`scipy.stats._stats_py:ttest_rel`

source: /scipy/stats/_stats_py.py :6850

Signature

   def     ttest_rel (    a    ,    b    ,    axis    =  0   ,    nan_policy    =  propagate   ,    alternative    =  two-sided   ,  * ,     keepdims    =  False     )

Summary

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

Extended Summary

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

Parameters

a, b : array_like

The arrays must have the same shape.

axis : int or None, default: 0

If an int, the axis of the input along which to compute the statistic. The statistic of each axis-slice (e.g. row) of the input will appear in a corresponding element of the output. If None, the input will be raveled before computing the statistic.

nan_policy : {'propagate', 'omit', 'raise'}

Defines how to handle input NaNs.

propagate: if a NaN is present in the axis slice (e.g. row) along which the statistic is computed, the corresponding entry of the output will be NaN.
omit: NaNs will be omitted when performing the calculation. If insufficient data remains in the axis slice along which the statistic is computed, the corresponding entry of the output will be NaN.
raise: if a NaN is present, a ValueError will be raised.

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.

keepdims : bool, default: False

If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the input array.

Returns

result : `~scipy.stats._result_classes.TtestResult`

An object with the following attributes:

statistic: statistic
pvalue: pvalue
df: df

The object also has the following method:

confidence_interval(confidence_level=0.95): Computes a confidence interval around the difference in population means for the given confidence level. The confidence interval is returned in a namedtuple with fields low and high.
versionadded 1.10.0

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 t-statistic is calculated as np.mean(a - b)/se, where se is the standard error. Therefore, the t-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.

Beginning in SciPy 1.9, np.matrix inputs (not recommended for new code) are converted to np.ndarray before the calculation is performed. In this case, the output will be a scalar or np.ndarray of appropriate shape rather than a 2D np.matrix. Similarly, while masked elements of masked arrays are ignored, the output will be a scalar or np.ndarray rather than a masked array with mask=False.

Array API Standard Support

ttest_rel has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

====================  ====================  ====================
Library               CPU                   GPU
====================  ====================  ====================
NumPy                 ✅                     n/a                 
CuPy                  n/a                   ✅                   
PyTorch               ✅                     ⛔                   
JAX                   ⚠️ no JIT             ⚠️ no JIT           
Dask                  ⚠️ computes graph     n/a                 
====================  ====================  ====================

See dev-arrayapi for more information.

Examples

import numpy as np
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)

✗

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)