[R-sig-ME] [External] Re: Help with interpreting one fixed-effect coefficient

Lenth, Russell V ru@@e||-|enth @end|ng |rom u|ow@@edu
Mon Sep 27 03:22:23 CEST 2021


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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