Can I use alternative pooling technique after multiple imputation? Research problem: Comparison of means (T-test, ANOVA) between 90 countries.
Analytical problem: I have a large data set of over 120 000 observations. Each observation was measures by 8 variables: 2 nominal and 6 continous. Nominal variables are complete. Continious variables have missing data. I wanted to use single median, by group (country), imputation to deal with NA. Although when I checked how the missing data is distributed among compared countries I discovered that in some countries NA reach 19% for a single continous variable.
Done/To do: I wanted to use state of art method thus I choosed Predictive Mean Matching. I created 5 imputation data sets and I came to the point of pooling results which complicate the analysis a little bit for me. Further procedures are a little bit above my statistical and programming skills. Thus I figured out that if I want to make my life simpler/easier I thought to compute from all created five simulation data sets median for each observation in all continous variables, thus reduce 5 artificial data sets to one.
Question: Is my apporach to the problem valid and would be accepted by other analyst's in the field?
Thank you for your opinions in advance!

No, your approach would not be valid. The estimate of the mean difference should be reasonably accurate in your approach. But your approach would be too certain about the estimate and would have incorrect confidence intervals, p-values, and hypothesis tests. For comparisons with little missing data and large sample sizes the difference will probably be negligible, but for countries/variables with ~19% missing data the difference could be important.
Although pooling the estimates is a complicated procedure, it is essential to correctly account for the extra uncertainty caused by the missing values. From the 5 imputed datasets we find 5 different estimates of the mean difference. The differences between those 5 estimates gives us an estimate of how much the missing data is affecting the estimate, and the rules for pooling take into account that extra source of variation. Reducing the 5 datasets into 1 loses the information about how large an effect the missing data has.
All of the statistical programming language I'm familiar with have the ability to automatically pool the different estimates and properly report the confidence intervals / p-values / hypothesis tests. Eg, the pool() function in R's mice package. I suggest looking into that option for whatever program you're using, to see if you can use the work others have done instead of trying to do it yourself.
