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I have a small dataset (n=39) with one dependent variable (y) and several independent variables (x1, x2, …, xn). For most of my independent variables I have 39 measurements. However, I am missing some measurements for some independent variables. For example, independent variable x1 is composed of 39 measurements but independent variable x2 and x3 are composed of 30 and 32 measurements, respectively.

I have built several simple- and multivariable-linear regression models to predict the dependent variable and I am having difficulty selecting the best model. Since some independent variables did not have 39 measurements, models were built different sample sizes. Most models had a sample size of 30. I had originally planned to use the corrected Akaike information criterion (AICc) for model selection (Wikipedia) but I recently learned that the AICc is only supposed to be used to compare models with the same sample size (Stack Exchange Link).

What model selecting criteria can I use to in this situation? I am using python for my analysis.

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Multiple imputation of the missing data is one accepted way to approach this problem. There are nearly 400 questions on this site tagged as multiple-imputation. That's valid provided that the data are missing at random in a specific way, often a reasonable assumption.

Instead of simply replacing missing values individually with means or medians or modes you create multiple copies of the data set, with each copy containing probabilistic estimates of the missing values. You then do your regression analyses on each of the imputed data sets and pool the results in a way that takes into account the uncertainty in the imputation. Then the problem of different sample sizes for the different models disappears.

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I would suggest you fix the number of data points before creating any model. You either remove all the data points which contain missing values or replace the missing values with the mean/median of that variable. You can also look for advanced methods like EM-algorithm to fill the missing values.

Once you are done with that, you can proceed with creating model. Apart from multiple regression, you can use best subset regression, ridge regression, lasso, etc. and compare them according to their AIC.

Hope this helped.

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