS:
   ttest_1samp = Dispatch ( (lttest_1samp, (list, tuple)),
                            (attest_1samp, (N.ndarray,)) )
   ttest_ind = Dispatch ( (lttest_ind, (list, tuple)),
                          (attest_ind, (N.ndarray,)) )
   ttest_rel = Dispatch ( (lttest_rel, (list, tuple)),
                          (attest_rel, (N.ndarray,)) )
   chisquare = Dispatch ( (lchisquare, (list, tuple)),
                          (achisquare, (N.ndarray,)) )
   ks_2samp = Dispatch ( (lks_2samp, (list, tuple)),
                         (aks_2samp, (N.ndarray,)) )
   mannwhitneyu = Dispatch ( (lmannwhitneyu, (list, tuple)),
                             (amannwhitneyu, (N.ndarray,)) )
   tiecorrect = Dispatch ( (ltiecorrect, (list, tuple)),
                           (atiecorrect, (N.ndarray,)) )
   ranksums = Dispatch ( (lranksums, (list, tuple)),
                         (aranksums, (N.ndarray,)) )
   wilcoxont = Dispatch ( (lwilcoxont, (list, tuple)),
                          (awilcoxont, (N.ndarray,)) )
   kruskalwallish = Dispatch ( (lkruskalwallish, (list, tuple)),
                               (akruskalwallish, (N.ndarray,)) )
   friedmanchisquare = Dispatch ( (lfriedmanchisquare, (list, tuple)),
                                  (afriedmanchisquare, (N.ndarray,)) )
   
  ## PROBABILITY CALCS:
   chisqprob = Dispatch ( (lchisqprob, (int, float)),
                          (achisqprob, (N.ndarray,)) )
   zprob = Dispatch ( (lzprob, (int, float)),
                      (azprob, (N.ndarray,)) )
   ksprob = Dispatch ( (lksprob, (int, float)),
                       (aksprob, (N.ndarray,)) )
   fprob = Dispatch ( (lfprob, (int, float)),
                      (afprob, (N.ndarray,)) )
   betacf = Dispatch ( (lbetacf, (int, float)),
                       (abetacf, (N.ndarray,)) )
   betai = Dispatch ( (lbetai, (int, float)),
                      (abetai, (N.ndarray,)) )
   erfcc = Dispatch ( (lerfcc, (int, float)),
                      (aerfcc, (N.ndarray,)) )
   gammln = Dispatch ( (lgammln, (int, float)),
                       (agammln, (N.ndarray,)) )
   
  ## ANOVA FUNCTIONS:
   F_oneway = Dispatch ( (lF_oneway, (list, tuple)),
                         (aF_oneway, (N.ndarray,)) )
   F_value = Dispatch ( (lF_value, (list, tuple)),
                        (aF_value, (N.ndarray,)) )
  
  ## SUPPORT FUNCTIONS:
   incr = Dispatch ( (lincr, (list, tuple, N.ndarray)), )
   sum = Dispatch ( (lsum, (list, tuple)),
                    (asum, (N.ndarray,)) )
   cumsum = Dispatch ( (lcumsum, (list, tuple)),
                       (acumsum, (N.ndarray,)) )
   ss = Dispatch ( (lss, (list, tuple)),
                   (ass, (N.ndarray,)) )
   summult = Dispatch ( (lsummult, (list, tuple)),
                        (asummult, (N.ndarray,)) )
   square_of_sums = Dispatch ( (lsquare_of_sums, (list, tuple)),
                               (asquare_of_sums, (N.ndarray,)) )
   sumdiffsquared = Dispatch ( (lsumdiffsquared, (list, tuple)),
                               (asumdiffsquared, (N.ndarray,)) )
   shellsort = Dispatch ( (lshellsort, (list, tuple)),
                          (ashellsort, (N.ndarray,)) )
   rankdata = Dispatch ( (lrankdata, (list, tuple)),
                         (arankdata, (N.ndarray,)) )
   findwithin = Dispatch ( (lfindwithin, (list, tuple)),
                           (afindwithin, (N.ndarray,)) )
  
  ######################  END OF NUMERIC FUNCTION BLOCK  #####################
  
  ######################  END OF STATISTICAL FUNCTIONS  ######################
  
  except ImportError:
   pass
  