Bradley Terry GLMM in R ?

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 at 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 at 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 at 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 at 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 at 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 at 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

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