The following uses data from Menec et al. 2004. The purpose of the analysis was to examine factors associated with health care costs at the end of life, including:

• Age at death - entered as individual years, or as age groups(19-44, 45-64, 65-74, 75-84, and 85+ years)
• Gender - males, females (reference)
• Region of residence - Northern Manitoba, Southern Manitoba, Central Manitoba, Winnipeg (reference)
• Cause of death - cardiovascular diseases, injuries, respiratory diseases, cancer, all other causes (reference)
• Location of death - hospital, LTC, homecare, other locations reference)

Heath care costs were derived for all individuals for each of the last six months prior to death. Analyses of trends over time were conducted for 7911 individuals who had non-zero health care costs in at least four of the six months preceding death.

A cross-sectional examination of costs revealed that they were moderately to highly skewed in each time period. Therefore a logarithmic transformation was applied to produce a more normal distribution. For each individual who had a zero monthly cost, this was replaced with a positive dollar value so that the logarithm could be computed. Once the logarithmic transformation was applied to the costs the data were approximately normally distributed.

Model #1 Random Intercept and No Fixed Effects

PROC MIXED DATA=data-set-name METHOD=reml covtest;
  CLASS id;
  MODEL Incost = / g gcorr;
  RANDOM INTERCEPT / SUBJECT=id;
  TITLE2 ' RANDOM INTERCEPT ONLY';
RUN;

This model, called the null model, is typically the first model that an analyst will run when deciding whether or not to select a random effects model for the data. It contains only one parameter, which is a random intercept. It partitions the total variation in the data into within-individual and between- individual components.

The intraclass correlation (ICC) computed from this null model is a useful tool for deciding whether a random effects model might be an appropriate choice for the data. The numeric formula for the ICC is

To illustrate:

Covariance Parameter Estimates
Cov Parm	Subject	Estimate	Standard Error	Z Value	Pr Z
Intercept	Id	1.7718	0.03435	51.58	<.0001
Residual		2.291	0.01629	140.63	<.0001

Here, =1.77 and =2.29. Therefore, ICC = 1.77/(1.77 + 2.29) = 0.44, indicating that 44% of the variation in the data is explained by allowing the intercept to vary across individuals. The statistically significant value for the within-individual variation suggests the data structure is best captured by using a random effects model

Model #2 Random Intercept and Slope

PROC MIXED DATA=data-set-name METHOD=reml covtest;
  CLASS id;
  MODEL Incost = time / solution g gcorr;
  RANDOM INTERCEPT time / SUBJECT=id;Type=un;
  TITLE2 ' RANDOM INTERCEPT and slope';
RUN;

The SAS output for the random effects variances and covariances is given below:

Covariance Parameter Estimates
Cov Parm	Subject	Estimate	Standard Error	Z Value	Pr Z
UN(1,1)	Id	3.3901	0.07487	42.49	<.0001
UN(2,1)	Id	-.5742	0.0145	-39.59	<.0001
UN(2,2)	Id	.1363	0.003553	38.37	<.0001
Residual		1.4562	0.01158	125.79	<.0001

The ICC is computed using variance estimates for both the intercept and slope, as well as their covariance. For this model, ICC = (3.3901 - .5742 + .1363) / (3.3901 - .5742 + .1363 + 1.4562) = .67, indicating that 67% of the variation in the data is accounted for by allowing the intercept and the slope to vary across individuals.

An additional piece of information that can be captured from models that contain both random slopes and intercepts is the correlation between the random effects. The correlation matrix given below shows us that there is a strong negative correlation between the slope and the intercept. This means that individuals who have higher initial health care costs tend to have lower rates of change over time, and individuals who have lower initial health care costs tend to have higher rates of change over time.

Estimated G Correlation Matrix
Row	Effect	Id	Col1	Col2
1	Intercept	1	1.0000	-0.7845
2	Time	1	-0.7845	1.0000

Model #3 Random Intercept and Slope plus Additional Fixed Effects

Given that a substantial proportion of the variation in the data could be explained by inclusion of both a random intercept plus a random slope, we retained these parameters in the model. The next model also contains all of the fixed effect predictors that we were interested in testing for statistical significance. Here, age is defined as a continuous variable that was centered at the mean.

PROC MIXED DATA=data-set-name METHOD=reml covtest;
  CLASS id sex cause locdth reg;
  MODEL Incost = time age sex cause locdth reg / solution g gcorr;
  RANDOM INTERCEPT time / SUBJECT=id TYPE=un;
  TITLE2 ' FULL MODEL _ MAIN EFFECTS';
RUN;

References

• Menec V, Lix L, Steinbach C, Ekuma O, Sirski M, Dahl M, Soodeen R. Patterns of Health Care Use and Cost at the End of Life. Winnipeg, MB: Manitoba Centre for Health Policy, 2004.