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

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 $\boldsymbol{X}\boldsymbol{\beta}$).

Note

This is a prototype function and the user interface is planned to change in upcoming versions of PlackettLuce.

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)))

}
#> Loading required package: prefmod
#> Loading required package: gnm
#> Loading required package: colorspace
#>          A          B          C          D 
#> 0.04060714 0.62839568 0.20874175 0.12224363