I have a question regarding the sign test when the individual measurements may be correlated. Let me start of with some background. Suppose we have 4 Organisms (a,b,c,d),and we make measurements in two separate ways, say A and B. Our data may look as follows
a = 3 for measurement A and 1 for measurement B
b = 4 for measurement A and 3 for measurement B
c = 0 for A and 4 for B
d = 2 for A and 0 for B
We now take the difference between A and B: $2,1,-4,2$. Looking at the signs we get the pattern $++-+$. We want to test if there is any difference between method A and method B. Take:
$H_0$(null hypothesis) = distribution for A is equal to the distribution for B
Under $H_0$ we would expect $\textrm{Pr}(A>B)=\textrm{Pr}(B>A)=.5$, therefore any pattern of $+$'s and $-$'s would be equally likely. i.e. $--+-$ is as likely to occur as $+-++$ etc. Let $U =$ number of $+$'s (in our case $U=3$). Assuming $H_0$ one can show that $\textrm{Pr}(U\ge3) = (1+4)/2^4 = 5/16=0.3125$.
Now, suppose the a and b are strongly positively correlated. Therefore not all combinations of $+$'s and $-$'s would be equally likely.For example one would not expect to have a > b for method A and a < b for method B. Therefore we would not expect sequences like $+-..$ or $-+..$ to occur. Taking this into account assuming $H_0$ it turns out that $\textrm{Pr}(U\ge3) = 3/8=0.375$, i.e. our p value increases.
Now I come to my question:
If instead of 4 organisms, I have say 100 organisms, and also suppose I have an upper bound on the number of correlations and the size of each correlation. Is there any way to construct an upper bound on the p value?
