Multiple imputation for non-parametric tests on small sample? I am new to multiple imputation, so apologies if the question is naive. I have been exploring multiple imputation for use on a small data set where there are just a couple of data points. I would like to run Wilcoxon signed-rank tests using the pooled imputations. However, SPSS doesn't provide pooled test statistics for non-parametrics because "Rubin's rules... don't address nonparametric tests". Consequently, SPSS calculates the statistics for each imputation, but not as a group. So I was wondering how useable this output is (e.g., should I report the statistic range) or is this indicative of a fundamental issue in trying to use multiple imputation with non-parametric tests. Because of these complexities and the small amount of missing data, I considered mean substitution instead but I appreciate that this approach is not optimal - although the overall findings remain the same either way.
Any advice would be greatly appreciated.
 A: Not a naive question at all. I'm not an expert, but I think this is actually a rather difficult question.
As you may be aware, Rubin's Rules assume that the full-data estimate, $\hat{\theta}_{full}$, would be normally distributed. Then from the $M$ imputed estimates, $\hat{\theta}_{1}, \ldots, \hat{\theta}_{M}$, we average them to find the point estimate $\hat{\theta}_{imp}$ and also combine the within- and between- variances to estimate the variance of $\hat{\theta}_{imp}$, $\hat{\text{V}}[\hat{\theta}_{imp}]$. Then we can use traditional methods for normally distributed values to find confidence intervals and perform tests. If the estimates $\hat{\theta}_{i}$ are not normally distributed then this will not work.
My understanding is that the Wilcoxon signed-rank test statistic, $W$, is asymptotically normally distributed. So in contrast to the SPSS manual, I think you could pool the estimates if you had a large sample. Although if it isn't supported by SPSS you'd probably have to program the calculations yourself.
But, it sounds like your sample isn't nearly large enough to consider this. $W$ is not close to normally distributed in small samples. An alternative you could consider would be pooling the p-values from the imputed datasets themselves. I don't know enough to offer guidance on it, but you can read about the procedure in section 5.3.2 of Flexible Imputation of Missing Data, https://stefvanbuuren.name/fimd/sec-multiparameter.html#sec:chi. The basic idea is to transform the imputed p-values to be approximately normally distributed, pool those transformed values, then transform back to find the final p-value. You can see some sample code in this answer, How to get pooled p-values on tests done in multiple imputed datasets?
Of course, if the amount of missing data is very small, you could consider using a complete-case analysis instead. The results would probably be very similar.
