G-estimation algorithm accounting for censoring by end

We now describe further the structural nested model used to estimate the time ratio per 100 WLM of exposure in the main text. We note that our choice of the model for the effect of exposure (repeated from the text) is shown in A.1.

T0¯=m+

T

Z

m

(1+ψX¯k)d k (A.1)

This SNAFT model differs from typical published examples (such as that shown in modelA.2 and e.g.Witteman et al.(1998);Chevrier et al.(2012);Naimi et al.(2014a)).

T0¯=m+

T

Z

m

e x p(ψX¯k)d k (A.2)

However, we show that our model is a valid form for a SNAFT model. Equation A.1 can be expressed equivalently using the function that relates the observed failure times to the potential failure times as:

T0¯=m+

T

Z

m

(1+ψX¯k)d k=h(T, ¯Xk,ψ) (A.3)

Thus, the functionh(T, ¯Xk,ψ)takes as its input the observed failure time, the expo-

sure of interest, and the value of 1-TR and yields the potential failure times we would observe under no exposure.

Robins notes that the functionh(T, ¯Xk,ψ)should satisfy three criteria:

2. consistent with null-hypothesis:h(t, ¯Xk,ψ=0) =t so thatψ=0 represents the

null hypothesis of no effect of exposure on the time of death

3. monotonicity:h(t, ¯Xk,ψ)>h(u, ¯Xk,ψ)ift >u

The model shown in (A.3) meets conditions 1 and 2 by noting that if eitherψ=0 or ¯

Xk is always 0, that the function reduces tom+

Rt

m1d k=t. In the case of radon and

lung cancer, (A.3) meets the third criteria by the linear-no threshold assumption under- lying the carcinogenic effects of radon. Practically, this implies that the TR will always be positive. In the case of parameter estimates that imply a protective effect of radon, we need only assume thatψ_∗M AX(X¯k)>−1, which guarantees monotonicity. In the

miner data, we observed a maximum cumulative exposure of around 60*100 WLM, implying that we would observe practical violations of monotonicity for estimates of the TR<0.983. Our analyses uniformly indicate TR>1, so this is not a concern in the miner data.

Censoring occurs in our analysis of the CPUM under three separate mechanisms: individuals are considered censored if they a) are lost to follow-up prior to death, b) die from a cause other than lung cancer or c) survive until the end of the study on 31 December 2005. Following other analyses of the CPUM, we consider a. and b. to be equivalent and we assume that there are no unmeasured risk factors for the censoring event. Under censoring events a, b or c, whereC is the time at censoring, analysis of time to event data typically considers estimation usingZ ≡M I N(T,C)and∆≡I(T < C), whereI(·)is the indicator function taking on values 1 or 0.

For the SNAFT model, we consider similar variablesZ0¯ _{≡M I N(T}¯0_,C0¯_)and∆¯0_≡

I(T0¯_<C¯0_{). Redefining the administrative censoring time C asC}0¯_{is necessary because} of one of the following reasons 1) if exposure is harmful (ψ >0), then we would expect that some of the events observed under no exposure would be unobserved if, in fact, the individual had been unexposed (i.e. life would be extended past C by remaining unexposed) and C0¯ _{should allow potential failure times to be observed only if they}

would have been observed under the minimum exposure; or 2) if exposure is beneficial

(ψ <0), then those with the shortest potential times under no exposure will (in general)

be the highest exposed, soC0¯_{should allow potential failure times to be observed only} if they would have been observed under the maximum exposure. If exposure has no effect on survival (ψ=0), then censoring times do not need to be adjusted.

Since radon is an established risk factor for lung cancer, we only need consider this calculation when ˜ψ >0. If we define the age at start of follow-up as m, then a possible function forC0¯_{under our SNAFT model is}

C0¯=m+ (1+ψ˜M I N(X¯k))(C −m) (A.4)

In other words, since the minimum possible cumulative radon exposure is 0,C0¯₌

C. This function results in wider confidence intervals than the optimal functions for

C0¯_{discussed by Robins and Tsiatis (1992), but it is much simpler to calculate.}

Because exposure may affect how many events are observed under administrative censoring,C0¯_{, rather than C is used to define the set(Z}0¯_,∆0¯_{)so that the identifying} assumption of no unmeasured confounding (see §C.3)an be written as

(Z0¯,∆0¯)_⊥Xk|L¯k, ¯Xk−1,V0,T >k (A.5) Under the assumption of no unmeasured confounding, we can test (A.5) using the methods given in A.2.3, which forms the basis of g-estimation.

To adjust for selection bias due to possibly informative censoring due to deaths from other causes or loss to follow-up. We utilized inverse probability of censoring weights, where, at each timek, we estimate the weightWk as

W_kc = P r(Ck=0|X¯k−1,V0,Ck−1=Dk−1=0) P r(Ck=0|L¯k−1, ¯Xk−1,V0,Ck−1=Dk−1=0)

(A.6)

due to end of follow-up as described above. While we do not include the effects of smoking in our exposure model, we include smoking status as of 12/31/1985 (never, former, <1 pk/day, 1 pk/day, >1pk/day) in V0 to allow for the effect of smoking on deaths due to other causes.