Consider the following SAS data set as an introductory example:
data heights; input Family Gender$ Height @@; datalines; 1 F 67 1 F 66 1 F 64 1 M 71 1 M 72 2 F 63 2 F 63 2 F 67 2 M 69 2 M 68 2 M 70 3 F 63 3 M 64 4 F 67 4 F 66 4 M 67 4 M 67 4 M 69 run;
The response variable Height measures the heights (in inches) of 18 individuals. The individuals are classified according to Family and Gender . You can perform a traditional two-way analysis of variance of these data with the following PROC MIXED code:
proc mixed; class Family Gender; model Height = Gender Family Family*Gender; run;
The PROC MIXED statement invokes the procedure. The CLASS statement instructs PROC MIXED to consider both Family and Gender as classification variables. Dummy (indicator) variables are, as a result, created corresponding to all of the distinct levels of Family and Gender . For these data, Family has four levels and Gender has two levels.
The MODEL statement first specifies the response (dependent) variable Height .The explanatory (independent) variables are then listed after the equal (=) sign. Here, the two explanatory variables are Gender and Family , and they comprise the main effects of the design. The third explanatory term , Family * Gender , models an interaction between the two main effects.
PROC MIXED uses the dummy variables associated with Gender , Family ,and Family * Gender to construct the X matrix for the linear model. A column of 1s is also included as the first column of X to model a global intercept. There are no Z or G matrices for this model, and R is assumed to equal ƒ ^{ 2 } I , where I is an 18 — 18 identity matrix.
The RUN statement completes the specification. The coding is precisely the same as with the GLM procedure. However, much of the output from PROC MIXED is different from that produced by PROC GLM.
The following is the output from PROC MIXED.
The Mixed Procedure Model Information Data Set WORK.HEIGHTS Dependent Variable Height Covariance Structure Diagonal Estimation Method REML Residual Variance Method Profile Fixed Effects SE Method Model-Based Degrees of Freedom Method Residual
The 'Model Information' table describes the model, some of the variables that it involves, and the method used in fitting it. This table also lists the method (profile, factor, or fit) for handling the residual variance.
The Mixed Procedure Class Level Information Class Levels Values Family 4 1 2 3 4 Gender 2 F M
The 'Class Level Information' table lists the levels of all variables specified in the CLASS statement. You can check this table to make sure that the data are correct.
The Mixed Procedure Dimensions Covariance Parameters 1 Columns in X 15 Columns in Z 0 Subjects 1 Max Obs Per Subject 18
The 'Dimensions' table lists the sizes of relevant matrices. This table can be useful in determining CPU time and memory requirements.
The Mixed Procedure Number of Observations Number of Observations Read 18 Number of Observations Used 18 Number of Observations Not Used 0
The 'Number of Observations' table displays information about the sample size being processed .
The Mixed Procedure Covariance Parameter Estimates Cov Parm Estimate Residual 2.1000
The 'Covariance Parameter Estimates' table displays the estimate of ƒ ^{ 2 } for the model.
The Mixed Procedure Fit Statistics 2 Res Log Likelihood 41.6 AIC (smaller is better) 43.6 AICC (smaller is better) 44.1 BIC (smaller is better) 43.9
The 'Fit Statistics' table lists several pieces of information about the fitted mixed model, including values derived from the computed value of the restricted/residual likelihood.
The Mixed Procedure Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F Gender 1 10 17.63 0.0018 Family 3 10 5.90 0.0139 Family*Gender 3 10 2.89 0.0889
The 'Type 3 Tests of Fixed Effects' table displays significance tests for the three effects listed in the MODEL statement. The Type III F -statistics and p -values are the same as those produced by the GLM procedure. However, because PROC MIXED uses a likelihood-based estimation scheme, it does not directly compute or display sums of squares for this analysis.
The Type 3 test for Family * Gender effect is not significantatthe5%level,butthe tests for both main effects are significant.
The important assumptions behind this analysis are that the data are normally distributed and that they are independent with constant variance. For these data, the normality assumption is probably realistic since the data are observed heights. However, since the data occur in clusters (families), it is very likely that observations from the same family are statistically correlated, that is, not independent.
The methods implemented in PROC MIXED are still based on the assumption of normally distributed data, but you can drop the assumption of independence by modeling statistical correlation in a variety of ways. You can also model variances that are heterogeneous, that is, nonconstant.
For the height data, one of the simplest ways of modeling correlation is through the use of random effects. Here the family effect is assumed to be normally distributed with zero mean and some unknown variance. This is in contrast to the previous model in which the family effects are just constants, or fixed effects. Declaring Family as a random effect sets up a common correlation among all observations having the same level of Family .
Declaring Family * Gender as a random effect models an additional correlation between all observations that have the same level of both Family and Gender .One interpretation of this effect is that a female in a certain family exhibits more correlation with the other females in that family than with the other males, and likewise for a male. With the height data, this model seems reasonable.
The code to fit this correlation model in PROC MIXED is as follows :
proc mixed; class Family Gender; model Height = Gender; random Family Family*Gender; run;
Note that Family and Family * Gender are now listed in the RANDOM statement. The dummy variables associated with them are used to construct the Z matrix in the mixed model. The X matrix now consists of a column of 1s and the dummy variables for Gender .
The G matrix for this model is diagonal, and it contains the variance components for both Family and Family * Gender . The R matrix is still assumed to equal ƒ ^{ 2 } I , where I is an identity matrix.
The output from this analysis is as follows.
The Mixed Procedure Model Information Data Set WORK.HEIGHTS Dependent Variable Height Covariance Structure Variance Components Estimation Method REML Residual Variance Method Profile Fixed Effects SE Method Model-Based Degrees of Freedom Method Containment
The 'Model Information' table shows that the containment method is used to compute the degrees of freedom for this analysis. This is the default method when a RANDOM statement is used; see the description of the DDFM= option on page 2693 for more information.
The Mixed Procedure Class Level Information Class Levels Values Family 4 1 2 3 4 Gender 2 F M
The 'Class Levels Information' table is the same as before.
The Mixed Procedure Dimensions Covariance Parameters 3 Columns in X 3 Columns in Z 12 Subjects 1 Max Obs Per Subject 18 Number of Observations Number of Observations Read 18 Number of Observations Used 18 Number of Observations Not Used 0
The 'Dimensions' table displays the new sizes of the X and Z matrices.
The Mixed Procedure Iteration History Iteration Evaluations -2 Res Log Like Criterion 0 1 74.11074833 1 2 71.51614003 0.01441208 2 1 71.13845990 0.00412226 3 1 71.03613556 0.00058188 4 1 71.02281757 0.00001689 5 1 71.02245904 0.00000002 6 1 71.02245869 0.00000000 Convergence criteria met.
The 'Iteration History' table displays the results of the numerical optimization of the restricted/residual likelihood. Six iterations are required to achieve the default convergence criterion of 1E ˆ’ 8.
The Mixed Procedure Covariance Parameter Estimates Cov Parm Estimate Family 2.4010 Family*Gender 1.7657 Residual 2.1668
The 'Covariance Parameter Estimates' table displays the results of the REML fit. The Estimate column contains the estimates of the variance components for Family and Family * Gender , as well as the estimate of ƒ ^{ 2 } .
The Mixed Procedure Fit Statistics -2 Res Log Likelihood 71.0 AIC (smaller is better) 77.0 AICC (smaller is better) 79.0 BIC (smaller is better) 75.2
The 'Fit Statistics' table contains basic information about the REML fit.
The Mixed Procedure Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F Gender 1 3 7.95 0.0667
The 'Type 3 Tests of Fixed Effects' table contains a significance test for the lone fixed effect, Gender . Note that the associated p -value is not nearly as significant as in the previous analysis. This illustrates the importance of correctly modeling correlation in your data.
An additional benefit of the random effects analysis is that it enables you to make inferences about gender that apply to an entire population of families, whereas the inferences about gender from the analysis where Family and Family * Gender are fixed effects apply only to the particular families in the data set.
PROC MIXED thus offers you the ability to model correlation directly and to make inferences about fixed effects that apply to entire populations of random effects.