Linear Mixed Models Theory
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The HPLMIXED Procedure

Formulation of the Mixed Model

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 $\epsilon $ is too restrictive. The mixed model extends the general linear model by allowing a more flexible specification of the covariance matrix of $\epsilon $. In other words, it allows for both correlation and heterogeneous variances, although you still assume normality.

The mixed model is written as

\[ \mb {y} = \mb {X}\bbeta + \mb {Z}\bgamma +\bepsilon \]

where everything is the same as in the general linear model except for the addition of the known design matrix, $\mb {Z}$, and the vector of unknown random-effects parameters, $\bgamma $. The matrix $\mb {Z}$ can contain either continuous or dummy variables, just like $\mb {X}$. The name mixed model comes from the fact that the model contains both fixed-effects parameters, $\bbeta $, and random-effects parameters, $\bgamma $. 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 $\bgamma $ and $\bepsilon $ are normally distributed with

\begin{align*} \mr {E}\left[ \begin{array}{c} \bgamma \\ \bepsilon \end{array} \right] & = \left[\begin{array}{c} \Strong{0} \\ \Strong{0} \end{array} \right] \\ \mr {Var}\left[ \begin{array}{c} \bgamma \\ \bepsilon \end{array} \right] & = \left[\begin{array}{cc} \mb {G} & \Strong{0} \\ \Strong{0} & \mb {R} \end{array} \right] \end{align*}

Therefore, the variance of $\mb {y}$ is $\mb {V} = \mb {ZGZ}’ + \mb {R}$. You can model $\mb {V}$ by setting up the random-effects design matrix $\mb {Z}$ and by specifying covariance structures for $\mb {G}$ and $\mb {R}$.

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 $\mb {Z}$ containing dummy variables, $\mb {G}$ containing variance components in a diagonal structure, and $\mb {R} = \sigma ^2\mb {I} _ n$, where $\mb {I} _ n$ denotes the $n \times n$ identity matrix. The general linear model is a further special case with $\mb {Z} = \mb {0} $ and $\mb {R} = \sigma ^2\mb {I} _ n$.

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

Example: Growth Curve with Compound Symmetry

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

\[ \mb {X} = \left[ \begin{array}{rr} 1 & 1 \\ 1 & 2 \\ 1 & 3 \\ \vdots & \vdots \\ 1 & 1 \\ 1 & 2 \\ 1 & 3 \\ \end{array} \right] \]

The first column (coded entirely with $1$s) fits an intercept, and the second column (coded with series of $1,2,3$) fits a slope. Here, $n = 3s$ and $p = 2$.

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 $\mb {Z}$ and $\mb {G}$ matrices and let the $\mb {R}$ 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 $\mb {R}$ would then be

\[ \mb {R} = \left[ \begin{array}{ccccccc} \sigma ^2_1 + \sigma ^2 & \sigma ^2_1 & \sigma ^2_1 & & & & \\ \sigma ^2_1 & \sigma ^2_1 + \sigma ^2 & \sigma ^2_1 & & & & \\ \sigma ^2_1 & \sigma ^2_1 & \sigma ^2_1 + \sigma ^2 & & & & \\ & & & \ddots & & & \\ & & & & \sigma ^2_1 + \sigma ^2 & \sigma ^2_1 & \sigma ^2_1 \\ & & & & \sigma ^2_1 & \sigma ^2_1 + \sigma ^2 & \sigma ^2_1 \\ & & & & \sigma ^2_1 & \sigma ^2_1 & \sigma ^2_1 + \sigma ^2 \\ \end{array} \right] \]

where blanks denote zeros. There are $3s$ rows and columns altogether, and the common correlation is $\sigma ^2_1/(\sigma ^2_1 + \sigma ^2)$.

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 $\mb {R}$ matrix: TYPE=CS specifies the compound symmetry structure, and SUBJECT=INDIV specifies the blocks of $\mb {R}$, 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

\begin{align*} \mb {Z} & = \left[ \begin{array}{cccc} 1 & & & \\ 1 & & & \\ 1 & & & \\ & 1 & & \\ & 1 & & \\ & 1 & & \\ & & \ddots & \\ & & & 1 \\ & & & 1 \\ & & & 1 \\ \end{array} \right] \\ \mb {G} & = \left[ \begin{array}{cccc} \sigma ^2_1 & & & \\ & \sigma ^2_1 & & \\ & & \ddots & \\ & & & \sigma ^2_1 \\ \end{array} \right] \end{align*}

and $\mb {R} = \sigma ^2\mb {I} _ n$. The $\mb {Z}$ matrix has $3s$ rows and $s$ columns, and $\mb {G}$ is $s \times s$.

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.

Example: Split-Plot Design

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

\[ \bR = \sigma ^2 \mb {I} _{24} \]

and $\bX $, $\bZ $, and $\bG $ have the following form:

\[ \bX = \left[ \begin{array}{cccccccccccc} 1 & 1 & & & 1 & & 1 & & & & & \\ 1 & 1 & & & & 1 & & 1 & & & & \\ 1 & & 1 & & 1 & & & & 1 & & & \\ 1 & & 1 & & & 1 & & & & 1 & & \\ 1 & & & 1 & 1 & & & & & & 1 & \\ 1 & & & 1 & & 1 & & & & & & 1 \\ \vdots & & \vdots & & \vdots & & & & & \vdots \\ 1 & 1 & & & 1 & & 1 & & & & & \\ 1 & 1 & & & & 1 & & 1 & & & & \\ 1 & & 1 & & 1 & & & & 1 & & & \\ 1 & & 1 & & & 1 & & & & 1 & & \\ 1 & & & 1 & 1 & & & & & & 1 & \\ 1 & & & 1 & & 1 & & & & & & 1 \\ \end{array} \right] \\ \]
\[ \bZ \, = \left[ \begin{array}{cccccccccccccccc} 1 & & & & 1 & & & & & & & & & & & \\ 1 & & & & 1 & & & & & & & & & & & \\ 1 & & & & & 1 & & & & & & & & & & \\ 1 & & & & & 1 & & & & & & & & & & \\ 1 & & & & & & 1 & & & & & & & & & \\ 1 & & & & & & 1 & & & & & & & & & \\ & 1 & & & & & & 1 & & & & & & & & \\ & 1 & & & & & & 1 & & & & & & & & \\ & 1 & & & & & & & 1 & & & & & & & \\ & 1 & & & & & & & 1 & & & & & & & \\ & 1 & & & & & & & & 1 & & & & & & \\ & 1 & & & & & & & & 1 & & & & & & \\ & & 1 & & & & & & & & 1 & & & & & \\ & & 1 & & & & & & & & 1 & & & & & \\ & & 1 & & & & & & & & & 1 & & & & \\ & & 1 & & & & & & & & & 1 & & & & \\ & & 1 & & & & & & & & & & 1 & & & \\ & & 1 & & & & & & & & & & 1 & & & \\ & & & 1 & & & & & & & & & & 1 & & \\ & & & 1 & & & & & & & & & & 1 & & \\ & & & 1 & & & & & & & & & & & 1 & \\ & & & 1 & & & & & & & & & & & 1 & \\ & & & 1 & & & & & & & & & & & & 1 \\ & & & 1 & & & & & & & & & & & & 1 \\ \end{array} \right] \\ \]
\[ \bG = \left[ \begin{array}{ccccccccc} \sigma ^2_ B & & & & & & & \\ & \sigma ^2_ B & & & & & & \\ & & \sigma ^2_ B & & & & & \\ & & & \sigma ^2_ B & & & & \\ & & & & \sigma ^2_{AB} & & & \\ & & & & & \sigma ^2_{AB} & & \\ & & & & & & \ddots & \\ & & & & & & & \sigma ^2_{AB} \\ \end{array} \right] \]

where $\sigma ^2_ B$ is the variance component for Block and $\sigma ^2_{AB}$ is the variance component for A*Block. Changing the RANDOM statement as follows fits the same model, but with $\mb {Z}$ and $\mb {G}$ sorted differently: 

random int a / subject=block;
\begin{align*} \bZ & = \left[ \begin{array}{cccccccccccccccc} 1 & 1 & & & & & & & & & & & & & & \\ 1 & 1 & & & & & & & & & & & & & & \\ 1 & & 1 & & & & & & & & & & & & & \\ 1 & & 1 & & & & & & & & & & & & & \\ 1 & & & 1 & & & & & & & & & & & & \\ 1 & & & 1 & & & & & & & & & & & & \\ & & & & 1 & 1 & & & & & & & & & & \\ & & & & 1 & 1 & & & & & & & & & & \\ & & & & 1 & & 1 & & & & & & & & & \\ & & & & 1 & & 1 & & & & & & & & & \\ & & & & 1 & & & 1 & & & & & & & & \\ & & & & 1 & & & 1 & & & & & & & & \\ & & & & & & & & 1 & 1 & & & & & & \\ & & & & & & & & 1 & 1 & & & & & & \\ & & & & & & & & 1 & & 1 & & & & & \\ & & & & & & & & 1 & & 1 & & & & & \\ & & & & & & & & 1 & & & 1 & & & & \\ & & & & & & & & 1 & & & 1 & & & & \\ & & & & & & & & & & & & 1 & 1 & & \\ & & & & & & & & & & & & 1 & 1 & & \\ & & & & & & & & & & & & 1 & & 1 & \\ & & & & & & & & & & & & 1 & & 1 & \\ & & & & & & & & & & & & 1 & & & 1 \\ & & & & & & & & & & & & 1 & & & 1 \\ \end{array} \right] \\ \bG & = \left[ \begin{array}{ccccccccccccccc} \sigma ^2_ B & & & & & & & & \\ & \sigma ^2_{AB} & & & & & & & \\ & & \sigma ^2_{AB} & & & & & & \\ & & & \sigma ^2_{AB} & & & & & \\ & & & & \ddots & & & & & \\ & & & & & \sigma ^2_ B & & & \\ & & & & & & \sigma ^2_{AB} & & \\ & & & & & & & \sigma ^2_{AB} & \\ & & & & & & & & \sigma ^2_{AB} \\ \end{array} \right] \end{align*}
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