ordmonoreg {monoreg}

Bayesian monotonic regression

Description

This function implements the non-parametric Bayesian monotonic regression procedure for ordinal outcomes described in Saarela, Rohrbeck & Arjas (2023).

Usage

ordmonoreg(niter=15000, burnin=5000, adapt=5000, refresh=10, thin=20, 
		   birthdeath=1, logit=FALSE, gam=FALSE, seed=1, rhoa=0.1, rhob=0.1, 
		   deltai=0.5, dlower=0, dupper=1, invprob=1.0, dc=0.0, 
		   predict, include, outcome, axes, covariates=NULL, 
		   cluster=NULL, ncluster=NULL, settozero)

Arguments

niter

Total number of MCMC iterations.

burnin

Number of iterations used for burn-in period.

adapt

Number of iterations used for adapting Metropolis-Hastings proposals.

refresh

Interval for producing summary output of the state of the MCMC sampler.

thin

Interval for saving the state of the MCMC sampler.

birthdeath

Number of birth-death proposals attempted at each iteration.

logit

Indicator for fitting the model on logit scale.

gam

Indicator for fitting non-monotonic generalized additive models for covariate effects.

seed

Seed for the random generator.

rhoa

Shape parameter of a Gamma hyperprior for the Poisson process rate parameters.

rhob

Scale parameter of a Gamma hyperprior for the Poisson process rate parameters.

deltai

Range for a uniform proposal for the function level parameters.

dlower

Lower bound for the allowed range for the monotonic function.

dupper

Upper bound for the allowed range for the monotonic function.

invprob

Probability with which to propose keeping the original direction of monotonicity.

dc

First parameter of the conditional beta prior to counter spiking behaviour near the origin - 1.

predict

Indicator vector for the observations for which absolute risks are calculated (not included in the likelihood expression).

include

Indicator vector for the observations to be included in the likelihood expression.

outcome

Vector of outcome variable values.

axes

A matrix where the columns specify the covariate axes in the non-parametrically specified regression functions. Each variable here must be scaled to zero-one interval.

covariates

A matrix of additional covariates to be included in the linear predictor of the events of interest (optional).

cluster

A vector of indicators for cluster membership, numbered from 0 to ncluster-1 (optional).

ncluster

Number of clusters (optional).

settozero

A zero-one matrix specifying the point process construction. Each row represents a point process, while columns correspond to the columns of the argument axes, indicating whether the column is one of the dimensions specifying the domain of the point process. (See function getcmat.)

Value

A list with elements

steptotal

A sample of total number of points in the marked point process construction.

steps

A sample of the number of points used per each additive component.

rho

A sample of the Poisson process rate parameters (one per each point process specified).

loglik

A sample of log-likelihood values.

tau

A sample of random intercept variances.

alpha

A sample of random intercepts.

beta

A sample of regression coefficients for the variables specified in the argument covariates.

lambda

A sample of non-parametric regression function levels for each observation specified in the argument predict.

pred

A sample of predicted means for each observation specified in the argument predict.

sigmasq

A sample of autoregressive prior variances.

Author(s)

Olli Saarela <[e-mail]>, Christian Rohrbeck <[e-mail]>

References

Saarela O., Rohrbeck C., Arjas E. (2023). Bayesian non-parametric ordinal regression under a monotonicity constraint. Bayesian Analysis, 18:193–221.

Examples

library(monoreg)
expit <- function(x) {1/(1+exp(-x))}
logit <- function(p) {log(p)-log(1-p)}
set.seed(1)
# nobs <- 500
nobs <- 200
x <- sort(runif(nobs))
ngrid <- 100
xgrid <- seq(1/ngrid, (ngrid-1)/ngrid, by=1/ngrid)
ngrid <- length(xgrid)
ncat <- 4
beta <- 0.75
disc <- c(Inf, Inf, 0.75, 0.5)
gamma <- c(0,0,0.25,0.5)

surv <- matrix(NA, nobs, ncat)
cdf <- matrix(NA, nobs, ncat)
cols <- c('black','red','blue','green')
for (i in 1:ncat) {
    surv[,i] <- expit(logit((ncat-(i-1))/ncat) + beta * x + 
	                  gamma[i] * (x > disc[i]) - gamma[i] * (x < disc[i]))
    if (i==1)
        plot(x, surv[,i], type='l', col=cols[i], ylim=c(0,1), lwd=2, ylab='S')
    else
        lines(x, surv[,i], col=cols[i], lwd=2)
}
head(surv)

for (i in 1:ncat) {
    if (i<ncat)
        cdf[,i] <- 1.0 - surv[,i+1]
    else
        cdf[,i] <- 1.0
    if (0) {
        if (i==1)
            plot(x, cdf[,i], type='l', col=cols[i], ylim=c(0,1), lwd=2, ylab='F')
        else
            lines(x, cdf[,i], col=cols[i], lwd=2)
    }
}
head(cdf)

u <- runif(nobs)
y <- rep(NA, nobs)
for (i in 1:nobs)
    y[i] <- findInterval(u[i], cdf[i,]) + 1
table(y)

xwindow <- 0.1/2
mw <- matrix(NA, nobs, ncat)
grid <- sort(x)
for (i in 1:nobs) {
    idx <- (x > grid[i] - xwindow) & (x < grid[i] + xwindow)
    for (j in 1:ncat)
        mw[i,j] <- sum(y[idx] >= j)/sum(idx)
}
for (j in 1:ncat) {
    lines(grid, mw[,j], lty='dashed', col=cols[j])
}

# results <- ordmonoreg(niter=15000, burnin=5000, adapt=5000, refresh=10, thin=5, 
results <- ordmonoreg(niter=3000, burnin=1000, adapt=1000, refresh=10, thin=4, 
           birthdeath=1, logit=FALSE, gam=FALSE, seed=1, rhoa=0.1, rhob=0.1, 
           deltai=0.2, dlower=0, dupper=1, invprob=1.0, dc=0.0, 
           predict=c(rep(0,nobs),rep(1,ngrid)), 
           include=c(rep(1,nobs),rep(0,ngrid)), 
           outcome=c(y, rep(1,ngrid)), 
           axes=c(x, xgrid), covariates=NULL, cluster=NULL, ncluster=NULL, 
           settozero=getcmat(1))
		   
s <- results$lambda
dim(s)
lines(xgrid, colMeans(subset(s, s[,1]==0)[,2:(ngrid+1)]))
lines(xgrid, colMeans(subset(s, s[,1]==1)[,2:(ngrid+1)]), col='red')
lines(xgrid, colMeans(subset(s, s[,1]==2)[,2:(ngrid+1)]), col='blue')
lines(xgrid, colMeans(subset(s, s[,1]==3)[,2:(ngrid+1)]), col='green')
legend('bottomright', legend=c(expression(P(Y>=1)), expression(P(Y>=2)), 
expression(P(Y>=3)), expression(P(Y>=4))), 
lwd=2, col=cols)