A statistical test and plotting function for time-series data in general,

# Time Series Test

Statistical testing and plotting functions for time-series data in general, and data from cognitive-pupillometry and electroencephalography (EEG) experiments in particular. Based on linear mixed effects modeling (or regular multiple linear regression), crossvalidation, and cluster-based permutation testing.

Sebastiaan Mathôt (@smathot)

## Citation

Mathôt, S., & Vilotijević, A. (2022). Methods in cognitive pupillometry: design, preprocessing, and analysis. Behavior Research Methods. https://doi.org/10.1101/2022.02.23.481628

This library provides two main functions for statistical testing of time-series data: lmer_crossvalidation_test() and lmer_permutation_test(). For a detailed description, see the manuscript above, but below a short introduction to both functions with their respective advantages and disadavantages.

### When to use crossvalidation?

In general terms, lmer_crossvalidation_test() 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, lmer_crossvalidation_test() 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 lmer_crossvalidation_test().

### When to use lmer_permutation_test()?

lmer_permutation_test() implements a fairly standard cluster-based permutation test, which differs from most other implementations in that it relies on linear mixed-effects modeling to calculate the test statistics. Therefore, this function tends to be extremely computationally intensive, but should also be more sensitive than cluster-based permutation tests that are based on average data. Its main advantage as compared to lmer_crossvalidation_test() is that it is also valid for data with multiple components, such as event-related potentials (ERPs).

### Can the tests also be based on regular multiple regression (instead of linear mixed effects modeling)?

Yes. If you pass groups=None to any of the functions, the analysis will be based on a regular multiple linear regression instead of linear mixed effects modeling.

## Installation

pip install time_series_test


## 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

from datamatrix import io


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',
sampling_freq=100)
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


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

plt.clf()
tst.plot(dm, dv='pupil', hue_factor='set_size', linestyle_factor='color_type',
results=results, sampling_freq=100)
plt.savefig('img/signal-plot-2.png') ## time_series_test.lmer_crossvalidation_test(dm, formula, groups, re_formula=None, winlen=1, split=4, split_method='interleaved', samples_fe=True, samples_re=True, localizer_re=False, fit_method=None, suppress_convergence_warnings=False, fit_kwargs=None, **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 None or list of str

The groups for the random effects, which should be regular (non-series) columns in dm. If None is specified, then all analyses are based on a regular multiple linear regression (instead of linear mixed effects model).

• 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.

• localizer_re: bool, optional

Indicates whether a random effects structure as specified using the re_formula keyword should also be used for the localizer models, or only for the final model.

• fit_kwargs: dict or None, optional

A dict that is passed as keyword arguments to mixedlm.fit(). For example, to specify the nm as the fitting method, specify fit_kwargs={'fit': 'nm'}.

• fit_method: str, list of str, or None, optional

Deprecated. Use fit_kwargs instead.

• 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().

### Returns

• dict

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

## time_series_test.lmer_permutation_test(dm, formula, groups, re_formula=None, winlen=1, suppress_convergence_warnings=False, fit_kwargs={}, iterations=1000, cluster_p_threshold=0.05, test_intercept=False, **kwargs)

Performs a cluster-based permutation test based on sample-by-sample linear-mixed-effects analyses. The permutation test identifies clusters based on p-value threshold and uses the absolute of the summed z-values of the clusters as test statistic.

If no clusters reach the threshold, the test is skipped right away. By default the Intercept is ignored for this criterion, because the intercept usually has significant clusters that we're not interested in. However, you can change this using the test_intercept keyword.

Warning: This is generally an extremely time-consuming analysis because it requires thousands of lmers to be run.

See lmer_crossvalidation() for an explanation of the arguments.

### Parameters

• dm: DataMatrix

• formula: str

• groups: str

• re_formula: str or None, optional

• winlen: int, optional

• suppress_convergence_warnings: bool, optional

• fit_kwargs: dict, optional

• iterations: int, optional

The number of permutations to run.

• cluster_p_threshold: float or None, optional

The maximum p-value for a sample to be considered part of a cluster.

• test_intercept: bool, optional

Indicates whether the intercept should be included when considering if there are any clusters, as described above.

• **kwargs: dict, optional

### Returns

• dict

A dict with effects as keys and lists of clusters defined by (start, end, z-sum, hit proportion) tuples. The p-value is 1 - hit proportion.

## time_series_test.lmer_series(dm, formula, winlen=1, fit_kwargs={}, **kwargs)

Performs a sample-by-sample linear-mixed-effects analysis. See lmer_crossvalidation() for an explanation of the arguments.

### Parameters

• dm: DataMatrix

• formula: str

• winlen: int, optional

• fit_kwargs: dict, optional

• **kwargs: dict, optional

### Returns

• DataMatrix

A DataMatrix with one row per effect, including the intercept, and three series columns with the same depth as the dependent measure specified in the formula:

• est: the slope
• p: the p value
• z: the z value
• se: the standard error

## 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=':', legend_kwargs=None, annotation_legend_kwargs=None, x0=0, sampling_freq=1)

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 lmer_crossvalidation().

• 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: str, list, or None, optional

The name of a matplotlib colormap or a list of hues to be used as line colors for the hue 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: str, list, or None, optional

The name of a matplotlib colormap or a list of hues to be used for the annotations if results is provided.

• annotation_linestyle: str, optional

The linestyle for the annotations.

• legend_kwargs: None or dict, optional

Optional keywords to be passed to plt.legend() for the factor legend.

• annotation_legend_kwargs: None or dict, optional

Optional keywords to be passed to plt.legend() for the annotation legend.

• x0: int, float

The starting value on the x-axis.

• sampling_freq: int, float

The sampling frequency.

## time_series_test.summarize(results, detailed=False)

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

### Parameters

• results: dict

A results dict as returned by lmer_crossvalidation().

• detailed: bool, optional

Indicates whether model details should be included in the summary.

### Returns

• str

time_series_test is licensed under the GNU General Public License v3.

## Project details

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