@xiaoshuxu wrote:
2) "that the distribution of the parameters to be pooled differs from that of the pooled parameter." "You can safely conclude that all of the pooled parameters in PROC MIANALYZE follow a t distribution, regardless of the distribution of the original parameters to be pooled." For my case, i.e., I am pooing my 'z' and my pooled 'z' follows a non-central t.
No, the last sentence is incorrect. Your z-statistics follow a standard normal distribution, which is not the same as a non-central t distribution.
@xiaoshuxu wrote:
So, I am still having question. The p-value from proc mianalyze for my pooled 'z' is what I should report?
Yes, that's correct.
@xiaoshuxu wrote:
What I meant is: we have missing data; we did MI; we got m imputed data sets; we got m 'z' from wilcoxon test; we got a pooled mean from proc mianalyze, which is just simple mean of all of my 'z' (but the pooled mean follows a non-central t); I got a p-value from proc mianalysis, which is for how the t is away from the non-central t (that is H0). Then, in the end, I should report this p-value (from proc mianalyze) as the 'best guess' p-value for our original data?
Your questions centers two topics: (1) Point estimate of the central tendency of the populations; (2) Hypothesis testing of populations. Regarding the first question, I don't think reporting the combined means is a good choice. Please check the normality of your epilepsy scores (via the complete cases). It is often because of violation of normality that the data analyst resort to Wilcoxon sum-of-rank tests to test intergroup differences. In this case, reporting the means of the two groups is inappropriate. Instead, you should report the medians. That is, you should calculate the medians of each imputed dataset and pool them as the eventual estimate of the central tendency of your populations. By the way, 1) according to central limit theorem, the means of any sample never follow a distribution other than the normal distribution, so your sentence "the pooled mean follows a non-central t" is incorrect; 2) there is currently no existing guideline as to how to pool the medians of the imputed dataset. Not long ago, I asked another user in the Community about that question. Here's the link of his/her answer: Missing value imputation. You can have a look.
Regarding the second question, as I had said before, you can directly report the result PROC MIANALYZE presented to you.
In short, you should report the pooled medians as a measure of central tendency and the P value PROC MIANALYZE presented to you as the result of hypothesis testing of intergroup difference. That is, your P value is not calculated from something regarding the median.
@xiaoshuxu wrote:
3) Explanation about my original thinking: I was thinking to get a ‘pooled’ statistic on my own from SAS output window, either from 'sumofscore" or "expectunderH0' (words from sas output window; maybe a little differences from sas output data set), also use Std Dev Under H0, etc . Later I thought I can directly pool 'z'. Till this moment, I am still not sure which method is the 'best'.
Pooling the z-statistic is the best among the three statistics you mentioned. As I had stated before, the sum-of-ranks and expected sum-of-rank under H0 do not follow a normal distribution, which is what violates Rubin's rules of pooling the results of each imputed dataset.
@xiaoshuxu wrote:
4) About stacking data directly: "Of course you can stack the datasets, but the correct way of dealing with missing data via multiple imputation (MI) is to calculate the statistics separately in each imputed dataset and combine (pool) them in some way." I know the generay way is not 'stacking'. But, for our data, what we imputed is the missing seizure count for missing days. Our endpoint is percent change of seizure frequency (averaged to per 28-day; there is some calculation after the imputation.). I read the book of "flexible imputation of missing data" by Buuren; it said a few sentences about stacking data directly. "If the scientific interest is solely restricted to the point estimate, then the stacked imputed data can be validly used to obtain a quick unbiased estimate for linear models. Be aware that routine methods for calculating test statistics, confidence intervals, or p-values will provide invalid answers if applied to the stacked imputed data." No time to think over and no mature idea. I just feel that, simple 'stacking' and getting 'averaged' seizure count of each day from m imputed data sets seems making some sense.
There are three concerns regarding these words. The first one is about van Buuren's words, he said that stacking the datasets and report the point estimate of the stacked dataset is a choice in linear models, but Wilcoxon sum-of-rank test is not a nonparametric method instead a linear model. Secondly, I don't know if this validity van Buuren stated concerning point estimate carries to hypothesis testing. Thirdly, you had stated that your endpoint was the percentage of change from baseline in seizure frequency, so it would be better to stick to your endpoint and conduct point estimate as well as hypothesis testing during the entire process of data analysis instead of deviating from it in the first place, try a different method and somehow try to reach your goal eventually in a detour as long as directly reaching your goal is a viable choice.
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