Section 2: An Introduction to the features of the MIXED procedure      

Most of pages found in this complete document will focus on the 
capabilities of SAS to analyze data with mixed effects linear models with 
occasional comments on the ability of SPSS to work with them as well. 
Other programs, such as Stata and S-Plus, also compute these models quite 
well, but will not be mentioned here.

The procedure MIXED was introduced by SAS in the early 1990s; "Mixed 
Models" were added to Version 12 of SPSS in the Advanced Regression 
module. Whereas the primary task of modeling data with the general linear 
model (e.g., PROC GLM with SAS) is to fit a linear ANOVA model with fixed 
effects, the mixed model can do this, plus broaden the choice of 
possibilities whenever you have two or more random effects (i.e., variance 
components).

The name of the procedure itself, MIXED, is derived from its enhanced 
ability to work with statistical designs that contain both fixed effects 
and random effects.  PROC MIXED fits linear models with the response 
variable modeled directly by linear combinations of fixed and random 
effects with the model of the form:

  y = X*B + Z*u + e

where

  y is a vector of observations
  B is a vector of unknown fixed effects
  X is a known design matrix that relates observations to fixed effects
  u is a vector of unknown random effects
  Z is a matrix that relates observations to random effects
  e is a vector of residuals

With this model the expected value of the response y is:

  E(y) = XB

Let G: the variance-covariance matrix for the random effects u
    R: the variance covariance matrix of the random residuals e

       so that u ~ N(0,G) and e ~ N(0,R)

The matrix G can be conceptualized as a between-subject 
variance/covariance matrix; R is a within-subject variance/covariance 
matrix. Since u and e are also assumed to be independent, the 
variance/covariance matrix of the vector of observations y from each 
subject has the form:

   VAR(y) = V = ZGZ' + R

Understanding the V matrix is a very important component of working with 
mixed models since it contains both sources of random variation and 
defines how these models differ from computations with Ordinary Least 
Squares (OLS). If you only have random effects models (such as a 
randomized block design) the G matrix is the primary focus. For 
within-subject effects, such as repeated measures with data collected over 
several within-subject treatment conditions, the R matrix is relevant. 
Both matrices are important components of the discussion on these pages 
and in many situations only one of them is needed; however, some data 
analysis situations require both matrices. If the G and R matrices are 
known, generalized least squares can estimate any linear combination of 
the fixed effects B. However, as usually is the case these matrices are 
not known, so a complex iterative algorithm for fitting linear models is 
invoked.


Benefits of Mixed Linear Models

One important example of the many benefits available to modelling data 
with this approach is that repeated measures data can be analyzed with a 
wider variety of correlation structures than available with PROC GLM. It 
also computes several tests of significance correctly that PROC GLM needed 
to be instructed how to compute with TEST statements, such as those 
required for split-plot designs.

There are a few data analysis situations where MIXED linear models and 
GLMs will provide identical results (e.g., fixed effects ANOVA with only 
one random component, the residual variance) and these designs are good 
starting points to demonstrate their similarities. However, it is the 
OTHER situations where you should consider the many features of PROC MIXED 
to add flexibility to data analysis. A mixed model provides you with more 
appropriate ways to work with clustered data, i.e., data that do not meet 
the three important assumptions for fixed effects ANOVA. Since they can 
handle multiple sources of variation better than GLM, mixed models have 
more flexibility to make broader and more appropriate inferences about 
data. If you have a randomized block design or any design that includes 
two or more random effects -- including repeated measures designs -- then 
you should consider learning about the versatility of MIXED models.

MIXED linear models are an extremely powerful tool for many types of 
statistical analyses involving linear models for continuous data because 
it has features built into it not previously available is other 
procedures. PROCs TTEST, ANOVA, and GLM of SAS (and the analogous 
procedures from other programs) were written in the early years of 
statistical computing when software and hardware technology was limited to 
the testing hypotheses of linear models with fixed effects. Computations 
with random effects are more feasible today with our high speed computers 
(even desktop and laptop models) and sophisticated software. In addition 
to the same ability to handle basic fixed effects analysis of variance 
problems, mixed modeling techniques is now the preferred approach for 
virtually all clustered or repeated measures data.

With clustered data (such as multiple measurements collected from each 
subject), GLM and MIXED will not necessarily provide the same results. The 
default procedure with MIXED estimates variance components through 
Restricted-Maximum Likelihood (REML) while PROC GLM employs moment 
estimates. For this reason, PROC MIXED is much more computationally 
intensive than GLM, which implies certain models may take a longer time to 
fit. It takes a different approach to statistical analysis to obtain 
estimates and confidence intervals for test statistics.

Despite the limitations PROC GLM reveals when compared to PROC MIXED, it 
should continue to find basic applications of linear models. For example, 
PROC GLM still is the choice to compute multivariate tests, such as Wilk's 
lambda, evaluate the estimability of linear combinations of factors (the E 
options), and for its wide range of post-hoc multiple comparison tests for 
independent groups ANOVA. However, you will quickly discover in the 
following pages (and many resources already based on PROC MIXED) that 
whatever PROC GLM can do, the MIXED model approach will mostly likely work 
better and offer more choices for fitting mixed linear models.

Since output from the general linear model for SPSS closely resembles PROC 
GLM of SAS, comparisons in this article will primarily be between the GLM 
and MIXED procedures of SAS. However, many of these same examples could be 
applied to the GLM and MIXED procedures from SPSS and other programs.

In closing this section, you need to consider applying PROC MIXED whenever 
you have:

* random effects
* any type of repeated measures
* covariance matrices which are not simple diagonal matrices.
* unbalanced designs
* split-plot designs (or any other sort of design where the
  experimental units for some factors are different from the experimental
  units for other factors)
* standard errors which are complex functions of model errors
* random coefficient models

And even if you don't know whether any of these situations describe your 
data analysis problem then you still ought to run PROC MIXED.

PROC MIXED is also a helpful study planning tool for complex designs 
(i.e., power analysis for a given effect size and a fixed number of 
subjects) as will be demonstrated in a subsequent section.


Where to Learn More?

Books and papers abound on running mixed linear models. Perhaps the best 
of them all is "The SAS System for Mixed Models", 2nd Ed. by Little, et. 
al. (2006) which provides many examples that an analyst is likely to 
encounter in real life.  Commands that fit appropriate models are 
presented and explained in detail. Everything is well organized, clear, 
and concise. For the mix of theory, examples, and code, you will not find 
a better treatment anywhere. It costs around $90 for 800+ pages of 
material and is well worth the price for a volume that provides much of 
the information that you would ever need to know about mixed linear 
models, GLIMMIX, and NLMIXED in one easily accessible source.

SUGI papers are a great source for some information; however, they do not 
provide a coherent, unified treatment on any specific subject, mixed 
models included. The same goes for SAS-L archives. One can easily search 
for and find a great deal of information on many topics from the SAS-L 
archives, but none of them are organized in a way that will contribute to 
an effective and comprehensive learning experience.