When trials are associated with continuous predictor variables, averaging is often a poor solution that loses part of the data. In such cases, a two-stage design can be employed that allows using the continuous predictor variable to test hypotheses at the group level. A two-stage analysis involves:
Stage 1: fit a regression model to each individual subject’s data
Stage 2: test regression coefficients at the group level
The example uses the same simulated data and design used in Multiple regression. The data are meant to vaguely resemble data from a word reading experiment, but not intended as a physiologically realistic simulation.
# sphinx_gallery_thumbnail_number = 1 from eelbrain import *
Generate simulated data: each function call to
generates a dataset for one subject (in a real experiment this would be
replaced with a function that loads data for this subject).
For each subject, a multiple regression model is fit using n characters and
cloze probability as continuous predictor variables.
lms =  for subject in range(10): # generate data for one subject ds = datasets.simulate_erp(seed=subject) # Re-reference EEG data ds['eeg'] -= ds['eeg'].mean(sensor=['M1', 'M2']) # Fit stage 1 model (samples=0 because we do not need permutations at stage 1) lm = testnd.LM('eeg', 'n_chars + cloze', data=ds, samples=0, subject=str(subject)) lms.append(lm)
Dataset with the first level statistic of interest.
rows =  for lm in lms: rows.append([lm.subject, lm.t('intercept'), lm.t('n_chars'), lm.t('cloze')]) # When creating the dataset for stage 2 analysis, declare subject as random factor; # this is only relevant if performing ANOVA as stage 2 test. data = Dataset.from_caselist(['subject', 'intercept', 'n_chars', 'cloze'], rows, random='subject') data
Now we can test whether the first stage estimates are consistent across subject.
Permutation test: 0%| | 0/1023 [00:00<?, ? permutations/s] Permutation test: 14%|█▍ | 148/1023 [00:00<00:00, 1476.45 permutations/s] Permutation test: 31%|███ | 315/1023 [00:00<00:00, 1586.08 permutations/s] Permutation test: 47%|████▋ | 484/1023 [00:00<00:00, 1627.56 permutations/s] Permutation test: 64%|██████▍ | 654/1023 [00:00<00:00, 1652.33 permutations/s] Permutation test: 81%|████████▏ | 832/1023 [00:00<00:00, 1693.20 permutations/s] Permutation test: 98%|█████████▊| 1003/1023 [00:00<00:00, 1693.47 permutations/s] Permutation test: 100%|██████████| 1023/1023 [00:00<00:00, 1660.64 permutations/s]
Instead of t-values, we might want to visualize regression coefficients:
rows =  for lm in lms: rows.append([lm.subject, lm.coefficient('n_chars')]) data_c = Dataset.from_caselist(['subject', 'n_chars'], rows, random='subject') # mask regression coefficients by significance to add outlines to plot masked_c = data_c['n_chars'].mean('case').mask(result.p > 0.05, missing=True) p = plot.TopoArray(masked_c, t=[0.120, 0.155, None], title=result, head_radius=0.35) p_cb = p.plot_colorbar(right_of=p.axes, label='µV', unit=1e-6)
- Of course, other tests could be applied at stage 2, for example
T-tests to compare coefficients for two different regressor, or two differen subject groups
ANOVA for multiple regressors and/or subject groups
Multiple regression models with subject variables to test for individual differnces