rscaleUsage implements an MCMC algorithm for multivariate ordinal data with scale usage heterogeniety.
rscaleUsage(Data, Prior, Mcmc)A list containing:
\(R/keep x p*p\) matrix of Sigma draws -- each row is the vector form of Sigma
\(R/keep x p\) matrix of mu draws
\(R/keep x n\) matrix of tau draws
\(R/keep x n\) matrix of sigma draws
\(R/keep x 4\) matrix of Lamda draws
\(R/keep x 1\) vector of e draws
list(x, k)
list(nu, V, mubar, Am, gs, Lambdanu, LambdaV)
list(R, keep, nprint, ndghk, e, y, mu, Sigma, sigma, tau, Lambda)
\(tau_i\), \(sigma_i\) are identified from the scale usage patterns in the \(p\) questions asked per respondent (# cols of \(x\)). Do not attempt to use this on datasets with only a small number of total questions.
Rob McCulloch (Arizona State University) and Peter Rossi (Anderson School, UCLA), perossichi@gmail.com.
\(n\) = nrow(x) individuals respond to \(p\) = ncol(x) questions;
all questions are on a scale \(1, \ldots, k\) for respondent \(i\) and question \(j\),
\(x_{ij} = d\) if \(c_{d-1} \le y_{ij} \le c_d\) where \(d = 1, \ldots, k\) and \(c_d = a + bd + ed^2\)
\(y_i = mu + tau_i*iota + sigma_i*z_i\) with \(z_i\) \(\sim\) \(N(0, Sigma)\)
\((tau_i, ln(sigma_i))\) \(\sim\) \(N(\phi, Lamda)\)
\(\phi = (0, lambda_{22})\)
\(mu\) \(\sim\) \(N(mubar, Am^{-1})\)
\(Sigma\) \(\sim\) \(IW(nu, V)\)
\(Lambda\) \(\sim\) \(IW(Lambdanu, LambdaV)\)
\(e\) \(\sim\) unif on a grid
It is highly recommended that the user choose the default prior settings. If you wish to change prior settings and/or the grids used, please carefully read the case study listed in the reference below.
Data = list(x, k)
x: | \(n x p\) matrix of discrete responses |
k: | number of discrete rating scale options |
Prior = list(nu, V, mubar, Am, gs, Lambdanu, LambdaV) [optional]
nu: | d.f. parameter for Sigma prior (def: p + 3) |
V: | scale location matrix for Sigma prior (def: nu*I) |
mubar: | \(p x 1\) vector of prior means (def: rep(k/2,p)) |
Am: | \(p x p\) prior precision matrix (def: 0.01*I) |
gs: | grid size for sigma (def: 100) |
Lambdanu: | d.f. parameter for Lambda prior (def: 20) |
LambdaV: | scale location matrix for Lambda prior (def: (Lambdanu - 3)*Lambda) |
Mcmc = list(R, keep, nprint, ndghk, e, y, mu, Sigma, sigma, tau, Lambda) [only R required]
R: | number of MCMC draws (def: 1000) |
keep: | MCMC thinning parameter -- keep every keepth draw (def: 1) |
nprint: | print the estimated time remaining for every nprint'th draw (def: 100, set to 0 for no print) |
ndghk: | number of draws for a GHK integration (def: 100) |
e: | initial value (def: 0) |
y: | initial values (def: x) |
mu: | initial values (def: apply(y,2,mean), a p-length vector) |
Sigma: | initial value (def: var(y)) |
sigma: | initial values (def: rep(1,n)) |
tau: | initial values (def: rep(0,n)) |
Lambda: | initial values (def: matrix(c(4,0,0,.5),ncol=2)) |
For further discussion, see Case Study 3 on Overcoming Scale Usage Heterogeneity, Bayesian Statistics and Marketing by Rossi, Allenby, and McCulloch.
if(nchar(Sys.getenv("LONG_TEST")) != 0) {R=1000} else {R=5}
set.seed(66)
data(customerSat)
surveydat = list(k=10, x=as.matrix(customerSat))
out = rscaleUsage(Data=surveydat, Mcmc=list(R=R))
summary(out$mudraw)
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