text: """
  .. _tut-timefreq-twoway-anova:
  
  ====================================================================
  Mass-univariate twoway repeated measures ANOVA on single trial power
  ====================================================================
  
  This script shows how to conduct a mass-univariate repeated measures
  ANOVA. As the model to be fitted assumes two fully crossed factors,
  we will study the interplay between perceptual modality
  (auditory VS visual) and the location of stimulus presentation
  (left VS right). Here we use single trials as replications
  (subjects) while iterating over time slices plus frequency bands
  for to fit our mass-univariate model. For the sake of simplicity we
  will confine this analysis to one single channel of which we know
  that it exposes a strong induced response. We will then visualize
  each effect by creating a corresponding mass-univariate effect
  image. We conclude with accounting for multiple comparisons by
  performing a permutation clustering test using the ANOVA as
  clustering function. The results final will be compared to multiple
  comparisons using False Discovery Rate correction.
  """
  # Authors: Denis Engemann <denis.engemann@gmail.com>
  #          Eric Larson <larson.eric.d@gmail.com>
  #          Alexandre Gramfort <alexandre.gramfort@inria.fr>
  #
  # License: BSD-3-Clause
  
  # %%
  
  import numpy as np
  import matplotlib.pyplot as plt
  
  import mne
  from mne.time_frequency import tfr_morlet
  from mne.stats import f_threshold_mway_rm, f_mway_rm, fdr_correction
  from mne.datasets import sample
  
  print(__doc__)
  
  # %%
  # Set parameters
  # --------------
  data_path = sample.data_path()
  raw_fname = data_path + '/MEG/sample/sample_audvis_raw.fif'
  event_fname = data_path + '/MEG/sample/sample_audvis_raw-eve.fif'
  tmin, tmax = -0.2, 0.5
  
  # Setup for reading the raw data
  raw = mne.io.read_raw_fif(raw_fname)
  events = mne.read_events(event_fname)
  
  include = []
  raw.info['bads'] += ['MEG 2443']  # bads
  
  # picks MEG gradiometers
  picks = mne.pick_types(raw.info, meg='grad', eeg=False, eog=True,
                         stim=False, include=include, exclude='bads')
  
  ch_name = 'MEG 1332'
  
  # Load conditions
  reject = dict(grad=4000e-13, eog=150e-6)
  event_id = dict(aud_l=1, aud_r=2, vis_l=3, vis_r=4)
  epochs = mne.Epochs(raw, events, event_id, tmin, tmax,
                      picks=picks, baseline=(None, 0), preload=True,
                      reject=reject)
  epochs.pick_channels([ch_name])  # restrict example to one channel
  
  # %%
  # We have to make sure all conditions have the same counts, as the ANOVA
  # expects a fully balanced data matrix and does not forgive imbalances that
  # generously (risk of type-I error).
  epochs.equalize_event_counts(event_id)
  
  # Factor to down-sample the temporal dimension of the TFR computed by
  # tfr_morlet.
  decim = 2
  freqs = np.arange(7, 30, 3)  # define frequencies of interest
  n_cycles = freqs / freqs[0]
  zero_mean = False  # don't correct morlet wavelet to be of mean zero
  # To have a true wavelet zero_mean should be True but here for illustration
  # purposes it helps to spot the evoked response.
  
  # %%
  # Create TFR representations for all conditions
  # ---------------------------------------------
  epochs_power = list()
  for condition in [epochs[k] for k in event_id]:
      this_tfr = tfr_morlet(condition, freqs, n_cycles=n_cycles,
                            decim=decim, average=False, zero_mean=zero_mean,
                            return_itc=False)
      this_tfr.apply_baseline(mode='ratio', baseline=(None, 0))
      this_power = this_tfr.data[:, 0, :, :]  # we only have one channel.
      epochs_power.append(this_power)
  
  # %%
  # Setup repeated measures ANOVA
  # -----------------------------
  #
  # We will tell the ANOVA how to interpret the data matrix in terms of factors.
  # This is done via the factor levels argument which is a list of the number
  # factor levels for each factor.
  
  n_conditions = len(epochs.event_id)
  n_replications = epochs.events.shape[0] // n_conditions
  
  factor_levels = [2, 2]  # number of levels in each factor
  effects = 'A*B'  # this is the default signature for computing all effects
  # Other possible options are 'A' or 'B' for the corresponding main effects
  # or 'A:B' for the interaction effect only (this notation is borrowed from the
  # R formula language)
  n_freqs = len(freqs)
  times = 1e3 * epochs.times[::decim]
  n_times = len(times)
  
  # %%
  # Now we'll assemble the data matrix and swap axes so the trial replications
  # are the first dimension and the conditions are the second dimension.
  data = np.swapaxes(np.asarray(epochs_power), 1, 0)
  # reshape last two dimensions in one mass-univariate observation-vector
  data = data.reshape(n_replications, n_conditions, n_freqs * n_times)
  
  # so we have replications * conditions * observations:
  print(data.shape)
  
  # %%
  # While the iteration scheme used above for assembling the data matrix
  # makes sure the first two dimensions are organized as expected (with A =
  # modality and B = location):
  #
  # .. table:: Sample data layout
  #
  #    ===== ==== ==== ==== ====
  #    trial A1B1 A1B2 A2B1 B2B2
  #    ===== ==== ==== ==== ====
  #    1     1.34 2.53 0.97 1.74
  #    ...   ...  ...  ...  ...
  #    56    2.45 7.90 3.09 4.76
  #    ===== ==== ==== ==== ====
  #
  # Now we're ready to run our repeated measures ANOVA.
  #
  # Note. As we treat trials as subjects, the test only accounts for
  # time locked responses despite the 'induced' approach.
  # For analysis for induced power at the group level averaged TRFs
  # are required.
  
  fvals, pvals = f_mway_rm(data, factor_levels, effects=effects)
  
  effect_labels = ['modality', 'location', 'modality by location']
  
  # let's visualize our effects by computing f-images
  for effect, sig, effect_label in zip(fvals, pvals, effect_labels):
      plt.figure()
      # show naive F-values in gray
      plt.imshow(effect.reshape(8, 211), cmap=plt.cm.gray, extent=[times[0],
                 times[-1], freqs[0], freqs[-1]], aspect='auto',
                 origin='lower')
      # create mask for significant Time-frequency locations
      effect[sig >= 0.05] = np.nan
      plt.imshow(effect.reshape(8, 211), cmap='RdBu_r', extent=[times[0],
                 times[-1], freqs[0], freqs[-1]], aspect='auto',
                 origin='lower')
      plt.colorbar()
      plt.xlabel('Time (ms)')
      plt.ylabel('Frequency (Hz)')
      plt.title(r"Time-locked response for '%s' (%s)" % (effect_label, ch_name))
      plt.show()
  
  # %%
  # Account for multiple comparisons using FDR versus permutation clustering test
  # -----------------------------------------------------------------------------
  #
  # First we need to slightly modify the ANOVA function to be suitable for
  # the clustering procedure. Also want to set some defaults.
  # Let's first override effects to confine the analysis to the interaction
  effects = 'A:B'
  
  # %%
  # A stat_fun must deal with a variable number of input arguments.
  # Inside the clustering function each condition will be passed as flattened
  # array, necessitated by the clustering procedure. The ANOVA however expects an
  # input array of dimensions: subjects X conditions X observations (optional).
  # The following function catches the list input and swaps the first and
  # the second dimension and finally calls the ANOVA function.
  
  
  def stat_fun(*args):
      return f_mway_rm(np.swapaxes(args, 1, 0), factor_levels=factor_levels,
                       effects=effects, return_pvals=False)[0]
  
  
  # The ANOVA returns a tuple f-values and p-values, we will pick the former.
  pthresh = 0.001  # set threshold rather high to save some time
  f_thresh = f_threshold_mway_rm(n_replications, factor_levels, effects,
                                 pthresh)
  tail = 1  # f-test, so tail > 0
  n_permutations = 256  # Save some time (the test won't be too sensitive ...)
  T_obs, clusters, cluster_p_values, h0 = mne.stats.permutation_cluster_test(
      epochs_power, stat_fun=stat_fun, threshold=f_thresh, tail=tail, n_jobs=1,
      n_permutations=n_permutations, buffer_size=None, out_type='mask')
  
  # %%
  # Create new stats image with only significant clusters:
  
  good_clusters = np.where(cluster_p_values < .05)[0]
  T_obs_plot = T_obs.copy()
  T_obs_plot[~clusters[np.squeeze(good_clusters)]] = np.nan
  
  plt.figure()
  for f_image, cmap in zip([T_obs, T_obs_plot], [plt.cm.gray, 'RdBu_r']):
      plt.imshow(f_image, cmap=cmap, extent=[times[0], times[-1],
                 freqs[0], freqs[-1]], aspect='auto',
                 origin='lower')
  plt.xlabel('Time (ms)')
  plt.ylabel('Frequency (Hz)')
  plt.title("Time-locked response for 'modality by location' (%s)\n"
            " cluster-level corrected (p <= 0.05)" % ch_name)
  plt.show()
  
  # %%
  # Now using FDR:
  
  mask, _ = fdr_correction(pvals[2])
  T_obs_plot2 = T_obs.copy()
  T_obs_plot2[~mask.reshape(T_obs_plot.shape)] = np.nan
  
  plt.figure()
  for f_image, cmap in zip([T_obs, T_obs_plot2], [plt.cm.gray, 'RdBu_r']):
      if np.isnan(f_image).all():
          continue  # nothing to show
      plt.imshow(f_image, cmap=cmap, extent=[times[0], times[-1],
                 freqs[0], freqs[-1]], aspect='auto',
                 origin='lower')
  
  plt.xlabel('Time (ms)')
  plt.ylabel('Frequency (Hz)')
  plt.title("Time-locked response for 'modality by location' (%s)\n"
            " FDR corrected (p <= 0.05)" % ch_name)
  plt.show()
  
  # %%
  # Both cluster level and FDR correction help get rid of
  # potential spots we saw in the naive f-images.
  