product.covar.weight {CaseCohortCoxSurvival}

product.covar.weight

Description

Computes the product of joint design weights and joint sampling indicators covariances, needed for the phase-two component of the variance (with design or calibrated weights).

Usage

product.covar.weight(casecohort, stratified = NULL)

Arguments

casecohort

if stratified = TRUE, data frame with status (case status), W (the J strata), strata.m (vector of length J with the numbers of sampled individuals in the strata) and strata.n (vector of length J with the strata sizes), for each individual in the stratified case-cohort data. If stratified = FALSE, data frame with status (case status), m (number of sampled individuals) and n (cohort size), for each individual in the un-stratified case-cohort data.

stratified

was the sampling of the case-cohort stratified on W? Default is FALSE.

Details

product.covar.weight creates the matrix with the products of joint design weights and joint sampling indicator covariances, for the non-cases in the case cohort. In other words, it has as many rows and columns as non-cases in the case cohort, and contains the w_{i,k,j} \sigma_{i,k,j}, with

w_{i,k,j} = \frac{n^{(j)}(n^{(j)} -1)}{m^{(j)}(m^{(j)} -1)} if individuals i and k in stratum j are both non-cases, and w_{i,k,j} = \left( \frac{n^{(j)}}{m^{(j)}} \right)^2 otherwise, i \neq k \in \lbrace 1, \dots, n^{(j)} \rbrace, j \in \lbrace 1, \dots, J \rbrace.

w_{i,i,j} = \frac{n^{(j)}}{m^{(j)}} if individuals i in stratum j is a non-case, i \in \lbrace 1, \dots, n^{(j)} \rbrace, j \in \lbrace 1, \dots, J \rbrace.

\sigma_{i,k,j} = \frac{m^{(j)}(m^{(j)} -1)}{n^{(j)}(n^{(j)} -1)} - \left( \frac{m^{(j)}}{n^{(j)}} \right)^2 if individuals i and k in stratum j are both non-cases, i \neq k \in \lbrace 1, \dots, n^{(j)} \rbrace, j \in \lbrace 1, \dots, J \rbrace.

\sigma_{i,i,j} = \frac{m^{(j)}}{n^{(j)}} - \left(1 - \frac{m^{(j)}}{n^{(j)}} \right) if individuals i in stratum j is a non-case, i \in \lbrace 1, \dots, n^{(j)} \rbrace, j \in \lbrace 1, \dots, J \rbrace.

See Section 3.3 in Etievant and Gail (2023).

Value

product.covar.weight: matrix with the products of joint design weights and joint sampling indicator covariances, for the non-cases in the case-cohort.

References

Etievant, L., Gail, M.H. (2023). Cox model inference for relative hazard and pure risk from stratified weight-calibrated case-cohort data. Submitted.

See Also

variance, that uses product.covar.weight to compute the variance estimate that follows the complete variance decomposition (superpopulation and phase-two variance components).

Examples

data(dataexample, package="CaseCohortCoxSurvival")

casecohort  <- dataexample$casecohort # a simulated stratified case-cohort

prod.covar.weight <- product.covar.weight(casecohort, stratified = TRUE)

nrow(prod.covar.weight)
ncol(prod.covar.weight)
sum(casecohort$status == 0) # number of non-cases in the case-cohort