Heterogeneity in Hedges
Homogeneity Test
Q   wi ( zi  z )
2
When the null (homogeneous rho) is true, Q is
distributed as chi-square with (k-1) df, where k is
the number of studies. This is a test of whether
Random Effects Variance Component is zero.
REVC   
2
Q
Q  Wi (Yi  M )
 Yi  M
Q   
 Si
2



2
Chi-square ( 2) is the sum of squared z scores, i.e., k
scores drawn from the unit normal, squared, and
summed. Q is the deviation of the observed effect
size from the mean over the standard error, squared
and summed. The expected value of Chi-square is its
degrees of freedom.
Estimating the REVC
Q   wi ( zi  z )
2
Q  (k  1)
REVC  ˆ 
2
 wi   wi /  wi
2
z


Q  df
T 
C
2
If REVC estimate is less than zero, set to zero.
Random-Effects Weights
Inverse variance weights give weight to each study
depending on the uncertainty for the true value of
that study. For fixed-effects, there is only sampling
error. For random-effects, there is also uncertainty
about where in the distribution the study came from,
so 2 sources of error. The InV weight is, therefore:
1
w* 
VYi  T 2
I-squared
 Q  df
I  
 Q
2

(100)

2

 
2
(100)
I  
 Vtotal 
Conceptually, Isquared is the
proportion of total
variation due to ‘true’
differences between
studies. Proportion
due to random effects.
Comparison
Depnds on k
Q
X
P
X
Depends on Scale
T-squared
X
T
X
I-squared
Confidence intervals for tau
and tau-squared
CI for  2
See Borenstein et al., p 122.
Formulas are long and tedious.
Small numbers of studies and a large
population value of tau-square make
for broad confidence intervals.
Prediction or Credibility
Intervals
Bounds M *  tdf T 2  VM *
Makes sense if random effects.
M is the random effects mean (summary effect).
The value of t is from the t table with your alpha
and df equal to (k-2) where k is the number of
independent effect sizes (studies). The variance is
the squared standard error of the RE summary
effect.