G-estimation using estimating equation methodology

In the SNAFT analysis from chapter 3.1, we estimated ψ using the g-estimation algorithm described byRobins and Tsiatis(1992) and in detail byHernán et al.(2005) and in the appendix ofJoffe et al.(2012). This algorithm uses a modified score function

U(ψ˜) similar to that described in Chapter 6 inKalbfleisch and Prentice (2002). The estimate ˆψis the value of ˜ψfor which the score function is zero. This is analogous to maximum likelihood methods, in which the score function (the partial derivative of the likelihood function) will equal zero at a local maximum in the likelihood function. This approach is advantageous in thatU(ψ˜)is easy to calculate and a full regression model is not needed for each candidate ˜ψ.

To find ˆψ, we used the estimating equation

U(ψ˜) = m i n(Ti,k_Li) X k=m N X i=1 [(Xi k−p(Xi k)]∆0_{i k}(ψ˜)W_{i k}c (A.10)

whereXi k is the exposure in monthk for individuali,p(Xi k)is the expected monthly

radon exposure (described in §A.2.2) in month k, given the exposure and covariate history and that the miner is still at risk for lung cancer. ∆0

i k(ψ˜) is the indicator of

whether or not the individual is not censored by the end of follow up or a competing risk (0=censored, 1=experiences event of interest), under the candidate value for the time ratio -1 ˜ψ. The weightWc

i k is the inverse probability of censoring (due to compet-

ing risks) weight shown in Equation A.6. The solution ˜ψ=ψis found by finding value of ˜ψat whichU(ψ˜) =0 either through a grid search, a non-gradient-based optimiza- tion algorithm, or any optimization algorithm that calculates an empirical gradient and can be run with multiple initial values. Gradient-based optimizers can be used effectively whenU(ψ˜) =0 is a smooth function, such as if a cohort is followed to ex- tinction, but due to our choice of estimating equation, our estimating equation is not a smooth function, as can be seen in Figures A.1 and A.2. Joffe et al. (2012) describe the

performance of several optimization routines. The lack of a smooth estimating func- tion also implies thatU(ψ˜) =0 will not equal 0 for any value of ˜ψ, instead the value closest to 0 is used. The exposure model forp(Xk)can typically be fit using standard

software for generalized linear models, but solving the estimating equation requires a separate procedure.

ψ~

χ

(

ψ

~

)

Figure A.1: G-function for SNAFT model for lung cancer mortality reported in §3.1

To estimate confidence intervals for ˆψ, we again followed the approach outlined by

Joffe et al.(2012). Briefly, this approach involves minimizing aχ2_{test statistic that can} be calculated by

χ2_{(ψ˜) =U(ψ˜)V(U(ψ˜))U(ψ˜) (A.11)} WhereV(U(ψ˜))is the empirical variance ofU(ψ˜), calculated as

V(U(ψ˜)) = m i n(Ti,k_Li) X k=m N X i=1 [(Xi k−p(Xi k)]∆0i k(ψ˜)W c i k [(Xi k−p(Xi k)]∆0i k(ψ˜)W c i k 0 (A.12)

ψ~

χ

(

ψ

~

)

Figure A.2: G-function for SNAFT model for all cause mortality reported in §3.1

equal to the dimension ofψ. SinceU(ψ˜)is a discrete function of potential survival in- dicators∆0¯_{, the minimumχ}2_{(ψ˜)will not reach 0,χ}2_{(ψ˜)will be a step function and may} not be monotonic. InsteadU(ψ˜)(andχ2_{(ψ˜)) were calculated over a grid of potential}

˜

ψvalues and the (mean) value(s) of ˜ψfor whichχ2_{(ψ˜)was smallest were accepted as} the estimate ˆψ, the log time-ratio for a 100 working level month increase in cumulative radon exposure. We perform g-estimation using a grid search for models with a single

ψparameter and utilize an active-set algorithm with multiple start points for models with multipleψparameters. The full g-functions for the grid searches for our primary analyses with lung cancer and all cause mortality are shown in Figures A.1 and A.2.

For our primary analysis, we calculated 95% confidence intervals as the set of ˜ψval- ues surrounding the point estimate for whichχ2_{(ψ˜)<3.84. Our method yields conser-} vative values for the 95% confidence intervals, but is considerably less computationally intensive than alternative methods, such as the bootstrap.

As noted above, a g-estimation algorithm can also be fit by adding∆0¯_{to the expo-} sure model, but this method is much more computationally intensive, since a model for exposure is fit for every value of ˜ψin a grid search, whereasU(ψ˜)is generated us-

ing relatively fast matrix calculations (Hernán et al.(2005);Joffe et al.(2012)), andU(ψ˜)

readily incorporates multi-dimensional ˜ψ (i.e. if we were interested in the effects of multiple exposures). Further, optimization routines may be implemented as an alter- native to grid searches as a way to estimateψ, though this approach may be problem- atic in practice (Joffe et al.(2012)).

A.3 Simulation studies: Fitting structural nested accelerated failure time models using g-estimation

To assess the success of our g-estimation algorithm in estimating causal parame- ters, We performed simulations under a data generating structure that produced ob- served data similar to that in the CPUM.