The previous general linear model is certainly a useful one (Searle, 1971), and it is the one fitted by the GLM procedure. However, many times the distributional assumption about is too restrictive. The mixed model extends the general linear model by allowing a more flexible specification of the covariance matrix of . In other words, it allows for both correlation and heterogeneous variances, although you still assume normality.

where everything is the same as in the general linear model except for the addition of the known design matrix, , and the vector of unknown random-effects parameters, . The matrix can contain either continuous or dummy variables, just like . The name *mixed model* comes from the fact that the model contains both fixed-effects parameters, , and random-effects parameters, . See Henderson (1990) and Searle, Casella, and McCulloch (1992) for historical developments of the mixed model.

A key assumption in the foregoing analysis is that and are normally distributed with

Therefore, the variance of is . You can model by setting up the random-effects design matrix and by specifying covariance structures for and .

Note that this is a general specification of the mixed model, in contrast to many texts and articles that discuss only simple random effects. Simple random effects are a special case of the general specification with containing dummy variables, containing variance components in a diagonal structure, and , where denotes the identity matrix. The general linear model is a further special case with and .

The following two examples illustrate the most common formulations of the general linear mixed model.

Suppose that you have three growth curve measurements for s individuals and that you want to fit an overall linear trend in time. Your matrix is as follows:

The first column (coded entirely with 1s) fits an intercept, and the second column (coded with series of ) fits a slope. Here, and .

Suppose further that you want to introduce a common correlation among the observations from a single individual, with correlation
being the same for all individuals. One way of setting this up in the general mixed model is to eliminate the and matrices and let the matrix be block-diagonal with blocks corresponding to the individuals and with each block having the *compound-symmetry* structure. This structure has two unknown parameters, one modeling a common covariance and the other modeling a residual
variance. The form for would then be

where blanks denote zeros. There are rows and columns altogether, and the common correlation is .

The following PROC HPLMIXED statements fit this model:

proc hplmixed; class indiv; model y = time; repeated morder/ type=cs subject=indiv; run;

Here, INDIV is a classification variable that indexes individuals. The MODEL statement fits a straight line for TIME ; the intercept is fit by default just as in PROC GLM. The REPEATED statement models the matrix: TYPE=CS specifies the compound symmetry structure, and SUBJECT= INDIV specifies the blocks of , and MORDER is the repeated effect that records the order of the measurements for each individual.

An alternative way of specifying the common intra-individual correlation is to let

and . The matrix has rows and s columns, and is .

You can set up this model in PROC HPLMIXED in two different but equivalent ways:

proc hplmixed; class indiv; model y = time; random indiv; run; proc hplmixed; class indiv; model y = time; random intercept / subject=indiv; run;

Both of these specifications fit the same model as the previous one that used the REPEATED statement. However, the RANDOM specifications constrain the correlation to be positive, whereas the REPEATED specification leaves the correlation unconstrained.

The split-plot design involves two experimental treatment factors, `A`

and `B`

, and two different sizes of experimental units to which they are applied (Winer, 1971; Snedecor and Cochran, 1980; Milliken and Johnson, 1992; Steel, Torrie, and Dickey, 1997). The levels of `A`

are randomly assigned to the larger-sized experimental units, called *whole plots*, whereas the levels of `B`

are assigned to the smaller-sized experimental units, the *subplots*. The subplots are assumed to be nested within the whole plots, so that a whole plot consists of a cluster of subplots and
a level of `A`

is applied to the entire cluster.

Such an arrangement is often necessary by nature of the experiment; the classical example is the application of fertilizer
to large plots of land and different crop varieties planted in subdivisions of the large plots. For this example, fertilizer
is the whole-plot factor `A`

and variety is the subplot factor `B`

.

The first example is a split-plot design for which the whole plots are arranged in a randomized block design. The appropriate PROC HPLMIXED statements are as follows:

proc hplmixed; class a b block; model y = a b a*b; random block a*block; run;

Here

and , , and have the following form:

where is the variance component for `Block`

and is the variance component for `A`

*`Block`

. Changing the RANDOM
statement as follows fits the same model, but with and sorted differently:

random int a / subject=block;