pingouin.pairwise_gameshowell

pingouin.pairwise_gameshowell(data=None, dv=None, between=None, alpha=0.05, tail='two-sided', effsize='hedges')[source]

Pairwise Games-Howell post-hoc test.

Parameters
datapandas.DataFrame

DataFrame

dvstring

Name of column containing the dependent variable.

between: string

Name of column containing the between factor.

alphafloat

Significance level

tailstring

Indicates whether to return the ‘two-sided’ or ‘one-sided’ p-values

effsizestring or None

Effect size type. Available methods are:

  • 'none': no effect size

  • 'cohen': Unbiased Cohen d

  • 'hedges': Hedges g

  • 'glass': Glass delta

  • 'r': Pearson correlation coefficient

  • 'eta-square': Eta-square

  • 'odds-ratio': Odds ratio

  • 'AUC': Area Under the Curve

  • 'CLES': Common Language Effect Size

Returns
statspandas.DataFrame

Stats summary:

  • 'A': Name of first measurement

  • 'B': Name of second measurement

  • 'mean(A)': Mean of first measurement

  • 'mean(B)': Mean of second measurement

  • 'diff': Mean difference (= mean(A) - mean(B))

  • 'se': Standard error

  • 'tail': indicate whether the p-values are one-sided or two-sided

  • 'T': T-values

  • 'df': adjusted degrees of freedom

  • 'pval': Games-Howell corrected p-values

  • 'hedges': effect size (or any effect size defined in effsize)

Notes

Games-Howell [1] is very similar to the Tukey HSD post-hoc test but is much more robust to heterogeneity of variances. While the Tukey-HSD post-hoc is optimal after a classic one-way ANOVA, the Games-Howell is optimal after a Welch ANOVA. Games-Howell is not valid for repeated measures ANOVA.

Compared to the Tukey-HSD test, the Games-Howell test uses different pooled variances for each pair of variables instead of the same pooled variance.

The T-values are defined as:

\[t = \frac{\overline{x}_i - \overline{x}_j} {\sqrt{(\frac{s_i^2}{n_i} + \frac{s_j^2}{n_j})}}\]

and the corrected degrees of freedom are:

\[v = \frac{(\frac{s_i^2}{n_i} + \frac{s_j^2}{n_j})^2} {\frac{(\frac{s_i^2}{n_i})^2}{n_i-1} + \frac{(\frac{s_j^2}{n_j})^2}{n_j-1}}\]

where \(\overline{x}_i\), \(s_i^2\), and \(n_i\) are the mean, variance and sample size of the first group and \(\overline{x}_j\), \(s_j^2\), and \(n_j\) the mean, variance and sample size of the second group.

The p-values are then approximated using the Studentized range distribution \(Q(\sqrt2*|t_i|, r, v_i)\).

Note that the p-values might be slightly different than those obtained using R or Matlab since the studentized range approximation is done using the Gleason (1999) algorithm [2], which is more efficient and accurate.

References

1

Games, Paul A., and John F. Howell. “Pairwise multiple comparison procedures with unequal n’s and/or variances: a Monte Carlo study.” Journal of Educational Statistics 1.2 (1976): 113-125.

2

Gleason, John R. “An accurate, non-iterative approximation for studentized range quantiles.” Computational statistics & data analysis 31.2 (1999): 147-158.

Examples

Pairwise Games-Howell post-hocs on the pain threshold dataset.

>>> from pingouin import pairwise_gameshowell, read_dataset
>>> df = read_dataset('anova')
>>> pairwise_gameshowell(data=df, dv='Pain threshold',
...                      between='Hair color')