[R-sig-ME] Collinearity diagnostics for (mixed) multinomial models
Sorkin, John
j@ork|n @end|ng |rom @om@um@ry|@nd@edu
Fri Feb 25 15:18:39 CET 2022
Juko,
It is my understanding, perhaps incorrect understanding, that collinearity of the independent variables is accessed independent of the nature of the dependent variable. Collinearity leads to inferential problems, i.e. determining if the predictors variables are independent predictors of the dependent variable. When the independent variables are collinear, the shared variance among the independent variables makes it difficult, or impossible, to determine (1) if the predictors variables are independent predictors of the dependent variable and (2) the magnitude of the contribution each independent variable makes to the prediction of the dependent variable�s value, regardless of the class of the dependent variable (continuous, binary, multinomial, etc.). On the other hand collinearity does not generally effect prediction. A set of collinear independent variables can predict a dependent variable accurately even if one can not separate the contribution of each of the collinear independent variables to the prediction of the dependent variable. I do not see why collinearity among independent variables would have a different effect on, or be looked for in multinomial models any differently than one would look for collinearity in the �usual� linear regression where the dependent variable is continuous. I would be happy to be disabused of an error in my understanding of collinearity.
John
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From: Juho Kristian Ruohonen<mailto:juho.kristian.ruohonen using gmail.com>
Sent: Friday, February 25, 2022 8:50 AM
To: Sorkin, John<mailto:jsorkin using som.umaryland.edu>
Cc: stevedrd using yahoo.com<mailto:stevedrd using yahoo.com>; John Fox<mailto:jfox using mcmaster.ca>; r-sig-mixed-models using r-project.org<mailto:r-sig-mixed-models using r-project.org>
Subject: Re: [R-sig-ME] Collinearity diagnostics for (mixed) multinomial models
I am indeed talking about collinearity of the predictors, not the response. A multinomial model consists of C-1 binary submodels, so it arguably doesn't make sense to measure collinearity in the entire dataset at once but, rather, it should be measured separately in the C-1 subdatasets to which the C-1 submodels are fit. My question is whether the way I propose to do this (in the original post) is sensible.
Best,
Juho
pe 25. helmik. 2022 klo 15.19 Sorkin, John (jsorkin using som.umaryland.edu<mailto:jsorkin using som.umaryland.edu>) kirjoitti:
I would agree with Steven. Collinearity is problem with the predictor variables, not the outcome variable. Given a multinomial model y = f(x1, x2, x3, . . . xn), one could run a simple linear regression x1 = f(x2,x3, . . .,xn) and look at vif to determine if x2 . . . xn are colinear and perhaps an additional regression x2=f(x1,x3, . . .xn) to determine if x1, x3, . . . xn are colinear. If I am missing something, I hope someone will correct me.
John (but not John Fox)
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Subject: Re: [R-sig-ME] Collinearity diagnostics for (mixed) multinomial models
This seems odd to me, but then I don't usually analyze multinomial models. Is there an issue with collinearity in the response variable in a multinomial model? I would think that the levels are collinear by definition. So then the issue, it seems to me, is whether there is collinearity in the fixed effects - and that should be independent of the response variables. Could you use the vif() function with a standard response (say = 1) to check collinearity in the fixed effects? I would think that your method on the sub datasets may not capture all of the collinearity in the full model.
But I could be waaaaaaay off base on this.
SteveDenham
On Friday, February 25, 2022, 03:24:15 AM EST, Juho Kristian Ruohonen <juho.kristian.ruohonen using gmail.com<mailto:juho.kristian.ruohonen using gmail.com>> wrote:
Dear John (and anyone else qualified to comment),
I fit lots of mixed-effects multinomial models in my research, and I would
like to see some (multi)collinearity diagnostics on the fixed effects, of
which there are over 30. My models are fit using the Bayesian *brms*
package because I know of no frequentist packages with multinomial GLMM
compatibility.
With continuous or dichotomous outcomes, my go-to function for calculating
multicollinearity diagnostics is of course *vif()* from the *car* package.
As expected, however, this function does not report sensible diagnostics
for multinomial models -- not even for standard ones fit by the *nnet*
package's *multinom()* function. The reason, I presume, is because a
multinomial model is not really one but C-1 regression models (where C is
the number of response categories) and the *vif()* function is not designed
to deal with this scenario.
Therefore, in order to obtain meaningful collinearity metrics, my present
plan is to write a simple helper function that uses *vif() *to calculate
and present (generalized) variance inflation metrics for the C-1
sub-datasets to which the C-1 component binomial models of the overall
multinomial model are fit. In other words, it will partition the data into
those C-1 subsets, and then apply *vif()* to as many linear regressions
using a made-up continuous response and the fixed effects of interest.
Does this seem like a sensible approach?
Best,
Juho
ma 27. syysk. 2021 klo 19.26 John Fox (jfox using mcmaster.ca<mailto:jfox using mcmaster.ca>) kirjoitti:
> Dear Simon,
>
> I believe that Russ's point is that the fact that the additive model
> allows you to estimate nonsensical quantities like a mean for girls in
> all-boys' schools implies a problem with the model. Why not do as I
> suggested and define two dichotomous factors: sex of student
> (male/female) and type of school (coed, same-sex)? The four combinations
> of levels then make sense.
>
> Best,
> John
>
> On 2021-09-27 12:09 p.m., Simon Harmel wrote:
> > 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<mailto: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<mailto:sim.harmel using gmail.com>>
> >> Cc: r-sig-mixed-models using r-project.org<mailto: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<mailto:sim.harmel using gmail.com>>
> >> Sent: Sunday, September 26, 2021 3:08 PM
> >> To: Lenth, Russell V <russell-lenth using uiowa.edu<mailto:russell-lenth using uiowa.edu>>
> >> Cc: r-sig-mixed-models using r-project.org<mailto: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://nam11.safelinks.protection.outlook.com/?url=https%3A%2F%2Fstat.ethz.ch%2Fpipermail%2Fr-sig-mixed-models%2F2021q3%2F029723.html&data=04%7C01%7Cjsorkin%40som.umaryland.edu%7C5fb7bcf6b8824a3109f708d9f85fa6f1%7C717009a620de461a88940312a395cac9%7C0%7C0%7C637813912584894963%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C0&sdata=kUR%2BudOSdu9gHZCsdimDJGEuheQLyI5pBlwqNctQu4A%3D&reserved=0<https://nam11.safelinks.protection.outlook.com/?url=https%3A%2F%2Fstat.ethz.ch%2Fpipermail%2Fr-sig-mixed-models%2F2021q3%2F029723.html&data=04%7C01%7Cjsorkin%40som.umaryland.edu%7C0f6e1d2cffb8442c60f308d9f865ba36%7C717009a620de461a88940312a395cac9%7C0%7C0%7C637813938549646032%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000&sdata=76bZe32xc%2FpHHb%2F26RpjZ9Pdri4F5ExSefe2giZ3IgU%3D&reserved=0> .
> >>
> >> 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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> >
> --
> John Fox, Professor Emeritus
> McMaster University
> Hamilton, Ontario, Canada
> web: https://nam11.safelinks.protection.outlook.com/?url=https%3A%2F%2Fsocialsciences.mcmaster.ca%2Fjfox%2F&data=04%7C01%7Cjsorkin%40som.umaryland.edu%7C5fb7bcf6b8824a3109f708d9f85fa6f1%7C717009a620de461a88940312a395cac9%7C0%7C0%7C637813912584894963%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C0&sdata=%2BAvoQotl3QBMkVTOWiHJtHPJ%2B79wFLAMF39m6Cgb01A%3D&reserved=0<https://nam11.safelinks.protection.outlook.com/?url=https%3A%2F%2Fsocialsciences.mcmaster.ca%2Fjfox%2F&data=04%7C01%7Cjsorkin%40som.umaryland.edu%7C0f6e1d2cffb8442c60f308d9f865ba36%7C717009a620de461a88940312a395cac9%7C0%7C0%7C637813938549646032%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000&sdata=V1%2FOZcoGRDJJk4jax%2BVcX%2FijE9KmusqCdoGBNzrYbrs%3D&reserved=0>
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