Help with interpreting one fixed-effect coefficient
Look at the model I fitted, which INCLUDES the interaction, and notice that those combinations are NOT estimated. Russ -----Original Message----- From: Simon Harmel <sim.harmel at gmail.com> Sent: Monday, September 27, 2021 11:10 AM To: Lenth, Russell V <russell-lenth at uiowa.edu> Cc: r-sig-mixed-models at r-project.org Subject: Re: [External] Re: [R-sig-ME] Help with interpreting one fixed-effect coefficient Thanks, Russ! There is one thing that I still don't understand. We have two completely empty cells (boys in girl-only & girls in boy-only schools). Then, how are the means of those empty cells computed (what data is used in their place in the additive model)? Let's' simplify the model for clarity: library(R2MLwiN) library(emmeans) Form3 <- normexam ~ schgend + sex ## + standlrt + (standlrt | school) model3 <- lm(Form3, data = tutorial) emmeans(model3, pairwise~sex+schgend)$emmeans sex schgend emmean SE df lower.CL upper.CL boy mixedsch -0.2160 0.0297 4055 -0.2742 -0.15780 girl mixedsch 0.0248 0.0304 4055 -0.0348 0.08437 boy boysch 0.0234 0.0437 4055 -0.0623 0.10897 girl boysch 0.2641 0.0609 4055 0.1447 0.38360<-how computed? boy girlsch -0.0948 0.0502 4055 -0.1931 0.00358<-how computed? girl girlsch 0.1460 0.0267 4055 0.0938 0.19829 On Sun, Sep 26, 2021 at 8:22 PM Lenth, Russell V
<russell-lenth at uiowa.edu> wrote:
By the way, returning to the topic of interpreting coefficients, you ought to have fun with the ones from the model I just fitted:
Fixed effects:
Estimate Std. Error t value
(Intercept) -0.18882 0.05135 -3.677
standlrt 0.55442 0.01994 27.807
schgendboysch 0.17986 0.09915 1.814
schgendgirlsch 0.17482 0.07877 2.219
sexgirl 0.16826 0.03382 4.975
One curious thing you'll notice is that there are no coefficients for the interaction terms. Why? Because those terms were "thrown out" of the model, and so they are not shown. I think it is unwise to not show what was thrown out (e.g., lm would have shown them as NAs), because in fact what we see is but one of infinitely many possible solutions to the regression equations. This is the solution where the last two coefficients are constrained to zero. There is another equally reasonable one where the coefficients for schgendboysch and schgendgirlsch are constrained to zero, and the two interaction effects would then be non-zero. And infinitely more where all 7 coefficients are non-zero, and there are two linear constraints among them.
Of course, since the particular estimate shown consists of all the main effects and interactions are constrained to zero, it does demonstrate that the additive model *could* have been used to obtain the same estimates and standard errors, and you can see that by comparing the results (and ignoring the invalid ones from the additive model). But it is just a lucky coincidence that it worked out this way, and the additive model did lead us down a primrose path containing silly results among the correct ones.
Russ
-----Original Message-----
From: Lenth, Russell V
Sent: Sunday, September 26, 2021 7:43 PM
To: Simon Harmel <sim.harmel at gmail.com>
Cc: r-sig-mixed-models at r-project.org
Subject: RE: [External] Re: [R-sig-ME] Help with interpreting one fixed-effect coefficient
I guess correctness is in the eyes of the beholder. But I think this illustrates the folly of the additive model. Having additive effects suggests a belief that you can vary one factor more or less independently of the other. In his comments, John Fox makes a good point that escaped my earlier cursory view of the original question, that you don't have data on girls attending all-boys' schools, nor boys attending all-girls' schools; yet the model that was fitted estimates a mean response for both those situations. That's a pretty clear testament to the failure of that model ? and also why the coefficients don't make sense. And finally why we have estimates of 15 comparisons (some of which are aliased with one another), when only 6 of them make sense.
If instead, a model with interaction were fitted, it would be a rank-deficient model because two cells are empty. Perhaps there is some sort of nesting structure that could be used to work around that. However, it doesn't matter much because emmeans assesses estimability, and the two combinations I mentioned above would be flagged as non-estimable. One could then more judiciously use the contrast function to test meaningful contrasts across this irregular array of cell means. Or even injudiciously asking for all pairwise comparisons, you will see 6 estimable ones and 9 non-estimable ones. See output below.
Russ
----- Interactive model -----
Form <- normexam ~ 1 + standlrt + schgend * sex + (standlrt | school) model <- lmer(Form, data = tutorial, REML = FALSE)
fixed-effect model matrix is rank deficient so dropping 2 columns / coefficients
emmeans(model, pairwise~schgend+sex)
... messages deleted ... $emmeans schgend sex emmean SE df asymp.LCL asymp.UCL mixedsch boy -0.18781 0.0514 Inf -0.2885 -0.0871 boysch boy -0.00795 0.0880 Inf -0.1805 0.1646 girlsch boy nonEst NA NA NA NA mixedsch girl -0.01955 0.0521 Inf -0.1216 0.0825 boysch girl nonEst NA NA NA NA girlsch girl 0.15527 0.0632 Inf 0.0313 0.2792 Degrees-of-freedom method: asymptotic Confidence level used: 0.95 $contrasts contrast estimate SE df z.ratio p.value mixedsch boy - boysch boy -0.1799 0.0991 Inf -1.814 0.4565 mixedsch boy - girlsch boy nonEst NA NA NA NA mixedsch boy - mixedsch girl -0.1683 0.0338 Inf -4.975 <.0001 mixedsch boy - boysch girl nonEst NA NA NA NA mixedsch boy - girlsch girl -0.3431 0.0780 Inf -4.396 0.0002 boysch boy - girlsch boy nonEst NA NA NA NA boysch boy - mixedsch girl 0.0116 0.0997 Inf 0.116 1.0000 boysch boy - boysch girl nonEst NA NA NA NA boysch boy - girlsch girl -0.1632 0.1058 Inf -1.543 0.6361 girlsch boy - mixedsch girl nonEst NA NA NA NA girlsch boy - boysch girl nonEst NA NA NA NA girlsch boy - girlsch girl nonEst NA NA NA NA mixedsch girl - boysch girl nonEst NA NA NA NA mixedsch girl - girlsch girl -0.1748 0.0788 Inf -2.219 0.2287 boysch girl - girlsch girl nonEst NA NA NA NA Degrees-of-freedom method: asymptotic P value adjustment: tukey method for comparing a family of 6 estimates --------------------------------------------------------- From: Simon Harmel <sim.harmel at gmail.com> Sent: Sunday, September 26, 2021 3:08 PM To: Lenth, Russell V <russell-lenth at uiowa.edu> Cc: r-sig-mixed-models at r-project.org Subject: [External] Re: [R-sig-ME] Help with interpreting one fixed-effect coefficient Dear Russ and the List Members, If we use Russ' great package (emmeans), we see that although meaningless, but "schgendgirl-only" can be interpreted using the logic I mentioned here: https://stat.ethz.ch/pipermail/r-sig-mixed-models/2021q3/029723.html . That is, "schgendgirl-only" can meaninglessly mean: ***diff. bet. boys in girl-only vs. mixed schools*** just like it can meaningfully mean: ***diff. bet. girls in girl-only vs. mixed schools*** Russ, have I used emmeans correctly? Simon Here is a reproducible code: library(R2MLwiN) # For the dataset library(lme4) library(emmeans) data("tutorial") Form <- normexam ~ 1 + standlrt + schgend + sex + (standlrt | school) model <- lmer(Form, data = tutorial, REML = FALSE) emmeans(model, pairwise~schgend+sex)$contrast contrast estimate SE df z.ratio p.value mixedsch boy - boysch boy -0.17986 0.0991 Inf -1.814 0.4565 mixedsch boy - girlsch boy -0.17482 0.0788 Inf -2.219 0.2287 <--This coef. equals mixedsch boy - mixedsch girl -0.16826 0.0338 Inf -4.975 <.0001 mixedsch boy - boysch girl -0.34813 0.1096 Inf -3.178 0.0186 mixedsch boy - girlsch girl -0.34308 0.0780 Inf -4.396 0.0002 boysch boy - girlsch boy 0.00505 0.1110 Inf 0.045 1.0000 boysch boy - mixedsch girl 0.01160 0.0997 Inf 0.116 1.0000 boysch boy - boysch girl -0.16826 0.0338 Inf -4.975 <.0001 boysch boy - girlsch girl -0.16322 0.1058 Inf -1.543 0.6361 girlsch boy - mixedsch girl 0.00656 0.0928 Inf 0.071 1.0000 girlsch boy - boysch girl -0.17331 0.1255 Inf -1.381 0.7388 girlsch boy - girlsch girl -0.16826 0.0338 Inf -4.975 <.0001 mixedsch girl - boysch girl -0.17986 0.0991 Inf -1.814 0.4565 mixedsch girl - girlsch girl -0.17482 0.0788 Inf -2.219 0.2287 <--This coef. boysch girl - girlsch girl 0.00505 0.1110 Inf 0.045 1.0000