scipy.stats.ttest_ind#

scipy.stats.ttest_ind(a, b, axis=0, equal_var=True, nan_policy='propagate', permutations=None, random_state=None, alternative='two-sided', trim=0, *, keepdims=False)[source]#

Calculate the T-test for the means of two independent samples of scores.

This is a test for the null hypothesis that 2 independent samples have identical average (expected) values. This test assumes that the populations have identical variances by default.

Parameters:

a, barray_like

The arrays must have the same shape, except in the dimension corresponding to axis (the first, by default).

axisint 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.

equal_varbool, optional

If True (default), perform a standard independent 2 sample test that assumes equal population variances [1]. If False, perform Welch’s t-test, which does not assume equal population variance [2].

Added in version 0.11.0.

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.

permutationsnon-negative int, np.inf, or None (default), optional

If 0 or None (default), use the t-distribution to calculate p-values. Otherwise, permutations is the number of random permutations that will be used to estimate p-values using a permutation test. If permutations equals or exceeds the number of distinct partitions of the pooled data, an exact test is performed instead (i.e. each distinct partition is used exactly once). See Notes for details.

Added in version 1.7.0.

random_state{None, int, numpy.random.Generator,

numpy.random.RandomState}, optional

If seed is None (or np.random), the numpy.random.RandomState singleton is used. If seed is an int, a new RandomState instance is used, seeded with seed. If seed is already a Generator or RandomState instance then that instance is used.

Pseudorandom number generator state used to generate permutations (used only when permutations is not None).

Added in version 1.7.0.

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.

Added in version 1.6.0.

trimfloat, optional

If nonzero, performs a trimmed (Yuen’s) t-test. Defines the fraction of elements to be trimmed from each end of the input samples. If 0 (default), no elements will be trimmed from either side. The number of trimmed elements from each tail is the floor of the trim times the number of elements. Valid range is [0, .5).

Added in version 1.7.

keepdimsbool, 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:

resultTtestResult

An object with the following attributes:

statisticfloat or ndarray: The t-statistic.
pvaluefloat or ndarray: The p-value associated with the given alternative.
dffloat or ndarray: The number of degrees of freedom used in calculation of the t-statistic. This is always NaN for a permutation t-test.

Added in version 1.11.0.

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. When a permutation t-test is performed, the confidence interval is not computed, and fields low and high contain NaN.

Added in version 1.11.0.

Notes

Suppose we observe two independent samples, e.g. flower petal lengths, and we are considering whether the two samples were drawn from the same population (e.g. the same species of flower or two species with similar petal characteristics) or two different populations.

The t-test quantifies the difference between the arithmetic means of the two samples. The p-value quantifies the probability of observing as or more extreme values assuming the null hypothesis, that the samples are drawn from populations with the same population means, is true. A p-value larger than a chosen threshold (e.g. 5% or 1%) indicates that our observation is not so unlikely to have occurred by chance. Therefore, we do not reject the null hypothesis of equal population means. If the p-value is smaller than our threshold, then we have evidence against the null hypothesis of equal population means.

By default, the p-value is determined by comparing the t-statistic of the observed data against a theoretical t-distribution. When 1 < permutations < binom(n, k), where

k is the number of observations in a,
n is the total number of observations in a and b, and
binom(n, k) is the binomial coefficient (n choose k),

the data are pooled (concatenated), randomly assigned to either group a or b, and the t-statistic is calculated. This process is performed repeatedly (permutation times), generating a distribution of the t-statistic under the null hypothesis, and the t-statistic of the observed data is compared to this distribution to determine the p-value. Specifically, the p-value reported is the “achieved significance level” (ASL) as defined in 4.4 of [3]. Note that there are other ways of estimating p-values using randomized permutation tests; for other options, see the more general permutation_test.

When permutations >= binom(n, k), an exact test is performed: the data are partitioned between the groups in each distinct way exactly once.

The permutation test can be computationally expensive and not necessarily more accurate than the analytical test, but it does not make strong assumptions about the shape of the underlying distribution.

Use of trimming is commonly referred to as the trimmed t-test. At times called Yuen’s t-test, this is an extension of Welch’s t-test, with the difference being the use of winsorized means in calculation of the variance and the trimmed sample size in calculation of the statistic. Trimming is recommended if the underlying distribution is long-tailed or contaminated with outliers [4].

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

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.

References

[1]

https://en.wikipedia.org/wiki/T-test#Independent_two-sample_t-test

[2]

https://en.wikipedia.org/wiki/Welch%27s_t-test

[3]

Efron and T. Hastie. Computer Age Statistical Inference. (2016).

[4]

Yuen, Karen K. “The Two-Sample Trimmed t for Unequal Population Variances.” Biometrika, vol. 61, no. 1, 1974, pp. 165-170. JSTOR, www.jstor.org/stable/2334299. Accessed 30 Mar. 2021.

[5]

Yuen, Karen K., and W. J. Dixon. “The Approximate Behaviour and Performance of the Two-Sample Trimmed t.” Biometrika, vol. 60, no. 2, 1973, pp. 369-374. JSTOR, www.jstor.org/stable/2334550. Accessed 30 Mar. 2021.

Examples

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

Test with sample with identical means:

>>> 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.ttest_ind(rvs1, rvs2)
Ttest_indResult(statistic=-0.4390847099199348, pvalue=0.6606952038870015)
>>> stats.ttest_ind(rvs1, rvs2, equal_var=False)
Ttest_indResult(statistic=-0.4390847099199348, pvalue=0.6606952553131064)

ttest_ind underestimates p for unequal variances:

>>> rvs3 = stats.norm.rvs(loc=5, scale=20, size=500, random_state=rng)
>>> stats.ttest_ind(rvs1, rvs3)
Ttest_indResult(statistic=-1.6370984482905417, pvalue=0.1019251574705033)
>>> stats.ttest_ind(rvs1, rvs3, equal_var=False)
Ttest_indResult(statistic=-1.637098448290542, pvalue=0.10202110497954867)

When n1 != n2, the equal variance t-statistic is no longer equal to the unequal variance t-statistic:

>>> rvs4 = stats.norm.rvs(loc=5, scale=20, size=100, random_state=rng)
>>> stats.ttest_ind(rvs1, rvs4)
Ttest_indResult(statistic=-1.9481646859513422, pvalue=0.05186270935842703)
>>> stats.ttest_ind(rvs1, rvs4, equal_var=False)
Ttest_indResult(statistic=-1.3146566100751664, pvalue=0.1913495266513811)

T-test with different means, variance, and n:

>>> rvs5 = stats.norm.rvs(loc=8, scale=20, size=100, random_state=rng)
>>> stats.ttest_ind(rvs1, rvs5)
Ttest_indResult(statistic=-2.8415950600298774, pvalue=0.0046418707568707885)
>>> stats.ttest_ind(rvs1, rvs5, equal_var=False)
Ttest_indResult(statistic=-1.8686598649188084, pvalue=0.06434714193919686)

When performing a permutation test, more permutations typically yields more accurate results. Use a np.random.Generator to ensure reproducibility:

>>> stats.ttest_ind(rvs1, rvs5, permutations=10000,
...                 random_state=rng)
Ttest_indResult(statistic=-2.8415950600298774, pvalue=0.0052994700529947)

Take these two samples, one of which has an extreme tail.

>>> a = (56, 128.6, 12, 123.8, 64.34, 78, 763.3)
>>> b = (1.1, 2.9, 4.2)

Use the trim keyword to perform a trimmed (Yuen) t-test. For example, using 20% trimming, trim=.2, the test will reduce the impact of one (np.floor(trim*len(a))) element from each tail of sample a. It will have no effect on sample b because np.floor(trim*len(b)) is 0.

>>> stats.ttest_ind(a, b, trim=.2)
Ttest_indResult(statistic=3.4463884028073513,
                pvalue=0.01369338726499547)