Multiple Imputation - calculating effect size and reporting results I'm analyzing the change of various psychometric features that are supposed to change during treatment and their relation to some other features (I don't think I have to go much into detail here). My dataset contains two waves of measurement on treated subjects (before and after treatment). Due to attrition, I handled missing data by multiple imputation. This worked out fine, but when I'm preparing my results for publication, several questions arise:


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*I included some sample characteristics (age, length of treatment etc.) in the imputation model, because they are related to missingness and are used in some analysis. When I'm reporting sample characteristics at the beginning of my results section, should I report those of my original, non-imputed sample (had some missing in those variables, too), or should I report the imputed data?

*In the first table, I want to show means and standard deviations of the various psychometric measures at the two measurement points, as well as the results of t-test and the effect size. Unfortunately, with MI I don't get any pooled standard deviation and thus no pooled effect size. Several journals as well as the APA publication manual say it's essential to report SD and effect size, what can I do?


Thanks for your help!
 A: Regarding your first question, I think you should report on the original sample at both time points. In the text, you would then explain that data was missing and how you handled it.
On your second question, it partly depends what software you are using. With SAS, you can certainly get effect sizes for any analysis done with imputed data. You can also do calculations on that imputed data set. Indeed, the imputation process is separate from the analysis. 
I haven't done much imputation work with R, but I am sure these things are available there, too. 
A: As far as your first question, you should report the multiply imputed data but be precise as to what the analysis refers. For example, do not say, "...the mean length of treatment was..." Instead, say, "... the multiply-imputed mean length of treatment was..." This is similar to reporting on transformations where you would say, "... the variance of the log-transformed length of treatment was..."
For your second question, in order to pool standard deviation you must use Rubin's rules (Rubin, 1987). Pooling the variable estimates is as simple as finding the mean across data sets. Finding the variance is a bit more complicated, but not out of reach for someone like you! There are 3 steps to pooling the variance across imputed data sets:
Step 1: Find $\bar U$, which is the within-imputation variance, where
$\bar U = \frac{1}{m} \sum_{i=1}^m\hat U_i$,
$m$ is the number of imputations, and $i$ is the observation.
Step 2: Find $B$, which is the between-imputation variance, where
$B=\frac{1}{m-1} \sum_{i=1}^m(\hat Q_i-\bar Q)^2$
Step 3: Find $T$, which is the variance of $Q$, where
$T=\bar U+(1+ \frac{1}{m})B$
Further, finding effect size such as $R^2$ of a multiple regression requires calculating the sum of squares of y-hat of the mean multiply-imputed variable(s). Then, divide that by the multiply-imputed dv.
A: Find detailed information about pooling and other subjects related to multiple imputations in this gem:
https://bookdown.org/mwheymans/bookmi/
