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T-test

Background

With a t-test one can investigate whether two collections of samples stem from different underlying populations. For instance, one can investigate whether P3 peaks in two experimental conditions are significantly different. The t-test is a parametric test, i.e. it assumes that data have come from a Gaussian probability distribution and make inferences about the parameters of the distribution. Assumptions of the t-test:

  • Population data from which the sample data are drawn are normally distributed.
  • The variances of the populations to be compared are equal.

{i} Paired-samples versus independent-samples t-test

  • In the paired-samples test with two conditions, the assumption is that all samples have been drawn pair-wise. In a typical BCI experiment, this means that every subject participated in condition 1 and condition 2. In other words, the samples in the two conditions are paired by the subjects.
  • In the independent samples test, the assumption is that all samples have been drawn independently. In a typical BCI experiment, this would mean that there are two groups of subjects. Group A was tested in condition 1, group B was tested in condition 2.
  • If applicable, the paired-samples is to be preferred because it accounts for the inter-subject variability and thus increases statistical power.

<!> Some pitfalls

  • Reaction times are not Gaussian-distributed. They have a skewed long-tail distribution. Therefore the arithmetic mean is not a good estimator of the mean. Use either the geometric mean (geomean() in Matlab) or calculate the reciprocal of reaction time, speed = 1/RT, which gives a more symmetric distribution and allows for the calculation of the arithmetic mean.
  • If a test does not yield a significant result (i.e., p > α), this does not necessarily mean that the distributions are the same. Measurements might not be accurate enough (when the difference between the distributions is subtle) or the number of data samples might be too small. The only thing you can say is that the distributions are not different.

Further reading

Toolbox

  • ttest
  • ttest2

If two conditions are compared, your data matrix is a Nx2 matrix. In a paired-samples t-test, the N subjects are arranged along the rows, with the first column corresponding to condition 1 and the second column corresponding to condition 2. In an independent-samples t-test, each column represents one group consisting of N subjects each; there is no specific order for the subjects in the two groups. This can be visualized as follows.

t-test

The following code gives an example on performing a paired-samples t-test in Matlab. First, two Gaussian distributed data columns are generated in d. Then, they are compared to each other using a t-test.

N = 20;
d = [randn(N,1) randn(N,1)+0.1];      % Each column is one dataset
[h,p,ci,stats] = ttest(d(:,1), d(:,2))

where h (0 or 1) specifies whether or not the null hypothesis "The means of the two distributions are equal" is rejected, p gives the according p-value, ci gives an 1-α confidence interval for the d1-d2 mean. stats is a struct giving the t-value, the degrees-of-freedom df, and the standard deviation sd.

You can also run a one-sample t-test on each of the two distributions, using

[h,p,ci,stats] = ttest(d(:,1))
[h2,p2,ci2,stats2] = ttest(d(:,2))

In this case, the t-test tests whether the mean of each distribution is significantly different from 0.

For an independent-samples t-test, the MATLAB code is virtually identical, but to use the MATLAB function ttest2

[h,p,ci,stats] = ttest2(d(:,1),d(:,2))

M-file

ttest_tutorial.m

Author(s)

Matthias Treder matthias.treder@tu-berlin.de