estimation.weights.phase3 {CaseCohortCoxSurvival}

estimation.weights.phase3

Description

Estimates the weights for the third phase of sampling (due to missingness).

Usage

estimation.weights.phase3(B.phase3, total.phase2, gamma0 = NULL, niter.max = NULL, 
epsilon.stop = NULL)

Arguments

B.phase3

matrix for the case-cohort (phase-three data), with phase-three sampling strata indicators. It should have as many columns as phase-three strata (J^{(3)}), with one 1 per row, to indicate the phase-three stratum position.

total.phase2

vector of length J^{(3)} with un-weighted column totals for B in the phase-two data (i.e., using all the individuals, even the ones with missing covariate data).

gamma0

vector of length J^{(3)} with initial values for \gamma (Lagrangian multipliers), to be used as seed in the iterative procedure. Default is (0,...,0).

niter.max

maximum number of iterations for the iterative optimization algorithm. Default is 10^4 iterations.

epsilon.stop

threshold for the difference between the estimated weighted total and the total in the whole cohort. If this difference is less than the value of epsilon.stop, no more iterations will be performed. Default is 10^(-10).

Details

estimation.weights.phase3 estimates the phase-three sampling weights by solving in \gamma

\sum_{j=1}^J \sum_{i=1}^{n^{(j)}} \lbrace \xi_{i,j} V_{i,j} \text{exp}( \gamma' B_{i,j}) B_{i,j} - \xi_{i,j} B_{i,j} \rbrace = 0,

with \xi_{i,j} the phase-two sampling indicator and V_{i,j} the phase-three sampling indicator of individual i in stratum j, and with \sum_{j=1}^J \sum_{i=1}^{n^{(j)}} \xi_{i,j} B_{i,j} the total in the phase-two data. See Section 5.2 in Etievant and Gail (2023). The Newton Raphson method is used to solve the optimization problem.

In the end, the estimated weights are given by \text{exp}(\hat \gamma' B_{i,j}), and \sum_{j=1}^J \sum_{i=1}^{n^{(j)}} \xi_{i,j} V_{i,j} \text{exp}(\hat \gamma' B_{i,j}) B_{i,j} gives the estimated total.

Value

gamma.hat: vector of length J^{(3)} with final gamma values.

estimated.weights: vector with the estimated phase-three weights for the individuals in the case-cohort (phase-three data), computed from B.phase3 and gamma.hat.

estimated.total: vector with the estimated totals, computed from the estimated.weights and B.phase3.

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

influences.missingdata, influences.RH.missingdata, influences.CumBH.missingdata and influences.PR.missingdata.

Examples

data(dataexample.missingdata, package="CaseCohortCoxSurvival")

casecohort      <- dataexample.missingdata$casecohort # a simulated stratified case-cohort
# phase-two data: dataexample.missingdata$casecohort.phase2 
B.phase2        <- dataexample.missingdata$B.phase2
B.phase3        <- dataexample.missingdata$B.phase3

total.B.phase2  <- colSums(B.phase2)
J3              <- ncol(B.phase3)

estimation.weights.p3 <- estimation.weights.phase3(B.phase3 = B.phase3, 
                                        total.phase2 = total.B.phase2, 
                                        gamma0 = rep(0, J3), 
                                        niter.max = 10^(4),
                                        epsilon.stop = 10^(-10))

# print estimated phase-three weights
#estimation.weights.p3$estimated.weights