Example 9.1: EM Algorithm for MLE

The MI Procedure

Example 9.1: EM Algorithm for MLE

This example uses the EM algorithm to compute the maximum likelihood estimates for the parameters of a multivariate normal distribution using data with missing values. The following statements invoke the MI procedure and request the EM algorithm to compute the MLE for $( {mu}, {\Sigma})$ of a multivariate normal distribution from the input data set FitMiss.

   proc mi data=FitMiss seed=55417 simple nimpute=0;
      em itprint outem=outem;   
      var Oxygen RunTime RunPulse; 
   run;

Note when you specify the option NIMPUTE=0, the missing values will not be imputed. The procedure generates the following output:

Output 9.1.1: Model Information

The MI Procedure

Model Information
Data Set	WORK.FITMISS
Method	MCMC
Multiple Imputation Chain	Single Chain
Initial Estimates for MCMC	EM Posterior Mode
Start	Starting Value
Prior	Jeffreys
Number of Imputations	0
Number of Burn-in Iterations	200
Number of Iterations	100
Seed for random number generator	55417

The "Model Information" table describes the method and options used in the procedure.

Output 9.1.2: Missing Data Patterns

The MI Procedure

Missing Data Patterns
Group	Oxygen	RunTime	RunPulse	Freq	Percent	Group Means
Group	Oxygen	RunTime	RunPulse	Freq	Percent	Oxygen	RunTime	RunPulse
1	X	X	X	21	67.74	46.353810	10.809524	171.666667
2	X	X	.	4	12.90	47.109500	10.137500	.
3	X	.	.	3	9.68	52.461667	.	.
4	.	X	X	1	3.23	.	11.950000	176.000000
5	.	X	.	2	6.45	.	9.885000	.

The "Missing Data Patterns" table lists distinct missing data patterns with corresponding frequencies and percents. Here, "X" means that the variable is observed in the corresponding group and "." means that the variable is missing. The table also displays group-specific variable means.

With the SIMPLE option, the procedure displays simple descriptive univariate statistics for available cases in the "Univariate Statistics" table and correlations from pairwise available cases in the "Pairwise Correlations" table.

Output 9.1.3: Univariate Statistics

The MI Procedure

Univariate Statistics
Variable	N	Mean	Std Dev	Minimum	Maximum
Oxygen	28	47.11618	5.41305	37.38800	60.05500
RunTime	28	10.68821	1.37988	8.63000	14.03000
RunPulse	22	171.86364	10.14324	148.00000	186.00000

Output 9.1.4: Pairwise Correlations

The MI Procedure

Pairwise Correlations
	Oxygen	RunTime	RunPulse
Oxygen	1.000000000	-0.849118562	-0.343961742
RunTime	-0.849118562	1.000000000	0.247258191
RunPulse	-0.343961742	0.247258191	1.000000000

With the EM statement, the procedure displays the initial parameter estimates for EM.

Output 9.1.5: Initial Parameter Estimates for EM

The MI Procedure

Initial Parameter Estimates for EM
_TYPE_	_NAME_	Oxygen	RunTime	RunPulse
MEAN		47.116179	10.688214	171.863636
COV	Oxygen	29.301078	0	0
COV	RunTime	0	1.904067	0
COV	RunPulse	0	0	102.885281

With the ITPRINT option, the "EM (MLE) Iteration History" table displays the iteration history for the EM algorithm.

Output 9.1.6: EM (MLE) Iteration History

The MI Procedure

EM (MLE) Iteration History
_Iteration_	-2 Log L	Oxygen	RunTime	RunPulse
0	289.544782	47.116179	10.688214	171.863636
1	263.549489	47.116179	10.688214	171.863636
2	255.851312	47.139089	10.603506	171.538203
3	254.616428	47.122353	10.571685	171.426790
4	254.494971	47.111080	10.560585	171.398296
5	254.483973	47.106523	10.556768	171.389208
6	254.482920	47.104899	10.555485	171.385257
7	254.482813	47.104348	10.555062	171.383345
8	254.482801	47.104165	10.554923	171.382424
9	254.482800	47.104105	10.554878	171.381992
10	254.482800	47.104086	10.554864	171.381796

The procedure then displays the EM (MLE) parameter estimates, the maximum likelihood estimates for and ${\Sigma}$ of a multivariate normal distribution from the data set FitMiss.

Output 9.1.7: EM (MLE) Parameter Estimates

The MI Procedure

EM (MLE) Parameter Estimates
_TYPE_	_NAME_	Oxygen	RunTime	RunPulse
MEAN		47.104086	10.554864	171.381796
COV	Oxygen	27.798014	-6.457929	-18.030790
COV	RunTime	-6.457929	2.015491	3.516092
COV	RunPulse	-18.030790	3.516092	97.766559

You can also output the EM (MLE) parameter estimates into an output data set with the OUTEM= option. The following statements list the observations in the output data set outem.

   proc print data=outem;
      title 'EM Estimates';
   run;

Output 9.1.8: EM Estimates

EM Estimates

Obs	_TYPE_	_NAME_	Oxygen	RunTime	RunPulse
1	MEAN		47.1041	10.5549	171.382
2	COV	Oxygen	27.7980	-6.4579	-18.031
3	COV	RunTime	-6.4579	2.0155	3.516
4	COV	RunPulse	-18.0308	3.5161	97.767

The output data set outem is a TYPE=COV data set. The observation with _TYPE_=`MEAN' contains the MLE for the parameter

and the observations with _TYPE_=`COV' contain the MLE for the parameter ${\Sigma}$ of a multivariate normal distribution from the data set FitMiss.

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