Jose, Thank you very much for the explanation! what about if I specify using "ML" instead of "REML" in the lme? I found that I still got different answers even I use "ML" in the lme call. And in any case, should I trust "lme" more than "aov", or vice versa? Thanks again --- jose.pinheiro at pharma.novartis.com wrote:> > I have a question about using "lme" and "aov" for > the > > following dataset. If I understand correctly, > using > > "aov" with an Error term in the formula is > equivalent > > to using "lme" with default settings, i.e. both > assume > > compound symmetry correlation structure. > > Not necessarily, only when the maximum of the REML > objective > function falls within the parameter space. This > obscure observation > translates, in your case, into saying that, when > the "Subject" > mean square error is larger than the "Within" MSE > (in the aov > notation), then the two fits should coincide. This > is the case for > your y response: > > Error: subject > Df Sum Sq Mean Sq F value Pr(>F) > . . . > Residuals 4 0.285268 0.071317 > > Error: Within > Df Sum Sq Mean Sq F value Pr(>F) > . . . > Residuals 8 0.231480 0.028935 > > However, when the Subject MSE is smaller than the > Within MSE, the > maximum of the REML objective occurs at a negative > value of the > between-subject variance. Because the lme algorithm > forces the > estimated parameters to remain within the parameter > space, the > solution you got probably corresponds to a boundary > of the parameter > space (the Subject Std. Dev must be fairly small). > This is what > happens for your "x" response. > > Error: subject > Df Sum Sq Mean Sq F value Pr(>F) > . . . > Residuals 4 0.122304 0.030576 > --- > > > Error: Within > Df Sum Sq Mean Sq F value Pr(>F) > . . . > Residuals 8 0.37469 0.04684 > > Hope this helps. > > Best, > --José > >---------------------------------------------------------> José Pinheiro > Biostatistics, Novartis Pharmaceuticals > One Health Plaza, Room 419/2115, East Hanover, NJ > 07936 >--------------------------------------------------------->