This function uses MCMC to sample from the posterior distribution of the Bradley–Terry model with ties.A multivariate normal prior distribution on the player quality parameters can be specified. An exponential prior distribution is placed on the tie parameter theta, and a Metropolis- Hasting random walk algorithm is used to update this parameter.
Usage
BBTm.ties(
n.objects,
outcome,
player1,
player2,
player.prior.var = NULL,
theta.initial = NULL,
lambda.initial = NULL,
n.iter = 1000,
hyperparameter = TRUE,
chi = 0.01,
psi = 0.01,
rw.sd = 0.1,
theta.rate = 0.01,
burn.in = 100,
n.thin = 1,
verbose = interactive()
)Arguments
- n.objects
The number of objects in the study.
- outcome
Vector of outcomes. 0 if player 1 is the winner, 1 if player 2 is the winner, and 2 if it is a tie.
- player1
Vector of first players.
- player2
Vector of second players.
- player.prior.var
(optional) Matrix specifying the prior covariance of the player correlation parameters.
- theta.initial
(optional) Value of the tied parameter there for the first MCMC iteration.
- lambda.initial
(optional) Vector containing the values of the player parameters for the first MCMC iteration.
- n.iter
Number of MCMC samples to be drawn.
- hyperparameter
Boolean indicating if inference should be performed for the prior variance hyperparameter. If TRUE the prior variance (main diagonal of the covariance matrix) must be set to 1.
- chi
(optional) Rate parameter for the inverse-gamma prior distribution on the hyperparameter.
- psi
(optional) Shape parameter for the inverse-gamma prior distribution on the hyperparameter.
- rw.sd
(optional) Number describing the standard deviation of normal distribution proposal distribution for theta.
- theta.rate
(optional) The rate parameter of the exponential prior distribution placed on theta.
- burn.in
(optional) The number of iterations to use as a burn-in period. Default is 100.
- n.thin
(optional) The number of iterations to thin the MCMC samples by. Default is 1.
- verbose
(optional) A boolean indicating if progress should be printed to the console. Default is
interactive()therefore progress is shown if run interactively, but disabled in non-interactive contexts.
Value
A "mcmc" object containing samples from the posterior distribution
Details
If player.prior.var is omitted, independent and identical
N(0, 5^2) prior distributions are placed on each object quality parameter.
If lambda.initial is omitted, it is set to a vector of zeroes.
Examples
############################################
## Deprivation in Dar es Salaam, Tanzania ##
## Seymour et al (2022) ##
############################################
# Construct covariance matrix based on spatial informartion
prior.var <- expm::expm(darEsSalaam$adjacencyMatrix)
prior.var <- diag(diag(prior.var)^-0.5) %*% prior.var %*% diag(diag(prior.var)^-0.5)
# Fit BT model with ties
# Using `n.iter = 3` here to reduce model runtime, you will need
# a larger number of iterations for valid inference.
darTiedModel <- BBTm.ties(
n.objects = nrow(darEsSalaam$adjacencyMatrix),
outcome = darEsSalaam$comparisons$outcome,
player1 = darEsSalaam$comparisons$subward1,
player2 = darEsSalaam$comparisons$subward2,
player.prior.var = prior.var,
hyperparameter = TRUE,
rw.sd = 0.005, n.iter = 3, burn.in = 0, verbose = FALSE
)
# Get posterior means
darTiedModelLambda <- parameter(darTiedModel, "lambda") -
colMeans(parameter(darTiedModel, "lambda"))
lambda.mean <- rowMeans(darTiedModelLambda)
# Generate trace plots
plot(lambda.mean)
plot(parameter(darTiedModel, "theta"), type = "l")
