code: def page_trend_test(
      data, ranked=False, predicted_ranks=None, method="auto"
  ):
      r"""
      Perform Page's Test, a measure of trend in observations between treatments.
  
      Page's Test (also known as Page's :math:`L` test) is useful when:
  
      * there are :math:`n \geq 3` treatments,
      * :math:`m \geq 2` subjects are observed for each treatment, and
      * the observations are hypothesized to have a particular order.
  
      Specifically, the test considers the null hypothesis that
  
      .. math::
  
          m_1 = m_2 = m_3 \cdots = m_n,
  
      where :math:`m_j` is the mean of the observed quantity under treatment
      :math:`j`, against the alternative hypothesis that
  
      .. math::
  
          m_1 \leq m_2 \leq m_3 \leq \cdots \leq m_n,
  
      where at least one inequality is strict.
  
      As noted by [4]_, Page's :math:`L` test has greater statistical power than
      the Friedman test against the alternative that there is a difference in
      trend, as Friedman's test only considers a difference in the means of the
      observations without considering their order. Whereas Spearman :math:`\rho`
      considers the correlation between the ranked observations of two variables
      (e.g. the airspeed velocity of a swallow vs. the weight of the coconut it
      carries), Page's :math:`L` is concerned with a trend in an observation
      (e.g. the airspeed velocity of a swallow) across several distinct
      treatments (e.g. carrying each of five coconuts of different weight) even
      as the observation is repeated with multiple subjects (e.g. one European
      swallow and one African swallow).
  
      Parameters
      ----------
      data : array-like
          A :math:`m \times n` array; the element in row :math:`i` and
          column :math:`j` is the observation corresponding with subject
          :math:`i` and treatment :math:`j`. By default, the columns are
          assumed to be arranged in order of increasing predicted mean.
  
      ranked : boolean, optional
          By default, `data` is assumed to be observations rather than ranks;
          it will be ranked with `scipy.stats.rankdata` along ``axis=1``. If
          `data` is provided in the form of ranks, pass argument ``True``.
  
      predicted_ranks : array-like, optional
          The predicted ranks of the column means. If not specified,
          the columns are assumed to be arranged in order of increasing
          predicted mean, so the default `predicted_ranks` are
          :math:`[1, 2, \dots, n-1, n]`.
  
      method : {'auto', 'asymptotic', 'exact'}, optional
          Selects the method used to calculate the *p*-value. The following
          options are available.
  
          * 'auto': selects between 'exact' and 'asymptotic' to
            achieve reasonably accurate results in reasonable time (default)
          * 'asymptotic': compares the standardized test statistic against
            the normal distribution
          * 'exact': computes the exact *p*-value by comparing the observed
            :math:`L` statistic against those realized by all possible
            permutations of ranks (under the null hypothesis that each
            permutation is equally likely)
  
      Returns
      -------
      res : PageTrendTestResult
          An object containing attributes:
  
          statistic : float
              Page's :math:`L` test statistic.
          pvalue : float
              The associated *p*-value
          method : {'asymptotic', 'exact'}
              The method used to compute the *p*-value
  
      See Also
      --------
      rankdata, friedmanchisquare, spearmanr
  
      Notes
      -----
      As noted in [1]_, "the :math:`n` 'treatments' could just as well represent
      :math:`n` objects or events or performances or persons or trials ranked."
      Similarly, the :math:`m` 'subjects' could equally stand for :math:`m`
      "groupings by ability or some other control variable, or judges doing
      the ranking, or random replications of some other sort."
  
      The procedure for calculating the :math:`L` statistic, adapted from
      [1]_, is:
  
      1. "Predetermine with careful logic the appropriate hypotheses
         concerning the predicted ording of the experimental results.
         If no reasonable basis for ordering any treatments is known, the
         :math:`L` test is not appropriate."
      2. "As in other experiments, determine at what level of confidence
         you will reject the null hypothesis that there is no agreement of
         experimental results with the monotonic hypothesis."
      3. "Cast the experimental material into a two-way table of :math:`n`
         columns (treatments, objects ranked, conditions) and :math:`m`
         rows (subjects, replication groups, levels of control variables)."
      4. "When experimental observations are recorded, rank them across each
         row", e.g. ``ranks = scipy.stats.rankdata(data, axis=1)``.
      5. "Add the ranks in each column", e.g.
         ``colsums = np.sum(ranks, axis=0)``.
      6. "Multiply each sum of ranks by the predicted rank for that same
         column", e.g. ``products = predicted_ranks * colsums``.
      7. "Sum all such products", e.g. ``L = products.sum()``.
  
      [1]_ continues by suggesting use of the standardized statistic
  
      .. math::
  
          \chi_L^2 = \frac{\left[12L-3mn(n+1)^2\right]^2}{mn^2(n^2-1)(n+1)}
  
      "which is distributed approximately as chi-square with 1 degree of
      freedom. The ordinary use of :math:`\chi^2` tables would be
      equivalent to a two-sided test of agreement. If a one-sided test
      is desired, *as will almost always be the case*, the probability
      discovered in the chi-square table should be *halved*."
  
      However, this standardized statistic does not distinguish between the
      observed values being well correlated with the predicted ranks and being
      _anti_-correlated with the predicted ranks. Instead, we follow [2]_
      and calculate the standardized statistic
  
      .. math::
  
          \Lambda = \frac{L - E_0}{\sqrt{V_0}},
  
      where :math:`E_0 = \frac{1}{4} mn(n+1)^2` and
      :math:`V_0 = \frac{1}{144} mn^2(n+1)(n^2-1)`, "which is asymptotically
      normal under the null hypothesis".
  
      The *p*-value for ``method='exact'`` is generated by comparing the observed
      value of :math:`L` against the :math:`L` values generated for all
      :math:`(n!)^m` possible permutations of ranks. The calculation is performed
      using the recursive method of [5].
  
      The *p*-values are not adjusted for the possibility of ties. When
      ties are present, the reported  ``'exact'`` *p*-values may be somewhat
      larger (i.e. more conservative) than the true *p*-value [2]_. The
      ``'asymptotic'``` *p*-values, however, tend to be smaller (i.e. less
      conservative) than the ``'exact'`` *p*-values.
  
      References
      ----------
      .. [1] Ellis Batten Page, "Ordered hypotheses for multiple treatments:
         a significant test for linear ranks", *Journal of the American
         Statistical Association* 58(301), p. 216--230, 1963.
  
      .. [2] Markus Neuhauser, *Nonparametric Statistical Test: A computational
         approach*, CRC Press, p. 150--152, 2012.
  
      .. [3] Statext LLC, "Page's L Trend Test - Easy Statistics", *Statext -
         Statistics Study*, https://www.statext.com/practice/PageTrendTest03.php,
         Accessed July 12, 2020.
  
      .. [4] "Page's Trend Test", *Wikipedia*, WikimediaFoundation,
         https://en.wikipedia.org/wiki/Page%27s_trend_test,
         Accessed July 12, 2020.
  
      .. [5] Robert E. Odeh, "The exact distribution of Page's L-statistic in
         the two-way layout", *Communications in Statistics - Simulation and
         Computation*,  6(1), p. 49--61, 1977.
  
      Examples
      --------
      We use the example from [3]_: 10 students are asked to rate three
      teaching methods - tutorial, lecture, and seminar - on a scale of 1-5,
      with 1 being the lowest and 5 being the highest. We have decided that
      a confidence level of 99% is required to reject the null hypothesis in
      favor of our alternative: that the seminar will have the highest ratings
      and the tutorial will have the lowest. Initially, the data have been
      tabulated with each row representing an individual student's ratings of
      the three methods in the following order: tutorial, lecture, seminar.
  
      >>> table = [[3, 4, 3],
      ...          [2, 2, 4],
      ...          [3, 3, 5],
      ...          [1, 3, 2],
      ...          [2, 3, 2],
      ...          [2, 4, 5],
      ...          [1, 2, 4],
      ...          [3, 4, 4],
      ...          [2, 4, 5],
      ...          [1, 3, 4]]
  
      Because the tutorial is hypothesized to have the lowest ratings, the
      column corresponding with tutorial rankings should be first; the seminar
      is hypothesized to have the highest ratings, so its column should be last.
      Since the columns are already arranged in this order of increasing
      predicted mean, we can pass the table directly into `page_trend_test`.
  
      >>> from scipy.stats import page_trend_test
      >>> res = page_trend_test(table)
      >>> res
      PageTrendTestResult(statistic=133.5, pvalue=0.0018191161948127822,
                          method='exact')
  
      This *p*-value indicates that there is a 0.1819% chance that
      the :math:`L` statistic would reach such an extreme value under the null
      hypothesis. Because 0.1819% is less than 1%, we have evidence to reject
      the null hypothesis in favor of our alternative at a 99% confidence level.
  
      The value of the :math:`L` statistic is 133.5. To check this manually,
      we rank the data such that high scores correspond with high ranks, settling
      ties with an average rank:
  
      >>> from scipy.stats import rankdata
      >>> ranks = rankdata(table, axis=1)
      >>> ranks
      array([[1.5, 3. , 1.5],
             [1.5, 1.5, 3. ],
             [1.5, 1.5, 3. ],
             [1. , 3. , 2. ],
             [1.5, 3. , 1.5],
             [1. , 2. , 3. ],
             [1. , 2. , 3. ],
             [1. , 2.5, 2.5],
             [1. , 2. , 3. ],
             [1. , 2. , 3. ]])
  
      We add the ranks within each column, multiply the sums by the
      predicted ranks, and sum the products.
  
      >>> import numpy as np
      >>> m, n = ranks.shape
      >>> predicted_ranks = np.arange(1, n+1)
      >>> L = (predicted_ranks * np.sum(ranks, axis=0)).sum()
      >>> res.statistic == L
      True
  
      As presented in [3]_, the asymptotic approximation of the *p*-value is the
      survival function of the normal distribution evaluated at the standardized
      test statistic:
  
      >>> from scipy.stats import norm
      >>> E0 = (m*n*(n+1)**2)/4
      >>> V0 = (m*n**2*(n+1)*(n**2-1))/144
      >>> Lambda = (L-E0)/np.sqrt(V0)
      >>> p = norm.sf(Lambda)
      >>> p
      0.0012693433690751756
  
      This does not precisely match the *p*-value reported by `page_trend_test`
      above. The asymptotic distribution is not very accurate, nor conservative,
      for :math:`m \leq 12` and :math:`n \leq 8`, so `page_trend_test` chose to
      use ``method='exact'`` based on the dimensions of the table and the
      recommendations in Page's original paper [1]_. To override
      `page_trend_test`'s choice, provide the `method` argument.
  
      >>> res = page_trend_test(table, method="asymptotic")
      >>> res
      PageTrendTestResult(statistic=133.5, pvalue=0.0012693433690751756,
                          method='asymptotic')
  
      If the data are already ranked, we can pass in the ``ranks`` instead of
      the ``table`` to save computation time.
  
      >>> res = page_trend_test(ranks,             # ranks of data
      ...                       ranked=True,       # data is already ranked
      ...                       )
      >>> res
      PageTrendTestResult(statistic=133.5, pvalue=0.0018191161948127822,
                          method='exact')
  
      Suppose the raw data had been tabulated in an order different from the
      order of predicted means, say lecture, seminar, tutorial.
  
      >>> table = np.asarray(table)[:, [1, 2, 0]]
  
      Since the arrangement of this table is not consistent with the assumed
      ordering, we can either rearrange the table or provide the
      `predicted_ranks`. Remembering that the lecture is predicted
      to have the middle rank, the seminar the highest, and tutorial the lowest,
      we pass:
  
      >>> res = page_trend_test(table,             # data as originally tabulated
      ...                       predicted_ranks=[2, 3, 1],  # our predicted order
      ...                       )
      >>> res
      PageTrendTestResult(statistic=133.5, pvalue=0.0018191161948127822,
                          method='exact')
  
      """
  
      # Possible values of the method parameter and the corresponding function
      # used to evaluate the p value
      methods = {
          "asymptotic": _l_p_asymptotic,
          "exact": _l_p_exact,
          "auto": None,
      }
      if method not in methods:
          raise ValueError(f"`method` must be in {set(methods)}")
  
      ranks = np.array(data, copy=False)
      if ranks.ndim != 2:  # TODO: relax this to accept 3d arrays?
          raise ValueError("`data` must be a 2d array.")
  
      m, n = ranks.shape
      if m < 2 or n < 3:
          raise ValueError(
              "Page's L is only appropriate for data with two "
              "or more rows and three or more columns."
          )
  
      if np.any(np.isnan(data)):
          raise ValueError(
              "`data` contains NaNs, which cannot be ranked "
              "meaningfully"
          )
  
      # ensure NumPy array and rank the data if it's not already ranked
      if ranked:
          # Only a basic check on whether data is ranked. Checking that the data
          # is properly ranked could take as much time as ranking it.
          if not (ranks.min() >= 1 and ranks.max() <= ranks.shape[1]):
              raise ValueError(
                  "`data` is not properly ranked. Rank the data or "
                  "pass `ranked=False`."
              )
      else:
          ranks = scipy.stats.rankdata(data, axis=-1)
  
      # generate predicted ranks if not provided, ensure valid NumPy array
      if predicted_ranks is None:
          predicted_ranks = np.arange(1, n + 1)
      else:
          predicted_ranks = np.array(predicted_ranks, copy=False)
          if predicted_ranks.ndim < 1 or (
              set(predicted_ranks) != set(range(1, n + 1))
              or len(predicted_ranks) != n
          ):
              raise ValueError(
                  f"`predicted_ranks` must include each integer "
                  f"from 1 to {n} (the number of columns in "
                  f"`data`) exactly once."
              )
  
      if type(ranked) is not bool:
          raise TypeError("`ranked` must be boolean.")
  
      # Calculate the L statistic
      L = _l_vectorized(ranks, predicted_ranks)
  
      # Calculate the p-value
      if method == "auto":
          method = _choose_method(ranks)
      p_fun = methods[
          method
      ]  # get the function corresponding with the method
      p = p_fun(L, m, n)
  
      page_result = PageTrendTestResult(
          statistic=L, pvalue=p, method=method
      )
      return page_result
  
file_path: catboost/contrib/python/scipy/py3/scipy/stats/_page_trend_test.py
imports: from itertools import permutations
  import numpy
  import math
  from ._continuous_distns import norm
  import scipy.stats
  from dataclasses import make_dataclass
label_desc: Use image node_id 1 for calling a global function with example usage: page_trend_test(data, ranked, predicted_ranks, method) and returns: page_result