1 Introduction

The main model-fitting function in the PlackettLuce package, PlackettLuce, directly models the worth of items with a separate parameter estimate for each item (see Introduction to PlackettLuce). This vignette introduces a new function, pladmm, that models the log-worth of items by a linear function of item covariates. This functionality is under development and provided for experimental use - the user interface is likely to change in upcoming versions of PlackettLuce.

pladmm supports partial rankings, but otherwise has limited functionality compared to PlackettLuce. In particular, ties, pseudo-rankings, prior information on log-worths, and ranker adherence parameters are not supported.

2 Plackett-Luce model with item covariates

The standard Plackett-Luce model specifies the probability of a ranking of \(J\) items, \({i_1 \succ \ldots \succ i_J}\), is given by

\[\prod_{j=1}^J \frac{\alpha_{i_j}}{\sum_{i \in A_j} \alpha_i}\]

where \(\alpha_{i_j}\) represents the worth of item \(i_j\) and \(A_j\) is the set of alternatives \(\{i_j, i_{j + 1}, \ldots, i_J\}\) from which item \(i_j\) is chosen.

pladmm models the log-worth as a linear function of item covariates:

\[\log \alpha_i = \beta_0 + \beta_1 x_{i1} + \ldots + \beta_p x_{ip}\]

where \(\beta_0\) is fixed by the constraint that \(\sum_i \alpha_i = 1\). The parameters are estimated using an Alternating Directions Method of Multipliers (ADMM) algorithm proposed by (Yildiz et al. 2020), hence the name pladmm.

ADMM alternates between estimating the worths \(\alpha_i\) and the linear coefficients \(\beta_k\), encapsulating them in a quadratic penalty on the likelihood:

\[L(\boldsymbol{\beta}, \boldsymbol{\alpha}, \boldsymbol{u}) = \mathcal{L}(\mathcal{D}|\boldsymbol{\alpha}) + \frac{\rho}{2}||\boldsymbol{X}\boldsymbol{\beta} - \log \boldsymbol{\alpha} + \boldsymbol{u}||^2_2 - \frac{\rho}{2}||\boldsymbol{u}||^2_2\] where \(\boldsymbol{u}\) is a dual variable that imposes the equality constraints (so that \(\log \boldsymbol{\alpha}\) converges to \(\boldsymbol{X}\boldsymbol{\beta}\)).

3 Salad Data

We shall illustrate the use of pladmm with a classic data set presented by (Critchlow and Fligner 1991) that is available as the salad data set in the prefmod package. The data are 32 full rankings of 4 salad dressings (A, B, C, D) by tartness, with 1 being the least tart and 4 being the most tart, according to the ranker.

library(prefmod)
head(salad, 4)
##   A B C D
## 1 1 2 3 4
## 2 1 2 3 4
## 3 2 1 3 4
## 4 2 1 4 3

The salad dressings were made with known quantities of acetic acid and gluconic acid, as specified in the following data frame:

features <- data.frame(salad = LETTERS[1:4],
                       acetic = c(0.5, 0.5, 1, 0),
                       gluconic = c(0, 10, 0, 10))

3.1 Standard Plackett-Luce model

We begin by using pladmm to fit a standard Plackett-Luce model, with a separate parameter for each salad dressing. The first three arguments are the rankings (a matrix or rankings object), a formula specifying the model for the log-worth (must include an intercept) and a data frame of item features containing variables in the model formula. rho is the penalty parameter determining the strength of penalty on the log-likelihood. As a rule of thumb, rho should be ~10% of the fitted log-likelihood.

library(PlackettLuce)
standardPL <- pladmm(salad, ~ salad, data = features, rho = 8)
summary(standardPL)
## Call: pladmm(rankings = salad, formula = ~salad, data = features, rho = 8)
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  -3.1740         NA      NA       NA    
## saladB        2.7305     0.4481   6.093 1.11e-09 ***
## saladC        1.5621     0.3965   3.939 8.17e-05 ***
## saladD        1.0275     0.3771   2.725  0.00644 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual deviance:  152.83 on 189 degrees of freedom
## AIC:  158.83 
## Number of iterations: 7

In this case, the intercept represents the log-worth of salad dressing A, which is fixed by the constraint that the worths sum to 1.

sum(exp(standardPL$x %*% coef(standardPL)))
## [1] 1

The remaining coefficients are the difference in log-worth between each salad dressing and salad dressing A. We can compare this to the results from PlackettLuce, which sets the log-worth of salad dressing A to zero:

standardPL_PlackettLuce <- PlackettLuce(salad, npseudo = 0)
summary(standardPL_PlackettLuce)
## Call: PlackettLuce(rankings = salad, npseudo = 0)
## 
## Coefficients:
##   Estimate Std. Error z value Pr(>|z|)    
## A   0.0000         NA      NA       NA    
## B   2.7299     0.4481   6.093 1.11e-09 ***
## C   1.5615     0.3965   3.939 8.20e-05 ***
## D   1.0268     0.3771   2.723  0.00646 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual deviance:  152.83 on 189 degrees of freedom
## AIC:  158.83 
## Number of iterations: 6

The differences in log-worth are the same to ~3 decimal places. We can improve the accuracy of pladmm by reducing rtol (by default 1e-4):

standardPL <- pladmm(salad, ~ salad, data = features, rho = 8, rtol = 1e-6)
summary(standardPL)
## Call: pladmm(rankings = salad, formula = ~salad, data = features, rho = 8, 
##     rtol = 1e-06)
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  -3.1735         NA      NA       NA    
## saladB        2.7299     0.4481   6.093 1.11e-09 ***
## saladC        1.5615     0.3965   3.939 8.20e-05 ***
## saladD        1.0268     0.3771   2.723  0.00646 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual deviance:  152.83 on 189 degrees of freedom
## AIC:  158.83 
## Number of iterations: 17

The itempar function can be used to obtain the worth estimates, e.g. 

itempar(standardPL)
## Item response item parameters (PLADMM):
##       A       B       C       D 
## 0.04186 0.64176 0.19950 0.11688

3.2 Plackett-Luce model with item covariates

To model the log-worth by item covariates, we simply update the model formula:

regressionPL <- pladmm(salad, ~ acetic + gluconic, data = features, rho = 8)
summary(regressionPL)
## Call: pladmm(rankings = salad, formula = ~acetic + gluconic, data = features, 
##     rho = 8)
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -4.84097         NA      NA       NA    
## acetic       3.27431    0.57650   5.680 1.35e-08 ***
## gluconic     0.27392    0.04505   6.081 1.20e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual deviance:  152.9 on 190 degrees of freedom
## AIC:  156.9 
## Number of iterations: 14

The model uses one less degree of freedom, but there is only a slight increase in the deviance:

deviance(regressionPL) - deviance(standardPL)
## [1] 0.07441099

So it is sufficient to model the log-worth by the concentration of acetic and gluconic acids.

An advantage of modelling log-worth by covariates is that we can predict the log-worth for new items. For example, suppose we have salad dressings with the following features:

features2 <- data.frame(salad = LETTERS[5:6],
                        acetic = c(0.5, 0),
                        gluconic = c(5, 5))

the predicted log-worth is given by

predict(regressionPL, features2)
##         1         2 
## -1.834198 -3.471352

Note that the names in features2$salad are unused as salad was not a variable in the model. The predicted log-worths have the same location as the original fitted values

fitted(regressionPL)
##          A          B          C          D 
## -3.2038115 -0.4645852 -1.5666574 -2.1017393

i.e. they are contrasts with the log-worth of salad dressing A. If we want to express the predictions as a new set of constrained item parameters, we can specify type = "itempar" (vs the default type = "lp" for linear predictor). The parameterization can then be specified by passing arguments on to itempar(), e.g. the following will compute the predicted worths constrained to sum to 1:

predict(regressionPL, features2, type = "itempar", log  = FALSE, ref = NULL)
##         1         2 
## 0.8371473 0.1628527

Standard errors can optionally be returned, by specifying se.fit = TRUE

predict(regressionPL, features2, type = "itempar", log  = FALSE, ref = NULL,
        se.fit = TRUE)
## $fit
##         1         2 
## 0.8371473 0.1628527 
## 
## $se.fit
##          1          2 
## 0.03929727 0.03929727

4 Cautionary notes

The PLADMM algorithm should in theory converge to the maximum likelihood estimates for the parameters. However, the algorithm may not behave well if the rankings are very sparse or if the penalty parameter rho is not set to a suitable value. Currently, pladmm does not provide checks/warnings to assist the user the validate the result. It is recommended that the standard Plackett-Luce model is fitted initially to give a reference of the expected log-likelihood and item parameters - pladmm should give broadly similar results.

pladmm also returns two estimates of the worths. The first set are the direct estimates from the last iteration of ADMM:

regressionPL$pi
##          A          B          C          D 
## 0.04061305 0.62842986 0.20872416 0.12223294

The second set are the estimates given by the estimates of \(\boldsymbol{\beta}\) from the last iteration:

regressionPL$tilde_pi
##          A          B          C          D 
## 0.04060714 0.62839568 0.20874175 0.12224363

These two sets of estimates should be approximately the same (but being approximately the same does not guarantee the solution is the global optimum).

References

Critchlow, Douglas, and Michael Fligner. 1991. “Paired comparison, triple comparison, and ranking experiments as generalized linear models, and their implementation on GLIM.” Psychometrika 56 (3): 517–33. https://doi.org/10.1007/BF02294488.

Yildiz, Ilkay, Jennifer Dy, Deniz Erdogmus, Jayashree Kalpathy-Cramer, Susan Ostmo, J. Peter Campbell, Michael F. Chiang, and Stratis Ioannidis. 2020. “Fast and Accurate Ranking Regression.” In Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, edited by Silvia Chiappa and Roberto Calandra, 108:77–88. Proceedings of Machine Learning Research. http://proceedings.mlr.press/v108/yildiz20a.html.