pingouin.ttest

pingouin.ttest(x, y, paired=False, tail='two-sided', correction='auto', r=0.707)[source]

T-test.

Parameters:

Parameters:	x : array_like First set of observations. y : array_like or float Second set of observations. If y is a single value, a one-sample T-test is computed. paired : boolean Specify whether the two observations are related (i.e. repeated measures) or independent. tail : string Specify whether to return two-sided or one-sided p-value. correction : string or boolean For unpaired two sample T-tests, specify whether or not to correct for unequal variances using Welch separate variances T-test. If ‘auto’, it will automatically uses Welch T-test when the sample sizes are unequal, as recommended by Zimmerman 2004. r : float Cauchy scale factor for computing the Bayes Factor. Smaller values of r (e.g. 0.5), may be appropriate when small effect sizes are expected a priori; larger values of r are appropriate when large effect sizes are expected (Rouder et al 2009). The default is 0.707 (= \(\sqrt{2} / 2\)).
Returns:	stats : pandas DataFrame T-test summary 'T' : T-value 'p-val' : p-value 'dof' : degrees of freedom 'cohen-d' : Cohen d effect size 'power' : achieved power of the test ( = 1 - type II error) 'BF10' : Bayes Factor of the alternative hypothesis

x : array_like: First set of observations.
y : array_like or float: Second set of observations. If y is a single value, a one-sample T-test is computed.
paired : boolean: Specify whether the two observations are related (i.e. repeated measures) or independent.
tail : string: Specify whether to return two-sided or one-sided p-value.
correction : string or boolean: For unpaired two sample T-tests, specify whether or not to correct for unequal variances using Welch separate variances T-test. If ‘auto’, it will automatically uses Welch T-test when the sample sizes are unequal, as recommended by Zimmerman 2004.
r : float: Cauchy scale factor for computing the Bayes Factor. Smaller values of r (e.g. 0.5), may be appropriate when small effect sizes are expected a priori; larger values of r are appropriate when large effect sizes are expected (Rouder et al 2009). The default is 0.707 (= \(\sqrt{2} / 2\)).

Returns:

stats : pandas DataFrame

T-test summary

'T' : T-value
'p-val' : p-value
'dof' : degrees of freedom
'cohen-d' : Cohen d effect size
'power' : achieved power of the test ( = 1 - type II error)
'BF10' : Bayes Factor of the alternative hypothesis

See also

mwu: non-parametric independent T-test
wilcoxon: non-parametric paired T-test
anova: One-way and two-way ANOVA
rm_anova: One-way and two-way repeated measures ANOVA
compute_effsize: Effect sizes

Notes

Missing values are automatically removed from the data. If x and y are paired, the entire row is removed.

The two-sample T-test for unpaired data is defined as:

\[t = \frac{\overline{x} - \overline{y}} {\sqrt{\frac{s^{2}_{x}}{n_{x}} + \frac{s^{2}_{y}}{n_{y}}}}\]

where \(\overline{x}\) and \(\overline{y}\) are the sample means, \(n_{x}\) and \(n_{y}\) are the sample sizes, and \(s^{2}_{x}\) and \(s^{2}_{y}\) are the sample variances. The degrees of freedom \(v\) are \(n_x + n_y - 2\) when the sample sizes are equal. When the sample sizes are unequal or when correction=True, the Welch–Satterthwaite equation is used to approximate the adjusted degrees of freedom:

\[v = \frac{(\frac{s^{2}_{x}}{n_{x}} + \frac{s^{2}_{y}}{n_{y}})^{2}} {\frac{(\frac{s^{2}_{x}}{n_{x}})^{2}}{(n_{x}-1)} + \frac{(\frac{s^{2}_{y}}{n_{y}})^{2}}{(n_{y}-1)}}\]

The p-value is then calculated using a T distribution with \(v\) degrees of freedom.

The T-value for paired samples is defined by:

\[t = \frac{\overline{x}_d}{s_{\overline{x}}}\]

where

\[s_{\overline{x}} = \frac{s_d}{\sqrt n}\]

where \(\overline{x}_d\) is the sample mean of the differences between the two paired samples, \(n\) is the number of observations (sample size), \(s_d\) is the sample standard deviation of the differences and \(s_{\overline{x}}\) is the estimated standard error of the mean of the differences.

The p-value is then calculated using a T-distribution with \(n-1\) degrees of freedom.

The scaled Jeffrey-Zellner-Siow (JZS) Bayes Factor is approximated using the pingouin.bayesfactor_ttest() function.

References

[1]	https://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm

[2]	Delacre, M., Lakens, D., & Leys, C. (2017). Why psychologists should by default use Welch’s t-test instead of Student’s t-test. International Review of Social Psychology, 30(1).

[3]	Zimmerman, D. W. (2004). A note on preliminary tests of equality of variances. British Journal of Mathematical and Statistical Psychology, 57(1), 173-181.

[4]	Rouder, J.N., Speckman, P.L., Sun, D., Morey, R.D., Iverson, G., 2009. Bayesian t tests for accepting and rejecting the null hypothesis. Psychon. Bull. Rev. 16, 225–237. https://doi.org/10.3758/PBR.16.2.225

Examples

One-sample T-test.

>>> from pingouin import ttest
>>> x = [5.5, 2.4, 6.8, 9.6, 4.2]
>>> ttest(x, 4)
            T     p-val  dof       tail  cohen-d  power   BF10
T-test  1.397  0.234824    4  two-sided    0.625  0.191  0.766

Paired two-sample T-test (one-tailed).

>>> pre = [5.5, 2.4, 6.8, 9.6, 4.2]
>>> post = [6.4, 3.4, 6.4, 11., 4.8]
>>> ttest(pre, post, paired=True, tail='one-sided')
            T     p-val  dof       tail  cohen-d  power   BF10
T-test -2.308  0.041114    4  one-sided    0.251  0.121  3.122

Paired two-sample T-test with missing values.

>>> import numpy as np
>>> pre = [5.5, 2.4, np.nan, 9.6, 4.2]
>>> post = [6.4, 3.4, 6.4, 11., 4.8]
>>> ttest(pre, post, paired=True)
            T     p-val  dof       tail  cohen-d  power   BF10
T-test -5.902  0.009713    3  two-sided    0.306  0.065  7.169

Independent two-sample T-test (equal sample size).

>>> np.random.seed(123)
>>> x = np.random.normal(loc=7, size=20)
>>> y = np.random.normal(loc=4, size=20)
>>> ttest(x, y, correction='auto')
            T         p-val  dof       tail  cohen-d  power          BF10
T-test  9.106  4.306971e-11   38  two-sided     2.88    1.0  1.365699e+08

Independent two-sample T-test (unequal sample size).

>>> np.random.seed(123)
>>> x = np.random.normal(loc=7, size=20)
>>> y = np.random.normal(loc=6.5, size=15)
>>> ttest(x, y, correction='auto')
            T     p-val  dof   dof-corr       tail  cohen-d  power   BF10
T-test  2.327  0.026748   33  30.745725  two-sided    0.792  0.614  2.454