Statistical inference

Here we will briefly cover multiple concepts of inferential statistics in an introductory manner, and demonstrate how to use some MNE statistical functions.

# Authors: Eric Larson <larson.eric.d@gmail.com>
# License: BSD (3-clause)

from functools import partial

import numpy as np
from scipy import stats
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D  # noqa, analysis:ignore

import mne
from mne.stats import (ttest_1samp_no_p, bonferroni_correction, fdr_correction,
                       permutation_t_test, permutation_cluster_1samp_test)

print(__doc__)

Hypothesis testing

Null hypothesis

From Wikipedia:

In inferential statistics, a general statement or default position that there is no relationship between two measured phenomena, or no association among groups.

We typically want to reject a null hypothesis with some probability level (e.g., p < 0.05). To think about what this means, let’s follow the illustrative example from [1] and construct a toy dataset consisting of a 40 x 40 square with a “signal” present in the center (at pixel [20, 20]) with white noise added and a 5-pixel-SD normal smoothing kernel applied.

width = 40
n_subjects = 10
signal_mean = 100
signal_sd = 100
noise_sd = 0.01
gaussian_sd = 5
sigma = 1e-3  # sigma for the "hat" method
threshold = -stats.distributions.t.ppf(0.05, n_subjects - 1)
n_permutations = 'all'  # run an exact test
n_src = width * width

# For each "subject", make a smoothed noisy signal with a centered peak
rng = np.random.RandomState(42)
X = noise_sd * rng.randn(n_subjects, width, width)
# Add a signal at the dead center
X[:, width // 2, width // 2] = signal_mean + rng.randn(n_subjects) * signal_sd
# Spatially smooth with a 2D Gaussian kernel
size = width // 2 - 1
gaussian = np.exp(-(np.arange(-size, size + 1) ** 2 / float(gaussian_sd ** 2)))
for si in range(X.shape[0]):
    for ri in range(X.shape[1]):
        X[si, ri, :] = np.convolve(X[si, ri, :], gaussian, 'same')
    for ci in range(X.shape[2]):
        X[si, :, ci] = np.convolve(X[si, :, ci], gaussian, 'same')

In this case, a null hypothesis we could test for each voxel is:

There is no difference between the mean value and zero (\(H_0: \mu = 0\)).

The alternative hypothesis, then, is that the voxel has a non-zero mean. This is a two-tailed test because the mean could be less than or greater than zero (whereas a one-tailed test would test only one of these possibilities, i.e. \(H_0: \mu \geq 0\) or \(H_0: \mu \leq 0\)).

Note

Here we will refer to each spatial location as a “voxel”. In general, though, it could be any sort of data value (e.g., cortical vertex at a specific time, pixel in a time-frequency decomposition, etc.).

Parametric tests

Let’s start with a 1-sample t-test, which is a standard test for differences in paired sample means. This test is parametric, as it assumes that the underlying sample distribution is Gaussian, and is only valid in this case. (This happens to be satisfied by our toy dataset, but is not always satisfied for neuroimaging data.)

In the context of our toy dataset, which has many voxels, applying the 1-sample t-test is called a mass-univariate approach as it treats each voxel independently.

titles = ['t-statistic']
out = stats.ttest_1samp(X, 0, axis=0)
ts = [out[0]]
ps = [out[1]]
mccs = [False]  # these are not multiple-comparisons corrected


# let's make a plotting function
def plot_t_p(t, p, title, mcc, axes=None):
    if axes is None:
        fig = plt.figure(figsize=(6, 3))
        axes = [fig.add_subplot(121, projection='3d'), fig.add_subplot(122)]
        show = True
    else:
        fig = axes[0].figure
        show = False
    p_lims = [0.05, 0.001]
    t_lims = -stats.distributions.t.ppf(p_lims, n_subjects - 1)
    p_lims = [-np.log10(0.05), -np.log10(0.001)]
    # t plot
    x, y = np.mgrid[0:width, 0:width]
    surf = axes[0].plot_surface(x, y, np.reshape(t, (width, width)),
                                rstride=1, cstride=1, linewidth=0,
                                vmin=t_lims[0], vmax=t_lims[1], cmap='viridis')
    axes[0].set(xticks=[], yticks=[], zticks=[],
                xlim=[0, width - 1], ylim=[0, width - 1])
    axes[0].view_init(30, 15)
    cbar = plt.colorbar(ax=axes[0], shrink=0.75, orientation='horizontal',
                        fraction=0.1, pad=0.025, mappable=surf)
    cbar.set_ticks(t_lims)
    cbar.set_ticklabels(['%0.1f' % t_lim for t_lim in t_lims])
    cbar.set_label('t-value')
    cbar.ax.get_xaxis().set_label_coords(0.5, -0.3)
    if not show:
        axes[0].set(title=title)
        if mcc:
            axes[0].title.set_weight('bold')
    # p plot
    use_p = -np.log10(np.reshape(np.maximum(p, 1e-5), (width, width)))
    img = axes[1].imshow(use_p, cmap='inferno', vmin=p_lims[0], vmax=p_lims[1],
                         interpolation='nearest')
    axes[1].set(xticks=[], yticks=[])
    cbar = plt.colorbar(ax=axes[1], shrink=0.75, orientation='horizontal',
                        fraction=0.1, pad=0.025, mappable=img)
    cbar.set_ticks(p_lims)
    cbar.set_ticklabels(['%0.1f' % p_lim for p_lim in p_lims])
    cbar.set_label('$-\log_{10}(p)$')
    cbar.ax.get_xaxis().set_label_coords(0.5, -0.3)
    if show:
        text = fig.suptitle(title)
        if mcc:
            text.set_weight('bold')
        plt.subplots_adjust(0, 0.05, 1, 0.9, wspace=0, hspace=0)
        mne.viz.utils.plt_show()


plot_t_p(ts[-1], ps[-1], titles[-1], mccs[-1])
../_images/sphx_glr_plot_background_statistics_001.png

“hat” variance adjustment

The “hat” technique regularizes the variance values used in the t-test calculation [1] to compensate for implausibly small variances.

ts.append(ttest_1samp_no_p(X, sigma=sigma))
ps.append(stats.distributions.t.sf(np.abs(ts[-1]), len(X) - 1) * 2)
titles.append('$\mathrm{t_{hat}}$')
mccs.append(False)
plot_t_p(ts[-1], ps[-1], titles[-1], mccs[-1])
../_images/sphx_glr_plot_background_statistics_002.png

Non-parametric tests

Instead of assuming an underlying Gaussian distribution, we could instead use a non-parametric resampling method. Under the null hypothesis, we have the principle of exchangeability, which means that, if the null is true, we should be able to exchange conditions and not change the distribution of the test statistic.

In the case of a 2-tailed paired t-test against 0 (or between two conditions where you have already subtracted them), exchangeability means that we can flip the signs of our data. Therefore, we can construct the null distribution values by taking random subsets of samples (subjects), flipping the sign of their data, and recording the resulting statistic absolute value. The absolute value of the statistic evaluated on the veridical data can then be compared to this distribution, and the p-value is simply the proportion of null distribution values that were smaller.

Note

In the case where n_permutations is large enough (or “all”) so that the complete set of unique resampling exchanges can be done (which is \(2^{N_{samp}}-1=1023\) for the one-tailed paired test here, not counting the veridical distribution), instead of randomly exchanging conditions the null is formed from using all possible exchanges. This is known as a permutation test (or exact test) form of a non-parametric resampling test.

# Here we have to do a bit of gymnastics to get our function to do
# a permutation test without correcting for multiple comparisons:

# Let's flatten the array for simplicity
X.shape = (n_subjects, n_src)
titles.append('Permutation')
ts.append(np.zeros(width * width))
ps.append(np.zeros(width * width))
mccs.append(False)
for ii in range(n_src):
    ts[-1][ii], ps[-1][ii] = \
        permutation_t_test(X[:, [ii]], verbose=True if ii == 0 else False)[:2]
plot_t_p(ts[-1], ps[-1], titles[-1], mccs[-1])
../_images/sphx_glr_plot_background_statistics_003.png

Out:

Permuting 511 times (exact test)...

Multiple comparisons

So far, we have done no correction for multiple comparisons. This is potentially problematic for these data because there are \(40 \times 40 = 1600\) tests being performed. If we just use a threshold p < 0.05 for all of our tests, we would expect many voxels to be declared significant even if there were no true effect. In other words, we would make many type I errors (adapted from here):

  Null hypothesis
True False
Reject Yes
Type I error
False positive
Correct
True positive
No
Correct
True Negative
Type II error
False negative

To see why, consider a standard \(\alpha = 0.05\). For a single test, our probability of making a type I error is 0.05. The probability of making at least one type I error in \(N_{\mathrm{test}}\) independent tests is then given by \(1 - (1 - \alpha)^{N_{\mathrm{test}}}\):

N = np.arange(1, 80)
alpha = 0.05
p_type_I = 1 - (1 - alpha) ** N
fig, ax = plt.subplots(figsize=(4, 3))
ax.scatter(N, p_type_I, 3)
ax.set(xlim=N[[0, -1]], ylim=[0, 1], xlabel='$N_{\mathrm{test}}$',
       ylabel=u'Probability of ≥ 1\ntype I error')
ax.grid(True)
fig.tight_layout()
mne.viz.utils.plt_show()
../_images/sphx_glr_plot_background_statistics_004.png

To combat this problem, multiple methods exist. Typically these provide control over either the:

  1. Familywise error rate (FWER)

    The probability of making one or more type I errors:

    \[\mathrm{P}(N_{\mathrm{type\ I}} >= 1 | H_0)\]
  2. False discovery rate (FDR)

    The expected proportion of rejected null hypotheses that are actually true:

    \[\mathrm{E}(N_{\mathrm{type\ I}} / N_{\mathrm{reject}} | N_{\mathrm{reject}} > 0) \mathrm{P}(N_{\mathrm{reject}} > 0 | H_0)\]

We cover some techniques that control FWER and FDR below.

Bonferroni correction

Perhaps the simplest way to deal with multiple comparisons, Bonferroni correction conservatively multiplies the p-values by the number of comparisons to control the FWER.

titles.append('Bonferroni')
ts.append(ts[-1])
ps.append(bonferroni_correction(ps[0])[1])
mccs.append(True)
plot_t_p(ts[-1], ps[-1], titles[-1], mccs[-1])
../_images/sphx_glr_plot_background_statistics_005.png

False discovery rate (FDR) correction

Typically FDR is performed with the Benjamini/Hochberg procedure, which is less restrictive than Bonferroni correction for large numbers of comparisons (fewer type II errors) but provides less strict control of errors (more type I errors).

titles.append('FDR')
ts.append(ts[-1])
ps.append(fdr_correction(ps[0])[1])
mccs.append(True)
plot_t_p(ts[-1], ps[-1], titles[-1], mccs[-1])
../_images/sphx_glr_plot_background_statistics_006.png

Non-parametric resampling test with a maximum statistic

Non-parametric resampling tests can also be used to correct for multiple comparisons. In its simplest form, we again do permutations using exchangeability under the null hypothesis, but this time we take the maximum statistic across all tests in each permutation to form the null distribution. The p-value for each voxel from the veridical data is then given by the proportion of null distribtion values that were smaller.

This method has two important features:

  1. It controls FWER.
  2. It is non-parametric. Even though our initial test statistic (here a 1-sample t-test) for clustering is parametric, the null distribution for the null hypothesis rejection (cluster size distribution is indistinguishable from zero) is obtained by permutations. This means that it makes no assumptions of Gaussianity (which do hold for this example but do not in general for some types of processed neuroimaging data).
titles.append('$\mathbf{Perm_{max}}$')
out = permutation_t_test(X)[:2]
ts.append(out[0])
ps.append(out[1])
mccs.append(True)
plot_t_p(ts[-1], ps[-1], titles[-1], mccs[-1])
../_images/sphx_glr_plot_background_statistics_007.png

Out:

Permuting 511 times (exact test)...

Clustering

Each of the aforementioned multiple comparisons corrections have the disadvantage of not fully incorporating the correlation structure of the data, i.e., that points close to one another (e.g., in space or time) tend to be correlated. However, by defining the connectivity/adjacency/neighbor structure in our data, we can use clustering to compensate.

To use this, we need to rethink our null hypothesis. Instead of thinking about a null hypothesis about means per voxel (with one independent test per voxel), we consider a null hypothesis about sizes of clusters in our data, which could be stated like:

The distribution of spatial cluster sizes observed in two experimental conditions are drawn from the same probability distribution.

Here we only have a single condition and we contrast to zero, which can be thought of as:

The distribution of spatial cluster sizes is independent of the sign of the data.

In this case, we again do permutations with a maximum statistic, but, under each permutation, we:

  1. Threshold the computed statistic with some initial threshold value.
  2. Cluster points that exceed this threshold (with the same sign) based on adjacency.
  3. Record the size of each cluster (measured, e.g., by a simple voxel count, or by the sum of voxel t-values within the cluster).

After doing these permutations, the cluster sizes in our veridical data are compared to this null distribution. The p-value associated with each cluster is again given by the proportion of smaller null distribution values. This can then be subjected to a standard p-value threshold (e.g., p < 0.05) to reject the null hypothesis (i.e., find an effect of interest).

This reframing to consider cluster sizes rather than individual means maintains the advantages of the standard non-parametric permutation test – namely controlling FWER and making no assumptions of parametric data distribution. Cricitally, though, it also accounts for the correlation structure in the data – which in this toy case is spatial but in general can be multidimensional (e.g., spatio-temporal) – because the null distribution will be derived from data in a way that preserves these correlations.

However, there is a drawback. If a cluster significantly deviates from the null, no further inference on the cluster (e.g., peak location) can be made, as the entire cluster as a whole is used to reject the null. Moreover, because the test statistic concerns the full data, the null hypothesis (and our rejection of it) refers to the structure of the full data. For more information, see also the excellent FieldTrip cluster interpretation tutorial.

Defining the connectivity/neighbor/adjacency matrix

First we need to define our connectivity/neighbor/adjacency matrix. This is a square array (or sparse matrix) of shape (n_src, n_src) that contains zeros and ones to define which spatial points are connected, i.e., which voxels are adjacent to each other. In our case this is quite simple, as our data are aligned on a rectangular grid.

Let’s pretend that our data were smaller – a 3x3 grid. Thinking about each voxel as being connected to the other voxels it touches, we would need a 9x9 connectivity matrix. The first row should contain the elements in the .ravel()’ed data that it touches. Since it touches the second element in the first row and the first element in the second row (and is also a neighbor to itself), this would be:

[1, 1, 0, 1, 0, 0, 0, 0, 0]

sklearn.feature_extraction provides a convenient function for this:

from sklearn.feature_extraction.image import grid_to_graph  # noqa: E402
mini_connectivity = grid_to_graph(3, 3).toarray()
assert mini_connectivity.shape == (9, 9)
print(mini_connectivity[0])
del mini_connectivity

Out:

[1 1 0 1 0 0 0 0 0]

In general the connectivity between voxels can be more complex, such as those between sensors in 3D space, or time-varying activation at brain vertices on a cortical surface. MNE provides several convenience functions for computing connectivity/neighbor/adjacency matrices, see the Statistics API.

Standard clustering

Here, since our data are on a grid, we can use connectivity=None to trigger optimized grid-based code, and run the clustering algorithm.

# Reshape data to what is equivalent to (n_samples, n_space, n_time)
titles.append('Clustering')
X.shape = (n_subjects, width, width)
t_clust, clusters, p_values, H0 = permutation_cluster_1samp_test(
    X, n_jobs=1, threshold=threshold, connectivity=None, tail=1,
    n_permutations=n_permutations)
# Put the cluster data in a viewable format
p_clust = np.ones((width, width))
for cl, p in zip(clusters, p_values):
    p_clust[cl] = p
ts.append(t_clust)
ps.append(p_clust)
mccs.append(True)
plot_t_p(ts[-1], ps[-1], titles[-1], mccs[-1])
../_images/sphx_glr_plot_background_statistics_008.png

Out:

stat_fun(H1): min=-2.676791 max=5.182156
Running initial clustering
Found 1 clusters
Permuting 1022 times...
Computing cluster p-values
Done.

“hat” variance adjustment

This method can also be used in this context to correct for small variances [1]:

titles.append(r'$\mathbf{C_{hat}}$')
stat_fun_hat = partial(ttest_1samp_no_p, sigma=sigma)
t_hat, clusters, p_values, H0 = permutation_cluster_1samp_test(
    X, n_jobs=1, threshold=threshold, connectivity=None, tail=1,
    n_permutations=n_permutations, stat_fun=stat_fun_hat, buffer_size=None)
p_hat = np.ones((width, width))
for cl, p in zip(clusters, p_values):
    p_hat[cl] = p
ts.append(t_hat)
ps.append(p_hat)
mccs.append(True)
plot_t_p(ts[-1], ps[-1], titles[-1], mccs[-1])
../_images/sphx_glr_plot_background_statistics_009.png

Out:

stat_fun(H1): min=-0.044294 max=4.600984
Running initial clustering
Found 1 clusters
Permuting 1022 times...
Computing cluster p-values
Done.

Threshold-free cluster enhancement (TFCE)

TFCE eliminates the free parameter initial threshold value that determines which points are included in clustering by approximating a continuous integration across possible threshold values with a standard Riemann sum [2]. This requires giving a starting threshold 'start' and a step size 'step', which in MNE is supplied as a dict. The smaller the 'step' and closer to 0 the 'start' value, the better the approximation, but the longer it takes).

A significant advantage of TFCE is that, rather than modifying the statistical null hypothesis under test (from one about individual voxels to one about the distribution of clusters in the data), it modifies the data under test while still controlling for multiple comparisons. The statistical test is then done at the level of individual voxels rather than clusters. This allows for evaluation of each point independently for significance rather than only as cluster groups.

titles.append(r'$\mathbf{C_{TFCE}}$')
threshold_tfce = dict(start=0, step=0.2)
t_tfce, _, p_tfce, H0 = permutation_cluster_1samp_test(
    X, n_jobs=1, threshold=threshold_tfce, connectivity=None, tail=1,
    n_permutations=n_permutations)
ts.append(t_tfce)
ps.append(p_tfce)
mccs.append(True)
plot_t_p(ts[-1], ps[-1], titles[-1], mccs[-1])
../_images/sphx_glr_plot_background_statistics_010.png

Out:

stat_fun(H1): min=-2.676791 max=5.182156
Running initial clustering
Using 26 thresholds from 0.00 to 5.00 for TFCE computation (h_power=2.00, e_power=0.50)
Found 1600 clusters
Permuting 1022 times...
Computing cluster p-values
Done.

We can also combine TFCE and the “hat” correction:

titles.append(r'$\mathbf{C_{hat,TFCE}}$')
t_tfce_hat, _, p_tfce_hat, H0 = permutation_cluster_1samp_test(
    X, n_jobs=1, threshold=threshold_tfce, connectivity=None, tail=1,
    n_permutations=n_permutations, stat_fun=stat_fun_hat, buffer_size=None)
ts.append(t_tfce_hat)
ps.append(p_tfce_hat)
mccs.append(True)
plot_t_p(ts[-1], ps[-1], titles[-1], mccs[-1])
../_images/sphx_glr_plot_background_statistics_011.png

Out:

stat_fun(H1): min=-0.044294 max=4.600984
Running initial clustering
Using 24 thresholds from 0.00 to 4.60 for TFCE computation (h_power=2.00, e_power=0.50)
Found 1600 clusters
Permuting 1022 times...
Computing cluster p-values
Done.

Visualize and compare methods

Let’s take a look at these statistics. The top row shows each test statistic, and the bottom shows p-values for various statistical tests, with the ones with proper control over FWER or FDR with bold titles.

fig = plt.figure(facecolor='w', figsize=(14, 3))
assert len(ts) == len(titles) == len(ps)
for ii in range(len(ts)):
    ax = [fig.add_subplot(2, 10, ii + 1, projection='3d'),
          fig.add_subplot(2, 10, 11 + ii)]
    plot_t_p(ts[ii], ps[ii], titles[ii], mccs[ii], ax)
fig.tight_layout(pad=0, w_pad=0.05, h_pad=0.1)
plt.show()
../_images/sphx_glr_plot_background_statistics_012.png

The first three columns show the parametric and non-parametric statistics that are not corrected for multiple comparisons:

  • t-test has jagged edges.
  • “hat” variance correction of the t-test has reduced peaky edges, correcting for sharpness in the statistic driven by low-variance voxels.
  • non-parametric resampling test is very similar to the t-test. This is to be expected: the data are drawn from a Gaussian distribution, and thus satisfy parametric assumptions.

The next three columns show multiple comparison corrections of the mass univariate tests (parametric and non-parametric). These too conservatively correct for multiple comparisons because neighboring voxels in our data are correlated:

  • Bonferroni correction eliminates any significant activity.
  • FDR correction is less conservative than Bonferroni.
  • Permutation test with a maximum statistic also eliminates any significant activity.

The final four columns show the non-parametric, cluster-based permutation tests with a maximum statistic:

  • Standard clustering identifies the correct region. However, the whole area must be declared significant, so no peak analysis can be done. Also, the peak is broad.
  • Clustering with “hat” tightens the estimate of significant activity.
  • Clustering with TFCE allows analyzing each significant point independently, but still has a broadened estimate.
  • Clustering with TFCE and “hat” tightens the area declared significant (again FWER corrected).

Statistical functions in MNE

The complete listing of statistical functions provided by MNE are in the Statistics API list, but we will give a brief overview here.

MNE provides several convenience parametric testing functions that can be used in conjunction with the non-parametric clustering methods. However, the set of functions we provide is not meant to be exhaustive.

If the univariate statistical contrast of interest to you is not listed here (e.g., interaction term in an unbalanced ANOVA), consider checking out the statsmodels package. It offers many functions for computing statistical contrasts, e.g., statsmodels.stats.anova.anova_lm(). To use these functions in clustering:

  1. Determine which test statistic (e.g., t-value, F-value) you would compute in a univariate context to compute your contrast of interest. In other words, if there were only a single output such as reaction times, what test statistic might you compute on the data?
  2. Wrap the call to that function within a function that takes an input of the same shape that is expected by your clustering function, and returns an array of the same shape without the “samples” dimension (e.g., mne.stats.permutation_cluster_1samp_test() takes an array of shape (n_samples, p, q) and returns an array of shape (p, q)).
  3. Pass this wrapped function to the stat_fun argument to the clustering function.
  4. Set an appropriate threshold value (float or dict) based on the values your statistical contrast function returns.

Parametric methods provided by MNE

  • mne.stats.ttest_1samp_no_p()
    Paired t-test, optionally with hat adjustment. This is used by default for contrast enhancement in paired cluster tests.
  • mne.stats.f_oneway()
    One-way ANOVA for independent samples. This can be used to compute various F-contrasts. It is used by default for contrast enhancement in non-paired cluster tests.
  • mne.stats.f_mway_rm()
    M-way ANOVA for repeated measures and balanced designs. This returns F-statistics and p-values. The associated helper function mne.stats.f_threshold_mway_rm() can be used to determine the F-threshold at a given significance level.
  • mne.stats.linear_regression()
    Compute ordinary least square regressions on multiple targets, e.g., sensors, time points across trials (samples). For each regressor it returns the beta value, t-statistic, and uncorrected p-value. While it can be used as a test, it is particularly useful to compute weighted averages or deal with continuous predictors.

Non-parametric methods

Warning

In most MNE functions, data has shape (..., n_space, n_time), where the spatial dimension can be e.g. sensors or source vertices. But for our spatio-temporal clustering functions, the spatial dimensions need to be last for computational efficiency reasons. For example, for mne.stats.spatio_temporal_cluster_1samp_test(), X needs to be of shape (n_samples, n_time, n_space). You can use numpy.transpose() to transpose axes if necessary.

References

[1](1, 2, 3) Ridgway et al. 2012, “The problem of low variance voxels in statistical parametric mapping; a new hat avoids a ‘haircut’”, NeuroImage. 2012 Feb 1;59(3):2131-41.
[2]Smith and Nichols 2009, “Threshold-free cluster enhancement: addressing problems of smoothing, threshold dependence, and localisation in cluster inference”, NeuroImage 44 (2009) 83-98.

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