Estimate_PLMODS {ODS}

Partial linear model for ODS data

Description

Estimate_PLMODS computes the estimate of parameters in a partial linear model in the setting of outcome-dependent sampling. See details in Zhou, Qin and Longnecker (2011).

Usage

Estimate_PLMODS(X, Y, Z, n_f, eta00, q_s, Cpt, mu_Y, sig_Y, degree, nknots,
  tol, iter)

Arguments

X

n by 1 matrix of the observed exposure variable

Y

n by 1 matrix of the observed outcome variable

Z

n by p matrix of the other covariates

n_f

n_f = c(n0, n1, n2), where n0 is the SRS sample size, n1 is the size of the supplemental sample chosen from (-infty, mu_Y-a*sig_Y), n2 is the size of the supplemental sample chosen from (mu_Y+a*sig_Y, +infty).

eta00

a column matrix. eta00 = (theta^T pi^T v^T sig0_sq)^T where theta=(alpha^T, gamma^T)^T. We refer to Zhou, Qin and Longnecker (2011) for details of these notations.

q_s

smoothing parameter

Cpt

cut point a

mu_Y

mean of Y in the population

sig_Y

standard deviation of Y in the population

degree

degree of the truncated power spline basis, default value is 2

nknots

number of knots of the truncated power spline basis, default value is 10

tol

convergence criteria, the default value is 1e-6

iter

maximum iteration number, the default value is 30

Details

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

E(Y|X,Z)=g(X)+Z^{T}\gamma

where X is the expensive exposure, Z are other covariates. 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, Qin and Longnecker (2011) describes a semiparametric empirical likelihood estimator for estimating the parameters in the partial linear model.

Value

Parameter estimates and standard errors for the partial linear model:

E(Y|X,Z)=g(X)+Z^{T}\gamma

where the unknown smooth function g is approximated by a spline function with fixed knots. The results contain the following components:

alpha

spline coefficient

gam

other linear regression coefficients

std_gam

standard error of gam

cov_mtxa

covariance matrix of alpha

step

numbers of iteration requied to acheive convergence

pi0

estimated notation pi

v0

estimated notation vtheta

sig0_sq0

estimated variance of error

Examples

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

nknots = 10
degree = 2

# get the initial value of the parameters from standard linear regression based on SRS data #
dataSRS = ods_data[1:200,]
YS = dataSRS[,1]
XS = dataSRS[,2]
ZS = dataSRS[,3:5]

knots = quantileknots(XS, nknots, 0)
# the power basis spline function
MS = Bfct(as.matrix(XS), degree, knots)
DS = cbind(MS, ZS)
theta00 = as.numeric(lm(YS ~ DS -1)$coefficients)
sig0_sq00 = var(YS - DS %*% theta00)
pi00 = c(0.15, 0.15)
v00 = c(0, 0)
eta00 = matrix(c(theta00, pi00, v00, sig0_sq00), ncol=1)
mu_Y = mean(YS)
sig_Y = sd(YS)

Y = matrix(ods_data[,1])
X = matrix(ods_data[,2])
Z = matrix(ods_data[,3:5], nrow=400)

# In this ODS data, the supplemental samples are taken from (-Infty, mu_Y-a*sig_Y) #
# and (mu_Y+a*sig_Y, +Infty), where a=1 #
n_f = c(200, 100, 100)
Cpt = 1

# GCV selection to find the optimal smoothing parameter #
q_s1 = logspace(-6, 7, 10)
gcv1 = rep(0, 10)

for (j in 1:10) {

  result = Estimate_PLMODS(X,Y,Z,n_f,eta00,q_s1[j],Cpt,mu_Y,sig_Y)
  etajj = matrix(c(result$alpha, result$gam, result$pi0, result$v0, result$sig0_sq0), ncol=1)
  gcv1[j] = gcv_ODS(X,Y,Z,n_f,etajj,q_s1[j],Cpt,mu_Y,sig_Y)
}

b = which(gcv1 == min(gcv1))
q_s = q_s1[b]
q_s

# Estimation of the partial linear model in the setting of outcome-dependent sampling #
result = Estimate_PLMODS(X, Y, Z, n_f, eta00, q_s, Cpt, mu_Y, sig_Y)
result