odsmle: MSELE estimator for analyzing the primary outcome in ODS...
In Yinghao-Pan/ODS: Statistical Methods for Outcome-Dependent Sampling Designs

Description Usage Arguments Details Value Examples

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).

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

`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).

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

Y = beta0 + beta1*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.

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

Y = beta0 + beta1*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

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)