# Estimate linear regression using items randomly selected from an item pool

I am asking this question against the background of a linear regression with single predicted variable $Y$ and multiple predictors $X$. $X$ comes from a survey using an "item pool" which suggests that not all items are presented to each respondents, but only a subset of items is. This subset is randomly selected (but in the current case $Y$ is always presented to all respondents). The objective is to estimate a linear regression model on all $X$ irrepesctive of the missing information due to randomized selection of items.

I have the following three questions:

1) Given randomization it seems to me that the data are missing completely at random (MCAR). Thus, estimates of sufficient statistics (correlation matrix and means) are unbiased. Hence, I infer that regression coeffcients should be unbiased. Is that correct?

2) Standard software will listwise delete cases resulting in an empty set of observations. Should I estimate coeffcieints using the sufficient statistics only? What would be suitable approaches in R (e.g., available functions)?

3) To add complexity to the problem, assume items vary in (known) probability to appear in a survey. Thus sample size varies per pair of items as a function of $np_1p_2$, where these are sample size and probabilities to appear in a survey for any two items. How should the varying sample sizes be included in estimating standard errors of estimate, and again how can available functions in R deal with this?

• What is the nature of your items and response variable? Likert scale? If this is psychometric test are there correlated items (covariates)? – berkorbay Aug 4 '14 at 20:53
• The variables are measured on likert scales, but there is not psyochometric scale underlying the variables. The covariates are correlated to some extent without explicit planing or knowledge about a scale. – tomka Aug 6 '14 at 12:29
• By the way, my current intuition is to multiply impute missing outcomes a large number of times. The fraction of missing information might be high, but given data are MCAR, estimates of paramters and their standard errors should be unbiased given large number of imputations. – tomka Aug 6 '14 at 14:12