Nonparametric sign test for correlated variables I have a question regarding the sign test when the individual measurements may be correlated. Let me start off 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?

 A: Under one interpretation of your situation there is no need to modify the p values at all.
For example, let's posit that a sequence of (unknown) bivariate distributions $p_i(x,y)$ govern $A$ and $B$ for each organism $i$.  That is, $\Pr(A=x, B=y) = p_i(x,y)$ for all possible outcomes $(x,y)$ of $(A,B)$.  To test whether the measurement procedures $A$ and $B$ differ, a reasonable null hypothesis is that these distributions are all symmetric:
$$H_0: p_i(x,y) = p_i(y,x) \text{ for all } i, x, y.$$
The sign statistic (difference between number of $+$ and number of $-$ results) is still a reasonable one to use in this test.  (It actually tests the null hypothesis $H_0: \Pr(A<B) = \Pr(B<A)$.)  Its distribution depends on the chances of ties; namely on the values $t_i = \sum_{x}p_i(x,x)$ (one for each organism $i$).  The question, which appears not to contemplate the possibility of ties at all, suggests their chances are fairly small.  In any case, the symmetry assumption in the null implies the chance of organism $i$ yielding a $+$ sign equals the chance of organism $i$ yielding a $-$ sign and the assumption that ties are unlikely implies both these chances are close to $1/2$.  This implies the distribution of the sign statistic is binomial, as usual, despite any correlation (or lack thereof) between $A$ and $B$.
If there is a substantial chance of ties, it looks like you cannot make any progress towards quantitative bounds until you specify something about those chances.  For example, if you provide an upper bound for the $t_i$ you can say something about the distribution of the sign statistic.
