FAQ/FVars/Fab162013-04-08 16:08:43PeterWatson152013-04-08 16:07:32PeterWatson142013-03-08 10:17:32localhostconverted to 1.6 markup132010-08-19 14:09:27PeterWatson122010-08-19 14:08:35PeterWatson112010-08-19 14:05:50PeterWatson102010-08-19 14:05:42PeterWatson92010-08-19 14:00:45PeterWatson82007-08-20 11:34:11PeterWatson72007-08-20 11:33:55PeterWatson62007-08-20 11:33:18PeterWatson52007-08-20 11:32:29PeterWatson42007-08-20 11:32:10PeterWatson32006-08-22 15:55:39PeterWatson22006-08-22 15:54:10PeterWatson12006-08-22 15:53:55PeterWatsonVariances 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)