time-series-test 0.3.0

A statistical test and plotting function for time-series data in general, and data from cognitive-pupillometry experiments in particular. Based on linear mixed effects modeling and crossvalidation.

Time Series Test

A statistical test and plotting function for time-series data in general, and data from cognitive-pupillometry experiments in particular. Based on linear mixed effects modeling and crossvalidation.

Sebastiaan Mathôt (@smathot)
Copyright 2021 - 2022

Contents

• Citation
• About
• Dependencies
• Usage
• Function reference
• License

Citation

Mathôt, S., & Vilotijević, A. (in prep). A Hands-on Guide to Cognitive Pupillometry: from Design to Analysis.

This is a work in progress. A preprint of this manuscript will be made available soon.

About

In general terms, this package implements a statistical test for a specific-yet-common question when analyzing time-series data:

Do one or more independent variables affect a continuously recorded dependent variable (a 'time series') at any point in time?

When to use this test:

• For time series consisting of only a single component, that is, when each independent variable has only a single effect on the time series. An example of this is the effect of stimulus intensity on pupil size, when presenting light flashes of different intensities.
• When you do not know a priori which time points to test.

When not to use this test:

• For time series that contain multiple components, that is, when each independent variable affects the time series in multiple ways that change over time. An example of this is the effect of visual attention on lateralized EEG recordings, where different EEG components emerge at different points in time.
• When you know a priori which time points to test.

More specifically, this package provides a function (find()) that locates and statistically tests effects in time-series data. It does so by using crossvalidation to identify time points to test, and then using a linear mixed effects model to actually perform the statistical test. More specifically, the data is subdivided in a number of subsets (by default 4). It takes one of the subsets (the test set) out of the full dataset, and conducts a linear mixed effects model on each sample of the remaining data (the training set). The sample with the highest absolute z value in the training set is used as the sample-to-be-tested for the test set. This procedure is repeated for all subsets of the data, and for all fixed effects in the model. Finally, a single linear mixed effects model is conducted for each fixed effects on the samples that were thus identified.

This packages also provides a function (plot()) to visualize time-series data to visually annotate the results of find().

For a more detailed description, see the manuscript above.

Dependencies

• Python 3
• datamatrix
• statsmodels
• matplotlib

Usage

We will use data from Zhou, Lorist, and Mathôt (2021). In brief, this is data from a visual-working-memory experiment in which participant memorized one or more colors (set size: 1, 2, 3 or 4) of two different types (color type: proto, nonproto) while pupil size was being recorded during a 3s retention interval.

This dataset contains the following columns:

• pupil, which is is our dependent measure. It is a baseline-corrected pupil time series of 300 samples, recorded at 100 Hz
• subject_nr, which we will use as a random effect
• set_size, which we will use as a fixed effect
• color_type, which we will use as a fixed effect

First, load the dataset:

from datamatrix import io
dm = io.readpickle('data/zhou_et_al_2021.pkl')

The plot() function provides a convenient way to plot pupil size over time as a function of one or two factors, in this case set size and color type:

import time_series_test as tst
from matplotlib import pyplot as plt

tst.plot(dm, dv='pupil', hue_factor='set_size', linestyle_factor='color_type')
plt.savefig('img/signal-plot-1.png')

From this plot, we can tell that there appear to be effects in the 1500 to 2000 ms interval. To test this, we could perform a linear mixed effects model on this interval, which corresponds to samples 150 to 200.

The model below uses mean pupil size during the 150 - 200 sample range as dependent measure, set size and color type as fixed effects, and a random by-subject intercept. In the more familiar notation of the R package lme4, this corresponds to mean_pupil ~ set_size * color_type + (1 | subject_nr). (To use more complex random-effects structures, you can use the re_formula argument to mixedlm().)

from statsmodels.formula.api import mixedlm
from datamatrix import series as srs, NAN

dm.mean_pupil = srs.reduce(dm.pupil[:, 150:200])
dm_valid_data = dm.mean_pupil != NAN
model = mixedlm(formula='mean_pupil ~ set_size * color_type',
                data=dm_valid_data, groups='subject_nr').fit()
print(model.summary())

Output:

                    Mixed Linear Model Regression Results
=============================================================================
Model:                    MixedLM       Dependent Variable:       mean_pupil 
No. Observations:         7300          Method:                   REML       
No. Groups:               30            Scale:                    38610.3390 
Min. group size:          235           Log-Likelihood:           -48952.3998
Max. group size:          248           Converged:                Yes        
Mean group size:          243.3                                              
-----------------------------------------------------------------------------
                              Coef.   Std.Err.   z    P>|z|  [0.025   0.975] 
-----------------------------------------------------------------------------
Intercept                    -144.024   17.438 -8.259 0.000 -178.202 -109.846
color_type[T.proto]           -24.133   11.299 -2.136 0.033  -46.278   -1.987
set_size                       49.979    2.906 17.200 0.000   44.284   55.675
set_size:color_type[T.proto]   10.176    4.120  2.470 0.014    2.101   18.251
subject_nr Var               7217.423    9.882                               
=============================================================================

The model summary shows that, assuming an alpha level of .05, there are significant main effects of color type (z = -2.136, p = .033), set size (z = 17.2, p < .001), and a significant color-type by set-size interaction (z = 2.47, p = .014). However, we have selectively analyzed a sample range that we knew, based on a visual inspection of the data, to show these effects. This means that our analysis is circular: we have looked at the data to decide where to look! The find() function improves this by splitting the data into training and tests sets, as described under About, thus breaking the circularity.

results = tst.find(dm,  'pupil ~ set_size * color_type',
                   groups='subject_nr', winlen=5)

The return value of find() is a dict, where keys are effect labels and values are named tuples of the following:

• model: a model as returned by mixedlm().fit()
• samples: a set with the sample indices that were used
• p: the p-value from the model
• z: the z-value from the model

The summarize() function is a convenient way to get the results in a human-readable format.

print(tst.summarize(results))

Output:

Intercept was tested at samples {95} → z = -13.1098, p = 2.892e-39
color_type[T.proto] was tested at samples {160, 170, 175} → z = -2.0949, p = 0.03618
set_size was tested at samples {185, 210, 195, 255} → z = 16.2437, p = 2.475e-59
set_size:color_type[T.proto] was tested at samples {165, 175} → z = 2.5767, p = 0.009974
None

We can pass the results to plot() to visualize the results:

tst.plot(dm, dv='pupil', hue_factor='set_size', linestyle_factor='color_type',
         results=results)
plt.savefig('img/signal-plot-2.png')

Function reference

time_series_test.find(dm, formula, groups, re_formula=None, winlen=1, split=4, split_method='interleaved', samples_fe=True, samples_re=True, fit_method=None, suppress_convergence_warnings=False, **kwargs)

Conducts a single linear mixed effects model to a time series, where the to-be-tested samples are determined through crossvalidation.

This function uses mixedlm() from the statsmodels package. See the statsmodels documentation for a more detailed explanation of the parameters.

Parameters

• dm: DataMatrix : The dataset
• formula: str : A formula that describes the dependent variable, which should be the name of a series column in dm, and the fixed effects, which should be regular (non-series) columns.
• groups: str or list of str : The groups for the random effects, which should be regular (non-series) columns in dm.
• re_formula: str or None : A formula that describes the random effects, which should be regular (non-series) columns in dm.
• winlen: int, optional : The number of samples that should be analyzed together, i.e. a downsampling window to speed up the analysis.
• split: int, optional : The number of splits that the analysis should be based on.
• split_method: str, optional : If 'interleaved', the data is split in a regular interleaved fashion, such that the first row goes to the first subset, the second row to the second subset, etc. If 'random', the data is split randomly in subsets. Interleaved splitting is deterministic (i.e. it results in the same outcome each time), but random splitting is not.
• samples_fe: bool, optional : Indicates whether sample indices are included as an additive factor to the fixed-effects formula. If all splits yielded the same sample index, this is ignored.
• samples_re: bool, optional : Indicates whether sample indices are included as an additive factor to the random-effects formula. If all splits yielded the same sample index, this is ignored.
• fit_method: str, list of str, or None, optional : The fitting method, which is passed as the method keyword to mixedlm.fit(). This can be a label or a list of labels, in which case different fitting methods are tried in case of convergence errors.
• suppress_convergence_warnings: bool, optional : Installs a warning filter to suppress conververgence (and other) warnings.
• **kwargs: dict, optional : Optional keywords to be passed to mixedlm(), such as groups and re_formula.

Returns

dict A dict where keys are effect labels, and values are named tuples of model, samples, p, and z.

time_series_test.plot(dm, dv, hue_factor, results=None, linestyle_factor=None, hues=None, linestyles=None, alpha_level=0.05, annotate_intercept=False, annotation_hues=None, annotation_linestyle=':')

Visualizes a time series, where the signal is plotted as a function of sample number on the x-axis. One fixed effect is indicated by the hue (color) of the lines. An optional second fixed effect is indicated by the linestyle. If the results parameter is used, significant effects are annotated in the figure.

Parameters

• dm: DataMatrix : The dataset
• dv: str : The name of the dependent variable, which should be a series column in dm.
• hue_factor: str : The name of a regular (non-series) column in dm that specifies the hue (color) of the lines.
• results: dict, optional : A results dict as returned by find().
• linestyle_factor: str, optional : The name of a regular (non-series) column in dm that specifies the linestyle of the lines for a two-factor plot.
• hues: list or None, optional : A list of hues to be used as line colors for the first factor.
• linestyles: list or None, optional : A list of linestyles to be used for the second factor.
• alpha_level: float, optional : The alpha level (maximum p value) to be used for annotating effects in the plot.
• annotate_intercept: bool, optional : Specifies whether the intercept should also be annotated along with the fixed effects.
• annotation_hues: list or None, optional : A list of hues to be used as line color for the annotations.
• annotation_linestyle: str, optional : The linestyle for the annotations.

time_series_test.summarize(results)

Generates a string with a human-readable summary of a results dict as returned by find().

Parameters

• results: dict : A results dict as returned by find().

Returns

str

License

time_series_test is licensed under the GNU General Public License v3.