Options relating to the model specification

Options relating to statistical tests

Options relating to output

SOLUTION	Solution for fixed effects is printed.
CL	Requests t-type confidence intervals for fixed effects given by SOLUTION option.
ALPHA=<p>	Specifies size of confidence intervals (default 0.05).
E3	Prints design matrix for all fixed effects.
OUTP=<dataset>	For each observation gives predicted values and residuals based on fixed and random effects. Approximate variances, standard errors and 95% confidence intervals for each predicted value are also listed.
OUTPM=<dataset>	Predicted values and residuals based on fixed effects only are given. Approximate variances, standard errors and 95% confidence intervals for each predicted value are also listed.

Options relating to model checking

Example

PROC MIXED DATA=a; CLASS centre treat;

MODEL dbp = dbp1 treat/ E3 SOLUTION DDFM=KR OUTP=pred

   OUTPM=predm;

RANDOM centre centre*treat;

PROC PRINT NOOBS DATA=pred;

PROC PRINT NOOBS DATA=predm;

Use of the SOLUTION and DDFM=KR options

Note that use of DDFM=KR has caused the standard errors to be based on an improved estimate (as described by Kenward and Roger, 1997), which helps alleviate bias (see Section 2.4.3). These standard errors are also used to calculate the t statistics and their approximate DF. (It is somewhat confusing that the DDFM=KR option in fact relates to the standard error calculation. The DF are still calculated using Satterthwaite’s approximation but based on these improved standard errors.)

Use of the E3 option

This shows that SAS has parameterised treatment effects by using differences from the last treatment (C).

Use of the DDFM=KR option

Note that these tests differ from the earlier analysis where the DDFM=KR option was not used.

Use of the OUTP= option

Use of the OUTPM= option

Options relating to model checking

Use of these options is illustrated in Section 2.5 and Section 6.3.

Output options

Options corresponding to the G and V matrices

Example

PROC MIXED DATA=a; CLASS centre treat;

MODEL dbp = dbp1 treat/ DDFM=KR;

RANDOM centre centre*treat/ V SOLUTION;

Use of the V option

This matrix has a dimension equal to the number of observations in the dataset and is hence very large in this example. (If a SUBJECT option is used in the RANDOM statement, the matrix will relate to only the first subject, otherwise the full variance matrix for all observations is given.) The diagonal terms are equal to the sum of the three variance components. Off-diagonal terms are equal to the sum of the centre and centre·treatment variance components, 10.6, when patients are at the same centre and receive the same treatment; they are equal to the centre variance component, 6.5, when patients are at the same centre but receive different treatments; they and are zero when patients are at different centres (denoted by blank entries).

Use of the SOLUTION option

The Estimate and Std Err Pred columns give the random effects estimates and their standard errors. The Wald t tests help to determine whether any centre is outlying.

Options for specifying the G matrix structure

GROUP=<effect>

This causes a separate G matrix to be estimated within each group. For example, if patient effects were fitted as random and the GROUP=treat option were used, then a separate variance patient component would be estimated for each treatment group. If patients 1 and 3 received treatment A and patients 2 and 4 received treatment B, then the G matrix for the first four patients would have the form

However, the statement should be used cautiously when there are many group levels, since a large number of parameters will then need to be estimated.

The G matrix is always diagonal in random effects models, and the following options given for the RANDOM statement are not required. However, a non-diagonal G matrix is necessary to fit a random coefficients model to allow the intercepts and slopes of the random coefficients to be correlated and also for fitting certain covariance structures in hierarchical repeated measures designs (see Section 8.1). The G matrix is blocked by the specified SUBJECT effect, and the covariance pattern is specified by the TYPE options in a very similar way to the REPEATED statement.

RANDOM <random effects or coefficients>/

   SUBJECT=<blocking effect> TYPE=<covariance pattern>;

Covariance patterns available for the TYPE= option will be listed under the REPEATED statement.

Random coefficients models In these models, the random coefficients on the same subject are assumed to be correlated (e.g. intercepts and slope effects are correlated for each subject in a repeated measures trial). This is achieved using a RANDOM statement of the following form:

RANDOM <random coefficients>/SUBJECT=<subject effect> TYPE=UN;

For example, to model random coefficients for the effect of time, the code might be

RANDOM INT time/SUBJECT=patient TYPE=UN;

This fits the random coefficients patient and patient·time. Patient effects are specified by the reserved SAS term INT, which fits an intercept for each patient.

Example

Examples of fitting random coefficients models are given in Section 6.6.

Options

Example

PROC MIXED; CLASS centre treat;

MODEL dbp = dbp1 treat/ DDFM=KR;

RANDOM centre centre*treat;

LSMEANS treat/ CL COV CORR DIFF PDIFF;

The last six columns show covariance and correlation matrices for the least squares means.

Note that the DF differ markedly between the least squares means and their differences because they are estimated using information from different combinations of error strata.

Options

When the effect of interest is contained within another fixed interaction effect, the coefficients for this effect are automatically included in the estimate by SAS. For example, the following code for a fixed effects analysis of the multi-centre hypertension data

PROC MIXED; CLASS centre treat;

MODEL dbp = dbp1 treat centre centre*treat/ DDFM=KR;

ESTIMATE ‘A−B’ treat 1 −1 0/ E CL;

would cause the following coefficients to be used by SAS for constructing the estimate.

The centre·treatment coefficients are equal to 1/n_i, where n_i is the number of centres at which treatment i is received (26 for treatment A and 25 for treatment B). Because treatments A and B are not received at every centre, this estimate is in fact ‘non-estimable’, and SAS gives the following output:

However, if the containing effect is fitted as random, SAS does not include any coefficients for this containing effect in the estimate. For example, the following code for a random effects analysis of the multi-centre hypertension data

PROC MIXED; CLASS centre treat;

MODEL dbp = dbp1 treat/DDFM=KR;

RANDOM centre centre*treat;

ESTIMATE ‘A−B’ treat 1 −1 0/ E CL;

would cause the following coefficients to be used by SAS for constructing the estimate:

The ESTIMATE statement can also be used to estimate shrunken random effects by specifying the coefficients of the random effects. For example, to estimate the shrunken difference between the first two treatments within the first two centres, the following ESTIMATE statements can be added to the previous code:

ESTIMATE ‘C1, A−B’ treat 1 −1 0| centre*treat 1 − 1 0/ E CL;

ESTIMATE ‘C2, A−B’ treat 1 −1 0| centre*treat

          0 0 0 1 − 1 0/ CL;

These cause the following coefficients to be used for the first estimate

and give the following estimates: