[R-sig-ME] Bradley Terry GLMM in R ?

[Shira Mitchell]

Hi JP !

This is all super helpful. Follow-up questions:

1. What is the difference between
anova(m_extra, m) as used in
http://bbolker.github.io/mixedmodels-misc/glmmFAQ.html#testing-significance-of-random-effects

and
lmtest::lrtest(m_extra, m) as used in
https://github.com/jvparidon/lmerMultiMember/blob/main/vignettes/lmermultimember_intro.Rmd
?

2. In summary(m) output, why is "Group memberships per observation for
multiple membership REs:" seem to always be all zeros ?
https://jvparidon.github.io/lmerMultiMember/articles/bradleyterry_models.html#sanity-check-on-random-intercepts-team-strength

3. We need to generate predictions for synthetic cases:
pr(message i beats a hypothetical average message | person with covariate
x) = logit^-1 ( lambda_i + beta_i x)
If I understand correctly, this isn't yet implemented ? So I wrote a clunky
ad hoc version for my model that pulls lambda_i and beta_i from ranef(m)
and multiplies by covariate x matrices and takes inverse logit to get the
above.

Thanks again,
Shira

On Fri, Dec 9, 2022 at 9:22 PM Jeroen van Paridon <vanparidon using wisc.edu>
wrote:

> Hi Shira,
>
> I'm glad you're finding it useful!
>
> If I understand correctly, the model you've run already includes
> continuous covariates (e.g. (x | indicators)), so I think your question
> is mostly about the categorical groups?
>
> Your intuition to specify them as random effects using the interaction
> notation (e.g. (1 | indicators:group)) would be correct for lme4, but
> unfortunately the lmerMultiMember way of specifying these interactions is a
> little bit messier (there's some backend complexities that prevent me from
> using the same syntax as lme4). Instead, you'll have to pre-generate the
> indicator matrices for the interaction groupings using the
> interaction_weights() function. There's a worked example of how to do
> this in the lmerMultiMember vignettes (here:
> https://jvparidon.github.io/lmerMultiMember/articles/lmermultimember_intro.html#using-nestedinteraction-multiple-membership-to-find-the-player-with-the-strongest-year-of-the-2010s)
> but the basic steps are as follows:
>
>    1. Generate a Bradley-Terry matrix Wbt for your messages using
>    bradleyterry_from_sparse() or whatever other method you're using.
>    2. Create a sparse matrix Wg for the grouping factor you want to nest
>    by, using Matrix::fac2sparse(group).
>    3. Generate the interaction matrix Wbtxg using interaction_weights(Wbt,
>    Wg) and then use that as your indicator matrix.
>
> I tend to name the interaction matrix dummies the same way I would specify
> interactions, with an X replacing the colon, so indicators:group would
> become indicatorsXgroup. This makes it a little easier to keep track of
> what the different dummies in the model formula mean.
>
> If you have multiple categorical factors that you want to use as random
> effects groupings (e.g. (1 | indicatorsXgender) + (1 |
> indicatorsXemployment_status)) you can create separate interaction matrices
> for those indicator:grouping interactions.
> In most cases, it would make sense to *also* include the indicators for
> the messages *only*, so your formula might end up looking something like
> (1 + age | indicators) + (1 | indicatorsXgender) + (1 |
> indicatorsXemployment_status), for example.
>
> I hope this explanation makes sense, but do let me know if it doesn't! (I
> don't talk about Bradley-Terry models very often, so my terminology and
> notation may be a bit off.)
> Two things to be aware of as you're doing this:
>
>    1. I haven't gotten around to optimizing the interaction_weights()
>    function, so it's a pretty slow implementation. Generating a matrix for a
>    large dataset could take a few minutes! (I'm hopeful I'll get around to
>    fixing that over the holidays...)
>    2. When you make interaction matrices and add covariates, the number
>    of random effects levels tends to explode, and it's very easy for the
>    number of random effects levels that would need to be estimated to exceed
>    the number of observations in your data. That tends to make the model
>    unidentifiable, meaning that lmerMultiMember can't fit it. I've programmed
>    the package so that it throws an error if this happens, just to be safe.
>
>
> As for the prediction question: The standard lme4 predict() method should
> work, so for getting predictions on the training data you can just call:
> m <- lmerMultiMember::glmer(...)
> predict(m)
>
> Which will return a vector of predictions, in the order of your original
> data. You can then use that for poststratification, etc.
>
> Unfortunately there is not yet a good method to generate predictions for
> synthetic cases (i.e. cases with a combination of predictor levels not
> observed in your training data). Statistically I think this should be
> possible, but unfortunately the technical implementation would be quite a
> bit of work, so I haven't gotten around to it.
>
>
> I hope this helps. Do let me know if any part of it was unclear!
>
>
> Cheers,
>
> JP
> ------------------------------
> *From:* Shira Mitchell <shiraqotj using gmail.com>
> *Sent:* Friday, December 9, 2022 8:53 AM
> *To:* JP van Paridon <jvparidon using gmail.com>; Jeroen van Paridon <
> vanparidon using wisc.edu>
> *Cc:* r-sig-mixed-models using r-project.org <r-sig-mixed-models using r-project.org>
> *Subject:* Re: [R-sig-ME] Bradley Terry GLMM in R ?
>
> Hi Dr van Paridon,
>
> Thank you so much !
>
> We are returning to this after our busy election season. We are using your
> awesome lmerMultiMember package and have questions.
>
> We have voter-specific variables x that influence which political message
> (i vs j) wins for them:
>
> logit[pr(i beats j | person with covariate x)] = lambda_i - lambda_j +
> (beta_i - beta_j) x
>
> We then model parameters as random effects:
> lambda_i ~ N(0, sigma_lambda)
> beta_i ~ N(0, sigma_beta)
>
> m <- lmerMultiMember::glmer(depvar ~ 1 + (1 | indicators) + (x |
> indicators),
>                                                     family = binomial,
>                                                     memberships =
> list(indicators = W),
>                                                     data = dat_train)
>
> This runs beautifully. :)
>
> Now suppose we want the strength of message i among people with covariates
> x (e.g. a specific age). In reality we have a few covariates, both
> continuous (x | indicators) and categorical groups (1 | indicators:group).
>
> pr(i beats a hypothetical average message | person with covariate x) =
> logit^-1 ( lambda_i + beta_i x)
>
> We have a data set that crosses all population x values with all messages,
> dat_population_all_messages.
>
> Also, if we want to predict specific match-ups from the training data
> dat_train, how do we do that ?
>
> Thanks again !!
>
> Shira
>
> ---------
>
> https://stat.ethz.ch/pipermail/r-sig-mixed-models/2022q4/030224.html
>
> In case it's helpful to anyone following this email thread: I wrote a vignette explaining how to fit a Bradley-Terry model in lme4 using lmerMultiMember. You can find it at https://jvparidon.github.io/lmerMultiMember/articles/bradleyterry_models.html
>
>
> Cheers,
>
> JP van Paridon (he/him)
> Research Associate, Lupyan Lab
> University of Wisconsin-Madisonhttps://github.com/jvparidon
> ________________________________
> From: Jeroen van Paridon <vanparidon using wisc.edu <https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models>>
> Sent: Thursday, October 20, 2022 1:42 AM
> To: r-sig-mixed-models using r-project.org <https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models> <r-sig-mixed-models using r-project.org <https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models>>
> Subject: Re: [R-sig-ME] Bradley Terry GLMM in R ?
>
> Hi,
>
> Just to expand on Ben's last email: In principle, lmerMultiMember allows you to pass arbitrary indicator/weight matrices for the random effects to lme4 for model fitting as long as they have the correct shape. The package contains helper functions for generating more conventional multiple membership matrices since that was my own use-case, but if you create your own matrix with opposed (1 and -1) weights I see no reason why it shouldn't work.
>
> Membership matrices need to be sparse matrices of class Matrix::dgCMatrix and shape n_groups x n_obs. You can probably just take whatever indicator matrix you already have, transpose it, and then cast it to the sparse format.
>
> If you're going this route and run into any issues, feel free to reach out to me, directly.
>
>
> Cheers,
>
> JP van Paridon (he/him)
> Research Associate, Lupyan Lab
> University of Wisconsin-Madisonhttps://github.com/jvparidon
> ________________________________
>
>
> On Mon, Oct 17, 2022 at 10:02 PM Shira Mitchell <shiraqotj using gmail.com>
> wrote:
>
> Questions for Ben Bolker about the excellent GLMM FAQ:
>
> Where does the hglm package fit into this very helpful table ?
>
> https://bbolker.github.io/mixedmodels-misc/glmmFAQ.html#what-methods-are-available-to-fit-estimate-glmms
>
> I wonder also about differences in model formula specifications, since
> some packages (e.g. lme4) don't seem to accommodate Bradley-Terry, whereas
> some packages (e.g. INLA, hglm, MCMCglmm) can accommodate.
> https://bbolker.github.io/mixedmodels-misc/glmmFAQ.html#model-specification
>
>
>
>
> On Mon, Oct 17, 2022 at 3:55 PM Shira Mitchell <shiraqotj using gmail.com>
> wrote:
>
> Thanks so much, Jarrod ! Not too late at all. Very interesting to compare
> MCMC with the approximations (INLA, hglm's extended quasi likelihood). I
> don't think I have the priors lined up yet across packages. The random
> effects seem more dispersed according to MCMCglmm than in INLA or hglm, but
> this could be due to priors not fit algorithm. Will look into the package
> prior defaults.
>
> On Mon, Oct 17, 2022 at 4:45 AM Jarrod Hadfield <j.hadfield using ed.ac.uk>
> wrote:
>
> Hi Shira,
>
> Perhaps a little late to be useful, but MCMCglmm also fits random-effect
> Bradley-Terry models. Just specify ~mm(opponent1-opponent2) in the random
> effect formula. The mm stands for multimembership - the BT model is like a
> multimembership model where some effects have been multiplied by -1, hence
> the ‘-' rather than ‘+’ in the mm model formula.
>
> Cheers,
>
> Jarrod
>
>
> > On 16 Oct 2022, at 22:49, Shira Mitchell <shiraqotj using gmail.com> wrote:
> >
> > This email was sent to you by someone outside the University.
> > You should only click on links or attachments if you are certain that
> the email is genuine and the content is safe.
> >
> > Update: Dr Heather Turner <https://www.heatherturner.net/> (author of
> > BradleyTerry2) suggested the hglm package, which unlike lme4 allows you
> to
> > specify generic design matrices (no longer constrained to lme4 formulas
> !)
> > Results look really similar to INLA so far. Yay !
> >
> > On Sun, Oct 16, 2022 at 2:19 PM Shira Mitchell <shiraqotj using gmail.com>
> wrote:
> >
> >> Super helpful !  Thank you so much !
> >>
> >> Out of curiosity, is there a way to fit this type of Bradley-Terry model
> >> in lme4 ? lme4 formulas include random effect via syntax:
> >> https://cran.r-project.org/web/packages/lme4/vignettes/lmer.pdf
> >> "(expr | factor). The expression expr is evaluated as a linear model
> >> formula, producing a model matrix following the same rules used in
> standard
> >> R modeling functions (e.g., `lm` or `glm`). The expression factor is
> >> evaluated as an `R` factor. One way to think about the vertical bar
> >> operator is as a special kind of interaction between the model matrix
> and
> >> the grouping factor. This interaction ensures that the columns of the
> model
> >> matrix have different effects for each level of the grouping factor."
> >>
> >> So (expr | factor) is X_expr * alpha_factor.
> >>
> >> So naively writing ~ (1 | m_1) + (1 | m_2) is alpha_{m_1}^{(1)} +
> >> alpha_{m_2}^{(2)}, twice as many parameters as what we want which is
> >> alpha_{m_1} - alpha_{m_2}.
> >>
> >> But then see this stackexchange:
> >>
> >>
> https://stats.stackexchange.com/questions/483833/opposing-effects-in-lme4-formulae-bradley-terry-model
> >> "I could just make a design matrix, where player 1 gets the value 1, and
> >> player 2 gets the value −1. However, unless I'm missing a trick, this
> would
> >> require having a separate column for each player, and plugging each
> player
> >> column's name into the formula"
> >>
> >> But suppose we create columns for all m = 1,...,M messages:
> >>
> >> A_m = 1 if m = m_1
> >>           -1 if m = m_2
> >>            0 otherwise
> >>
> >> I think then ~ (A_1 + ... + A_M  | m_1) is alpha_{m_1}^{(m_1)} -
> >> alpha_{m_1}^{(m_2)}, also not what we would want.
> >>
> >> Back to INLA. Suppose we now want to add random message-specific slopes
> >> for variable X_i in addition to random message-specific intercepts:
> >>
> >> P[i chooses m_1] = logit^-1 (beta_0 + (alpha_{m_1} - alpha_{m_2}) +
> >> (beta_{m_1} - beta_{m_2})X_i)
> >>
> >> alpha_1,...,alpha_M ~ N(0,sigma_intercept)
> >> beta_1,...,beta_M ~ N(0,sigma_slope)
> >>
> >> I see some resources about this, but nothing super comprehensive. Any
> >> advice where to look for complete documentation ?
> >>
> >>
> https://groups.google.com/g/r-inla-discussion-group/c/iQELaQF8M9Q/m/q7f4-W8YQksJ
> >> (
> >>
> https://becarioprecario.bitbucket.io/inla-gitbook/ch-multilevel.html#multilevel-models-for-longitudinal-data
> >> https://rpubs.com/corey_sparks/431920
> >> https://avianecologist.com/2016/10/05/multilevel-models/
> >>
> >> Here is what we did:
> >>
> >> data$w_X = -data$X
> >> data$m_1_beta = data$m_1
> >> data$m_2_beta = data$m_2
> >>
> >> inla(depvar ~  f(m_1, model="iid", values = issues) +
> >>                           f(m_2, w, copy = "m_1") +
> >>                           f(m_1_beta, X, model="iid", values = issues) +
> >>                           f(m_2_beta, w_X, copy = "m_1_beta"),
> >>                         family="binomial",
> >>                         data=data)
> >>
> >>
> >>
> >>
> >> On Mon, Oct 10, 2022 at 4:24 AM Thierry Onkelinx <
> thierry.onkelinx using inbo.be>
> >> wrote:
> >>
> >>> Dear Shira,
> >>>
> >>> - in a formula object means remove that object from the formula. Use a
> >>> weight of -1 instead.
> >>>
> >>> f(home, model = "iid")) + f(away, w = -1, copy = "home")
> >>>
> >>> Best regards,
> >>>
> >>> ir. Thierry Onkelinx
> >>> Statisticus / Statistician
> >>>
> >>> Vlaamse Overheid / Government of Flanders
> >>> INSTITUUT VOOR NATUUR- EN BOSONDERZOEK / RESEARCH INSTITUTE FOR NATURE
> >>> AND FOREST
> >>> Team Biometrie & Kwaliteitszorg / Team Biometrics & Quality Assurance
> >>> thierry.onkelinx using inbo.be
> >>> Havenlaan 88 bus 73, 1000 Brussel
> >>> www.inbo.be
> >>>
> >>>
> >>>
> ///////////////////////////////////////////////////////////////////////////////////////////
> >>> To call in the statistician after the experiment is done may be no more
> >>> than asking him to perform a post-mortem examination: he may be able
> to say
> >>> what the experiment died of. ~ Sir Ronald Aylmer Fisher
> >>> The plural of anecdote is not data. ~ Roger Brinner
> >>> The combination of some data and an aching desire for an answer does
> not
> >>> ensure that a reasonable answer can be extracted from a given body of
> data.
> >>> ~ John Tukey
> >>>
> >>>
> ///////////////////////////////////////////////////////////////////////////////////////////
> >>>
> >>> <https://www.inbo.be>
> >>>
> >>>
> >>> Op vr 7 okt. 2022 om 23:36 schreef Shira Mitchell <shiraqotj using gmail.com
> >:
> >>>
> >>>> Thanks so much, Thierry ! This is great.
> >>>>
> >>>> This works except that I cannot subtract because:
> >>>> f(home, model = "iid")) - f(away, copy = "home")
> >>>>
> >>>> just drops the second term. Apologies that I'm not super familiar with
> >>>> INLA syntax yet.
> >>>>
> >>>>
> >>>>
> >>>> On Fri, Oct 7, 2022 at 10:19 AM Thierry Onkelinx <
> >>>> thierry.onkelinx using inbo.be> wrote:
> >>>>
> >>>>> Hi Shira,
> >>>>>
> >>>>> I fit such models with the INLA package (https://www.r-inla.org/).
> The
> >>>>> trick is to define two random effects but force their parameter
> estimates
> >>>>> to be identical.
> >>>>>
> >>>>> The code would contain something like f(home, model = "iid")) +
> f(away,
> >>>>> copy = "home"). Meaning home ~ N(0, sigma_beta_i) and home[i] =
> away[i]
> >>>>>
> >>>>> Best regards,
> >>>>>
> >>>>> ir. Thierry Onkelinx
> >>>>> Statisticus / Statistician
> >>>>>
> >>>>> Vlaamse Overheid / Government of Flanders
> >>>>> INSTITUUT VOOR NATUUR- EN BOSONDERZOEK / RESEARCH INSTITUTE FOR
> NATURE
> >>>>> AND FOREST
> >>>>> Team Biometrie & Kwaliteitszorg / Team Biometrics & Quality Assurance
> >>>>> thierry.onkelinx using inbo.be
> >>>>> Havenlaan 88 bus 73, 1000 Brussel
> >>>>> www.inbo.be
> >>>>>
> >>>>>
> >>>>>
> ///////////////////////////////////////////////////////////////////////////////////////////
> >>>>> To call in the statistician after the experiment is done may be no
> more
> >>>>> than asking him to perform a post-mortem examination: he may be able
> to say
> >>>>> what the experiment died of. ~ Sir Ronald Aylmer Fisher
> >>>>> The plural of anecdote is not data. ~ Roger Brinner
> >>>>> The combination of some data and an aching desire for an answer does
> >>>>> not ensure that a reasonable answer can be extracted from a given
> body of
> >>>>> data. ~ John Tukey
> >>>>>
> >>>>>
> ///////////////////////////////////////////////////////////////////////////////////////////
> >>>>>
> >>>>> <https://www.inbo.be>
> >>>>>
> >>>>>
> >>>>> Op vr 7 okt. 2022 om 15:00 schreef Shira Mitchell <
> shiraqotj using gmail.com
> >>>>>> :
> >>>>>
> >>>>>> We want to fit Bradley-Terry-style GLMM models in R. We looked into:
> >>>>>>
> >>>>>>
> >>>>>>
> https://cran.r-project.org/web/packages/BradleyTerry2/vignettes/BradleyTerry.pdf
> >>>>>> and
> >>>>>> http://bbolker.github.io/mixedmodels-misc/glmmFAQ.html
> >>>>>>
> >>>>>> We have voter-specific variables x that influence which political
> >>>>>> message
> >>>>>> (i vs j) wins for them:
> >>>>>>
> >>>>>> logit[pr(i beats j | person with covariate x)] = lambda_i -
> lambda_j +
> >>>>>> (beta_i - beta_j) x
> >>>>>>
> >>>>>> We then model parameters as random effects:
> >>>>>> lambda_i ~ N(0, sigma_lambda)
> >>>>>> beta_i ~ N(0, sigma_beta)
> >>>>>>
> >>>>>> Is there a way to do this in R ? We do this in TensorFlow in Python
> by
> >>>>>> directly specifying design matrices with the 0,-1,1 or 0,-x,x
> entries.
> >>>>>> However, I do not see how to do this in R using lme4, BradleyTerry2,
> >>>>>> mgcv,
> >>>>>> etc.
> >>>>>>
> >>>>>> Thanks so much,
> >>>>>> Shira
> >>>>>>
> >>>>>>        [[alternative HTML version deleted]]
> >>>>>>
> >>>>>> _______________________________________________
> >>>>>> R-sig-mixed-models using r-project.org mailing list
> >>>>>> https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
> >>>>>>
> >>>>>
> >
> >        [[alternative HTML version deleted]]
> >
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