How do I run multiple linear regression with a limited data set for each subject?

Question

I'm trying to use limited data across a range of variables to make predictions. There are ten variables and each subject has three of the ten variables defined. It's approximately random which three variables each subject has defined, but not completely.

If each subject had all ten variables defined I'd know easily how to run a least squares linear regression.

I have one subject who has all ten variables defined - using this data, I could try to fill in what the other seven variables would be for each subject, but it doesn't seem right. What's the best way to calculate the coefficients for a linear regression model in the case where all subjects only have three variables of the ten? Thanks!

Answer 1

With the variables that were observed or not being pretty much random you should be able to use standard missing data tools. These include the EM algorithm and Multiple imputation. There are existing tools that help with Multiple imputation and make a regression analysis fairly straight forward.

Another option is to do a Bayesian regression fit and specifying a prior on each of the missing values.