Two-level models for repeated measures on persons can readily be adapted t o incorporate the clustering of persons within social settings. Bryk and
"Note also that exposure is essentially continuous. Thus, the HLM results cannot be duplicated by construction of a finite set of covariance matrices.
0.035 0.011 0.014 0.015 0.014 0.035 0.018 0.016 0.016 sion
Model 1 : Model 2:
Complete Data have Dispersion Depending
Unstructured but o n Exposure
Homogeneous Dispersion to Deviant Peers
Faxed Effects Coeff SE Ratio Coeff SE Ratio
0.054 0.034 0.028 0.062 0.042
A = I 0.062 -
Intercept, BOO 0.3252 0.0127 25.67 0.3251 0.0125 25.86
Linear, PIO 0.0487 0.0045 10.74 0.0466 0.0047 10.00
Quadratic, ,I320 -0.0006 0.0030 -0.21 0.0006 0.0030 0.21
Exposure, p30 0.3186 0.0244 13.07 0.3430 0.0295 11.62
Variance-covariance Component
T =
0.0236 0.0034 -0.0016 0.0072 0.0021 0.0000 -0.0029
0.0038 0.0000 0.0457
2 = 0.0210
i?
Deviance -517.26 -520.63 E-
19 15 E-
Model Fit c
cr Df
A1 t erna tive Covariance Structures 47
Table 2.7
Robustness of Model-Based Standard Errors for
Y z j = POO + P10aij + P z o ~ : ~ + E i j
HLM with Robust
Eij = uoj + u1jaij + EZj
homogeneous level-1 variance
Standard Errors
Parameter Coeff S E t S E t
Mean, Po0 0.3272 0.0153 21.38 0.0153 23.38 Linear, Pl0 0.0647 0.0049 13.14 0.0249 13.14 Quadratic, PZO 0.0002 0.0032 0.05 0.0032 0.05
Raudenbush (1988) studied school differences in children’s growth by adding a third level t o the standard two-level model for individual change. The first level thus represented individual change over time, the second level represented individual differences in change within schools, and the third level represented variation between schools.
Given time-structured data, the three-level HLMs are submodels of a two-level general multivariate linear model (cf. Thum, 1997). Thus, by adding a level t o Jennrich and Schlucter’s (1986) multivariate model, we can estimate a range of covariance structures within a model that incorporates clustering.
We shall reanalyze the Sustaining Effects Data earlier analyzed by (Bryk
& Raudenbush, 1988). Mathematics achievement, measured on a vertically equated scale t o reflect growth, is the outcome. Children were observed at spring of kindergarten and twice annually during first and second grades.
Although the aim was t o obtain five repeated measures, some children were absent at various testing times. Thus, the data were time-structured in design but incomplete in practice. The 618 participants were nested within 86 schools.
Three-Level HLM Model
Bryk and Raudenbush (1988) formulated a three-level model, where, at level 1, the mathematics outcome for student j in school Ic was represented as depending linearly on time plus the effect of a time-varying covariate:
Here time = 0,1,2,3,4 at times 1,2,3,4,5. Thus, 7rojk represents the initial status of student j in school k . “summer” takes on a value of 1 if the previous time was summer and 0 if not. It can readily be shown, then, t ha t
7 r 1 j k is the calendar year growth rate for student J’ in school k ; T l j k + T 2 j k
is the summer growth rate and nljk - 7 r 2 j k is the academic year growth rate. Thus, three 7r’s capture the growth of each student as a function of an initial status, a calendar year growth rate and a summer effect.
At level-2, the 7r’s become outcome variables in a model that explains variation between students within schools. For example, we might have
n p j k = P p O k f P p l k ( c h i l d p0v)jk + u p j k (2.35)
Here (child pov)jk is an indicator taking on a value of 1 if child j k is in poverty (as indicated by eligibility for a free lunch) and 0 if not.
The variances and covariances among the u p j k are collected in matrix T.
Combining these two models, we have
The model implies, then, t h at , within a school, math achievement at a given time depends upon the “time,” “summer,” and “child poverty,”
two-way interactions of child poverty with “time” and “summer,” plus a random error:
The ”standard” three-level HLM can test alternative covariance struc- The level-3 model accounts for variation between schools. For example, tures by setting one or two of the u’s in Equation 2.35 t o zero.
we might simply estimate
book = Yo00 + WOOk
P l O k = 7 1 0 0 + W l O k
P 2 0 k = 7 2 0 0 + W 2 0 k P O P k = Y O p 0 , P > 0
(2.38)
Here the variances and covariances of the u p 0 k are collected in a matrix R. In subsequent analyses, the random effect of summer is dropped from Equations 2.33 and 2.36.
A1 t erna tive Covariance Structures 49
Reformulation as a Hierarchical Multivariate Model with Incomplete Data
Level-1 Model
The first level of the model again represents the relationship between the observed and complete data. Schools are incorporated simply by adding a subscript t o Equation 2.13:
T=5
(2.39) Thus x j k is the math achievement score on the ith occasion (i = 1, ..., njk) for student j in school k and q;k is the score that would have been observed if student j within school k had been present a t time t (t = 1, ... 5 ) . The indicator rntijk takes on a value of 1 if occasion i corresponds to time t.
Level-2 Model
The level-2 model is a multivariate, within-school model for the complete data:
q * j k = POOk + PlOk(time)tjk + P20k(summer)tjk
+POlk(childpov)jk + Pllk(childpov)jk * (time)ijk (2.40) +Pzlk(child p 0 v ) j k * (SUmmer)ijk -k E t j k
This level-2 model has the same form as the level-2 model of HLM, but now the residuals may have an arbitrary variance-covariance matrix, A , composed of elements
Alternatively, the covariance matrix A may be structured as in the two- level case. The level-3 model has the same form as Equation 2.38.
Results
Table 2.8 compares results from three models. The first two models are three- level hierarchical models with randomly varying intercepts and an- nual growth rates a t level 2 (between children within schools) and level 3 (between schools). They differ in that the first model assumes homogenous level-1 variance whereas the second model allows a separate level-1 vari- ance for each time point. This second model fits better than the first, x 2 =
29998.90 - 29966.80 = 32.10, df = 4, p = ,000. However, neither fits as well as the model with unrestricted variances and covariances a t level 2. De- spite the poorer fit of the simpler models, inferences about the fixed effects
are remarkably similar. Of most importance in the original analysis was the extent of variation between schools in the annual rates of growth. T he estimate of 14.78 based on the model with unrestricted level-:! covariance structure is similar t o t ha t based on the hierarchical model with heteroge- neous variances, 14.33, and a bit smaller than the estimate based on the hierarchical model with homogenous level-1 variance. As in the two-level case, it is possible and often useful t o estimate key parameters under a variety of plausible alternative specifications as a sensitivity analysis. It will also be useful and straightforward t o compare standard errors for fixed effects based on robust estimation, especially when the number of level-3 units is reasonably large.