DA_AllEE5_inside: Internal calculation of estimating equation for DA_AllEE5

Description

This is the internal function to solve the estimating equation constructed by pair (N1,M1), (N1,M2), (N2,M1) and (N2,M2) with selection bias probility $\pi(s,\eta)$ included. Since it's a internal function for function DA_AllEE5, thus it's not a necessary or important function.

Usage

DA_AllEE5_inside(beta, CASEZ_1, CASEZ_2, CASEZhat_1, CASEZhat_2, CASEZhat_22, CONTZ_1, CONTZ_2, CONTZhat_1, CONTZhat_2, CONTZhat_22, prob_case_1, prob_case_11, prob_case_2, prob_case_22, prob_cont_1, prob_cont_2, p, pi_case_1, pi_case_1_t, pi_case_2, pi_case_2_t, pi_cont_1, pi_cont_1_t, pi_cont_2, Z_case_pi_1, Z_case_pi_1_t, Z_case_pi_2, Z_case_pi_2_t, Z_cont_pi_1, Z_cont_pi_1_t, Z_cont_pi_2, J_step3, V_step3, pwt_cont_2, subset_2, subset_3, subset_4)

Arguments

beta

Parameter $\beta$.

CASEZ_1, CASEZhat_1

case data(N1) from case-control study, details please see definition in the help of realdata_covariates.

CASEZ_2, CASEZhat_2, CASEZhat_22

CTR data(N2), details please see definition in the help of realdata_covariates.

CONTZ_1, CONTZhat_1

control data(M1) from case-control study, details please see definition in the help of realdata_covariates.

CONTZ_2, CONTZhat_2, CONTZhat_22

BRFSS data(M2), details please see definition in the help of realdata_covariates.

prob_cont_1, prob_cont_2, prob_case_1, prob_case_11, prob_case_2, prob_case_22, pwt_cont_2

please see definition in the help of realdata_alpha.

Number of parameters, a constant value of 8.

pi_case_1, pi_case_1_t, pi_case_2, pi_case_2_t, pi_cont_1, pi_cont_1_t, pi_cont_2

selection bias

Z_case_pi_1, Z_case_pi_1_t, Z_case_pi_2, Z_case_pi_2_t, Z_cont_pi_1, Z_cont_pi_1_t, Z_cont_pi_2

part of variables from covariates, used for the estiamtion of variance.

J_step3

Derivative of the estimating equation.

V_step3

Variance of the estimating equation.

subset_2

A vector of 1:(p-2).

subset_3

A vector of 1:p.

subset_4

A vector of 1:(p-2).

Value

A list of (f,J,V)

f The final form of the estimating equation after adjusting eta.
J_step3 The derivative of the estimating equation.
V_step3 The variance of the estimating equation.

Details

The function solves estimating equation based on GMM combined estimating equations with handling selection bias. It also accounts for the uncertainty due to the estimated value of eta. The function will output the estimating equation at current input value beta. Hence it can be used in "nleqslv" to solve for $\beta$. Because the function also outputs J and V, the asymptotic variance of $\beta$ can be calculated in a straightforward way. $\hat{Z}_l$ may be highly correlated with Z_d, so it is removed in the estimation. And it has to be careful in constructing f, J and V.