1. Use SAS to perform EM on a real data of your choice or the example dataset
given in the class.  If you use the example dataset given in the class,
estimate the mean and variance of MH1 and MH1 when time=1.

"MH1 and MH1" is a typo, obviously.  Any two variables that make sense (not ID) will be acceptable for this question.
I'm going to show the requist

/* data file is exam05 in the "work" library */
/* first, subset to time=1 */
data exam05_time1;
set exam05;
if time=1;
run;

/* now, let's run PROC MI for the EM algorithm */
proc mi data=exam05_time1 seed=24634 simple nimpute=0;
	em itprint outem=outem;
var MH1 MH2;
run;


The output is available as "hw2_ProcMI_output.lst"  To view it, save it then File -> Open -> [List Files]




2. Answer the following questions:
 a) What is the missing-data pattern?
    This is easily seen in the output file.  In this case, 941 cases had complete data, while 25 were missing MH1.

 b) Is multiple imputation like EM?
    It's somewhat similar.  EM focuses on finding the maximum likelihood of parameters.  In this case, its parameters are those of a multivariate normal distribution.
    MI focuses on the imputation of data.  In MCMC methods, rather than finding the expected value of the missing cases (E step), we impute the missing cases (I step).  Rather than just maximizing the likelihood (M step), we maximize then draw a new set of parameters (P step).

 c) Is multiple imputation just making up data?
    No!  If the assumptions are met (MAR and distinct), multiple imputation allows us to make unbiased predictions as to the values of the missing data points, but without ignoring the fact that we have increased variability due to the missing data.

 d) What is the difference between single imputation and multiple imputation 
    methods?
    Single imputation methods do not account for the extra variability associated with the imputation of the data.  Additionally, single imputation methods are much less robust to a single bad imputation, whereas MI averages out over multiple datasets.  However, the variability is the primary concern.

 f) If the analyst's model is a special case of the imputer's one and if the 
the imputer's extra assumption is true, will the MI inference be valid?
    Yes; the important thing to recognize is that the imputer's model must be correct.  If it is not, then the imputations of the missing data will be biased, and analysis from the imputations will be incorrect.

 f) Will the ML estimate be more efficient than an observed data estimate derived 
purely from the analyst's model? Why?
    Most likely, but not certainly.  If the data is MCAR, then a complete-case analysis on the observed data is valid.  The increase in sample size from MI usually offsets the increase in standard errors (between-imputation variance), but not always.  The importance of MI is that it deals with MAR data, which complete-case methods generally cannot handle.