[R-sig-ME] Help with interpreting one fixed-effect coefficient
Simon Harmel
@|m@h@rme| @end|ng |rom gm@||@com
Mon Sep 27 19:45:25 CEST 2021
Sure, many thanks.
Simon
On Mon, Sep 27, 2021 at 11:32 AM Lenth, Russell V <russell-lenth using uiowa.edu>
wrote:
> I guess I misinterpreted Simon's last question... Those coefficients in
> the additive model are just estimated using data from other cells. Under a
> model that the effects are additive, there are fewer parameters and that
> makes it possible -- just like having an empty cell or two in a block
> experiment, it is still possible to impute the missing data based on an
> additive model for a block design. Or yet simpler, a straight-line model
> for y vs. x allows us to estimate E(y|x) for infinitely many x's not
> included in the dataset. A simple enough model can estimate anything.
>
> Russ
>
> -----Original Message-----
> From: Simon Harmel <sim.harmel using gmail.com>
> Sent: Monday, September 27, 2021 11:10 AM
> To: Lenth, Russell V <russell-lenth using uiowa.edu>
> Cc: r-sig-mixed-models using 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 using 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 using gmail.com>
> > Cc: r-sig-mixed-models using 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 using gmail.com>
> > Sent: Sunday, September 26, 2021 3:08 PM
> > To: Lenth, Russell V <russell-lenth using uiowa.edu>
> > Cc: r-sig-mixed-models using 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
> >
> >
>
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