Multilevel Models. Other names for the same basic thing. hierarchical linear models multilevel models mixed-effects models mixed models variance-components models random-effects regression models random-coefficients regression models . Multilevel models. Common situations:
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…and traditional data analytic techniques are appropriateHierarchical Data Structures in a Group Randomized Trial
In many research studies, we start by drawing a sample of individuals…
and randomly assign them to either treatment or control
However, we are not always able to separate people from their contexts.
Students learn in schools
Children grow up in neighborhoods
Patients are treated in hospitals
When the cluster is a necessary part of a research design, the resultant data will be nested, or hierarchically structured.Hierarchical Data Structures in a Group Randomized Trial
Different groups are allocated to each condition.
The units of observation are members of the groups.
The number of groups allocated to each condition is usually limited.Characteristics of Hierarchical Data Structures in a Group Randomized Trial
Source: I made this up.
Are the points evenly scattered around the line???
The points are closer to the regression line if you have a separate regression line for each school.
Y = β0 + β1 X + ε
β0j = ψ00 + μ0j
β1j = ψ10 + μ1j
Type I errors
(Conclude that an effect is significant when it’s really not)
σ2g(variance due to the group)
σ2m + σ2g (total variance: member + group)
English: The proportion of the total variance that’s due to the grouping variable
PROC MIXED METHOD = ML COVTEST ;
CLASS school ;
MODEL dv= / SOLUTION ;
RANDOM INT / TYPE=UN SUBJECT=school ;
Cov Parm Subject Estimate Std Error Z Pr > |Z|
UN(1,1) School 129.19 25.48 5.07 0.0001
Residual 321.56 10.63 30.25 0.0001
Variance component for school is 129.19
Variance component left over after variance due to school has been explained is 321.56.
ICC = variance due to clustering variable /
(variance due to clustering variable + variance remaining)
129 / (129 + 321) = .29
Real power of cluster randomized trials according to discrepancy between a priori postulated and a posteriori estimated intraclass correlation coefficients
g=number of clusters
M=average cluster size
N=total number of subjects
Source: Guittet, L., Giraudeau, B., & Ravaud, P. (2005). A priori postulated and real power in cluster randomized trials: mind the gap. BMC Medical Research Methodology 2005, 5:25
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 12.53389 12.53389 30.12 <.0001
Error 1902 791.60096 0.41619
Corrected Total 1903 804.13485
Root MSE 0.64513 R-Square 0.0156
Dependent Mean 0.20623 Adj R-Sq 0.0151
Coeff Var 312.82168
Parameter Standard Standardized
Variable DF Estimate Error t Value Pr > |t| Estimate
Intercept 1 0.21036 0.01480 14.21 <.0001 0
income1000 1 -0.08135 0.01482 -5.49 <.0001 -0.12485
random intercept/sub=sch3 solution;
Covariance Parameter Estimates within schools.
Cov Parm Subject Estimate
Intercept sch3 0.003753
(If you had run the unconditional means model, this is where you would get the numbers to calculate the ICC.)
procmixed method=ml covtest;
model smkscale3= /solution;
random intercept/type=un sub=sch3;
Covariance Parameter Estimates
Cov Parm Subject Estimate Error Value Pr Z
UN(1,1) sch3 0.0078 0.0037 2.11 0.0174
Residual 0.4276 0.0132 32.52 <.0001
ICC= .0078 / (.0078+.4276) = .0179
David Murray was right—it is around .02!
Solution for Fixed Effects within schools.
Effect Estimate Error DF t Value Pr > |t|
Intercept 0.1891 0.02277 23 8.30 <.0001
income1000 -0.05732 0.02129 1879 -2.69 0.0072
Type 3 Tests of Fixed Effects
Effect DF DF F Value Pr > F
income1000 1 1879 7.25 0.0072
OLS model would have overestimated the effect of income on smoking. Part of the effect is due to the fact that low-income schools have high smoking, and high-income schools have low smoking.