Let T1,…, Tn be the observations in the treatment group and let C1,…, Cm be the observations in the control group. Assume that the data are independent samples from two normal distributions,
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(B.1) |
Let Bw
and Bo be the sample means of the treatment and control groups,
respectively. Then Bw and Bo are independently
normally distributed with means 
and 
and variances 
T2/n
and C2/m,
respectively. The variables
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(B.2) |
follow a bivariate normal
distribution, with means 
1
= –
and 2 = + ,
respectively, variance 2
= T2/n
+ C2/m
(for both X1 and X2), and correlation
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(B.3) |
Let W = X1/X2.
As the ratio 2/2
increases and the probability that X2 is negative tends to
zero (or if this is impossible for practical reasons, as in the current situation),
then the sampling distribution of W (the probability density function)
approaches (Hinkley 1969, 1970)
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(B.4) |
with
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(B.5) |
Furthermore, the variable
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(B.6) |
has a standard normal
distribution (Hinkley 1969). A Taylor’s development around
1/2
shows that
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(B.7) |
It therefore follows
that the distribution of W can be approximated by a normal distribution with
mean 1/2
and variance
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(B.8) |
Literature cited
Hinkley, D. V. 1969. On the ratio of two correlated normal random variables. Biometrika 56:635–639.
Hinkley, D. V. 1970. Correction. Biometrika 57:683.
