# Partial eta squared calculation with multiple imputation data

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!

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 $$eta^2 = SS_{effect} / SS_{total}$$