Is it possible for Statsmodel's pairwise Tukey's test to produce a p-value that is negative?

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No, it is not possible for Statsmodels' pairwise Tukey's test to produce a p-value that is negative. P-values represent the probability of obtaining the observed test statistic, or a more extreme test statistic, under the null hypothesis. As probabilities cannot be negative, p-values cannot be negative. If a p-value appears to be negative, it may be due to rounding errors or incorrect calculations.