se.spmle: standard error for MSELE estimator
In ODS: Statistical Methods for Outcome-Dependent Sampling Designs

Description Usage Arguments Value Examples

se.spmle calculates the standard error for MSELE estimator in Zhou et al. 2002

1	se.spmle(y, x, beta, sig, pis, a, N.edf, rhos, strat, size.nc)

`y`	vector of the primary response
`x`	the design matrix with a column of 1's for the intercept
`beta`	final estimates of the regression coefficients obtained from odsmle
`sig`	final estimate of the error variance obtained from odsmle
`pis`	final estimates of the stratum probabilities obtained from odsmle
`a`	vector of cutpoints for the primary response (e.g., a = c(-2.5,2))
`N.edf`	should be the size of the SRS (simple random sample)
`rhos`	which is size/pis, where size is a vector representing the stratum sizes of supplemental samples. e.g. size = c(100, 0, 100), and pis are the final estimates obtained from odsmle.
`strat`	vector that indicates the stratum numbers of supplemental samples, except that you should only list stratum with size > 0. (e.g. if the supplemental size is c(100, 0, 100), then the strat vector should be c(1,3))
`size.nc`	total size of the validation sample (SRS plus supplemental samples)

A list which contains the standard error estimates for betas in the model :

Y = beta0 + beta1*X + epsilon,

where epsilon has variance sig.

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)

# obtain the parameter estimates
ODS.model = odsmle(Y,X,beta,sig,pis,a,rs.size,size,strat)

# calculate the standard error estimate
y <- Y
x <- X
beta <- ODS.model$beta
sig <- ODS.model$sig
pis <- ODS.model$pis
a <- c(0.162,2.59)
N.edf <- rs.size
rhos <- size/pis
strat <- c(1,3)
size.nc <- length(y)

se = se.spmle(y, x, beta, sig, pis, a, N.edf, rhos, strat, size.nc)

# summarize the result
ODS.tvalue <- ODS.model$beta / se
ODS.pvalue <- 2 * pt( - abs(ODS.tvalue), sum(rs.size, size)-2)

ODS.results <- cbind(ODS.model$beta, se, ODS.tvalue, ODS.pvalue)
dimnames(ODS.results)[[2]] <- c("Beta","SEbeta","tvalue","Pr(>|t|)")
row.names(ODS.results) <- c("(Intercept)","X","Z1","Z2","Z3")

ODS.results