pingouin.welch_anova(dv=None, between=None, data=None, export_filename=None)[source]

One-way Welch ANOVA.


Name of column containing the dependant variable.


Name of column containing the between factor.

datapandas DataFrame

DataFrame. Note that this function can also directly be used as a Pandas method, in which case this argument is no longer needed.


Filename (without extension) for the output file. If None, do not export the table. By default, the file will be created in the current python console directory. To change that, specify the filename with full path.


ANOVA summary

'Source' : Factor names
'SS' : Sums of squares
'DF' : Degrees of freedom
'MS' : Mean squares
'F' : F-values
'p-unc' : uncorrected p-values
'np2' : Partial eta-square effect sizes

See also


One-way ANOVA


One-way and two-way repeated measures ANOVA


Two way mixed ANOVA


Non-parametric one-way ANOVA


The classic ANOVA is very powerful when the groups are normally distributed and have equal variances. However, when the groups have unequal variances, it is best to use the Welch ANOVA that better controls for type I error (Liu 2015). The homogeneity of variances can be measured with the homoscedasticity function. The two other assumptions of normality and independance remain.

The main idea of Welch ANOVA is to use a weight \(w_i\) to reduce the effect of unequal variances. This weight is calculated using the sample size \(n_i\) and variance \(s_i^2\) of each group \(i=1,...,r\):

\[w_i = \frac{n_i}{s_i^2}\]

Using these weights, the adjusted grand mean of the data is:

\[\overline{Y}_{welch} = \frac{\sum_{i=1}^r w_i\overline{Y}_i} {\sum w}\]

where \(\overline{Y}_i\) is the mean of the \(i\) group.

The treatment sums of squares is defined as:

\[SS_{treatment} = \sum_{i=1}^r w_i (\overline{Y}_i - \overline{Y}_{welch})^2\]

We then need to calculate a term lambda:

\[\Lambda = \frac{3\sum_{i=1}^r(\frac{1}{n_i-1}) (1 - \frac{w_i}{\sum w})^2}{r^2 - 1}\]

from which the F-value can be calculated:

\[F_{welch} = \frac{SS_{treatment} / (r-1)} {1 + \frac{2\Lambda(r-2)}{3}}\]

and the p-value approximated using a F-distribution with \((r-1, 1 / \Lambda)\) degrees of freedom.

When the groups are balanced and have equal variances, the optimal post-hoc test is the Tukey-HSD test (pairwise_tukey). If the groups have unequal variances, the Games-Howell test is more adequate.

Results have been tested against R.



Liu, Hangcheng. “Comparing Welch’s ANOVA, a Kruskal-Wallis test and traditional ANOVA in case of Heterogeneity of Variance.” (2015).


Welch, Bernard Lewis. “On the comparison of several mean values: an alternative approach.” Biometrika 38.3/4 (1951): 330-336.


  1. One-way Welch ANOVA on the pain threshold dataset.

>>> from pingouin import welch_anova, read_dataset
>>> df = read_dataset('anova')
>>> aov = welch_anova(dv='Pain threshold', between='Hair color',
...                   data=df, export_filename='pain_anova.csv')
>>> aov
       Source  ddof1  ddof2     F     p-unc
0  Hair color      3   8.33  5.89  0.018813