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?