pingouin.power_anova#

pingouin.power_anova(eta_squared=None, k=None, n=None, power=None, alpha=0.05)[source]#

Evaluate power, sample size, effect size or significance level of a one-way balanced ANOVA.

Parameters:
eta_squaredfloat

ANOVA effect size (eta-squared, \(\eta^2\)).

kint

Number of groups

nint

Sample size per group. Groups are assumed to be balanced (i.e. same sample size).

powerfloat

Test power (= 1 - type II error).

alphafloat

Significance level \(\alpha\) (type I error probability). The default is 0.05.

Notes

Exactly ONE of the parameters eta_squared, k, n, power and alpha must be passed as None, and that parameter is determined from the others.

alpha has a default value of 0.05 so None must be explicitly passed if you want to compute it.

This function is a Python adaptation of the pwr.anova.test function implemented in the pwr R package.

Statistical power is the likelihood that a study will detect an effect when there is an effect there to be detected. A high statistical power means that there is a low probability of concluding that there is no effect when there is one. Statistical power is mainly affected by the effect size and the sample size.

For one-way ANOVA, eta-squared is the same as partial eta-squared. It can be evaluated from the F-value (\(F^*\)) and the degrees of freedom of the ANOVA (\(v_1, v_2\)) using the following formula:

\[\eta^2 = \frac{v_1 F^*}{v_1 F^* + v_2}\]

GPower uses the \(f\) effect size instead of the \(\eta^2\). The formula to convert from one to the other are given below:

\[f = \sqrt{\frac{\eta^2}{1 - \eta^2}}\]
\[\eta^2 = \frac{f^2}{1 + f^2}\]

Using \(\eta^2\) and the total sample size \(N\), the non-centrality parameter is defined by:

\[\delta = N * \frac{\eta^2}{1 - \eta^2}\]

Then the critical value of the non-central F-distribution is computed using the percentile point function of the F-distribution with:

\[q = 1 - \alpha\]
\[v_1 = k - 1\]
\[v_2 = N - k\]

where \(k\) is the number of groups.

Finally, the power of the ANOVA is calculated using the survival function of the non-central F-distribution using the previously computed critical value, non-centrality parameter, and degrees of freedom.

scipy.optimize.brenth() is used to solve power equations for other variables (i.e. sample size, effect size, or significance level). If the solving fails, a nan value is returned.

Results have been tested against GPower and the pwr R package.

Examples

  1. Compute achieved power

>>> from pingouin import power_anova
>>> print('power: %.4f' % power_anova(eta_squared=0.1, k=3, n=20))
power: 0.6082

  1. Compute required number of groups

>>> print('k: %.4f' % power_anova(eta_squared=0.1, n=20, power=0.80))
k: 6.0944

  1. Compute required sample size

>>> print('n: %.4f' % power_anova(eta_squared=0.1, k=3, power=0.80))
n: 29.9256

  1. Compute achieved effect size

>>> print('eta-squared: %.4f' % power_anova(n=20, k=4, power=0.80, alpha=0.05))
eta-squared: 0.1255

  1. Compute achieved alpha (significance)

>>> print('alpha: %.4f' % power_anova(eta_squared=0.1, n=20, k=4, power=0.80, alpha=None))
alpha: 0.1085