PlackettLuce (version 0.4.3)

pladmm: Fit a Plackett-Luce Model with Linear Predictor for Log-worth

Description

Fit a Plackett-Luce model where the log-worth is predicted by a linear function of covariates. The rankings may be partial (each ranking completely ranks a subset of the items), but ties are not supported.

Usage

pladmm(
  rankings,
  formula,
  data = NULL,
  weights = freq(rankings),
  start = NULL,
  contrasts = NULL,
  rho = 1,
  n_iter = 500,
  rtol = 1e-04
)

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.

formula

a formula specifying the linear model for log-worth.

data

a data frame containing the variables in the model.

weights

weights for the rankings.

start

starting values for the coefficients.

contrasts

an optional list specifying contrasts for the factors in formula. See the contrasts.arg of model.matrix().

rho

the penalty parameter in the penalized likelihood, see details.

n_iter

the maximum number of iterations (also for inner loops).

rtol

the convergence tolerance (also for inner loops)

Details

The log-worth is modelled 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 (2020). 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 \(\boldsymbol{X}\boldsymbol{\beta}\)).

References

Yildiz, I., Dy, J., Erdogmus, D., Kalpathy-Cramer, J., Ostmo, S., Campbell, J. P., Chiang, M. F. and Ioannidis, S. (2020) Fast and Accurate Ranking Regression In Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, 108, 77–-88.

Examples

if (require(prefmod)){
  data(salad)
  # data.frame of rankings for salad dressings A B C D
  # 1 = most tart, 4 = least tart
  salad[1:3,]

  # create data frame of corresponding features
  # (acetic and gluconic acid concentrations in salad dressings)
  features <- data.frame(salad = LETTERS[1:4],
                         acetic = c(0.5, 0.5, 1, 0),
                         gluconic = c(0, 10, 0, 10))

  # fit Plackett-Luce model based on covariates
  res_PLADMM <- pladmm(salad, ~ acetic + gluconic, data = features, rho = 8)
  ## coefficients
  coef(res_PLADMM)
  ## worth
  res_PLADMM$pi
  ## worth as predicted by linear function
  res_PLADMM$tilde_pi
  ## equivalent to
  drop(exp(res_PLADMM$x %*% coef(res_PLADMM)))

}