Variances of combinations of estimates, a and b
(based upon the application of Taylor Series expansions).
For estimates a and b with variances V(a) and V(b) respectively and, if not independent, a non-zero covariance, Cov(a,b) we have
V(a+b) = V(a) + V(b) + 2 Cov(a,b) = V(a) + V(b) + 2 rho(a,b) sd(a) sd(b)
V(a-b) = V(a) + V(b) - 2 Cov(a,b) = V(a) + V(b) - 2 rho(a,b) sd(a) sd(b)
where rho(a,b) is the correlation between a and b. This is the variance which is used in a paired t-test (Howell, 1997 p.189-190).
Application to a paired t-test
Since for n subjects with each with a paired observation of form (ai,bi) for the i-th subject
mean(a) - mean(b) = (a1 + a2 + ... an)/n - (b1 + b2 + ... + bn)/n
= ((a1 - b1) + (a2 - b2) + ... + (an - bn)) / n
= mean(a-b)
It follows from the two relationships for means and variances of differences shown above that a paired t-test on the differences between two paired observations such as time points on individuals is equivalent to a one sample t-test on the differences. Both have n-1 degrees of freedom.
V(a/b) = [ $$ \mbox{a}2 V(b) - 2ab Cov(a,b) + b2 V(a) ] / b4 $$
V(ab) = $$ \mbox{b}2 V(a) + a2 V(b) + 2ab Cov(a,b)