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I have missing data that I have done multiple imputation with. I want to then use the means or 'pooled' data from the five imputations to do a repeated measures ANOVA. It seems I can't do this in SPSS. So my question is: Do I manually find the means of the five values I have for each missing number and put this into the dataset to make up one complete data set to work with? Or do I let the repeated measures function use all five data sets?

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  • $\begingroup$ All 5 datasets, if you want the benefits of multiple imputation. 5 is a bit low by the way, usually GPS enough for okay coverage, but often losing power/making CIs too wide. $\endgroup$ – Björn Jan 10 '16 at 21:37
  • $\begingroup$ Thanks. My understanding was that spss did about 100 imputations in the background, but gave details from the 5 'best fit'? $\endgroup$ – Vonni Jan 16 '16 at 15:57
  • $\begingroup$ When doing MI, the imputed data depend on the observed data, so the imputed analyses have a correlation. With independent data, you combined them with Rubin's Rules. I'm not aware of any form/expression for dependent data analysis; even if there were such a thing, it may be quite complicated. $\endgroup$ – AdamO Dec 28 '18 at 15:54
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    $\begingroup$ @Björn it is a bit of a convention to use 5 multiply imputed datasets. Rubin's book "Missing Data Analysis" unfortunately touted that 5 was usually sufficient. Now 5 is the default for most MI implementations. In some simulations I agree with you that the result of too few imputations is CIs that are a bit too wide (as compared with an asymptotically efficient approach like EM algorithm). $\endgroup$ – AdamO Dec 28 '18 at 16:08
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The short answer is that you shouldn't have to do any part of multiple imputation manually and that you certainly don't want to let repeated measures use the 5 individual stochastic imputations, as that would be missing the point of using multiple imputation in the first place.

Before using a method such as multiple imputation, you'll want to think about the pattern of "missingness" in your data. In your mind, you'll want to distinguish between three such patterns: Type 1) Non-missing values of other variables can predict missing values of the variable of interest, Type 2) Missing values of the variable itself predict missingness, and Type 3) If it is neither Type 1 nor Type 2. Your missing data is Type 2 if say your variable is a self-reported Age and older people are more likely to not report Age. It is Type 1 if say your variable is a self-reported Age and Women are more likely than men to not report Age. Finally, your data is Type 3 if it is neither of the previous two, in which case there is little hope for meaningful "regression-based" imputation, which multiple imputation is. Realize that by opting for Multiple Imputation you're assuming your pattern of missingness is Type 1. This need not always be the case and if it is not, the use of Multiple Imputation may be questionable. With repeated measures there is a good chance that your missingness has both Type 1 and Type 2 components, suggesting that you may have to do something more than Multiple Imputation to address the Type 2 component as well.

With regards to Multiple Imputation and the pattern of missingness we referred to as Type 1 above, the Multiple Imputation software implementations (other than SPSS) that I've seen all have a built-in pooling function. You shouldn't have to pool the coefficients from each individual stochastic imputation (sounds like you have 5 of those), whose repeated iteration make up your multiple imputation. In the pooling stage, the statistical software should pool together the parameters and standard errors from each individual stochastic imputation. Note that your focus on the imputed values and their average is a bit misguided as the main point of multiple imputation isn't to impute "correct" values but to do so in a way that does not downward bias the variance of the imputed values. If you didn't care about not downward biasing the variance of the imputed values, why bother with multiple imputation (several iterations of stochastic imputation) or even stochastic imputation in the first place? A deterministic (conditional mean) imputation without a stochastic component would do. Stochastic imputation adds a random component to the conditional mean imputation for the express purpose of increasing the variance of your imputed values and multiple imputation adds to stochastic imputation a measure of uncertainty with regards to your imputation.

In choosing the variables that you include in your multiple imputation you'll want to think about all the variables that carry informational content as to the true value of the missing values (you should have already have done much of this thinking at the outset when you determined the pattern of your missingness to be Type 1). There is an added layer of complexity with repeated measures, because you'll want to think about how your repeated measures for the same subject are related over time and reflect that in the regression included in the multiple imputation.

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I tried to do multiple imputation on a dataset of repeated measures (health testing done over several time points during recovery). When I looked at the results, they don't make sense. I expect the values to increase over time, but the MI would give much lower results on the 2nd and 3rd repetitions of testing. Or much higher results than clinically possible. So, I don't think that the MI procedure takes into account repeated measures, which by definition are linked in some way. Therefore, you cannot do a RM ANOVA with the MI dataset!

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