Solved – Paired t-test and correlation

The coefficient of correlation and paired t-test are getting at different things. The two tests don't need to align in terms of statistical significance. Consider the following four scenarios, coded in R.

# same mean, no correlation # t.test Not significant # cor.test Not significant options(scipen = 99) set.seed(1) s1 <- rnorm(20,0,1) s2 <- rnorm(20,0,1) t.test(s1,s2,paired=T)$p.value     cor.test(s1,s2)$p.value  # different means, no correlation # t.test Significant # cor.test Not significant set.seed(2) s1 <- rnorm(20,0,1) s2 <- rnorm(20,2,1) t.test(s1,s2,paired=T)$p.value     cor.test(s1,s2)$p.value  # different means, high correlation # t.test Significant # cor.test Significant set.seed(3) s1 <- rnorm(20) s2 <- s1+2+rnorm(20,0,0.5) t.test(s1,s2,paired=T)$p.value     cor.test(s1,s2)$p.value  # same means, high correlation # t.test Not significant # cor.test Significant set.seed(4) s1 <- rnorm(20) s2 <- s1+rnorm(20,0,0.5) t.test(s1,s2,paired=T)$p.value     cor.test(s1,s2)$p.value 

Not seeing a significant correlation between your two tests may be a sign the measurement error of your tests is high for your context. You want the standard deviations of your samples to be close to what you would see in practice and you need them to be much greater than your measurement error to detect a correlation in only 20 samples. Consider this final example where the measurement error is high in sample 2.

# same means, low correlation because # high measurement error in sample 2 # t.test Not significant # cor.test Not significant set.seed(5) s1 <- rnorm(20) s2 <- s1+rnorm(20,0,3) t.test(s1,s2,paired=T)$p.value     cor.test(s1,s2)$p.value