Fit a Plackett-Luce Model

Fit a Plackett-Luce model to a set of rankings. The rankings may be partial (each ranking completely ranks a subset of the items) and include ties of arbitrary order.

Usage

PlackettLuce(
  rankings,
  npseudo = 0.5,
  normal = NULL,
  gamma = NULL,
  adherence = NULL,
  weights = freq(rankings),
  na.action = getOption("na.action"),
  start = NULL,
  method = c("iterative scaling", "BFGS", "L-BFGS"),
  epsilon = 1e-07,
  steffensen = 0.1,
  maxit = c(500, 10),
  trace = FALSE,
  verbose = TRUE,
  ...
)

Arguments

rankings: a "rankings" object, or an object that can be coerced by as.rankings. An "aggregated_rankings" object can be used to specify rankings and weights simultaneously. A "grouped_rankings" object should be used when estimating adherence for rankers with multiple rankings per ranker.
npseudo: when using pseudodata: the number of wins and losses to add between each object and a hypothetical reference object.
normal: a optional list with elements named mu and Sigma specifying the mean and covariance matrix of a multivariate normal prior on the log worths.
gamma: a optional list with elements named shape and rate specifying parameters of a gamma prior on adherence parameters for each ranker (use grouped_rankings to group multiple rankings by ranker). The short-cut TRUE may be used to specify a Gamma(10, 10) prior. If NULL (or FALSE), adherence is fixed to adherence for all rankers.
adherence: an optional vector of adherence values for each ranker. If missing, adherence is fixed to 1 for all rankers. If gamma is not NULL, this specifies the starting values for the adherence.
weights: an optional vector of weights for each ranking.
na.action: a function to handle any missing rankings, see na.omit().
start: starting values for the worth parameters and the tie parameters on the raw scale (worth parameters need not be scaled to sum to 1). If normal is specified, exp(normal$mu) is used as starting values for the worth parameters. Coefficients from a previous fit can be passed as the result of a call to coef.PlackettLuce, or the coefficients element of a "PlackettLuce" object.
method: the method to be used for fitting: "iterative scaling" (iterative scaling to sequentially update the parameter values), "BFGS" (the BFGS optimisation algorithm through the optim interface), "L-BFGS" (the limited-memory BFGS optimisation algorithm as implemented in the lbfgs package). Iterative scaling is used by default, unless a prior is specified by normal or gamma, in which case the default is "BFGS".
epsilon: the maximum absolute difference between the observed and expected sufficient statistics for the ability parameters at convergence.
steffensen: a threshold defined as for epsilon after which to apply Steffensen acceleration to the iterative scaling updates.
maxit: a vector specifying the maximum number of iterations. If gamma is NULL, only the first element is used and specifies the maximum number of iterations of the algorithm specified by method. If gamma is not NULL, a second element may be supplied to specify the maximum number of iterations of an alternating algorithm, where the adherence parameters are updated alternately with the other parameters. The default is to use 10 outer iterations.
trace: logical, if TRUE show trace of iterations.
verbose: logical, if TRUE show messages from validity checks on the rankings.
...: additional arguments passed to optim or lbfgs. In particular the convergence tolerance may be adjusted using e.g. control = list(reltol = 1e-10).

Value

An object of class "PlackettLuce", which is a list containing the following elements:

call: The matched call.
coefficients: The model coefficients.
loglik: The maximized log-likelihood.
null.loglik: The maximized log-likelihood for the null model (all alternatives including ties have equal probability).
df.residual: The residual degrees of freedom.
df.null: The residual degrees of freedom for the null model.
rank: The rank of the model.
logposterior: If a prior was specified, the maximised log posterior.
gamma: If a gamma prior was specified, the list of parameters.
normal: If a normal prior was specified, the list of parameters.
iter: The number of iterations run.
rankings: The rankings passed to rankings, converted to a "rankings" object if necessary.
weights: The weights applied to each ranking in the fitting.
adherence: The fixed or estimated adherence per ranker.
ranker: The ranker index mapping rankings to rankers (the "index" attribute of rankings if specified as a "grouped_rankings" object.)
ties: The observed tie orders corresponding to the estimated tie parameters.
conv: The convergence code: 0 for successful convergence; 1 if reached maxit (outer) iterations without convergence; 2 if Steffensen acceleration cause log-likelihood to increase; negative number if L-BFGS algorithm failed for other reason.

Note

As the maximum tie order increases, the number of possible choices for each rank increases rapidly, particularly when the total number of items is high. This means that the model will be slower to fit with higher $D$. In addition, due to the current implementation of the vcov() method, computation of the standard errors (as by summary()) can take almost as long as the model fit and may even become infeasible due to memory limits. As a rule of thumb, for > 10 items and > 1000 rankings, we recommend PlackettLuce() for ties up to order 4. For higher order ties, a rank-ordered logit model, see ROlogit::rologit() or generalized Mallows Model as in BayesMallows::compute_mallows() may be more suitable, as they do not model tied events explicitly.

Model definition

A single ranking is given by $$R = \{C_1, C_2, \ldots, C_J\}$$ where the items in set $C_1$ are ranked higher than (better than) the items in $C_2$, and so on. If there are multiple objects in set $C_j$ these items are tied in the ranking.

For a set if items $S$, let $$f(S) = \delta_{|S|} \left(\prod_{i \in S} \alpha_i \right)^\frac{1}{|S|}$$ where $|S|$ is the cardinality (size) of the set, $\delta_n$ is a parameter related to the prevalence of ties of order $n$ (with $\delta_1 \equiv 1$), and $\alpha_i$ is a parameter representing the worth of item $i$. Then under an extension of the Plackett-Luce model allowing ties up to order $D$, the probability of the ranking $R$ is given by $$\prod_{j = 1}^J \frac{f(C_j)}{ \sum_{k = 1}^{\min(D_j, D)} \sum_{S \in {A_j \choose k}} f(S)}$$ where $D_j$ is the cardinality of $A_j$, the set of alternatives from which $C_j$ is chosen, and $A_j \choose k$ is all the possible choices of $k$ items from $A_j$. The value of $D$ can be set to the maximum number of tied items observed in the data, so that $\delta_n = 0$ for $n > D$.

When the worth parameters are constrained to sum to one, they represent the probability that the corresponding item comes first in a ranking of all items, given that first place is not tied.

The 2-way tie prevalence parameter $\delta_2$ is related to the probability that two items of equal worth tie for first place, given that the first place is not a 3-way or higher tie. Specifically, that probability is $\delta_2/(2 + \delta_2)$.

The 3-way and higher tie-prevalence parameters are similarly interpretable, in terms of tie probabilities among equal-worth items.

When intermediate tie orders are not observed (e.g. ties of order 2 and order 4 are observed, but no ties of order 3), the maximum likelihood estimate of the corresponding tie prevalence parameters is zero, so these parameters are excluded from the model.

Pseudo-rankings

In order for the maximum likelihood estimate of an object's worth to be defined, the network of rankings must be strongly connected. This means that in every possible partition of the objects into two nonempty subsets, some object in the second set is ranked higher than some object in the first set at least once.

If the network of rankings is not strongly connected then pseudo-rankings may be used to connect the network. This approach posits a hypothetical object with log-worth 0 and adds npseudo wins and npseudo losses to the set of rankings.

The parameter npseudo is the prior strength. With npseudo = 0 the MLE is the posterior mode. As npseudo approaches infinity the log-worth estimates all shrink towards 0. The default, npseudo = 0.5, is sufficient to connect the network and has a weak shrinkage effect. Even for networks that are already connected, adding pseudo-rankings typically reduces both the bias and variance of the estimators of the worth parameters.

Incorporating prior information on log-worths

Prior information can be incorporated by using normal to specify a multivariate normal prior on the log-worths. The log-worths are then estimated by maximum a posteriori (MAP) estimation. Model summaries (deviance, AIC, standard errors) are based on the log-likelihood evaluated at the MAP estimates, resulting in a finite sample bias that should disappear as the number of rankings increases. Inference based on these model summaries is valid as long as the prior is considered fixed and not tuned as part of the model.

Incorporating a prior is an alternative method of penalization, therefore npseudo is set to zero when a prior is specified.

Incorporating ranker adherence parameters

When rankings come from different rankers, the model can be extended to allow for varying reliability of the rankers, as proposed by Raman and Joachims (2014). In particular, replacing $f(S)$ by $$h(S) = \delta_{|S|} \left(\prod_{i \in S} \alpha_i \right)^\frac{\eta_g}{|S|}$$ where $\eta_g > 0$ is the adherence parameter for ranker $g$. In the standard model, all rankers are assumed to have equal reliability, so $\eta_g = 1$ for all rankers. Higher $\eta_g = 1$ increases the distance between item worths, giving greater weight' to the ranker's choice. Conversely, lower $\eta_g = 1$ shrinks the item worths towards equality so the ranker's choice is less relevant.

The adherence parameters are not estimable by maximum likelihood, since for given item worths the maximum likelihood estimate of adherence would be infinity for rankers that give rankings consistent with the items ordered by worth and zero for all other rankers. Therefore it is essential to include a prior on the adherence parameters when these are estimated rather than fixed. Setting gamma = TRUE specifies the default $\Gamma(10,10)$ prior, which has a mean of 1 and a probability of 0.99 that the adherence is between 0.37 and 2. Alternative parameters can be specified by a list with elements shape and rate. Setting scale and rate to a common value $\theta$ specifies a mean of 1; $\theta \ge$ 2 will give low prior probability to near-zero adherence; as $\theta$ increases the density becomes more concentrated (and more symmetrical) about 1.

Since the number of adherence parameters will typically be large and it is assumed the worth and tie parameters are of primary interest, the adherence parameters are not included in model summaries, but are included in the returned object.

Controlling the fit

For models without priors, using nspseudo = 0 will use standard maximum likelihood, if the network is connected (and throw an error otherwise).

The fitting algorithm is set by the method argument. The default method "iterative scaling" is a slow but reliable approach. In addition, this has the most control on the accuracy of the final fit, since convergence is determined by direct comparison of the observed and expected values of the sufficient statistics for the worth parameters, rather than a tolerance on change in the log-likelihood.

The "iterative scaling" algorithm is slow because it is a first order method (does not use derivatives of the likelihood). From a set of starting values that are 'close enough' to the final solution, the algorithm can be accelerated using Steffensen's method. PlackettLuce attempts to apply Steffensen's acceleration when all differences between the observed and expected values of the sufficient statistics are less than steffensen. This is an ad-hoc rule defining 'close enough' and in some cases the acceleration may produce negative worth parameters or decrease the log-likelihood. PlackettLuce will only apply the update when it makes an improvement.

The "BFGS" and "L-BFGS" algorithms are second order methods, therefore can be quicker than the default method. Control parameters can be passed on to optim or lbfgs.

For models with priors, the iterative scaling method cannot be used, so BFGS is used by default.

References

Raman, K. and Joachims, T. (2014) Methods for Ordinal Peer Grading. arXiv:1404.3656.

Examples

# Six partial rankings of four objects, 1 is top rank, e.g
# first ranking: item 1, item 2
# second ranking: item 2, item 3, item 4, item 1
# third ranking: items 2, 3, 4 tie for first place, item 1 second
R <- matrix(c(1, 2, 0, 0,
              4, 1, 2, 3,
              2, 1, 1, 1,
              1, 2, 3, 0,
              2, 1, 1, 0,
              1, 0, 3, 2), nrow = 6, byrow = TRUE)
colnames(R) <- c("apple", "banana", "orange", "pear")

# create rankings object
R <- as.rankings(R)

# Standard maximum likelihood estimates
mod_mle <- PlackettLuce(R, npseudo = 0)
coef(mod_mle)
#>      apple     banana     orange       pear       tie2       tie3 
#>  0.0000000  0.2942875 -0.7335113 -0.1190960 -1.8619467 -0.7369735 

# Fit with default settings
mod <- PlackettLuce(R)
# log-worths are shrunk towards zero
coef(mod)
#>       apple      banana      orange        pear        tie2        tie3 
#>  0.00000000  0.25287379 -0.61350684 -0.08688475 -2.15068111 -0.79245358 

# independent N(0, 9) priors on log-worths, as in Raman and Joachims
prior <- list(mu = rep(0, ncol(R)),
              Sigma = diag(rep(9, ncol(R))))
mod_normal <- PlackettLuce(rankings = R, normal = prior)
# slightly weaker shrinkage effect vs pseudo-rankings,
# with less effect on tie parameters (but note small number of rankings here)
coef(mod_normal)
#>      apple     banana     orange       pear       tie2       tie3 
#>  0.0000000  0.2753696 -0.6772960 -0.1030173 -1.8679502 -0.7453151 

# estimate adherence assuming every ranking is from a separate ranker
mod_separate <- PlackettLuce(rankings = R, normal = prior, gamma = TRUE)
coef(mod_separate)
#>      apple     banana     orange       pear       tie2       tie3 
#>  0.0000000  0.2310944 -0.8436801 -0.1654011 -1.8637839 -0.7405627 
# gives more weight to rankers 4 & 6 which rank apple first,
# so worth of apple increased relative to banana
mod_separate$adherence
#> [1] 0.8889778 0.8732159 0.8876840 0.9431440 0.8818594 0.9506536

# estimate adherence based on grouped rankings
#  - assume two rankings from each ranker
G <- group(R, rep(1:3, each = 2))
mod_grouped <- PlackettLuce(rankings = G, normal = prior, gamma = TRUE)
coef(mod_grouped)
#>      apple     banana     orange       pear       tie2       tie3 
#>  0.0000000  0.2855958 -0.7611189 -0.1201862 -1.8678374 -0.7453447 
# first ranker is least consistent so down-weighted
mod_grouped$adherence
#> [1] 0.8834957 0.9137346 0.9144347