I have 10 multiple imputation datasets ($N = 97$, two groups) and am running ANCOVA (controlling for pre-test values) to look at post-test group differences. Working in SPSS and can't really invest the time to switch to (i.e. learn from scratch how to do the analyses in) R at this point.

What I need is to obtain pooled test statistics. I've figured out how to pool the values for $F$ and $p$ using R miceadds (yes, I got that far). I also know I can derive partial eta squared from $\frac{F \cdot df_1}{F \cdot df_1 + df_2}$. However, the $df_2$ for the individual analyses on which the pooled $F$ is based ($df_{Error} = 94$) differs from the $df_2$ obtained from the pooling procedure (151.55). My question now is, what figure for $df_2$ am I supposed to use for calculating the pooled effect size to go with the pooled $F$: the $df_{Error} = 94$ from the actual analyses or the weird 151.55 that came with the pooling procedure?

So thankful for any help I can get!


1 Answer 1


You cannot use the pooled $df_{2}$ value.

The pooled $df_{2}$ is calculated as follows (see enders 2010 p239 and following)

$df_{2Pooled} = 4 + (km - k - 4)[1+(1-\frac{2}{km-k})\frac{1}{ARIV}]^2$

where k = number of parameters, m = number of imputation data, ARIV = average relative increase in variance and km - m > 4. It is hard to see from the equation but if you plot it, you see a linear increase in $df_2$ for m (see here an example with k=4 and reduction in variance of 5). Because $eta^2$ is inverse proportional to $df_2$, your $eta^2$ will shrink if you use the pooled $df_2$ version and increase m (the number of imputations). You can also just try it out, use the pooled $df_2$ value and calculate the $eta^2$ values for different values of m.

I would suggest to calculate $eta^2$ by bootstrapping the sum of squares. In the end $eta^2 = SS_{effect} / SS_{total}$


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