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I have a dataset with 141 observations and 8 corresponding variables and I mean to apply a GLM to this dataset. However, a lot of observations lack either one or multiple variable values. So if I want to apply the dredge function on the full model, there only remain about 20 observations with complete data. I do not know beforehand if I could exclude any variables as to increase my complete observation count. Does anyone have an idea how I could approach this?

Kind regards.

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Your sample size is limited even if you had complete data, considering that you want to analyze more than 1 variable (note that n=400 may be required to well estimate a single correlation coefficient). So you need to work hard to avoid making the sample size any smaller. Multiple imputation using predictive mean matching is the best default choice for this. This algorithm has been implemented in multiple R packages, Stata, and other software. See for example the R Hmisc package aregImpute function. I cover this in the Missing Data chapter of the RMS book and course notes.

Multiple imputation has been shown to work better than either excluding observations or excluding variables. You need to have some confidence that the missing values are missing at random, i.e., the reasons they are missing are either completely random or are described by non-missing data values.

Do avoid variable selection even with complete data.

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