To complement this two-stage conceptual paradigm, a two-level multivariate hierarchical linear model (MHLM) is developed. In this model, performance parameters of individual functioning are assumed to be drawn from a population model of human variation in performance. Although this model for studying behavior maintains a clear analytic separation between the behavioral fluctuations of individuals and the variation of individual performance in the population, it provides a description of the observable features of individual functioning within an integrated individual difference framework. For the experimenter, the MHLM approach offers a formal mechanism for generalizing over subjects (aggregating individual models). For the correlationist, it retains the requisite focus on within-subject processes. But what is important to both is that this approach meets headon the major interpretative and estimation problems associated with either the aggregation of observations or the aggregation of results.
With repeated measurement, a dataset in the behavioral sciences
typically displays a two-level nested structure. We consider the fairly
general situation in which a set of k related outcome measures,
,
is observed for each of 
replications for individual j,
( 
and 
). In particular, we pose the
multivariate linear response model for each individual as follows:
(1) 
is the 
matrix of responses,
thus 
is 
. Corresponding to each observed
outcome vector is a matrix of outcome-varying covariates which may be
partitioned into a column subset of predictors, 
, with
fixed coefficients 
( 
), and a complementary
column subset of predictors, 
, with random coefficients
( 
). This Stage 1 model relates the
observed outcome matrix to certain features of individual performance. One
useful interpretation of the distinction between
and is that represents
the degree of regression on aspects of performance which individuals seem
to share whereas represents the extent of
regression for those aspects on which individuals differ. To complete the
model, we assume that
where 
is a 
nonsingular error
variance-covariance matrix.
To assess how individual performance relates to particular
between-subject treatment manipulations, or context variables, 
,
we consider an individual difference model which is
for the large-sample normal case at Stage 2. Here,
is a 
vector fixed regression
parameters and 
is a 
nonsingular matrix of residual
parameter variance.
It is fair to say that, in recent years, univariate versions of this two-stage model, estimated under large sample assumptions, have attained near-paradigmatic proportions in quantitative educational research (Bock, 1989; DeLeeuw & Kreft, 1986; Goldstein, 1987; Longford, 1987; Rogosa & Willett, 1985a; Raudenbush, 1988; Raudenbush & Bryk, 1986). But because behavior is most often multivariate - both in its conceptualization and in the way it is reported, multi-level analyses employing these models do not take proper advantage of the richly textured information about the behavior in a formal way. For example, a teacher's attitudes towards school reform proposals and his efforts in this connection are obviously related. Yet, reports most frequently analyse each aspect of the teacher's attitudes and his efforts separately, a practice that may conceal informative patterns in the data from the researcher's view. Put differently, studying a single criterion separately is unsatisfactory because it ignores important relationships among outcomes.
Behavioral experiments also involve far fewer subjects than is
typical in, say, surveys. When m is small, the MHLM replaces the prior
in equation (3) with a multivariate t density, with
degrees of freedom:
The last equation reveals another key feature of the MHLM, which is that it deals with multivariate data with modest to large sample sizes. This important member of the MHLM is defined within the same general analytic framework. We compare this simple variant with a more fully Bayesian approach implemented by Gibbs sampling (e.g., Seltzer, 1993).
