scipy.stats.ttest_rel#

scipy.stats.ttest_rel(a, b, axis=0, nan_policy='propagate', alternative='two-sided')[source]#

Calculate the t-test on TWO RELATED samples of scores, a and b.

This is a test for the null hypothesis that two related or repeated samples have identical average (expected) values.

Parameters
a, barray_like

The arrays must have the same shape.

axisint or None, optional

Axis along which to compute test. If None, compute over the whole arrays, a, and b.

nan_policy{‘propagate’, ‘raise’, ‘omit’}, optional

Defines how to handle when input contains nan. The following options are available (default is ‘propagate’):

  • ‘propagate’: returns nan

  • ‘raise’: throws an error

  • ‘omit’: performs the calculations ignoring nan values

alternative{‘two-sided’, ‘less’, ‘greater’}, optional

Defines the alternative hypothesis. The following options are available (default is ‘two-sided’):

  • ‘two-sided’: the means of the distributions underlying the samples are unequal.

  • ‘less’: the mean of the distribution underlying the first sample is less than the mean of the distribution underlying the second sample.

  • ‘greater’: the mean of the distribution underlying the first sample is greater than the mean of the distribution underlying the second sample.

New in version 1.6.0.

Returns
statisticfloat or array

t-statistic.

pvaluefloat or array

The p-value.

Notes

Examples for use are scores of the same set of student in different exams, or repeated sampling from the same units. The test measures whether the average score differs significantly across samples (e.g. exams). If we observe a large p-value, for example greater than 0.05 or 0.1 then we cannot reject the null hypothesis of identical average scores. If the p-value is smaller than the threshold, e.g. 1%, 5% or 10%, then we reject the null hypothesis of equal averages. Small p-values are associated with large t-statistics.

The statistic is calculated as np.mean(a - b)/se, where se is the standard error. Therefore, the statistic will be positive when the sample mean of a - b is greater than zero and negative when the sample mean of a - b is less than zero.

References

https://en.wikipedia.org/wiki/T-test#Dependent_t-test_for_paired_samples

Examples

>>> from scipy import stats
>>> rng = np.random.default_rng()

>>> rvs1 = stats.norm.rvs(loc=5, scale=10, size=500, random_state=rng)
>>> rvs2 = (stats.norm.rvs(loc=5, scale=10, size=500, random_state=rng)
...         + stats.norm.rvs(scale=0.2, size=500, random_state=rng))
>>> stats.ttest_rel(rvs1, rvs2)
Ttest_relResult(statistic=-0.4549717054410304, pvalue=0.6493274702088672)
>>> rvs3 = (stats.norm.rvs(loc=8, scale=10, size=500, random_state=rng)
...         + stats.norm.rvs(scale=0.2, size=500, random_state=rng))
>>> stats.ttest_rel(rvs1, rvs3)
Ttest_relResult(statistic=-5.879467544540889, pvalue=7.540777129099917e-09)