Hi,

I am comparing two paired groups. **Should I follow 9.4.1 paired samples in the user guide or 9.7 in the user guide?** [I know 9.7 is for mixed of independent and dependent factors. But two paired group is a special case of 0 independent and 1 dependent factor, so 9.7 should be right also for two paired groups.] I actually think 9.7 is the mixed effect model, thus 9.4.1 (paired t test) should be a special case of 9.7. But from the following test, I am apparently wrong somewhere.

My small simulation study is following,

y = matrix(rnorm(2*10), nrow=2) treat = factor(rep(c("T1","T2"),5)) subject_id = rep(c("A","B","C","D","E"), each = 2) design <- model.matrix(~0+treat) colnames(design) <- levels(treat) # 1 corfit <- duplicateCorrelation(y,design,block=subject_id) fit <- lmFit(y,design, block = subject_id, correlation = corfit$consensus) cm <- makeContrasts( Diff = T2-T1, levels=design) fit2 <- contrasts.fit(fit, cm) fit2 <- eBayes(fit2) topTable(fit2, coef="Diff") logFC AveExpr t P.Value adj.P.Val B 1 -0.7933632 -0.1601572 -1.837463 0.08478521 0.1695704 -4.118469 2 -0.3660406 0.1586736 -1.025867 0.32021504 0.3202150 -4.955690 # 2 design <- model.matrix(~subject_id+treat) fit <- lmFit(y, design) fit <- eBayes(fit) topTable(fit, coef="treatT2") logFC AveExpr t P.Value adj.P.Val B 1 -0.7933632 -0.1601572 -1.668410 0.1345112 0.2690224 -4.295977 2 -0.3660406 0.1586736 -1.166903 0.2774725 0.2774725 -4.686874 # 3 t.test(y[1,]~treat, paired = TRUE) t = 1.4203, df = 4, p-value = 0.2286

They return different p value. Why? And which one is more appropriate?

Thank you Aaron. If I understand this correctly, the model X is not the mixed effect model?

Well, X is analogous to a mixed effect model. There are some technical differences relating to how the model is ultimately fitted (e.g., GLS vs REML), and also in how the variance of the random batch effect is estimated.

However, these differences are irrelevant to your original question.

Even ifX was a fully fledged mixed effect model, there is no reason that it would yield the same p-value as model Y. The fact remains that the batch effect is being modelled in an entirely different manner between X and Y, so the p-values should still be different. As far as I can remember, the theory for hypothesis testing in mixed effect models is a lot messier than that for standard linear models, necessitating different tests entirely.Don't believe me? Let's whip out an example with

lme4:So while the

conceptof a fixed effects model is a specialization of the concept a mixed effects model, the specific fixed effects model used in`lm`

above is not a specialized case of the model used in`lmer`

.Thanks a lot, Aaron. I think I need to go to a library and study my linear regression textbook again.