odsmle {ODS}

MSELE estimator for analyzing the primary outcome in ODS design

Description

odsmle provides a maximum semiparametric empirical likelihood estimator (MSELE) for analyzing the primary outcome Y with respect to expensive exposure and other covariates in ODS design (Zhou et al. 2002).

Usage

odsmle(Y, X, beta, sig, pis, a, rs.size, size, strat)

Arguments

Y

vector for the primary response

X

the design matrix with a column of 1's for the intercept

beta

starting parameter values for the regression coefficients that relate Y to X.

sig

starting parameter values for the error variance of the regression.

pis

starting parameter values for the stratum probabilities (the probability that Y belongs to certain stratum) e.g. pis = c(0.1, 0.8, 0.1).

a

vector of cutpoints for the primary response (e.g., a = c(-2.5,2))

rs.size

size of the SRS (simple random sample)

size

vector of the stratum sizes of the supplemental samples (e.g. size = c(50,0,50) represents that two supplemental samples each of size 50 are taken from the upper and lower tail of Y.)

strat

vector that indicates the stratum numbers (e.g. strat = c(1,2,3) represents that there are three stratums).

Details

We assume that in the population, the primary outcome variable Y follows the following model:

Y=\beta_{0}+\beta_{1}X+\epsilon,

where X are the covariates, and epsilon has variance sig. In ODS design, a simple random sample is taken from the full cohort, then two supplemental samples are taken from two tails of Y, i.e. (-Infty, mu_Y - a*sig_Y) and (mu_Y + a*sig_Y, +Infty). Because ODS data has biased sampling nature, naive regression analysis will yield biased estimates of the population parameters. Zhou et al. (2002) describes a semiparametric empirical likelihood estimator for estimating the parameters in the primary outcome model.

Value

A list which contains the parameter estimates for the primary response model:

Y=\beta_{0}+\beta_{1}X+\epsilon,

where epsilon has variance sig. The list contains the following components:

beta

parameter estimates for beta

sig

estimates for sig

pis

estimates for the stratum probabilities

grad

gradient

hess

hessian

converge

whether the algorithm converges: True or False

i

Number of iterations

Examples

library(ODS)
# take the example data from the ODS package
# please see the documentation for details about the data set ods_data

Y <- ods_data[,1]
X <- cbind(rep(1,length(Y)), ods_data[,2:5])

# use the simple random sample to get an initial estimate of beta, sig #
# perform an ordinary least squares #
SRS <- ods_data[1:200,]
OLS.srs <- lm(SRS[,1] ~ SRS[,2:5])
OLS.srs.summary <- summary(OLS.srs)

beta <- coefficients(OLS.srs)
sig <- OLS.srs.summary$sigma^2
pis <- c(0.1,0.8,0.1)

# the cut points for this data is Y < 0.162, Y > 2.59.
a <- c(0.162,2.59)
rs.size <- 200
size <- c(100,0,100)
strat <- c(1,2,3)

odsmle(Y,X,beta,sig,pis,a,rs.size,size,strat)