Semiparametric Estimation in the Secondary Analysis of Case-Control Studies

. Author manuscript; available in PMC: 2016 Jan 28.

Published in final edited form as: J R Stat Soc Series B Stat Methodol. 2015 Feb 15;78(1):127–151. doi: 10.1111/rssb.12107

Algorithm 1: Computing the Locally Efficient Score Function

The first two steps are done only once.

Posit a model for η₂(ε, x) which has mean zero, and calculate (5), calling the result S*(X, Y, D). Use S*(·) in place of S(·) in (6)-(7).
Estimate f_X|D(x, d) by a kernel density estimate among the data with D_i = d, with result f̂_x|D(x, d).

The rest of the steps are done iteratively in the estimation algorithm.

Solve ${\hat{π}}_{0} = \sum_{i = 1}^{N} H (0, X_{i}, Y_{i}) {N_{0} H (0, X_{i}, Y_{i}) / {\hat{π}}_{0} + N_{1} H (1, X_{i}, Y_{i}) / (1 - {\hat{π}}_{0})}^{- 1}$ to obtain π̂₀ and set π̂₁ = 1 − π̂₀.
In the definition of κ(x, y) in (6), form κ̂(x, y) by replacing π_d by π̂_d. Define κ̂_i = κ̂(X_i, Y_i).
Define f̂_di = f̂_{D|X, Y}(d, X_i, Y_i) = N_dH(d, X_i, Y_i)κ̂_i/(Nπ̂_d).
For any function h(d, x, y) in (6), estimate E{h(D, X, Y) | X, D = d) by nonparametric regression among observations with D_i = d.
For any function h(d, x, y) in (6), estimate E{h(D, X, Y) | D = d) as $\hat{E} {h (D, X, Y) | D = d) = \sum_{i = 1}^{N} h (d, X_{i}, Y_{i}) {\hat{f}}_{di} / \sum_{i = 1}^{N} {\hat{f}}_{di}$ .
For any function h(d, x, y) in (6), estimate E{h(D, Y, X)|ε, X} by $\hat{E} {h (D, Y, X) | ε, X} = \sum_{d = 0}^{1} N_{d} H (d, X, Y) h (d, Y, X) \hat{κ} (X, Y) / (N {\hat{π}}_{d})$ .
For any function h(d, x, y) in (6), estimate E_true{h(D, X, Y) | X) by ${\hat{E}}_{true} {h (D, X, Y) | X) = \sum_{d = 0}^{1} {\hat{π}}_{d} \hat{E} {h (d, X, Y) | X, D = d) {\hat{f}}_{X | D} (X, d) / \sum_{d = 0}^{1} {\hat{π}}_{d} {\hat{f}}_{X | D} (X, d)$ .

Application to the terms in (6) yields ĝ(ε_i, X_i) and v̂_d, and we then form

{\hat{S}}_{eff}^{*} (D, X_{i}, Y_{i}) = S^{*} (X, Y, D) - \hat{g} (ε, X) - (1 - D) {\hat{v}}_{0} - D {\hat{v}}_{1}

We have described the algorithm when X is continuous. When X is discrete, one simply replaces the density estimators and various nonparametric regressions with the corresponding averages associated with the different x values.