We denote Bw
and 
w the
mean and standard deviation of plant biomass in the treatment group, and Bo
and o the
mean and standard deviation of plant biomass in the control group, based on
n and m plants, respectively. The sample index, RII = (Bw – Bo)/(Bw + Bo),
is an estimate of the population index. RII is approximately normally distributed
(Appendix B), with mean equal to the true population
index and variance, 
, approximately
equal to
|
|
(A.1) |
where
|
|
(A.2) |
It is often the
case that the variance in the control and treatment groups are approximately
equal and the experimental design is balanced (n = m). Then 
= 0 and the variance of RII is
|
|
(A.3) |
It is worth noticing
that the variance of the RII index can be substantially smaller than the variance
of the lnRR index. Thus, when the experimental effect is small and Bw
= Bo + 
, to
the first order in the variance
of the RII index is
|
|
(A.4) |
while, from Hedges et al. (1999), the variance of the lnRR index would be
|
|
(A.5) |
The 100ˇ(1 – 
)%
confidence interval for the RII index, r, is approximately given by
|
|
(A.6) |
where 
is the 100ˇ(1 – )%
point of the standard normal distribution. While RII has a small-sample bias
and the sampling distribution of RII is somewhat skewed (also suggested for
the sampling distribution of lnRR, Hedges et al. [1999]),
both bias and skew disappear rapidly as (w2/n + o2/m)0.5/(Bw + Bo)
decreases, and the approximation is excellent when (w2/n + o2/m)0.5/(Bw + Bo)
is smaller than 0.1. For values of RII (-0.5 < RII < 0.5), a very good
approximation is obtained when (w2/n + o2/m)0.5/(Bw + Bo)
< 0.2. Hedges et al. (1999) showed that lnRR achieved
a distribution close to normal for variance values greater than these. However,
as vlnRR 
4
vRII, the sampling distribution of lnRR
and RII behave approximately the same.
Literature cited
Hedges, L. V., J. Gurevitch, and P. S. Curtis. 1999. The meta-analysis of response ratios in experimental ecology. Ecology 80:1150–1156.
