Variance of a mean of correlated measures.
Consider the mean of four measures (j=1,2,3,4) within group i, xi1, xi2, xi3, xi4, mutually
correlated ICC=r because they share a common environment,
and the variance of that mean.
mean xi. = (xi1 + xi2 + xi3 + xi4)/4
var(xi.) =
var([xi1 + xi2 + xi3 + xi4]/4)
= (1/16)* var(xi1 + xi2 + xi3 + xi4)
= (1/16)*{var(xi1)+var(xi2)+var(xi3)+var(xi4)
+cov(xi1,xi2) + cov(xi1,xi3)
cov(xi2,xi1)
+ cov(xi2,xi3)
cov(xi3,xi1)
+cov(xi3,xi2)
cov(xi4,xi1)
+cov(xi4,xi2) + cov(xi4,xi3)
and assuming homoscedasticity of variances and covariances
= (1/16)*{4xvar(X) + (42-4)xcov(Xij,Xik) }
= var(X)/4 * { 1 + (4-1) xcov(Xij,Xik)/var(X) }
= var(X)/4 * { 1 + (4-1) x ICC }
or more generally, for m members in the group,
var(xi.) = var(X)/4 * { 1 + (m-1) x ICC}
+ cov(xi1,xi4)
+ cov(xi2,xi4)
+ cov(xi3,xi4)
}
Common model (compound symmetry, exchangeability)
xij
= ui + eij
xik
= ui + eik where ui~N(0,τ2) and eij, eik ~ indepN(0,σ2res)
so var(X)= (σ2res + τ2), cov(xij,xik) = τ2, and ICC = τ2/(σ2res + τ2),
DEFF = 1+(m-1)ICC for Design Effect
From the analysis of variance perspective, the natural measure of homogeneity is the
variance component ratio, VCR = τ2/σ2res. Some algebra shows
var(xi.) = σ2res/4 * { 1 + m x VCR)
RVIF = 1 + m VCR
for Residual Variance Inflation Factor.
In the analysis of the school MATH data the MIXED model analysis of 311 students in
20 schools (mean number students=15.6) gave estimates of the components of variance
school(cond)
262.67
residual
2150.33 with the ration (VCR) = 0.1222
and so Total variance 2413.00
ICC = 0.1222/(1+0.1222) = 0.1089
DEFF = 1 + 14.6*0.1089 = 2.590
inflating total variance 2413
so variance of the mean is
Var(Xavg) =2413*2.59/311 = 20.0950
RVIF= 1+15.6*0.1222 = 2.906
inflating residual variance 2150.33
Var(Xavg) = 2150.33*2.906/311 = 20.0950