Plackett-Luce Models with Item Covariates
In PlackettLuce: Plackett-Luce Models for Rankings

library(knitr)
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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.

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 [@Yildiz2020], 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 $\boldsymbol{X}\boldsymbol{\beta}$).

Salad Data

We shall illustrate the use of pladmm with a classic data set presented by [@Critchlow1991] 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)

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))

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)

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)))

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)

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)

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

itempar(standardPL)

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)

The model uses one less degree of freedom, but there is only a slight increase in the deviance, that is not significant:

anova(standardPL, regressionPL)

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)

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)

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)

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

predict(regressionPL, features2, type = "itempar", log  = FALSE, ref = NULL,
        se.fit = TRUE)

Plackett-Luce tree with item covariates

The Plackett-Luce model with item covariates can also be used in model-based partitioning. To illustrate, we shall simulate some covariate data for the judges than ranked the four salads, based on their ranking of salad A

set.seed(1)
judge_features <- data.frame(varC = rpois(nrow(salad), lambda = salad$C^2))

This simulates the scenario where some characteristic of the judge affects how they rank salad A, so we expect the item worth to depend on this variable.

Now we group the rankings by judge in preparation to fit a Plackett-Luce tree:

grouped_salad <- group(as.rankings(salad), 1:nrow(salad))

We specify the Plackett-Luce tree to partition the grouped rankings by any of the judge features (grouped_salad ~ .), with the log-worth of the salads modelled by a linear function of the acetic and gluconic acid concentrations (~acetic + gluconic). The corresponding variables are found in data, which should be a list of two data frames, the first containing the group covariates and the second containing the item covariates. We set a minimum group size of 10 and reduce the rho parameter accordingly.

tree <- pltree(grouped_salad ~ .,
               worth = ~acetic + gluconic,
               data = list(judge_features, features),
               rho = 2, minsize = 10)
plot(tree, ylines = 2)

The result is a tree with two nodes; both groups prefer salad B, but the first group (varC ≤ 7) places salad C in second place, while the second group (varC > 7) prefer salad D. This is as we might expect, since we simulated the judge covariate varC to correlate with the ranking of C, so a higher value of this variable correlates to a lower preference for C. We can see the difference in the coefficients of the item features:

tree

From the first group to the second group, the coefficient for acetic acid concentration reduces from 4.3 to 2.7. Since the acetic acid concentration for salad C is 1, with 0 gluconic acid, this reduces the worth of salad C in the second group. At the same time, the coefficient for gluconic acid concentration increases 0.28 to 0.34 between the first and second groups. Since the gluconic acid concentration for salad D is 1, with 0 acetic acid, this increases the worth of salad D in the second group.

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

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

regressionPL$tilde_pi

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