Survivor bias in Mendelian randomization analysis

Survivor bias in Mendelian randomization analysis Summary Mendelian randomization studies employ genotypes as experimental handles to infer the effect of genetically modified exposures (e.g. vitamin D exposure) on disease outcomes (e.g. mortality). The statistical analysis of these studies makes use of the standard instrumental variables framework. Many of these studies focus on elderly populations, thereby ignoring the problem of left truncation, which arises due to the selection of study participants being conditional upon surviving up to the time of study onset. Such selection, in general, invalidates the assumptions on which the instrumental variables analysis rests. We show that Mendelian randomization studies of adult or elderly populations will therefore, in general, return biased estimates of the exposure effect when the considered genotype affects mortality; in contrast, standard tests of the causal null hypothesis that the exposure does not affect the mortality rate remain unbiased, even when they ignore this problem of left truncation. To eliminate “survivor bias” or “truncation bias” from the effect of exposure on mortality, we next propose various simple strategies under a semi-parametric additive hazard model. We examine the performance of the proposed methods in simulation studies and use them to infer the effect of vitamin D on all-cause mortality based on the Monica10 study with the genetic variant filaggrin as instrumental variable. 1. Introduction The genetic revolution has prompted a revival of Mendelian randomization studies (Katan, 1986; Davey Smith and Ebrahim, 2003). These employ genetic variants (e.g. filaggrin) to infer the effect of exposures, known to be affected by those variants (e.g. Vitamin D), on health outcomes (e.g. all-cause mortality), building on the notion that an association between these genetic variants and the outcome of interest can only be explained by an effect of the exposure on the outcome. The statistical analysis of Mendelian randomization studies elaborates on the instrumental variables design, which utilizes so-called instrumental variables $$Z$$ as experimental handles to infer the effect of an exposure $$X$$ on an outcome $$T$$ in the presence of confounding by possibly unmeasured variables $$U$$. Here, an instrumental variable is a variable (a) that is associated with the exposure, (b) that has no direct effect on the outcome other than through the exposure, so that $$T\perp\!\!\!\perp Z|X,U$$, and (c) whose association with the endpoint of interest is not confounded by unmeasured variables in the sense that $$Z\perp\!\!\!\perp U$$ (see e.g. Hernán and Robins, 2006). Assumption (a) is testable, and assumption (c) can, in general, reasonably well be justified on the basis of random Mendelian inheritance. Assumption (b) is typically harder to justify (Didelez and Sheehan, 2007; Lawlor and others, 2008). Instrumental variables analysis is well developed for prospective designs which randomly sample subjects, and next “prospectively” assess the instrumental variables, exposure, and outcome, each in turn, in order to deliver a random population sample of data on instrumental variables, exposure, and outcome. A typical example of such design is a randomized experiment in which, for instance, smokers are randomly encouraged to quit smoking or not ($$Z$$), and next smoking behavior ($$X$$), and health outcomes ($$T$$) are recorded (Permutt and Hebel, 1989). The Mendelian randomization design is different in that, typically, adult or elderly subjects are randomly sampled in the study. Here, at least the genetic instrument, but often also the exposure, reflects conditions that were already present before study onset (e.g. at birth). For instance, Tchetgen and others (2015) evaluate the effect of diabetes on mortality using data from the Health and retirement Study, a representative sample of persons aged 50 years or older and their spouses in the United States. Likewise, the Monica10 study, which we will analyze in Section 5, includes 2656 individuals of Danish origin between 40 and 71 years of age, who had been previously recruited from the Danish Central Personal Register into the Monica I study, as a random population sample in this age range (Olsen and others, 2007; Skaaby and others, 2013). Entry in these studies is obviously conditional on surviving the onset of the study, thereby resulting in left truncation or delayed entry. In Section 2 of this article, we show that this does not induce bias in standard tests of the causal null hypothesis that the considered exposure does not affect all-cause mortality, even when these tests ignore the problem of left truncation. However, when the exposure was present prior to entry in the study, then it generally induces (selection) bias in exposure effect estimates and may, moreover, bias tests of the causal null hypothesis that the considered exposure does not affect given nonmortality endpoints or cause-specific mortality. This is a major concern (Boef and others, 2015), especially in Mendelian randomization studies with mortality endpoints, because genetic instruments are typically weak and it is well known that even minor violations of the instrumental variables assumptions can induce sizeable biases when the instruments are weak (Bound and others, 1995). In Section 3, we propose simple strategies to correct this bias under studies with an all-cause mortality endpoint. This will be done under a class of semi-parametric additive hazard models introduced in Section 3. We examine the performance of the proposed methods in simulation studies in Section 4 and use them to infer the effect of vitamin D on all cause mortality based on the Monica10 study with the genetic variant filaggrin as instrumental variable in Section 5. We discuss limitations of the considered proposals, and the analysis of nonmortality endpoints in Section 6. 2. Delayed entry and survivor bias Let us use $$T$$ to denote the lifetime, $$T_0$$ the truncation time and $$C$$ the censoring time, all measured since birth. Consider a Mendelian randomization design which intends to collect independent and identically distributed data on $$Z_i,X_i,T_{i0}$$, the observed event time $$V_i=\min(T_i,C_i)$$ and the censoring indicator $$\Delta_i=I(T_i<C_i)$$ for a random sample of subjects $$i=1,...,n$$ who are alive at study onset, i.e. for whom $$T_i>T_{i0}$$. The study may in particular be designed to sample (with replacement) birth cohorts according to some distribution (possibly different from the actual population distribution); here, the truncation time $$T_0$$ refers to the age of people in a given birth cohort at the start of the study (which is well defined, even for individuals who died). For instance, we may consider randomly sampling people who were born 40–71 years ago from a birth register. However, the study merely retains alive subjects; that is, people for whom $$T\geq T_0$$. For each individual in the sample, we assess (at their entry time $$T_0$$) the genetic instruments (e.g. the filaggrin genotype), and record the exposure (e.g. Vitamin D). Each participant is subsequently followed until death, or censoring. Throughout, we will assume that $$X$$ captures the entire exposure history since birth. This assumption is implicit in all Mendelian randomization analyses since failure of it would generally imply failure of the so-called exclusion restriction (b). We will furthermore assume that $$(T,X,Z)\perp\!\!\!\perp T_0|U$$. This is guaranteed to hold when there is no birth cohort effect on the genotype distribution in the sense that $$Z\perp\!\!\!\perp T_0|U$$ (i.e. when the population allele frequencies are the same across birth cohorts). Indeed, in that case, one can augment the vector of variables $$U$$ to include $$T_0$$, in which case $$(T,X,Z)\perp\!\!\!\perp T_0|U$$ is trivially satisfied (and the instrumental variables assumption (a) continues to be satisfied). The above design is visualized in the causal diagram of Figure 1. It is seen via d-separation (Pearl, 2000) on this diagram that if Vitamin D has an effect on survival, then conditioning on surviving up until the point of study entry may induce an artificial association between the genotype and the unmeasured confounders as a result of collider bias (Boef and others, 2015). More formally, conditioning on a descendant, $$T>T_0$$, of a collider $$X$$ may render the causes $$Z$$ and $$U$$ dependent, thereby violating the instrumental variables assumption (c). Similar biases have been described in Mendelian randomization analyses of case-control and case-only studies (Bowden and Vansteelandt, 2011). The dependence of the event $$T>T_0$$ on $$Z$$ is likely weak in Mendelian randomization analyses that use weak instruments, in which case the instrumental variables assumption (c) is only mildly violated due to left truncation. However, as we will show later, the resulting bias remains potentially substantive because even minor violations of the instrumental variables assumptions may induce large bias when the instruments are weak. Fig. 1. View largeDownload slide Causal diagram with delayed entry; $$t$$ is an arbitrary fixed time point. Fig. 1. View largeDownload slide Causal diagram with delayed entry; $$t$$ is an arbitrary fixed time point. One exception occurs when the exposure has no effect on surviving study entry (e.g. when there are only long term effects or when the exposure only manifests itself after study entry), which would justify removal of the edge from $$X$$ to $$(T>T_0)$$ in Figure 1. This problem of delayed entry can thus in particular be ignored when testing for the presence of an exposure effect on all-cause mortality by assessing whether the survival risks are dependent on the instrumental variable $$Z$$ within the study population. Such tests can be performed using standard techniques (e.g. score tests under a Cox proportional hazards model, accounting for left truncation) from survival analysis. This finding is in line with the recommendations in VanderWeele and others (2014) that Mendelian randomization analyses are less vulnerable to bias when it comes to testing the null hypothesis of no exposure effect. Suppose now that interest lies in the exposure effect on an endpoint $$Y$$ (e.g. the effect of diabetes on inflammation markers or on cardiovascular mortality), other than all-cause mortality, recorded at time $$T_0$$. Then by applying the same reasoning as in the previous paragraph on the causal diagram of Figure 2, the instrumental variables assumption (c) is seen to be violated whenever there is an exposure effect on the risk of surviving study onset (or the truncation time and instrument are dependent). Standard instrumental variables analyses will thus generally return biased estimates of the effect of exposure on arbitrary outcomes $$Y$$, regardless of whether or not there is such effect, when the considered exposure affects mortality. Interestingly, tests of the causal null hypothesis that the considered exposure does not affect the outcome $$Y$$ are also generally biased in this case, except when the exposure has no effect on surviving study entry. Fig. 2. View largeDownload slide Causal diagram with delayed entry and nonmortality endpoint $$Y$$. Fig. 2. View largeDownload slide Causal diagram with delayed entry and nonmortality endpoint $$Y$$. Because the problem of left truncation may thus be a concern for all Mendelian randomization studies, we recommend that routine analyses incorporate an assessment of the exposure effect on mortality, where possible, regardless of whether all-cause mortality is the primary endpoint. At a minimum, we recommend an investigation of the association between the instrumental variable and the mortality endpoint. 3. Eliminating survivor bias In this section, we will develop methods for estimating the effect of some exposure $$X$$ on time to death $$T$$ from all causes, accounting for the above suggested selection bias. 3.1. Semiparametric additive hazard model Throughout we will parameterize the exposure effect on the hazard scale using the semiparametric additive hazard model defined as \begin{equation}\label{sah} \lambda(t|X,Z,U)=\omega(t,U)+\psi X, \end{equation} (3.1) for a range of time points $$t>0$$, where $$\lambda(t|X,Z,U)$$ is the conditional hazard of death at time $$t$$, given exposure, instrument and confounders. Here, $$\omega(t,U)$$ captures the effects of time and confounders on the hazard; unlike in the Aalen additive hazard model (Aalen, 1980), we leave it unspecified because hazards can show a complex time pattern, and moreover, parameterizing the effect of confounders on the hazard is difficult since they may be unmeasured. Martinussen and others (2017) consider the following further relaxation of the above model \begin{equation}\label{nsah} \lambda(t|X,Z,U)=\omega(t,U)+\psi(t) X, \end{equation} (3.2) for a range of time points $$t>0$$, where the exposure coefficient $$\psi(t)$$ is an unknown (integrable) function of time. In model (3.1) (with time coded in years), the parameter $$100\psi$$ essentially expresses how many more subjects out of a total of 100 would die by the end of the year if their exposure were increased with a unit. Alternatively, interpretation can be made on the relative risk scale upon noting that model (3.1) is equivalent with the following restrictions for each time $$t>0$$: \begin{equation}\label{rsah} \frac{P(T>t|X,U)}{P(T>t|X=0,U)}=\exp(-\psi t X). \end{equation} (3.3) Importantly, we thus assume this exposure effect on the relative risk scale to be the same at all levels of $$U$$; similar no-effect-modification assumptions are commonly imposed in the instrumental variables literature (Hernán and Robins, 2006). Under this assumption, we will achieve identification of the exposure effect $$\psi$$ and can moreover interpret $$\exp(-\psi t x)$$ as the marginalized or standardized relative risk \[\frac{E\left\{P(T>t|X=x,U)\right\}}{E\left\{P(T>t|X=0,U)\right\}}, \] as a result of collapsibility of the relative risk (Martinussen and Vansteelandt, 2013). The latter is indeed easier to interpret as it does not involve conditioning on unmeasured variables $$U$$. Moreover, letting $$T(x)$$ denote the counterfactual event time when $$X$$ is set to $$x$$, and assuming that $$U$$ is sufficient to adjust for confounding of the effect of $$X$$ on $$T$$ (in the sense that $$T(x)\perp\!\!\!\perp X|U$$ for all $$x$$), this can also be interpreted as the causal relative chance of surviving time $$t$$, $$P\{T(x)>t\}/P\{T(0)>t\}$$, at exposure level $$x$$ versus 0. Although we will let the time origin be time at birth, note that the multiplicative structure of the model enables immediate translation to other (fixed) time origins. 3.2. Estimation under restrictions on the exposure distribution Under the semiparametric additive hazard model, the failure of the instrumental variables assumption (c) alluded to in Section 2, can be understood more formally as follows. By Bayes’ rule, for each fixed $$t>0$$, the joint distribution of the instrumental variable and the unmeasured confounders amongst individuals who have entered the study and are still at risk at time $$t$$ equals \begin{eqnarray*} f(Z,U|T_0<t \leq T) &=& \frac{P(T_0<t \leq T|Z,U)}{P(T_0<t \leq T)}f(Z)f(U)\\ &=& \frac{P(T_0<t|U)}{P(T_0<t \leq T)}P(T\geq t|Z,U)f(Z)f(U)\\ &=&\frac{P(T_0<t|U)}{P(T_0<t \leq T)}E\left[\exp\left\{-\int_0^t \omega(s,U){\rm{d}}s-\psi t X\right\}|Z,U\right]f(Z)f(U), \end{eqnarray*} where we use that $$T_0\perp\!\!\!\perp (T,Z)|U$$, as previously assumed. In general, the moment generating function $$E\left[\exp(-\psi t X)|Z,U\right]$$ does not factorize into terms involving either $$Z$$ or $$U$$. This then renders $$Z$$ and $$U$$ dependent within the risk sets of subjects who are in the study and alive at a given time $$t$$, thus violating the instrumental variables assumptions. One exception occurs, however, when the instrument and unmeasured confounders do not interact (on the additive scale) in their effect on the exposure, in the following sense \begin{equation}\label{ls} X=s(Z,\epsilon_z)+t(U,\epsilon_u) \end{equation} (3.4) for specific functions $$s(.)$$ and $$t(.)$$ and residual errors $$\epsilon_z$$ and $$\epsilon_u$$ that satisfy $$\epsilon_z\perp\!\!\!\perp \epsilon_u|Z,U$$, $$\epsilon_z\perp\!\!\!\perp U|Z$$ and $$\epsilon_u\perp\!\!\!\perp Z|U$$. This is for instance satisfied when the exposure is normally distributed with mean $$E(X|Z,U)=s^*(Z)+t^*(U)$$ for specific functions $$s^*(.)$$ and $$t^*(.)$$, and residual variance that either depends on $$Z$$ or $$U$$, but not both; it holds much more generally as the residual errors $$\epsilon_z$$ and $$\epsilon_u$$ need not be normally distributed. Under assumption (3.4), the moment generating function $$E\left[\exp(-\psi t X)|Z,U\right]$$ factorizes into terms involving either $$Z$$ or $$U$$, but not both. From the above calculations, it then follows that the instrumental variables conditions are satisfied within each of the observed risk sets; that is, for all $$t$$, \begin{equation}\label{riskset} Z\perp\!\!\!\perp U|T_0<t \leq T. \end{equation} (3.5) In the remainder of this section, we will show how this independence can be exploited to develop inference for the cumulative effect $$\Psi(t)-\Psi(t_0),t>t_0$$ under model (3.2) for all $$t>t_0$$, where \[ \Psi(t)=\int_0^t \psi(s){\rm{d}}s, \] and $$t_0$$ is such that $$P(T_0<t_0<T)>\sigma>0$$. Identity (3.5), along with model restriction (3.2), implies that the estimating function \begin{equation}\label{ef} \left\{Z-E(Z|T_0<t \leq T)\right\}R(t)R_{0}(t)\left\{{\rm d}N(t)-\psi(t)X{\rm{d}}t\right\}\!, \end{equation} (3.6) is unbiased at each time $$t$$; here, $$R(t)=I(T>t)$$ is the at-risk indicator and $$R_0(t)=I(T_0<t)$$ is an indicator whether or not the considered subject had entered the study by time $$t$$. Indeed, the mean of (3.6) is readily seen to equal \[E\left[\left\{Z-E(Z|T_0<t \leq T)\right\}R(t)R_{0}(t)\omega(t,U){\rm{d}}t\right]=0,\] when (3.2) and (3.5) hold for the given $$t$$. This suggests the estimator \[\hat{\Psi}(t)-\hat{\Psi}(t_0)=\int_{t_0}^t \frac{\sum_{i=1}^n \left\{Z_i-\bar{Z}(s)\right\}R_{i0}(s){\rm{d}}N_i(s)}{\sum_{i=1}^n \left\{Z_i-\bar{Z}(s)\right\}R_{i0}(s)R_i(s)X_i},\] where \[ \bar{Z}(t)=\frac{\sum_{i=1}^n R_i(t)R_{0i}(t)Z_i}{\sum_{i=1}^n R_i(t)R_{0i}(t)}, \] is a nonparametric estimator of $$E(Z|T_{0}<t \leq T)$$. In Section 2 of Supplementary materials available at Biostatistics online, we show that this estimator is equivalent to the two-stage estimator briefly mentioned in the discussion of Tchetgen and others (2015), where in the first stage, the exposure is linearly regressed on the instrument within the risk set, and in the second stage a standard additive hazard model is fitted with the (time-varying) fitted value from the first stage as the only predictor. We moreover show that the estimator $$\hat{\Psi}(t)-\hat{\Psi}(t_0)$$ admits the following i.i.d. expansion \[\sqrt{n}\left(\hat{\Psi}(t)-\hat{\Psi}(t_0)-\Psi(t)+\Psi(t_0)\right)=\frac{1}{\sqrt{n}} \sum_{i=1}^n\epsilon_i(t) + o_p(1),\] for mean zero errors \[\epsilon_i(t)= \int_{t_0}^t \frac{ \left\{Z_i-E(Z|T_0<s\leq T)\right\}R_i(s)R_{i0}(s)}{E\left[\left\{Z_i-E(Z|T_0<s\leq T)\right\}R_{i0}(s)R_i(s)X_i\right]}\left[{\rm{d}}N_i(s)-{\rm{d}}\Psi(s)X_i-{\rm{d}}\Psi_0(s)\right]\!,\] where \[{\rm{d}}\Psi_0(s)=E\left\{{\rm d}N(s)-{\rm{d}}\Psi(s)X|T_0<s\leq T\right\}=\frac{E\left[R_{0}(s)R(s)\left\{{\rm d}N(s)-{\rm{d}}\Psi(s)X\right\}\right]}{E\left[R_{0}(s)R(s)\right]}.\] The variance of the limit distribution at a given time $$t$$ can thus be consistently estimated by the sample variance of $$\epsilon_i(t)$$ (upon substituting population expectations by sample analogs and $${\rm{d}}\Psi(s)$$ by $${\rm{d}}\hat{\Psi}(s)$$ at each time $$s\in[t_0,t]$$). To study temporal changes, it is more useful to consider a uniform confidence band. This can be derived based on the above i.i.d. representation as also outlined in Lin and others (2000) and Martinussen (2010). Let $$Q_1^m,\ldots ,Q_n^m$$ be independent standard normal variates. Then, given the data, \[W^m_n(t)\equiv \sqrt{n}\left(\hat{\Psi}^m(t)-\hat{\Psi}^m(t_0)-\Psi(t)+\Psi(t_0)\right)=\frac{1}{\sqrt{n}} \sum_{i=1}^n\epsilon_i(t)Q_i^m\] also converges in distribution to a zero-mean Gaussian process with variance given by the variance of $$\epsilon_i(t)$$. The limit distribution can thus be evaluated by generating a large number, $$M$$, of replicates of $$W^m_n(t),m=1,\ldots,M$$. Uniform confidence bands can now be based on the distribution of $$\sup_{t_0\leq t\leq \tau}|W^m_n(t)|$$, $$m=1, \ldots ,M$$, where $$\tau$$ is some finite time point. It is readily verified that the above results continue to hold in the presence of censoring, when $$N_i(t)$$ is redefined as $$N_i(t)=I(V_i\leq t,\Delta_i=1)$$ and $$R_i(t)$$ as $$R_i(t)=I(t\leq V_i)$$, provided that $$C_i\perp\!\!\!\perp (T_i,T_{i0},X_i,Z_i)|U_i$$. Under the assumption of a constant exposure effect (CST), i.e. in model (3.1) for $$t>t_0$$, the exposure effect $$\psi$$ can be estimated by solving an estimating equation based on the integrated estimating functions (3.6), to obtain: \[\hat{\psi}_{\rm ADD/EXP}=\frac{\sum_{i=1}^n \int_{T_{i0}}^{\infty} \left\{Z_i-\bar{Z}(t)\right\}{\rm{d}}N_i(t)}{\sum_{i=1}^n \int_{T_{i0}}^{\infty} \left\{Z_i-\bar{Z}(t)\right\}R_i(t)X_i{\rm{d}}t}, \] where the subscript “ADD/EXP” is a reminder that we are relying on the assumption (3.4) that instrument and confounders act additively on the exposure. It follows from the results in the Appendix that $$\sqrt{n}(\hat{\psi}_{\rm ADD/EXP}-\psi)$$ converges to a normal variate with mean zero and variance which can be consistently estimated by the variance of \[ \epsilon_i= \frac{\int_{t_0}^{\infty} \left\{Z_i-E(Z|T_0<s\leq T)\right\}R_i(s)R_{i0}(s)\left[{\rm{d}}N_i(s)-{\rm{d}}\Psi(s)X_i-{\rm{d}}\Psi_0(s)\right]}{\int_{t_0}^{\infty} E\left[\left\{Z_i-E(Z|T_0<s\leq T)\right\}R_{i0}(s)R_i(s)X_i\right]{\rm{d}}s}. \] 3.3. Estimation under restrictions on the exposure effect When the additivity assumption (3.4) on the exposure distribution is believed not to hold, then progress can still be made under the (partially untestable) assumption that the additive hazard model (3.1) holds for all $$t>0$$ (rather than just $$t>t_0$$). In particular, by a similar reasoning as in Martinussen and others (2011, 2017), the exposure effect under this model can be removed (in expectation) from the at-risk indicators $$R(t)$$, by calculating $$R(t)\exp\left(-\psi t X\right)$$. What remains must then be mean independent of the instrument $$Z$$ by the instrumental variables assumptions. This suggests estimating $$\psi$$ as the solution to the estimating equation \begin{equation}\label{knownz} 0=\sum_{i=1}^n \int_{T_{i0}}^{\infty} e(t)\left\{Z_i-E(Z)\right\}R_i(t)\exp\left(\psi X_it\right){\rm{d}}t, \end{equation} (3.7) for arbitrary (scalar) function $$e(t)$$, e.g. $$e(t)=1$$. That the solution to (3.7) delivers a consistent estimator of $$\psi$$ when $$E(Z)$$ equals the population expectation of the instrumental variable “at birth” (i.e. before deaths may occur), follows formally from the standard theory of M-estimators and the fact that the contributions to the above estimating equation then have expectation zero. Indeed, the estimating functions $$e(t)\left\{Z-E(Z)\right\}R_0(t)R(t)\exp\left(\psi Xt\right)$$ in (3.7) have mean \begin{eqnarray*} e(t)E\left[\left\{Z-E(Z)\right\}P(T_0<t|U)\exp\left\{-\int_{0}^t \omega(s,U){\rm{d}}s\right\}\right]&=&0, \end{eqnarray*} for each choice of $$e(t)$$, where we again use that $$(T,X,Z)\perp\!\!\!\perp T_0|U$$. It is readily verified that the above identity continues to hold in the presence of censoring, when $$R_i(t)$$ is redefined as $$R_i(t)=I(t\leq V_i)$$, provided that $$C_i\perp\!\!\!\perp (T_i,T_{i0},X_i,Z_i)|U_i$$. Solving (3.7) demands knowledge of the population expectation $$E(Z)$$, which may be available in certain Mendelian randomization studies that have external information on the genotype distribution (e.g. in the form of allele frequencies). The resulting solution will be denoted $$\hat{\psi}_{\rm CST/K}$$, where the subscript “CST/K” is a reminder that we assume a constant treatment effect (CST) and the population expectation of the instrumental variable to be a priori known (K). However, caution is needed when using this estimator, since choosing an incorrect value of $$E(Z)$$ generally leads to biased estimators of the exposure effect, even under the null hypothesis of no exposure effect, as we will confirm in the simulation studies later in this article. One may alternatively substitute $$E(Z)$$ by an estimate of the expectation $$E(Z|T>T_{0})$$ of $$Z$$ amongst people who have been recruited into the study. The resulting solution will be denoted $$\hat{\psi}_{\rm CST/N}$$, where the subscript “CST/N” is a reminder that we assume a CST and use a naïve (N) estimate of the population expectation of the instrumental variable. The estimator $$\hat{\psi}_{\rm CST/N}$$ has the advantage of being unbiased under the null hypothesis of no exposure effect, for then $$E(Z|T>T_{0})$$ equals $$E(Z)$$, but the disadvantage of being biased when the null hypothesis does not hold. It follows from the standard theory of M-estimators that, asymptotically, this bias equals $$E\left\{U_i(\psi)\right\}/E\left\{\partial U_i(\psi)/\partial \psi\right\}$$, evaluated at the population value $$\psi$$, with $$U_i(\psi)\equiv \int_{T_{i0}}^{\infty} e(t)\left\{Z_i-E(Z)\right\}R_i(t)\exp\left(\psi X_it\right){\rm{d}}t$$ the estimating function in (3.7); that is, \begin{equation}\label{bias} \frac{\int e(t)E\left[\left\{E(Z)-E(Z|T>T_{0})\right\}P(T_0<t|U)\exp\left\{-\int_{0}^t \omega(s,U){\rm{d}}s\right\}\right]{\rm{d}}t}{\int e(t)E\left[\left\{Z-E(Z|T>T_{0})\right\}P(T_0<t|U)E(X|Z,U)t\exp\left\{-\int_{0}^t \omega(s,U){\rm{d}}s\right\}\right]{\rm{d}}t}. \end{equation} (3.8) Letting $$X$$ be normal with mean $$E(X|Z,U)=\alpha_0+\alpha_1Z+\alpha_2U$$ and constant variance, $$Z$$ be normal with mean 0 and constant variance, and $$Z\perp\!\!\!\perp T_0|U$$, one can see after some algebra that \[E(Z|T>T_{0})=\frac{E\left\{ZP(T> T_0|Z)\right\}}{E\left\{P(T> T_0|Z)\right\}} = \frac{E\left\{Z\exp{\left(-\psi\alpha_1 ZT_0\right)}w(T_0)\right\}}{E\left\{\exp{\left(-\psi\alpha_1 ZT_0\right)}w(T_0)\right\}} =-\psi\alpha_1\frac{E\left\{T_0w^*(T_0)\right\}}{E\left\{w^*(T_0)\right\}}, \] for specific functions $$w(T_0),w^*(T_0)$$ that do not depend on $$\alpha_1$$. It follows that both the numerator and denominator of (3.8) are proportional to $$\alpha_1$$ (see the detailed calculations in the Supplementary materials available at Biostatistics online), and thus that the degree of collider bias is (to first order) the same regardless of the strength of the instruments. This formalizes our earlier suggestion that the problem of left truncation is important, even when the instruments are weak (i.e. when $$\alpha_1$$ is close to zero). This phenomenon has been noted recently in the epidemiological literature (Munafo and others, 2016) and will also be confirmed by simulation studies in the next section. In view of the above concern about bias, we propose to substitute $$E(Z_i)$$ in (3.7) by \begin{equation}\label{newz} \hat{E}(Z;\psi)\equiv \frac{\sum_{i=1}^n Z_i\exp\left(\psi T_{i0} X_i\right)}{\sum_{i=1}^n \exp\left(\psi T_{i0} X_i\right)}, \end{equation} (3.9) when the population allele frequencies are unknown. This is justified upon noting that \begin{eqnarray*} \frac{E\left\{Z\exp\left(\psi T_0 X\right)|T\geq T_0\right\}}{E\left\{\exp\left(\psi T_0 X\right)|T\geq T_0\right\}} &=&\frac{E\left\{I\left(T\geq T_0\right)Z\exp\left(\psi T_0 X\right)\right\}}{E\left\{I\left(T\geq T_0\right)\exp\left(\psi T_0 X\right)\right\}}\\ &=&\frac{E\left\{\int_0^{\infty} P\left(T\geq t_0|Z,X,U\right)Z\exp\left(\psi t_0 X\right)f_{T_0|Z,X,U}(t_0){\rm{d}}t_0\right\}}{E\left\{\int_0^{\infty} P\left(T\geq t_0|Z,X,U\right)\exp\left(\psi t_0 X\right)f_{T_0|Z,X,U}(t_0){\rm{d}}t_0\right\}}\\ &=&\frac{E\left\{\int_0^{\infty} \exp\left(-\int_{0}^{t_0} \omega(t,U){\rm{d}}t \right)Zf_{T_0|Z,X,U}(t_0){\rm{d}}t_0\right\}}{E\left\{\int_0^{\infty} \exp\left(-\int_{0}^{t_0} \omega(t,U)dt \right)f_{T_0|Z,X,U}(t_0){\rm{d}}t_0\right\}}\\ &=&\frac{E\left\{\exp\left(-\int_{0}^{T_0} \omega(t,U){\rm{d}}t \right)Z\right\}}{E\left\{\exp\left(-\int_{0}^{T_0} \omega(t,U){\rm{d}}t \right)\right\}}=E(Z), \end{eqnarray*} where we again use that $$Z\perp\!\!\!\perp (T_0,U)$$. Throughout, we will use $$\hat{\psi}_{\rm CST/U}$$ to denote the solution to (3.7) with $$E(Z)$$ substituted by $$\hat{E}(Z_i;\psi)$$. Here, the subscript “CST/U” is a reminder that we assume a CST and the population expectation of the instrumental variable to be unknown (U). The choice of index function $$e(t)$$ affects the efficiency of the solution to (3.7). To simplify integration over time, we here recommend to set $$e(t)=1$$ and defer a more detailed study on semi-parametric efficiency to future work. We conjecture that reasonable efficiency will then be obtained when mean-centering the exposure $$X_i$$, and choosing $$Z_i$$ to be the fitted value for subject $$i$$ as obtained from a model for the exposure given the instrumental variables(s). For this choice, $$e(t)=1$$ for all $$t>0$$, (3.7) reduces to \[0=\sum_{i=1}^n U_i(\psi),\] where \[U_i(\psi)=\left\{\begin{array}{@{}ccc} \left\{Z_i-E(Z)\right\}\left\{\exp\left(\psi X_iT_i\right)-\exp\left(\psi X_iT_{i0}\right)\right\}/(\psi X_i) & {\rm{when}} & \psi X_i\ne 0\\ \left\{Z_i-E(Z)\right\}(T_i-T_{i0}) & {\rm{when}} & \psi X_i= 0. \end{array}\right.\] It follows from the theory of M-estimators that, under weak regularity conditions, $$\sqrt{n}(\hat{\psi}-\psi)$$ converges to a normal mean zero variate with variance which can be consistently estimated by \[\frac{\frac{1}{n}\sum_{i=1}^n \left[U_i(\psi)-c(\psi)\left\{Z_i-\hat{E}(Z_i;\psi)\right\}\exp\left({\psi} X_iT_{i0}\right)\right]^2}{\left[\frac{1}{n}\sum_{i=1}^n V_i(\psi)-W_i(\psi)\right]^2},\] where \begin{eqnarray*} V_i(\psi)&=&\left\{\begin{array}{@{}ccc} \left\{Z_i-E(Z)\right\}\left\{\exp\left(\psi X_iT_i\right)T_i-\exp\left(\psi X_iT_{i0}\right)T_{i0}\right\}/\psi & {\rm{when}} & \psi \ne 0\\ \left\{Z_i-E(Z)\right\}(T_i^2-T_{i0}^2)X_i & {\rm{when}} & \psi = 0 \end{array}\right. \\ W_i(\psi)&=&\left\{\begin{array}{@{}ccc} U_i(\psi)/\psi & {\rm{when}} & \psi X_i \ne 0\\ \left\{Z_i-E(Z)\right\}(T_i^2-T_{i0}^2)X_i/2 & {\rm{when}} & \psi X_i = 0. \end{array}\right. \\ c(\psi)&=& \frac{\sum_{i=1}^n \left\{\exp\left(\psi X_iT_i\right)-\exp\left(\psi X_iT_{i0}\right)\right\}/(\psi X_i)}{\sum_{i=1}^n \exp\left(\psi T_{i0} X_i\right)}, \end{eqnarray*} and where $$E(Z)$$ is substituted by $$\hat{E}(Z_i;\psi)$$ in all expressions, and $$\psi$$ by $$\hat{\psi}_{\rm CST/U}$$. Also $$\sqrt{n}(\hat{\psi}_{\rm CST/K}-\psi)$$ converges to a normal variate with mean zero and variance which is likewise obtained, but upon setting $$c(\psi)$$ to zero, $$E(Z)$$ to the known population expectation, and $$\psi$$ to $$\hat{\psi}_{\rm CST/K}$$. 4. Simulation study To gain insight into the extent of the bias of instrumental variables analyses that ignore left truncation, as well as the performance of the novel proposals, we conducted a number of simulation experiments. The settings were chosen to mimic the situation observed in the Monica10 study, which we will analyze in Section 5. In particular, we first generated 10 000 samples of a dichotomous instrumental variable $$Z$$, coded 0 or 1, with mean 0.067, and independently generated a standard normal confounder $$U$$. We next generated the exposure $$X$$ to be normal with mean $$Z-2U$$ and variance 1 in the first simulation experiment, normal with mean $$Z-4U$$ and variance 1 in the second experiment, normal with mean $$0.5Z-2U$$ and variance 1 in the third experiment, Bernoulli distributed with mean expit$$(Z+U)$$ in the fourth experiment, and finally normal with mean $$0.75Z-4U$$ corresponding with a weak IV. We then standardized the exposure to give it mean zero and variance 1. In all cases, the event time was exponentially distributed with hazard $$0.24-0.1X+|U|/5$$. Entry ages were generated uniformly between 0 and 4, and 2571 subjects whose event time exceeds the entry time were randomly sampled from the generated population of 10 000. Twenty percent were potentially censored according to a uniform distribution on (0,10), and the rest were censored at $$t = 10$$, corresponding to the study being closed at this time point, leading to a cumulative censoring rate in the range of 11–13% across the 4 experiments. We next evaluated four estimators: the estimator $$\hat{\psi}_{\rm ADD/EXP}$$ which assumes that instrument and confounder act additively on the exposure, and the three estimators that assume CSTs since birth. In particular, we evaluated the naïve estimator $$\hat{\psi}_{\rm{CST/N}}$$ which ignores left truncation by solving (3.7) with the sample average of $$Z_i$$ in the observed sample in lieu of $$E(Z_i)$$; the estimator $$\hat{\psi}_{\rm{CST/K}}$$ which assumes known $$E(Z_i)$$ equal to 0.067; and the estimator $$\hat{\psi}_{\rm{CST/U}}$$ which makes no assumptions on the mean of the instrument. In rare occasions, the estimating equation (3.7) did not reach zero at any value of $$\psi$$, although the estimating equation reached a minimum or maximum close to zero. In that case, we have set the estimator to be the value of $$\psi$$ that corresponds with this minimum or maximum. In the Appendix, we show that this strategy still delivers a consistent estimator provided that this minimum or maximum of the estimating equation converges to zero as the sample size goes to infinity (van der Vaart, 1998). We moreover show how the asymptotic variance of the estimator can be calculated in that case. The simulation results are summarized in Table 1. They show that the naive estimator which ignores left truncation can be severely biased (relative bias of about 40%), and that the three proposed estimators are unbiased. This is even so for the estimator $$\hat{\psi}_{\rm ADD/EXP}$$ when the exposure is dichotomous, even though the additivity assumption (3.4) fails in that case; however, the estimator becomes more variable now. As theoretically predicted, the amount of bias does not change with the strength of the instrument. The sandwich estimators generally perform adequately in the sense of delivering confidence intervals with close to nominal—albeit slightly conservative—coverage, but they can be quite variable when the instrument is weak. This sometimes lead to discrepancies between the mean and median standard errors, which were partly due to 1 (or just a few) extreme outlier(s). This in turn has implications for the evaluation of coverage; e.g. while the naive estimator $$\hat{\psi}_{\rm CST/N}$$ reaches a coverage of 95.1% in Experiment 4, this reduces to 82.7% when the sample size is doubled. Also the estimator $$\hat{\psi}_{\rm ADD/EXP}$$ itself tends to be more variable when the instrument is weak. Limiting integration in the calculation of the estimator to the 80% percentile of the observed event times improved the results (e.g. it reduced the empirical standard deviation of $$\hat{\psi}_{\rm ADD/EXP}$$ from 0.253 to 0.206), by preventing highly variable estimates of the mean $$E(Z|T_0<t<T)$$ of the instrument in the observed risk set. Finally, using the known mean $$E(Z)$$ rather than an estimate of it, typically resulted in some minor precision loss, as is typical for instrumental variable estimators (see e.g. Vansteelandt and Goetghebeur, 2003, Section 2.4). Table 1. Average of the estimates (Mean); empirical standard error (ESE); median (mean) of the sandwich standard errors (SSE); coverage of 95% Wald confidence intervals (Coverage) Experiment Estimator Mean ESE Median (mean) SSE Coverage 1 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.061 0.040 0.038 (0.042) 80.2 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.100 0.064 0.059 (0.069) 95.1 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.105 0.071 0.064 (0.088) 95.6 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.100 0.061 0.064 (0.064) 97.4 2 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.063 0.082 0.073 (0.101) 91.6 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.095 0.112 0.105 (0.140) 94.7 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.095 0.111 0.110 (0.337) 94.4 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.097 0.253 0.122 (0.233) 98.3 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.062 0.084 0.079 (0.161) 90.0 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.089 0.111 0.109 (0.199) 93.0 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.090 0.111 0.113 (0.288) 94.9 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.104 0.230 0.135 (0.242) 98.5 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.061 0.057 0.048 (0.067) 95.1 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.098 0.081 0.070 (0.163) 99.4 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.096 0.078 0.083 (0.204) 98.9 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.104 0.091 0.089 (0.097) 97.9 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.063 0.101 0.094 (0.15) 90.7 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.089 0.134 0.128 (0.22) 91.1 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.088 0.122 0.131 (0.25) 93.2 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.0024$$^*$$ 3.05 0.214 (64.0) 99.3 Experiment Estimator Mean ESE Median (mean) SSE Coverage 1 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.061 0.040 0.038 (0.042) 80.2 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.100 0.064 0.059 (0.069) 95.1 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.105 0.071 0.064 (0.088) 95.6 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.100 0.061 0.064 (0.064) 97.4 2 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.063 0.082 0.073 (0.101) 91.6 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.095 0.112 0.105 (0.140) 94.7 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.095 0.111 0.110 (0.337) 94.4 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.097 0.253 0.122 (0.233) 98.3 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.062 0.084 0.079 (0.161) 90.0 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.089 0.111 0.109 (0.199) 93.0 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.090 0.111 0.113 (0.288) 94.9 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.104 0.230 0.135 (0.242) 98.5 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.061 0.057 0.048 (0.067) 95.1 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.098 0.081 0.070 (0.163) 99.4 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.096 0.078 0.083 (0.204) 98.9 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.104 0.091 0.089 (0.097) 97.9 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.063 0.101 0.094 (0.15) 90.7 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.089 0.134 0.128 (0.22) 91.1 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.088 0.122 0.131 (0.25) 93.2 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.0024$$^*$$ 3.05 0.214 (64.0) 99.3 $$^*$$The median estimate was $$-0.099$$ Table 1. Average of the estimates (Mean); empirical standard error (ESE); median (mean) of the sandwich standard errors (SSE); coverage of 95% Wald confidence intervals (Coverage) Experiment Estimator Mean ESE Median (mean) SSE Coverage 1 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.061 0.040 0.038 (0.042) 80.2 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.100 0.064 0.059 (0.069) 95.1 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.105 0.071 0.064 (0.088) 95.6 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.100 0.061 0.064 (0.064) 97.4 2 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.063 0.082 0.073 (0.101) 91.6 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.095 0.112 0.105 (0.140) 94.7 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.095 0.111 0.110 (0.337) 94.4 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.097 0.253 0.122 (0.233) 98.3 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.062 0.084 0.079 (0.161) 90.0 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.089 0.111 0.109 (0.199) 93.0 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.090 0.111 0.113 (0.288) 94.9 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.104 0.230 0.135 (0.242) 98.5 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.061 0.057 0.048 (0.067) 95.1 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.098 0.081 0.070 (0.163) 99.4 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.096 0.078 0.083 (0.204) 98.9 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.104 0.091 0.089 (0.097) 97.9 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.063 0.101 0.094 (0.15) 90.7 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.089 0.134 0.128 (0.22) 91.1 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.088 0.122 0.131 (0.25) 93.2 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.0024$$^*$$ 3.05 0.214 (64.0) 99.3 Experiment Estimator Mean ESE Median (mean) SSE Coverage 1 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.061 0.040 0.038 (0.042) 80.2 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.100 0.064 0.059 (0.069) 95.1 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.105 0.071 0.064 (0.088) 95.6 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.100 0.061 0.064 (0.064) 97.4 2 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.063 0.082 0.073 (0.101) 91.6 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.095 0.112 0.105 (0.140) 94.7 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.095 0.111 0.110 (0.337) 94.4 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.097 0.253 0.122 (0.233) 98.3 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.062 0.084 0.079 (0.161) 90.0 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.089 0.111 0.109 (0.199) 93.0 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.090 0.111 0.113 (0.288) 94.9 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.104 0.230 0.135 (0.242) 98.5 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.061 0.057 0.048 (0.067) 95.1 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.098 0.081 0.070 (0.163) 99.4 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.096 0.078 0.083 (0.204) 98.9 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.104 0.091 0.089 (0.097) 97.9 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.063 0.101 0.094 (0.15) 90.7 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.089 0.134 0.128 (0.22) 91.1 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.088 0.122 0.131 (0.25) 93.2 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.0024$$^*$$ 3.05 0.214 (64.0) 99.3 $$^*$$The median estimate was $$-0.099$$ 5. Analysis of the Monica10 study Vitamin D helps with the absorption of calcium in the body (besides playing a number of other roles), as a result of which vitamin D deficiencies may be linked to an increased risk for heart disease and subsequent mortality. This motivates our analysis, which aims to assess the effect of vitamin D status on all-cause mortality, using the filaggrin genotype as an instrument according to the principles of Mendelian randomization. The data source is the population based study Monica10, and vitamin D is measured as the serum 25-OH-D (nmol/L). The Monica I study was conducted in 1982–1984 and included examinations of 3785 individuals of Danish origin recruited from the Danish Central Personal Register as a random sample of the population. The ten-year follow-up study, Monica10, included 2656 individuals between 40 and 71 years. It was carried out in 1993–1994; for further details see Olsen and others (2007) and Skaaby and others (2013). We considered the following variables: $$Z$$ is an indicator of filaggrin mutation; $$X$$ is the standardized natural logarithm of the Vitamin D (nmol/l) measurements. The time-scale is time since birth, and the number of individuals with complete information on the variables examined was 2571. The odds of having a normal Vitamin D value ($$>$$30 nmol/L) was estimated to be 3.13 (95% CI: 1.28 to 7.76, P = 0.01) times larger for those with the filaggrin mutation compared to those without. The $$F$$-test statistic, when regressing Vitamin D on the instrument, is 7.34 (P = 0.007), demonstrating that the instrument is weak. The association of vitamin D in a standard Aalen additive hazards analysis, accounting for left truncation, corresponds with a protective additive effect on the hazard of $$-$$0.002 (95% confidence interval $$-$$0.0029 to $$-$$0.0011, $${\rm P}=7\times10^{-6}$$). Judging from the result, there seems to be an effect of Vitamin D on overall survival, but note that this is under the assumption of no unmeasured confounders. This assumption is likely violated because, for instance, people with poor health conditions might be less likely exposed to sunlight and thus have lower Vitamin D levels. A naïve Mendelian randomisation analysis—namely, the analysis in Section 3.3 with $$E(Z)$$ set to the sample average 0.0692 of the observed genotypes—gave a nearly identical effect size, corresponding with an additive effect on the hazard of $$\hat{\psi}_{\rm CST/N}=-0.0016$$ (95% confidence interval $$-$$0.0046 to 0.0014, P = 0.28). Also this analysis is likely invalid as a result of survivor bias. Assuming that the population allele frequency equals 0.046 Barker and others (2007), we obtain an additive effect $$\hat{\psi}_{\rm CST/K}=-0.026$$ (95% confidence interval $$-$$0.040 to $$-$$0.012, P = 0.0002). Figure 3 shows additional results for other allele frequencies. All of these results have the drawback, however, that they are biased and that corresponding tests of the null hypothesis are invalid when the wrong allele frequency is chosen. Relative to $$\hat{\psi}_{\rm CST/K}$$, we therefore prefer the estimator $$\hat{\psi}_{\rm CST/U}=-0.010$$ (95% confidence interval $$-$$0.026 to 0.005, P = 0.19), which does not pre-specify the allele frequency, but estimates it as $$\hat{E}(Z_i;\hat{\psi})=0.067$$. Fig. 3. View largeDownload slide Sensitivity analysis: $$\hat{\psi}_{\rm CST/K}$$ in function of $$E(Z)$$ (solid line) and 95% confidence intervals (dashed lines), along with the estimates $$\hat{\psi}_{\rm CST/K}$$ (corresponding to $$E(Z)=0.046$$), $$\hat{\psi}_{\rm CST/N}$$, $$\hat{\psi}_{\rm CST/U}$$, $$\hat{\psi}_{\rm ADD/EXP}$$, and their 95% confidence intervals. Fig. 3. View largeDownload slide Sensitivity analysis: $$\hat{\psi}_{\rm CST/K}$$ in function of $$E(Z)$$ (solid line) and 95% confidence intervals (dashed lines), along with the estimates $$\hat{\psi}_{\rm CST/K}$$ (corresponding to $$E(Z)=0.046$$), $$\hat{\psi}_{\rm CST/N}$$, $$\hat{\psi}_{\rm CST/U}$$, $$\hat{\psi}_{\rm ADD/EXP}$$, and their 95% confidence intervals. However, a major limitation of both estimators $$\hat{\psi}_{\rm CST/K}$$ and $$\hat{\psi}_{\rm CST/U}$$ is that their validity relies on the assumption that the exposure effect is constant over time since birth. This assumption is not innocent as it determines the degree of adjustment in (3.9), thereby enabling us to back-calculate the expectation of the instrumental variable “at birth”. It is also partially untestable in many Mendelian randomization studies where e.g. data are only available for an elderly population. Assuming that instrument and confounders act additively on the exposure, Figure 4 shows the nonparametric estimate proposed in Section 2.2. It suggests no exposure effect until roughly age 60, and lack of information from age 80 onwards, due to the IV being very weak from those ages onwards; overall it suggests that the assumption of a CST since birth is less likely. In view of this, we recommend the estimator $$\hat{\psi}_{\rm ADD/EXP}$$ for our conclusive data analysis, as its consistency only relies on correct specification of model (3.1) at times $$t$$ at which $$P(T_0<t \leq T)>0$$ (along with the stated assumptions on the exposure distribution). This is important as it implies that the estimator continues to be valid when the constant effect assumption fails prior to the (minimum) age of study-onset (e.g. when the exposure has no effect at young age). This estimator equals $$\hat{\psi}_{\rm ADD/EXP}=-0.025$$ (95% confidence interval $$-$$0.057 to 0.0056, P = 0.11), indicating that, over the course of follow-up, a standard deviation increase in the natural logarithm of Vitamin D is associated with on average 2.5 fewer deaths for each year of follow-up in each 100 persons alive at the start of the year; re-estimating the effect by limiting integration up to age 80 resulted in an effect estimate of $$\hat{\psi}_{\rm ADD/EXP}=-0.020$$. Fig. 4. View largeDownload slide Cumulative exposure effect: nonparametric (solid step function) and parametric (dashed straight line) estimator. Fig. 4. View largeDownload slide Cumulative exposure effect: nonparametric (solid step function) and parametric (dashed straight line) estimator. Our analysis above is a simplification of a complex reality and must therefore be viewed as an illustration, for various reasons. We only have information on the Vitamin D exposure at one point in time, so that we could not accommodate its time-varying nature. Furthermore, filaggrin deficiency is associated with an increased risk of ichtyosis vulgaris, atopic dermatitis, allergic rhinitis, and asthma (van den Oord and Sheikh, 2009), with asthma itself being linked with mortality; this may imply a violation of the IV assumptions. 6. Discussion The problem of left truncation and related survivor bias is pertinent to all Mendelian randomization studies of exposures that affect mortality. Assessing the exposure effect on mortality should thus be a key component of all Mendelian randomization studies, even those in which mortality is a secondary outcome. In this article, we have presented simple strategies as to how this can be done. The proposed strategies will enable future Mendelian randomization studies to gain power by focusing even more on selective populations of individuals at high risk. Our development considered a scalar instrument for pedagogic purposes. However, multivariate instruments can be efficiently incorporated by substituting $$Z$$ by the fitted value from a model for the exposure, given those instruments. Misspecification of such model would not invalidate our proposals, but at worst lead to a loss of efficiency. Furthermore, standard errors for the exposure effect proposed below are valid, regardless of whether $$Z$$ was directly observed or indirectly obtained as the fitted value from some regression model. Our development, like the rest of the Mendelian randomization literature, considered the exposure to be fixed. When $$X$$ is just one manifestation of a time-varying exposure process, it is very likely that not all of the instrument’s effect is mediated by the single-observed exposure assessment, as some of it may also be mediated by more proximal or distal exposures VanderWeele and others (2014). Mendelian randomization studies in which exposures are longitudinally collected over long periods of time, would provide a valuable source of information to account for this. We have furthermore focused on the relative chance of survival. In some applications, especially those in which survival chances are very high, it may be more interesting to focus on the relative risk of failure. The estimators of Section 3.3 are readily modified to this setting by redefining $$R(t)$$ as the failure indicator at time $$t$$; more work is needed to evaluate how to best adapt the estimators of Section 3.2 to this effect estimand. Section 3.3 assumed the exposure effect to be constant over time since birth. This assumption can be critical, and unfortunately is largely untestable in most Mendelian randomization studies. In view of this, in Section 3.2 we have developed an approach that makes no assumptions on the exposure effect at times before participants entered the study. We find this approach preferable, even though it relies on assumptions about the exposure distribution instead and appears more susceptible to weak instruments. In Section 4 of Supplementary materials available at Biostatistics online, we suggest an alternative sensitivity analysis approach. The proposal in Section 3.2 is also readily extended to incorporate adjustment for measured covariates $$C$$ under the assumption that $$Z\perp\!\!\!\perp U|C$$, upon substituting $$\bar Z(t)$$ by the fitted value from a regression of $$Z$$ on $$C$$ in subjects with $$R(t)=R_0(t)=1$$. Munafo and others (2016) have also highlighted how selection into Mendelian randomization studies can lead to biased effect estimates with general outcomes. In simulation studies, they found that small influences on selection could yield misleading results. Our work shows how unbiased inference about effects on mortality can be obtained in the presence of selection, but this does not generally extend to nonmortality outcomes. For instance, in model \[E(Y|X,Z,U)=\omega^*(U)+\gamma X, \] it is tempting to estimate the exposure effect $$\gamma$$ as the solution to estimating equation \[0=\sum_{i=1}^n \left(Z_i-\mu\right)(Y_i-\gamma X_i), \] with $$\mu$$ set to a nonparametric estimate of $$E(Z|T_0<T)$$ under the additivity assumption (3.4), or to $$\hat{E}(Z_i;\hat{\psi})$$ under the assumption of a CST since birth. However, even when assumption (3.5) holds for all $$t$$, it is not generally true that $$Z\perp\!\!\!\perp U|T_0<T$$, thus invalidating the choice $$\mu=E(Z|T_0<T)$$. More importantly, the outcome $$Y$$ is generally ill defined for people who did not survive up until study entry. As a result, the exposure effect is only well defined for the principal stratum of individuals who would have survived the time of study entry regardless of their exposure (Jemiai and others, 2007); our results are not directly applicable to estimation of the exposure effect for this latent subgroup. For the same reason, our results are not immediately applicable to handle problems of selection bias in Mendelian randomization studies that evaluate the effects of particular exposures on outcomes (e.g. disease progression) in individuals with chronic disease (e.g. multiple sclerosis or Parkinson’s disease), since a change in exposure may cause a change in disease status for some individuals, making the outcome ill-defined for them. Supplementary material Supplementary material is available at http://biostatistics.oxfordjournals.org. Acknowledgments The authors would like to thank Eric Tchetgen Tchetgen (Harvard University) for helpful discussions, and Stephen Burgess (University of Cambridge) and an anonymous reviewer for excellent comments that helped us to improve the article. The third author’s work is part of the Dynamical Systems Interdisciplinary Network, University of Copenhagen. Funding Strategic Basic Research PhD grant from the Research Foundation - Flanders (FWO) to O.D. Appendix Asymptotic distribution of $$\sqrt{n}\left(\hat{\Psi}(t)-\hat{\Psi}(t_0)\right)$$ Define \[\hat{\Psi}(t,\mu)-\hat{\Psi}(t_0,\mu)=\int_{t_0}^t \frac{\sum_{i=1}^n \left\{Z_i-\mu(s)\right\}R_{0i}(s){\rm{d}}N_i(s)}{\sum_{i=1}^n \left\{Z_i-\mu(s)\right\}R_i(s)R_{0i}(s)X_i}. \] Then, under standard regularity conditions, \begin{eqnarray*} \sqrt{n}\left(\hat{\Psi}(t)-\hat{\Psi}(t_0)-\Psi(t)+\Psi(t_0)\right)&=& \sqrt{n}\left(\hat{\Psi}(t,\mu)-\hat{\Psi}(t_0,\mu)-\Psi(t)+\Psi(t_0)\right)\\ &&+\sqrt{n}\left(\hat{\Psi}(t)-\hat{\Psi}(t_0)-\hat{\Psi}(t,\mu)-\hat{\Psi}(t_0,\mu)\right)\\ &=&\frac{1}{\sqrt{n}} \sum_{i=1}^n \int_{t_0}^t \frac{ \left\{Z_i-\mu(s)\right\}R_{0i}(s)R_i(s)\left\{{\rm{d}}N_i(s)-{\rm{d}}\Psi(s)X_i\right\}}{E\left[\left\{Z_i-\mu(s)\right\}R_i(s)R_{0i}(s)X_i\right]}\\ &&-\frac{1}{\sqrt{n}} \sum_{i=1}^n \int_{t_0}^t \frac{E\left[R_{0i}(s)R_i(s)\left\{{\rm{d}}N_i(s)-{\rm{d}}\Psi(s)X_i\right\}\right]}{E\left[R_{0i}(s)R_i(s)\right]}\\ &&\times \frac{ \left\{Z_i-\mu(s)\right\}R_i(s)R_{0i}(s)}{E\left[\left\{Z_i-\mu(s)\right\}R_i(s)R_{0i}(s)X_i\right]}+o_p(1) \end{eqnarray*} Asymptotic distribution of the minimizer of the squared estimating equation Let $$U_i(\psi)$$ be the estimating function given in (3.7) and let the estimate $$\hat{\psi}$$ be the minimizer of $$\left\{n^{-1}\sum_{i=1}^n U_i(\psi)\right\}^2$$. When $$\left\{n^{-1}\sum_{i=1}^n U_i(\hat{\psi})\right\}^2=o_p(1)$$, then it follows from Theorem 5.1 in van der Vaart (1998) (under the usual identifiability and uniform convergence criteria) that $$\hat{\psi}$$ is a consistent estimator of $$\psi$$. To find its asymptotic distribution, note that $$\hat{\psi}$$ solves an estimating equation of the form \begin{eqnarray*} 0=n^{-1}\sum_{i=1}^n U_i(\psi)\sum_{j=1}^n \frac{\partial U_j}{\partial \psi}(\psi). \end{eqnarray*} The sandwich variance estimator previously derived is not valid for this solution when $$n^{-1}\left\{\sum_{i=1}^n U_i(\psi)\right\}^2$$ reaches a minimum away from zero, for then the sandwich estimator’s denominator $$n^{-1}\sum_{j=1}^n \frac{\partial U_j}{\partial \psi}(\hat{\psi})$$ is zero. A modified standard error calculation can be made as follows. Let $$O_i$$ be the observed data recorded for subject $$i$$. Applying Hajek projections to the above estimating equation (van der Vaart, 1998), we then find that \[\sqrt{n}(\hat{\psi}-\psi)=\frac{1}{\sqrt{n}}\sum_{i=1}^n E\left\{\frac{\partial W_i}{\partial \psi}(\psi)\right\}^{-1}W_i(\psi)+o_p(1), \] where \[W_i(\psi)\equiv E\left\{U_i(\psi)\frac{\partial U_j}{\partial \psi}(\psi)|O_i\right\}+E\left\{U_j(\psi)\frac{\partial U_i}{\partial \psi}(\psi)|O_i\right\}\!. \] It follows that the asymptotic variance of $$\hat{\psi}$$ can be consistently estimated as the sample variance of \[\tilde{W}_i(\psi)\equiv U_i(\hat{\psi})\frac{1}{n}\sum_{j=1}^n \frac{\partial U_j}{\partial \psi}(\hat{\psi}) +\frac{\partial U_i}{\partial \psi}(\hat{\psi})\frac{1}{n}\sum_{j=1}^n U_j(\hat{\psi}),\] divided by \[\left[\left\{\frac{1}{n}\sum_{i=1}^n \frac{\partial U_i}{\partial \psi}(\hat{\psi})\right\}^2 +\frac{1}{n}\sum_{i=1}^n U_i(\hat{\psi})\frac{1}{n}\sum_{i=1}^n \frac{\partial^2 U_i}{\partial \psi^2}(\hat{\psi}) \right]^2.\] This reduces to the usual sandwich variance estimator (as previously derived) when $$\sum_{i=1}^n U_i(\hat{\psi})=0$$, but not otherwise. 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Survivor bias in Mendelian randomization analysis

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Abstract

Summary Mendelian randomization studies employ genotypes as experimental handles to infer the effect of genetically modified exposures (e.g. vitamin D exposure) on disease outcomes (e.g. mortality). The statistical analysis of these studies makes use of the standard instrumental variables framework. Many of these studies focus on elderly populations, thereby ignoring the problem of left truncation, which arises due to the selection of study participants being conditional upon surviving up to the time of study onset. Such selection, in general, invalidates the assumptions on which the instrumental variables analysis rests. We show that Mendelian randomization studies of adult or elderly populations will therefore, in general, return biased estimates of the exposure effect when the considered genotype affects mortality; in contrast, standard tests of the causal null hypothesis that the exposure does not affect the mortality rate remain unbiased, even when they ignore this problem of left truncation. To eliminate “survivor bias” or “truncation bias” from the effect of exposure on mortality, we next propose various simple strategies under a semi-parametric additive hazard model. We examine the performance of the proposed methods in simulation studies and use them to infer the effect of vitamin D on all-cause mortality based on the Monica10 study with the genetic variant filaggrin as instrumental variable. 1. Introduction The genetic revolution has prompted a revival of Mendelian randomization studies (Katan, 1986; Davey Smith and Ebrahim, 2003). These employ genetic variants (e.g. filaggrin) to infer the effect of exposures, known to be affected by those variants (e.g. Vitamin D), on health outcomes (e.g. all-cause mortality), building on the notion that an association between these genetic variants and the outcome of interest can only be explained by an effect of the exposure on the outcome. The statistical analysis of Mendelian randomization studies elaborates on the instrumental variables design, which utilizes so-called instrumental variables $$Z$$ as experimental handles to infer the effect of an exposure $$X$$ on an outcome $$T$$ in the presence of confounding by possibly unmeasured variables $$U$$. Here, an instrumental variable is a variable (a) that is associated with the exposure, (b) that has no direct effect on the outcome other than through the exposure, so that $$T\perp\!\!\!\perp Z|X,U$$, and (c) whose association with the endpoint of interest is not confounded by unmeasured variables in the sense that $$Z\perp\!\!\!\perp U$$ (see e.g. Hernán and Robins, 2006). Assumption (a) is testable, and assumption (c) can, in general, reasonably well be justified on the basis of random Mendelian inheritance. Assumption (b) is typically harder to justify (Didelez and Sheehan, 2007; Lawlor and others, 2008). Instrumental variables analysis is well developed for prospective designs which randomly sample subjects, and next “prospectively” assess the instrumental variables, exposure, and outcome, each in turn, in order to deliver a random population sample of data on instrumental variables, exposure, and outcome. A typical example of such design is a randomized experiment in which, for instance, smokers are randomly encouraged to quit smoking or not ($$Z$$), and next smoking behavior ($$X$$), and health outcomes ($$T$$) are recorded (Permutt and Hebel, 1989). The Mendelian randomization design is different in that, typically, adult or elderly subjects are randomly sampled in the study. Here, at least the genetic instrument, but often also the exposure, reflects conditions that were already present before study onset (e.g. at birth). For instance, Tchetgen and others (2015) evaluate the effect of diabetes on mortality using data from the Health and retirement Study, a representative sample of persons aged 50 years or older and their spouses in the United States. Likewise, the Monica10 study, which we will analyze in Section 5, includes 2656 individuals of Danish origin between 40 and 71 years of age, who had been previously recruited from the Danish Central Personal Register into the Monica I study, as a random population sample in this age range (Olsen and others, 2007; Skaaby and others, 2013). Entry in these studies is obviously conditional on surviving the onset of the study, thereby resulting in left truncation or delayed entry. In Section 2 of this article, we show that this does not induce bias in standard tests of the causal null hypothesis that the considered exposure does not affect all-cause mortality, even when these tests ignore the problem of left truncation. However, when the exposure was present prior to entry in the study, then it generally induces (selection) bias in exposure effect estimates and may, moreover, bias tests of the causal null hypothesis that the considered exposure does not affect given nonmortality endpoints or cause-specific mortality. This is a major concern (Boef and others, 2015), especially in Mendelian randomization studies with mortality endpoints, because genetic instruments are typically weak and it is well known that even minor violations of the instrumental variables assumptions can induce sizeable biases when the instruments are weak (Bound and others, 1995). In Section 3, we propose simple strategies to correct this bias under studies with an all-cause mortality endpoint. This will be done under a class of semi-parametric additive hazard models introduced in Section 3. We examine the performance of the proposed methods in simulation studies in Section 4 and use them to infer the effect of vitamin D on all cause mortality based on the Monica10 study with the genetic variant filaggrin as instrumental variable in Section 5. We discuss limitations of the considered proposals, and the analysis of nonmortality endpoints in Section 6. 2. Delayed entry and survivor bias Let us use $$T$$ to denote the lifetime, $$T_0$$ the truncation time and $$C$$ the censoring time, all measured since birth. Consider a Mendelian randomization design which intends to collect independent and identically distributed data on $$Z_i,X_i,T_{i0}$$, the observed event time $$V_i=\min(T_i,C_i)$$ and the censoring indicator $$\Delta_i=I(T_i<C_i)$$ for a random sample of subjects $$i=1,...,n$$ who are alive at study onset, i.e. for whom $$T_i>T_{i0}$$. The study may in particular be designed to sample (with replacement) birth cohorts according to some distribution (possibly different from the actual population distribution); here, the truncation time $$T_0$$ refers to the age of people in a given birth cohort at the start of the study (which is well defined, even for individuals who died). For instance, we may consider randomly sampling people who were born 40–71 years ago from a birth register. However, the study merely retains alive subjects; that is, people for whom $$T\geq T_0$$. For each individual in the sample, we assess (at their entry time $$T_0$$) the genetic instruments (e.g. the filaggrin genotype), and record the exposure (e.g. Vitamin D). Each participant is subsequently followed until death, or censoring. Throughout, we will assume that $$X$$ captures the entire exposure history since birth. This assumption is implicit in all Mendelian randomization analyses since failure of it would generally imply failure of the so-called exclusion restriction (b). We will furthermore assume that $$(T,X,Z)\perp\!\!\!\perp T_0|U$$. This is guaranteed to hold when there is no birth cohort effect on the genotype distribution in the sense that $$Z\perp\!\!\!\perp T_0|U$$ (i.e. when the population allele frequencies are the same across birth cohorts). Indeed, in that case, one can augment the vector of variables $$U$$ to include $$T_0$$, in which case $$(T,X,Z)\perp\!\!\!\perp T_0|U$$ is trivially satisfied (and the instrumental variables assumption (a) continues to be satisfied). The above design is visualized in the causal diagram of Figure 1. It is seen via d-separation (Pearl, 2000) on this diagram that if Vitamin D has an effect on survival, then conditioning on surviving up until the point of study entry may induce an artificial association between the genotype and the unmeasured confounders as a result of collider bias (Boef and others, 2015). More formally, conditioning on a descendant, $$T>T_0$$, of a collider $$X$$ may render the causes $$Z$$ and $$U$$ dependent, thereby violating the instrumental variables assumption (c). Similar biases have been described in Mendelian randomization analyses of case-control and case-only studies (Bowden and Vansteelandt, 2011). The dependence of the event $$T>T_0$$ on $$Z$$ is likely weak in Mendelian randomization analyses that use weak instruments, in which case the instrumental variables assumption (c) is only mildly violated due to left truncation. However, as we will show later, the resulting bias remains potentially substantive because even minor violations of the instrumental variables assumptions may induce large bias when the instruments are weak. Fig. 1. View largeDownload slide Causal diagram with delayed entry; $$t$$ is an arbitrary fixed time point. Fig. 1. View largeDownload slide Causal diagram with delayed entry; $$t$$ is an arbitrary fixed time point. One exception occurs when the exposure has no effect on surviving study entry (e.g. when there are only long term effects or when the exposure only manifests itself after study entry), which would justify removal of the edge from $$X$$ to $$(T>T_0)$$ in Figure 1. This problem of delayed entry can thus in particular be ignored when testing for the presence of an exposure effect on all-cause mortality by assessing whether the survival risks are dependent on the instrumental variable $$Z$$ within the study population. Such tests can be performed using standard techniques (e.g. score tests under a Cox proportional hazards model, accounting for left truncation) from survival analysis. This finding is in line with the recommendations in VanderWeele and others (2014) that Mendelian randomization analyses are less vulnerable to bias when it comes to testing the null hypothesis of no exposure effect. Suppose now that interest lies in the exposure effect on an endpoint $$Y$$ (e.g. the effect of diabetes on inflammation markers or on cardiovascular mortality), other than all-cause mortality, recorded at time $$T_0$$. Then by applying the same reasoning as in the previous paragraph on the causal diagram of Figure 2, the instrumental variables assumption (c) is seen to be violated whenever there is an exposure effect on the risk of surviving study onset (or the truncation time and instrument are dependent). Standard instrumental variables analyses will thus generally return biased estimates of the effect of exposure on arbitrary outcomes $$Y$$, regardless of whether or not there is such effect, when the considered exposure affects mortality. Interestingly, tests of the causal null hypothesis that the considered exposure does not affect the outcome $$Y$$ are also generally biased in this case, except when the exposure has no effect on surviving study entry. Fig. 2. View largeDownload slide Causal diagram with delayed entry and nonmortality endpoint $$Y$$. Fig. 2. View largeDownload slide Causal diagram with delayed entry and nonmortality endpoint $$Y$$. Because the problem of left truncation may thus be a concern for all Mendelian randomization studies, we recommend that routine analyses incorporate an assessment of the exposure effect on mortality, where possible, regardless of whether all-cause mortality is the primary endpoint. At a minimum, we recommend an investigation of the association between the instrumental variable and the mortality endpoint. 3. Eliminating survivor bias In this section, we will develop methods for estimating the effect of some exposure $$X$$ on time to death $$T$$ from all causes, accounting for the above suggested selection bias. 3.1. Semiparametric additive hazard model Throughout we will parameterize the exposure effect on the hazard scale using the semiparametric additive hazard model defined as \begin{equation}\label{sah} \lambda(t|X,Z,U)=\omega(t,U)+\psi X, \end{equation} (3.1) for a range of time points $$t>0$$, where $$\lambda(t|X,Z,U)$$ is the conditional hazard of death at time $$t$$, given exposure, instrument and confounders. Here, $$\omega(t,U)$$ captures the effects of time and confounders on the hazard; unlike in the Aalen additive hazard model (Aalen, 1980), we leave it unspecified because hazards can show a complex time pattern, and moreover, parameterizing the effect of confounders on the hazard is difficult since they may be unmeasured. Martinussen and others (2017) consider the following further relaxation of the above model \begin{equation}\label{nsah} \lambda(t|X,Z,U)=\omega(t,U)+\psi(t) X, \end{equation} (3.2) for a range of time points $$t>0$$, where the exposure coefficient $$\psi(t)$$ is an unknown (integrable) function of time. In model (3.1) (with time coded in years), the parameter $$100\psi$$ essentially expresses how many more subjects out of a total of 100 would die by the end of the year if their exposure were increased with a unit. Alternatively, interpretation can be made on the relative risk scale upon noting that model (3.1) is equivalent with the following restrictions for each time $$t>0$$: \begin{equation}\label{rsah} \frac{P(T>t|X,U)}{P(T>t|X=0,U)}=\exp(-\psi t X). \end{equation} (3.3) Importantly, we thus assume this exposure effect on the relative risk scale to be the same at all levels of $$U$$; similar no-effect-modification assumptions are commonly imposed in the instrumental variables literature (Hernán and Robins, 2006). Under this assumption, we will achieve identification of the exposure effect $$\psi$$ and can moreover interpret $$\exp(-\psi t x)$$ as the marginalized or standardized relative risk \[\frac{E\left\{P(T>t|X=x,U)\right\}}{E\left\{P(T>t|X=0,U)\right\}}, \] as a result of collapsibility of the relative risk (Martinussen and Vansteelandt, 2013). The latter is indeed easier to interpret as it does not involve conditioning on unmeasured variables $$U$$. Moreover, letting $$T(x)$$ denote the counterfactual event time when $$X$$ is set to $$x$$, and assuming that $$U$$ is sufficient to adjust for confounding of the effect of $$X$$ on $$T$$ (in the sense that $$T(x)\perp\!\!\!\perp X|U$$ for all $$x$$), this can also be interpreted as the causal relative chance of surviving time $$t$$, $$P\{T(x)>t\}/P\{T(0)>t\}$$, at exposure level $$x$$ versus 0. Although we will let the time origin be time at birth, note that the multiplicative structure of the model enables immediate translation to other (fixed) time origins. 3.2. Estimation under restrictions on the exposure distribution Under the semiparametric additive hazard model, the failure of the instrumental variables assumption (c) alluded to in Section 2, can be understood more formally as follows. By Bayes’ rule, for each fixed $$t>0$$, the joint distribution of the instrumental variable and the unmeasured confounders amongst individuals who have entered the study and are still at risk at time $$t$$ equals \begin{eqnarray*} f(Z,U|T_0<t \leq T) &=& \frac{P(T_0<t \leq T|Z,U)}{P(T_0<t \leq T)}f(Z)f(U)\\ &=& \frac{P(T_0<t|U)}{P(T_0<t \leq T)}P(T\geq t|Z,U)f(Z)f(U)\\ &=&\frac{P(T_0<t|U)}{P(T_0<t \leq T)}E\left[\exp\left\{-\int_0^t \omega(s,U){\rm{d}}s-\psi t X\right\}|Z,U\right]f(Z)f(U), \end{eqnarray*} where we use that $$T_0\perp\!\!\!\perp (T,Z)|U$$, as previously assumed. In general, the moment generating function $$E\left[\exp(-\psi t X)|Z,U\right]$$ does not factorize into terms involving either $$Z$$ or $$U$$. This then renders $$Z$$ and $$U$$ dependent within the risk sets of subjects who are in the study and alive at a given time $$t$$, thus violating the instrumental variables assumptions. One exception occurs, however, when the instrument and unmeasured confounders do not interact (on the additive scale) in their effect on the exposure, in the following sense \begin{equation}\label{ls} X=s(Z,\epsilon_z)+t(U,\epsilon_u) \end{equation} (3.4) for specific functions $$s(.)$$ and $$t(.)$$ and residual errors $$\epsilon_z$$ and $$\epsilon_u$$ that satisfy $$\epsilon_z\perp\!\!\!\perp \epsilon_u|Z,U$$, $$\epsilon_z\perp\!\!\!\perp U|Z$$ and $$\epsilon_u\perp\!\!\!\perp Z|U$$. This is for instance satisfied when the exposure is normally distributed with mean $$E(X|Z,U)=s^*(Z)+t^*(U)$$ for specific functions $$s^*(.)$$ and $$t^*(.)$$, and residual variance that either depends on $$Z$$ or $$U$$, but not both; it holds much more generally as the residual errors $$\epsilon_z$$ and $$\epsilon_u$$ need not be normally distributed. Under assumption (3.4), the moment generating function $$E\left[\exp(-\psi t X)|Z,U\right]$$ factorizes into terms involving either $$Z$$ or $$U$$, but not both. From the above calculations, it then follows that the instrumental variables conditions are satisfied within each of the observed risk sets; that is, for all $$t$$, \begin{equation}\label{riskset} Z\perp\!\!\!\perp U|T_0<t \leq T. \end{equation} (3.5) In the remainder of this section, we will show how this independence can be exploited to develop inference for the cumulative effect $$\Psi(t)-\Psi(t_0),t>t_0$$ under model (3.2) for all $$t>t_0$$, where \[ \Psi(t)=\int_0^t \psi(s){\rm{d}}s, \] and $$t_0$$ is such that $$P(T_0<t_0<T)>\sigma>0$$. Identity (3.5), along with model restriction (3.2), implies that the estimating function \begin{equation}\label{ef} \left\{Z-E(Z|T_0<t \leq T)\right\}R(t)R_{0}(t)\left\{{\rm d}N(t)-\psi(t)X{\rm{d}}t\right\}\!, \end{equation} (3.6) is unbiased at each time $$t$$; here, $$R(t)=I(T>t)$$ is the at-risk indicator and $$R_0(t)=I(T_0<t)$$ is an indicator whether or not the considered subject had entered the study by time $$t$$. Indeed, the mean of (3.6) is readily seen to equal \[E\left[\left\{Z-E(Z|T_0<t \leq T)\right\}R(t)R_{0}(t)\omega(t,U){\rm{d}}t\right]=0,\] when (3.2) and (3.5) hold for the given $$t$$. This suggests the estimator \[\hat{\Psi}(t)-\hat{\Psi}(t_0)=\int_{t_0}^t \frac{\sum_{i=1}^n \left\{Z_i-\bar{Z}(s)\right\}R_{i0}(s){\rm{d}}N_i(s)}{\sum_{i=1}^n \left\{Z_i-\bar{Z}(s)\right\}R_{i0}(s)R_i(s)X_i},\] where \[ \bar{Z}(t)=\frac{\sum_{i=1}^n R_i(t)R_{0i}(t)Z_i}{\sum_{i=1}^n R_i(t)R_{0i}(t)}, \] is a nonparametric estimator of $$E(Z|T_{0}<t \leq T)$$. In Section 2 of Supplementary materials available at Biostatistics online, we show that this estimator is equivalent to the two-stage estimator briefly mentioned in the discussion of Tchetgen and others (2015), where in the first stage, the exposure is linearly regressed on the instrument within the risk set, and in the second stage a standard additive hazard model is fitted with the (time-varying) fitted value from the first stage as the only predictor. We moreover show that the estimator $$\hat{\Psi}(t)-\hat{\Psi}(t_0)$$ admits the following i.i.d. expansion \[\sqrt{n}\left(\hat{\Psi}(t)-\hat{\Psi}(t_0)-\Psi(t)+\Psi(t_0)\right)=\frac{1}{\sqrt{n}} \sum_{i=1}^n\epsilon_i(t) + o_p(1),\] for mean zero errors \[\epsilon_i(t)= \int_{t_0}^t \frac{ \left\{Z_i-E(Z|T_0<s\leq T)\right\}R_i(s)R_{i0}(s)}{E\left[\left\{Z_i-E(Z|T_0<s\leq T)\right\}R_{i0}(s)R_i(s)X_i\right]}\left[{\rm{d}}N_i(s)-{\rm{d}}\Psi(s)X_i-{\rm{d}}\Psi_0(s)\right]\!,\] where \[{\rm{d}}\Psi_0(s)=E\left\{{\rm d}N(s)-{\rm{d}}\Psi(s)X|T_0<s\leq T\right\}=\frac{E\left[R_{0}(s)R(s)\left\{{\rm d}N(s)-{\rm{d}}\Psi(s)X\right\}\right]}{E\left[R_{0}(s)R(s)\right]}.\] The variance of the limit distribution at a given time $$t$$ can thus be consistently estimated by the sample variance of $$\epsilon_i(t)$$ (upon substituting population expectations by sample analogs and $${\rm{d}}\Psi(s)$$ by $${\rm{d}}\hat{\Psi}(s)$$ at each time $$s\in[t_0,t]$$). To study temporal changes, it is more useful to consider a uniform confidence band. This can be derived based on the above i.i.d. representation as also outlined in Lin and others (2000) and Martinussen (2010). Let $$Q_1^m,\ldots ,Q_n^m$$ be independent standard normal variates. Then, given the data, \[W^m_n(t)\equiv \sqrt{n}\left(\hat{\Psi}^m(t)-\hat{\Psi}^m(t_0)-\Psi(t)+\Psi(t_0)\right)=\frac{1}{\sqrt{n}} \sum_{i=1}^n\epsilon_i(t)Q_i^m\] also converges in distribution to a zero-mean Gaussian process with variance given by the variance of $$\epsilon_i(t)$$. The limit distribution can thus be evaluated by generating a large number, $$M$$, of replicates of $$W^m_n(t),m=1,\ldots,M$$. Uniform confidence bands can now be based on the distribution of $$\sup_{t_0\leq t\leq \tau}|W^m_n(t)|$$, $$m=1, \ldots ,M$$, where $$\tau$$ is some finite time point. It is readily verified that the above results continue to hold in the presence of censoring, when $$N_i(t)$$ is redefined as $$N_i(t)=I(V_i\leq t,\Delta_i=1)$$ and $$R_i(t)$$ as $$R_i(t)=I(t\leq V_i)$$, provided that $$C_i\perp\!\!\!\perp (T_i,T_{i0},X_i,Z_i)|U_i$$. Under the assumption of a constant exposure effect (CST), i.e. in model (3.1) for $$t>t_0$$, the exposure effect $$\psi$$ can be estimated by solving an estimating equation based on the integrated estimating functions (3.6), to obtain: \[\hat{\psi}_{\rm ADD/EXP}=\frac{\sum_{i=1}^n \int_{T_{i0}}^{\infty} \left\{Z_i-\bar{Z}(t)\right\}{\rm{d}}N_i(t)}{\sum_{i=1}^n \int_{T_{i0}}^{\infty} \left\{Z_i-\bar{Z}(t)\right\}R_i(t)X_i{\rm{d}}t}, \] where the subscript “ADD/EXP” is a reminder that we are relying on the assumption (3.4) that instrument and confounders act additively on the exposure. It follows from the results in the Appendix that $$\sqrt{n}(\hat{\psi}_{\rm ADD/EXP}-\psi)$$ converges to a normal variate with mean zero and variance which can be consistently estimated by the variance of \[ \epsilon_i= \frac{\int_{t_0}^{\infty} \left\{Z_i-E(Z|T_0<s\leq T)\right\}R_i(s)R_{i0}(s)\left[{\rm{d}}N_i(s)-{\rm{d}}\Psi(s)X_i-{\rm{d}}\Psi_0(s)\right]}{\int_{t_0}^{\infty} E\left[\left\{Z_i-E(Z|T_0<s\leq T)\right\}R_{i0}(s)R_i(s)X_i\right]{\rm{d}}s}. \] 3.3. Estimation under restrictions on the exposure effect When the additivity assumption (3.4) on the exposure distribution is believed not to hold, then progress can still be made under the (partially untestable) assumption that the additive hazard model (3.1) holds for all $$t>0$$ (rather than just $$t>t_0$$). In particular, by a similar reasoning as in Martinussen and others (2011, 2017), the exposure effect under this model can be removed (in expectation) from the at-risk indicators $$R(t)$$, by calculating $$R(t)\exp\left(-\psi t X\right)$$. What remains must then be mean independent of the instrument $$Z$$ by the instrumental variables assumptions. This suggests estimating $$\psi$$ as the solution to the estimating equation \begin{equation}\label{knownz} 0=\sum_{i=1}^n \int_{T_{i0}}^{\infty} e(t)\left\{Z_i-E(Z)\right\}R_i(t)\exp\left(\psi X_it\right){\rm{d}}t, \end{equation} (3.7) for arbitrary (scalar) function $$e(t)$$, e.g. $$e(t)=1$$. That the solution to (3.7) delivers a consistent estimator of $$\psi$$ when $$E(Z)$$ equals the population expectation of the instrumental variable “at birth” (i.e. before deaths may occur), follows formally from the standard theory of M-estimators and the fact that the contributions to the above estimating equation then have expectation zero. Indeed, the estimating functions $$e(t)\left\{Z-E(Z)\right\}R_0(t)R(t)\exp\left(\psi Xt\right)$$ in (3.7) have mean \begin{eqnarray*} e(t)E\left[\left\{Z-E(Z)\right\}P(T_0<t|U)\exp\left\{-\int_{0}^t \omega(s,U){\rm{d}}s\right\}\right]&=&0, \end{eqnarray*} for each choice of $$e(t)$$, where we again use that $$(T,X,Z)\perp\!\!\!\perp T_0|U$$. It is readily verified that the above identity continues to hold in the presence of censoring, when $$R_i(t)$$ is redefined as $$R_i(t)=I(t\leq V_i)$$, provided that $$C_i\perp\!\!\!\perp (T_i,T_{i0},X_i,Z_i)|U_i$$. Solving (3.7) demands knowledge of the population expectation $$E(Z)$$, which may be available in certain Mendelian randomization studies that have external information on the genotype distribution (e.g. in the form of allele frequencies). The resulting solution will be denoted $$\hat{\psi}_{\rm CST/K}$$, where the subscript “CST/K” is a reminder that we assume a constant treatment effect (CST) and the population expectation of the instrumental variable to be a priori known (K). However, caution is needed when using this estimator, since choosing an incorrect value of $$E(Z)$$ generally leads to biased estimators of the exposure effect, even under the null hypothesis of no exposure effect, as we will confirm in the simulation studies later in this article. One may alternatively substitute $$E(Z)$$ by an estimate of the expectation $$E(Z|T>T_{0})$$ of $$Z$$ amongst people who have been recruited into the study. The resulting solution will be denoted $$\hat{\psi}_{\rm CST/N}$$, where the subscript “CST/N” is a reminder that we assume a CST and use a naïve (N) estimate of the population expectation of the instrumental variable. The estimator $$\hat{\psi}_{\rm CST/N}$$ has the advantage of being unbiased under the null hypothesis of no exposure effect, for then $$E(Z|T>T_{0})$$ equals $$E(Z)$$, but the disadvantage of being biased when the null hypothesis does not hold. It follows from the standard theory of M-estimators that, asymptotically, this bias equals $$E\left\{U_i(\psi)\right\}/E\left\{\partial U_i(\psi)/\partial \psi\right\}$$, evaluated at the population value $$\psi$$, with $$U_i(\psi)\equiv \int_{T_{i0}}^{\infty} e(t)\left\{Z_i-E(Z)\right\}R_i(t)\exp\left(\psi X_it\right){\rm{d}}t$$ the estimating function in (3.7); that is, \begin{equation}\label{bias} \frac{\int e(t)E\left[\left\{E(Z)-E(Z|T>T_{0})\right\}P(T_0<t|U)\exp\left\{-\int_{0}^t \omega(s,U){\rm{d}}s\right\}\right]{\rm{d}}t}{\int e(t)E\left[\left\{Z-E(Z|T>T_{0})\right\}P(T_0<t|U)E(X|Z,U)t\exp\left\{-\int_{0}^t \omega(s,U){\rm{d}}s\right\}\right]{\rm{d}}t}. \end{equation} (3.8) Letting $$X$$ be normal with mean $$E(X|Z,U)=\alpha_0+\alpha_1Z+\alpha_2U$$ and constant variance, $$Z$$ be normal with mean 0 and constant variance, and $$Z\perp\!\!\!\perp T_0|U$$, one can see after some algebra that \[E(Z|T>T_{0})=\frac{E\left\{ZP(T> T_0|Z)\right\}}{E\left\{P(T> T_0|Z)\right\}} = \frac{E\left\{Z\exp{\left(-\psi\alpha_1 ZT_0\right)}w(T_0)\right\}}{E\left\{\exp{\left(-\psi\alpha_1 ZT_0\right)}w(T_0)\right\}} =-\psi\alpha_1\frac{E\left\{T_0w^*(T_0)\right\}}{E\left\{w^*(T_0)\right\}}, \] for specific functions $$w(T_0),w^*(T_0)$$ that do not depend on $$\alpha_1$$. It follows that both the numerator and denominator of (3.8) are proportional to $$\alpha_1$$ (see the detailed calculations in the Supplementary materials available at Biostatistics online), and thus that the degree of collider bias is (to first order) the same regardless of the strength of the instruments. This formalizes our earlier suggestion that the problem of left truncation is important, even when the instruments are weak (i.e. when $$\alpha_1$$ is close to zero). This phenomenon has been noted recently in the epidemiological literature (Munafo and others, 2016) and will also be confirmed by simulation studies in the next section. In view of the above concern about bias, we propose to substitute $$E(Z_i)$$ in (3.7) by \begin{equation}\label{newz} \hat{E}(Z;\psi)\equiv \frac{\sum_{i=1}^n Z_i\exp\left(\psi T_{i0} X_i\right)}{\sum_{i=1}^n \exp\left(\psi T_{i0} X_i\right)}, \end{equation} (3.9) when the population allele frequencies are unknown. This is justified upon noting that \begin{eqnarray*} \frac{E\left\{Z\exp\left(\psi T_0 X\right)|T\geq T_0\right\}}{E\left\{\exp\left(\psi T_0 X\right)|T\geq T_0\right\}} &=&\frac{E\left\{I\left(T\geq T_0\right)Z\exp\left(\psi T_0 X\right)\right\}}{E\left\{I\left(T\geq T_0\right)\exp\left(\psi T_0 X\right)\right\}}\\ &=&\frac{E\left\{\int_0^{\infty} P\left(T\geq t_0|Z,X,U\right)Z\exp\left(\psi t_0 X\right)f_{T_0|Z,X,U}(t_0){\rm{d}}t_0\right\}}{E\left\{\int_0^{\infty} P\left(T\geq t_0|Z,X,U\right)\exp\left(\psi t_0 X\right)f_{T_0|Z,X,U}(t_0){\rm{d}}t_0\right\}}\\ &=&\frac{E\left\{\int_0^{\infty} \exp\left(-\int_{0}^{t_0} \omega(t,U){\rm{d}}t \right)Zf_{T_0|Z,X,U}(t_0){\rm{d}}t_0\right\}}{E\left\{\int_0^{\infty} \exp\left(-\int_{0}^{t_0} \omega(t,U)dt \right)f_{T_0|Z,X,U}(t_0){\rm{d}}t_0\right\}}\\ &=&\frac{E\left\{\exp\left(-\int_{0}^{T_0} \omega(t,U){\rm{d}}t \right)Z\right\}}{E\left\{\exp\left(-\int_{0}^{T_0} \omega(t,U){\rm{d}}t \right)\right\}}=E(Z), \end{eqnarray*} where we again use that $$Z\perp\!\!\!\perp (T_0,U)$$. Throughout, we will use $$\hat{\psi}_{\rm CST/U}$$ to denote the solution to (3.7) with $$E(Z)$$ substituted by $$\hat{E}(Z_i;\psi)$$. Here, the subscript “CST/U” is a reminder that we assume a CST and the population expectation of the instrumental variable to be unknown (U). The choice of index function $$e(t)$$ affects the efficiency of the solution to (3.7). To simplify integration over time, we here recommend to set $$e(t)=1$$ and defer a more detailed study on semi-parametric efficiency to future work. We conjecture that reasonable efficiency will then be obtained when mean-centering the exposure $$X_i$$, and choosing $$Z_i$$ to be the fitted value for subject $$i$$ as obtained from a model for the exposure given the instrumental variables(s). For this choice, $$e(t)=1$$ for all $$t>0$$, (3.7) reduces to \[0=\sum_{i=1}^n U_i(\psi),\] where \[U_i(\psi)=\left\{\begin{array}{@{}ccc} \left\{Z_i-E(Z)\right\}\left\{\exp\left(\psi X_iT_i\right)-\exp\left(\psi X_iT_{i0}\right)\right\}/(\psi X_i) & {\rm{when}} & \psi X_i\ne 0\\ \left\{Z_i-E(Z)\right\}(T_i-T_{i0}) & {\rm{when}} & \psi X_i= 0. \end{array}\right.\] It follows from the theory of M-estimators that, under weak regularity conditions, $$\sqrt{n}(\hat{\psi}-\psi)$$ converges to a normal mean zero variate with variance which can be consistently estimated by \[\frac{\frac{1}{n}\sum_{i=1}^n \left[U_i(\psi)-c(\psi)\left\{Z_i-\hat{E}(Z_i;\psi)\right\}\exp\left({\psi} X_iT_{i0}\right)\right]^2}{\left[\frac{1}{n}\sum_{i=1}^n V_i(\psi)-W_i(\psi)\right]^2},\] where \begin{eqnarray*} V_i(\psi)&=&\left\{\begin{array}{@{}ccc} \left\{Z_i-E(Z)\right\}\left\{\exp\left(\psi X_iT_i\right)T_i-\exp\left(\psi X_iT_{i0}\right)T_{i0}\right\}/\psi & {\rm{when}} & \psi \ne 0\\ \left\{Z_i-E(Z)\right\}(T_i^2-T_{i0}^2)X_i & {\rm{when}} & \psi = 0 \end{array}\right. \\ W_i(\psi)&=&\left\{\begin{array}{@{}ccc} U_i(\psi)/\psi & {\rm{when}} & \psi X_i \ne 0\\ \left\{Z_i-E(Z)\right\}(T_i^2-T_{i0}^2)X_i/2 & {\rm{when}} & \psi X_i = 0. \end{array}\right. \\ c(\psi)&=& \frac{\sum_{i=1}^n \left\{\exp\left(\psi X_iT_i\right)-\exp\left(\psi X_iT_{i0}\right)\right\}/(\psi X_i)}{\sum_{i=1}^n \exp\left(\psi T_{i0} X_i\right)}, \end{eqnarray*} and where $$E(Z)$$ is substituted by $$\hat{E}(Z_i;\psi)$$ in all expressions, and $$\psi$$ by $$\hat{\psi}_{\rm CST/U}$$. Also $$\sqrt{n}(\hat{\psi}_{\rm CST/K}-\psi)$$ converges to a normal variate with mean zero and variance which is likewise obtained, but upon setting $$c(\psi)$$ to zero, $$E(Z)$$ to the known population expectation, and $$\psi$$ to $$\hat{\psi}_{\rm CST/K}$$. 4. Simulation study To gain insight into the extent of the bias of instrumental variables analyses that ignore left truncation, as well as the performance of the novel proposals, we conducted a number of simulation experiments. The settings were chosen to mimic the situation observed in the Monica10 study, which we will analyze in Section 5. In particular, we first generated 10 000 samples of a dichotomous instrumental variable $$Z$$, coded 0 or 1, with mean 0.067, and independently generated a standard normal confounder $$U$$. We next generated the exposure $$X$$ to be normal with mean $$Z-2U$$ and variance 1 in the first simulation experiment, normal with mean $$Z-4U$$ and variance 1 in the second experiment, normal with mean $$0.5Z-2U$$ and variance 1 in the third experiment, Bernoulli distributed with mean expit$$(Z+U)$$ in the fourth experiment, and finally normal with mean $$0.75Z-4U$$ corresponding with a weak IV. We then standardized the exposure to give it mean zero and variance 1. In all cases, the event time was exponentially distributed with hazard $$0.24-0.1X+|U|/5$$. Entry ages were generated uniformly between 0 and 4, and 2571 subjects whose event time exceeds the entry time were randomly sampled from the generated population of 10 000. Twenty percent were potentially censored according to a uniform distribution on (0,10), and the rest were censored at $$t = 10$$, corresponding to the study being closed at this time point, leading to a cumulative censoring rate in the range of 11–13% across the 4 experiments. We next evaluated four estimators: the estimator $$\hat{\psi}_{\rm ADD/EXP}$$ which assumes that instrument and confounder act additively on the exposure, and the three estimators that assume CSTs since birth. In particular, we evaluated the naïve estimator $$\hat{\psi}_{\rm{CST/N}}$$ which ignores left truncation by solving (3.7) with the sample average of $$Z_i$$ in the observed sample in lieu of $$E(Z_i)$$; the estimator $$\hat{\psi}_{\rm{CST/K}}$$ which assumes known $$E(Z_i)$$ equal to 0.067; and the estimator $$\hat{\psi}_{\rm{CST/U}}$$ which makes no assumptions on the mean of the instrument. In rare occasions, the estimating equation (3.7) did not reach zero at any value of $$\psi$$, although the estimating equation reached a minimum or maximum close to zero. In that case, we have set the estimator to be the value of $$\psi$$ that corresponds with this minimum or maximum. In the Appendix, we show that this strategy still delivers a consistent estimator provided that this minimum or maximum of the estimating equation converges to zero as the sample size goes to infinity (van der Vaart, 1998). We moreover show how the asymptotic variance of the estimator can be calculated in that case. The simulation results are summarized in Table 1. They show that the naive estimator which ignores left truncation can be severely biased (relative bias of about 40%), and that the three proposed estimators are unbiased. This is even so for the estimator $$\hat{\psi}_{\rm ADD/EXP}$$ when the exposure is dichotomous, even though the additivity assumption (3.4) fails in that case; however, the estimator becomes more variable now. As theoretically predicted, the amount of bias does not change with the strength of the instrument. The sandwich estimators generally perform adequately in the sense of delivering confidence intervals with close to nominal—albeit slightly conservative—coverage, but they can be quite variable when the instrument is weak. This sometimes lead to discrepancies between the mean and median standard errors, which were partly due to 1 (or just a few) extreme outlier(s). This in turn has implications for the evaluation of coverage; e.g. while the naive estimator $$\hat{\psi}_{\rm CST/N}$$ reaches a coverage of 95.1% in Experiment 4, this reduces to 82.7% when the sample size is doubled. Also the estimator $$\hat{\psi}_{\rm ADD/EXP}$$ itself tends to be more variable when the instrument is weak. Limiting integration in the calculation of the estimator to the 80% percentile of the observed event times improved the results (e.g. it reduced the empirical standard deviation of $$\hat{\psi}_{\rm ADD/EXP}$$ from 0.253 to 0.206), by preventing highly variable estimates of the mean $$E(Z|T_0<t<T)$$ of the instrument in the observed risk set. Finally, using the known mean $$E(Z)$$ rather than an estimate of it, typically resulted in some minor precision loss, as is typical for instrumental variable estimators (see e.g. Vansteelandt and Goetghebeur, 2003, Section 2.4). Table 1. Average of the estimates (Mean); empirical standard error (ESE); median (mean) of the sandwich standard errors (SSE); coverage of 95% Wald confidence intervals (Coverage) Experiment Estimator Mean ESE Median (mean) SSE Coverage 1 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.061 0.040 0.038 (0.042) 80.2 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.100 0.064 0.059 (0.069) 95.1 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.105 0.071 0.064 (0.088) 95.6 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.100 0.061 0.064 (0.064) 97.4 2 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.063 0.082 0.073 (0.101) 91.6 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.095 0.112 0.105 (0.140) 94.7 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.095 0.111 0.110 (0.337) 94.4 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.097 0.253 0.122 (0.233) 98.3 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.062 0.084 0.079 (0.161) 90.0 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.089 0.111 0.109 (0.199) 93.0 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.090 0.111 0.113 (0.288) 94.9 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.104 0.230 0.135 (0.242) 98.5 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.061 0.057 0.048 (0.067) 95.1 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.098 0.081 0.070 (0.163) 99.4 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.096 0.078 0.083 (0.204) 98.9 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.104 0.091 0.089 (0.097) 97.9 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.063 0.101 0.094 (0.15) 90.7 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.089 0.134 0.128 (0.22) 91.1 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.088 0.122 0.131 (0.25) 93.2 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.0024$$^*$$ 3.05 0.214 (64.0) 99.3 Experiment Estimator Mean ESE Median (mean) SSE Coverage 1 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.061 0.040 0.038 (0.042) 80.2 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.100 0.064 0.059 (0.069) 95.1 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.105 0.071 0.064 (0.088) 95.6 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.100 0.061 0.064 (0.064) 97.4 2 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.063 0.082 0.073 (0.101) 91.6 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.095 0.112 0.105 (0.140) 94.7 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.095 0.111 0.110 (0.337) 94.4 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.097 0.253 0.122 (0.233) 98.3 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.062 0.084 0.079 (0.161) 90.0 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.089 0.111 0.109 (0.199) 93.0 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.090 0.111 0.113 (0.288) 94.9 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.104 0.230 0.135 (0.242) 98.5 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.061 0.057 0.048 (0.067) 95.1 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.098 0.081 0.070 (0.163) 99.4 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.096 0.078 0.083 (0.204) 98.9 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.104 0.091 0.089 (0.097) 97.9 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.063 0.101 0.094 (0.15) 90.7 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.089 0.134 0.128 (0.22) 91.1 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.088 0.122 0.131 (0.25) 93.2 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.0024$$^*$$ 3.05 0.214 (64.0) 99.3 $$^*$$The median estimate was $$-0.099$$ Table 1. Average of the estimates (Mean); empirical standard error (ESE); median (mean) of the sandwich standard errors (SSE); coverage of 95% Wald confidence intervals (Coverage) Experiment Estimator Mean ESE Median (mean) SSE Coverage 1 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.061 0.040 0.038 (0.042) 80.2 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.100 0.064 0.059 (0.069) 95.1 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.105 0.071 0.064 (0.088) 95.6 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.100 0.061 0.064 (0.064) 97.4 2 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.063 0.082 0.073 (0.101) 91.6 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.095 0.112 0.105 (0.140) 94.7 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.095 0.111 0.110 (0.337) 94.4 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.097 0.253 0.122 (0.233) 98.3 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.062 0.084 0.079 (0.161) 90.0 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.089 0.111 0.109 (0.199) 93.0 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.090 0.111 0.113 (0.288) 94.9 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.104 0.230 0.135 (0.242) 98.5 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.061 0.057 0.048 (0.067) 95.1 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.098 0.081 0.070 (0.163) 99.4 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.096 0.078 0.083 (0.204) 98.9 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.104 0.091 0.089 (0.097) 97.9 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.063 0.101 0.094 (0.15) 90.7 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.089 0.134 0.128 (0.22) 91.1 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.088 0.122 0.131 (0.25) 93.2 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.0024$$^*$$ 3.05 0.214 (64.0) 99.3 Experiment Estimator Mean ESE Median (mean) SSE Coverage 1 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.061 0.040 0.038 (0.042) 80.2 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.100 0.064 0.059 (0.069) 95.1 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.105 0.071 0.064 (0.088) 95.6 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.100 0.061 0.064 (0.064) 97.4 2 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.063 0.082 0.073 (0.101) 91.6 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.095 0.112 0.105 (0.140) 94.7 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.095 0.111 0.110 (0.337) 94.4 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.097 0.253 0.122 (0.233) 98.3 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.062 0.084 0.079 (0.161) 90.0 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.089 0.111 0.109 (0.199) 93.0 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.090 0.111 0.113 (0.288) 94.9 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.104 0.230 0.135 (0.242) 98.5 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.061 0.057 0.048 (0.067) 95.1 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.098 0.081 0.070 (0.163) 99.4 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.096 0.078 0.083 (0.204) 98.9 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.104 0.091 0.089 (0.097) 97.9 $$\hat{\psi}_{\rm CST/N}$$ $$-$$0.063 0.101 0.094 (0.15) 90.7 $$\hat{\psi}_{\rm CST/K}$$ $$-$$0.089 0.134 0.128 (0.22) 91.1 $$\hat{\psi}_{\rm CST/U}$$ $$-$$0.088 0.122 0.131 (0.25) 93.2 $$\hat{\psi}_{\rm ADD/EXP}$$ $$-$$0.0024$$^*$$ 3.05 0.214 (64.0) 99.3 $$^*$$The median estimate was $$-0.099$$ 5. Analysis of the Monica10 study Vitamin D helps with the absorption of calcium in the body (besides playing a number of other roles), as a result of which vitamin D deficiencies may be linked to an increased risk for heart disease and subsequent mortality. This motivates our analysis, which aims to assess the effect of vitamin D status on all-cause mortality, using the filaggrin genotype as an instrument according to the principles of Mendelian randomization. The data source is the population based study Monica10, and vitamin D is measured as the serum 25-OH-D (nmol/L). The Monica I study was conducted in 1982–1984 and included examinations of 3785 individuals of Danish origin recruited from the Danish Central Personal Register as a random sample of the population. The ten-year follow-up study, Monica10, included 2656 individuals between 40 and 71 years. It was carried out in 1993–1994; for further details see Olsen and others (2007) and Skaaby and others (2013). We considered the following variables: $$Z$$ is an indicator of filaggrin mutation; $$X$$ is the standardized natural logarithm of the Vitamin D (nmol/l) measurements. The time-scale is time since birth, and the number of individuals with complete information on the variables examined was 2571. The odds of having a normal Vitamin D value ($$>$$30 nmol/L) was estimated to be 3.13 (95% CI: 1.28 to 7.76, P = 0.01) times larger for those with the filaggrin mutation compared to those without. The $$F$$-test statistic, when regressing Vitamin D on the instrument, is 7.34 (P = 0.007), demonstrating that the instrument is weak. The association of vitamin D in a standard Aalen additive hazards analysis, accounting for left truncation, corresponds with a protective additive effect on the hazard of $$-$$0.002 (95% confidence interval $$-$$0.0029 to $$-$$0.0011, $${\rm P}=7\times10^{-6}$$). Judging from the result, there seems to be an effect of Vitamin D on overall survival, but note that this is under the assumption of no unmeasured confounders. This assumption is likely violated because, for instance, people with poor health conditions might be less likely exposed to sunlight and thus have lower Vitamin D levels. A naïve Mendelian randomisation analysis—namely, the analysis in Section 3.3 with $$E(Z)$$ set to the sample average 0.0692 of the observed genotypes—gave a nearly identical effect size, corresponding with an additive effect on the hazard of $$\hat{\psi}_{\rm CST/N}=-0.0016$$ (95% confidence interval $$-$$0.0046 to 0.0014, P = 0.28). Also this analysis is likely invalid as a result of survivor bias. Assuming that the population allele frequency equals 0.046 Barker and others (2007), we obtain an additive effect $$\hat{\psi}_{\rm CST/K}=-0.026$$ (95% confidence interval $$-$$0.040 to $$-$$0.012, P = 0.0002). Figure 3 shows additional results for other allele frequencies. All of these results have the drawback, however, that they are biased and that corresponding tests of the null hypothesis are invalid when the wrong allele frequency is chosen. Relative to $$\hat{\psi}_{\rm CST/K}$$, we therefore prefer the estimator $$\hat{\psi}_{\rm CST/U}=-0.010$$ (95% confidence interval $$-$$0.026 to 0.005, P = 0.19), which does not pre-specify the allele frequency, but estimates it as $$\hat{E}(Z_i;\hat{\psi})=0.067$$. Fig. 3. View largeDownload slide Sensitivity analysis: $$\hat{\psi}_{\rm CST/K}$$ in function of $$E(Z)$$ (solid line) and 95% confidence intervals (dashed lines), along with the estimates $$\hat{\psi}_{\rm CST/K}$$ (corresponding to $$E(Z)=0.046$$), $$\hat{\psi}_{\rm CST/N}$$, $$\hat{\psi}_{\rm CST/U}$$, $$\hat{\psi}_{\rm ADD/EXP}$$, and their 95% confidence intervals. Fig. 3. View largeDownload slide Sensitivity analysis: $$\hat{\psi}_{\rm CST/K}$$ in function of $$E(Z)$$ (solid line) and 95% confidence intervals (dashed lines), along with the estimates $$\hat{\psi}_{\rm CST/K}$$ (corresponding to $$E(Z)=0.046$$), $$\hat{\psi}_{\rm CST/N}$$, $$\hat{\psi}_{\rm CST/U}$$, $$\hat{\psi}_{\rm ADD/EXP}$$, and their 95% confidence intervals. However, a major limitation of both estimators $$\hat{\psi}_{\rm CST/K}$$ and $$\hat{\psi}_{\rm CST/U}$$ is that their validity relies on the assumption that the exposure effect is constant over time since birth. This assumption is not innocent as it determines the degree of adjustment in (3.9), thereby enabling us to back-calculate the expectation of the instrumental variable “at birth”. It is also partially untestable in many Mendelian randomization studies where e.g. data are only available for an elderly population. Assuming that instrument and confounders act additively on the exposure, Figure 4 shows the nonparametric estimate proposed in Section 2.2. It suggests no exposure effect until roughly age 60, and lack of information from age 80 onwards, due to the IV being very weak from those ages onwards; overall it suggests that the assumption of a CST since birth is less likely. In view of this, we recommend the estimator $$\hat{\psi}_{\rm ADD/EXP}$$ for our conclusive data analysis, as its consistency only relies on correct specification of model (3.1) at times $$t$$ at which $$P(T_0<t \leq T)>0$$ (along with the stated assumptions on the exposure distribution). This is important as it implies that the estimator continues to be valid when the constant effect assumption fails prior to the (minimum) age of study-onset (e.g. when the exposure has no effect at young age). This estimator equals $$\hat{\psi}_{\rm ADD/EXP}=-0.025$$ (95% confidence interval $$-$$0.057 to 0.0056, P = 0.11), indicating that, over the course of follow-up, a standard deviation increase in the natural logarithm of Vitamin D is associated with on average 2.5 fewer deaths for each year of follow-up in each 100 persons alive at the start of the year; re-estimating the effect by limiting integration up to age 80 resulted in an effect estimate of $$\hat{\psi}_{\rm ADD/EXP}=-0.020$$. Fig. 4. View largeDownload slide Cumulative exposure effect: nonparametric (solid step function) and parametric (dashed straight line) estimator. Fig. 4. View largeDownload slide Cumulative exposure effect: nonparametric (solid step function) and parametric (dashed straight line) estimator. Our analysis above is a simplification of a complex reality and must therefore be viewed as an illustration, for various reasons. We only have information on the Vitamin D exposure at one point in time, so that we could not accommodate its time-varying nature. Furthermore, filaggrin deficiency is associated with an increased risk of ichtyosis vulgaris, atopic dermatitis, allergic rhinitis, and asthma (van den Oord and Sheikh, 2009), with asthma itself being linked with mortality; this may imply a violation of the IV assumptions. 6. Discussion The problem of left truncation and related survivor bias is pertinent to all Mendelian randomization studies of exposures that affect mortality. Assessing the exposure effect on mortality should thus be a key component of all Mendelian randomization studies, even those in which mortality is a secondary outcome. In this article, we have presented simple strategies as to how this can be done. The proposed strategies will enable future Mendelian randomization studies to gain power by focusing even more on selective populations of individuals at high risk. Our development considered a scalar instrument for pedagogic purposes. However, multivariate instruments can be efficiently incorporated by substituting $$Z$$ by the fitted value from a model for the exposure, given those instruments. Misspecification of such model would not invalidate our proposals, but at worst lead to a loss of efficiency. Furthermore, standard errors for the exposure effect proposed below are valid, regardless of whether $$Z$$ was directly observed or indirectly obtained as the fitted value from some regression model. Our development, like the rest of the Mendelian randomization literature, considered the exposure to be fixed. When $$X$$ is just one manifestation of a time-varying exposure process, it is very likely that not all of the instrument’s effect is mediated by the single-observed exposure assessment, as some of it may also be mediated by more proximal or distal exposures VanderWeele and others (2014). Mendelian randomization studies in which exposures are longitudinally collected over long periods of time, would provide a valuable source of information to account for this. We have furthermore focused on the relative chance of survival. In some applications, especially those in which survival chances are very high, it may be more interesting to focus on the relative risk of failure. The estimators of Section 3.3 are readily modified to this setting by redefining $$R(t)$$ as the failure indicator at time $$t$$; more work is needed to evaluate how to best adapt the estimators of Section 3.2 to this effect estimand. Section 3.3 assumed the exposure effect to be constant over time since birth. This assumption can be critical, and unfortunately is largely untestable in most Mendelian randomization studies. In view of this, in Section 3.2 we have developed an approach that makes no assumptions on the exposure effect at times before participants entered the study. We find this approach preferable, even though it relies on assumptions about the exposure distribution instead and appears more susceptible to weak instruments. In Section 4 of Supplementary materials available at Biostatistics online, we suggest an alternative sensitivity analysis approach. The proposal in Section 3.2 is also readily extended to incorporate adjustment for measured covariates $$C$$ under the assumption that $$Z\perp\!\!\!\perp U|C$$, upon substituting $$\bar Z(t)$$ by the fitted value from a regression of $$Z$$ on $$C$$ in subjects with $$R(t)=R_0(t)=1$$. Munafo and others (2016) have also highlighted how selection into Mendelian randomization studies can lead to biased effect estimates with general outcomes. In simulation studies, they found that small influences on selection could yield misleading results. Our work shows how unbiased inference about effects on mortality can be obtained in the presence of selection, but this does not generally extend to nonmortality outcomes. For instance, in model \[E(Y|X,Z,U)=\omega^*(U)+\gamma X, \] it is tempting to estimate the exposure effect $$\gamma$$ as the solution to estimating equation \[0=\sum_{i=1}^n \left(Z_i-\mu\right)(Y_i-\gamma X_i), \] with $$\mu$$ set to a nonparametric estimate of $$E(Z|T_0<T)$$ under the additivity assumption (3.4), or to $$\hat{E}(Z_i;\hat{\psi})$$ under the assumption of a CST since birth. However, even when assumption (3.5) holds for all $$t$$, it is not generally true that $$Z\perp\!\!\!\perp U|T_0<T$$, thus invalidating the choice $$\mu=E(Z|T_0<T)$$. More importantly, the outcome $$Y$$ is generally ill defined for people who did not survive up until study entry. As a result, the exposure effect is only well defined for the principal stratum of individuals who would have survived the time of study entry regardless of their exposure (Jemiai and others, 2007); our results are not directly applicable to estimation of the exposure effect for this latent subgroup. For the same reason, our results are not immediately applicable to handle problems of selection bias in Mendelian randomization studies that evaluate the effects of particular exposures on outcomes (e.g. disease progression) in individuals with chronic disease (e.g. multiple sclerosis or Parkinson’s disease), since a change in exposure may cause a change in disease status for some individuals, making the outcome ill-defined for them. Supplementary material Supplementary material is available at http://biostatistics.oxfordjournals.org. Acknowledgments The authors would like to thank Eric Tchetgen Tchetgen (Harvard University) for helpful discussions, and Stephen Burgess (University of Cambridge) and an anonymous reviewer for excellent comments that helped us to improve the article. The third author’s work is part of the Dynamical Systems Interdisciplinary Network, University of Copenhagen. Funding Strategic Basic Research PhD grant from the Research Foundation - Flanders (FWO) to O.D. Appendix Asymptotic distribution of $$\sqrt{n}\left(\hat{\Psi}(t)-\hat{\Psi}(t_0)\right)$$ Define \[\hat{\Psi}(t,\mu)-\hat{\Psi}(t_0,\mu)=\int_{t_0}^t \frac{\sum_{i=1}^n \left\{Z_i-\mu(s)\right\}R_{0i}(s){\rm{d}}N_i(s)}{\sum_{i=1}^n \left\{Z_i-\mu(s)\right\}R_i(s)R_{0i}(s)X_i}. \] Then, under standard regularity conditions, \begin{eqnarray*} \sqrt{n}\left(\hat{\Psi}(t)-\hat{\Psi}(t_0)-\Psi(t)+\Psi(t_0)\right)&=& \sqrt{n}\left(\hat{\Psi}(t,\mu)-\hat{\Psi}(t_0,\mu)-\Psi(t)+\Psi(t_0)\right)\\ &&+\sqrt{n}\left(\hat{\Psi}(t)-\hat{\Psi}(t_0)-\hat{\Psi}(t,\mu)-\hat{\Psi}(t_0,\mu)\right)\\ &=&\frac{1}{\sqrt{n}} \sum_{i=1}^n \int_{t_0}^t \frac{ \left\{Z_i-\mu(s)\right\}R_{0i}(s)R_i(s)\left\{{\rm{d}}N_i(s)-{\rm{d}}\Psi(s)X_i\right\}}{E\left[\left\{Z_i-\mu(s)\right\}R_i(s)R_{0i}(s)X_i\right]}\\ &&-\frac{1}{\sqrt{n}} \sum_{i=1}^n \int_{t_0}^t \frac{E\left[R_{0i}(s)R_i(s)\left\{{\rm{d}}N_i(s)-{\rm{d}}\Psi(s)X_i\right\}\right]}{E\left[R_{0i}(s)R_i(s)\right]}\\ &&\times \frac{ \left\{Z_i-\mu(s)\right\}R_i(s)R_{0i}(s)}{E\left[\left\{Z_i-\mu(s)\right\}R_i(s)R_{0i}(s)X_i\right]}+o_p(1) \end{eqnarray*} Asymptotic distribution of the minimizer of the squared estimating equation Let $$U_i(\psi)$$ be the estimating function given in (3.7) and let the estimate $$\hat{\psi}$$ be the minimizer of $$\left\{n^{-1}\sum_{i=1}^n U_i(\psi)\right\}^2$$. When $$\left\{n^{-1}\sum_{i=1}^n U_i(\hat{\psi})\right\}^2=o_p(1)$$, then it follows from Theorem 5.1 in van der Vaart (1998) (under the usual identifiability and uniform convergence criteria) that $$\hat{\psi}$$ is a consistent estimator of $$\psi$$. To find its asymptotic distribution, note that $$\hat{\psi}$$ solves an estimating equation of the form \begin{eqnarray*} 0=n^{-1}\sum_{i=1}^n U_i(\psi)\sum_{j=1}^n \frac{\partial U_j}{\partial \psi}(\psi). \end{eqnarray*} The sandwich variance estimator previously derived is not valid for this solution when $$n^{-1}\left\{\sum_{i=1}^n U_i(\psi)\right\}^2$$ reaches a minimum away from zero, for then the sandwich estimator’s denominator $$n^{-1}\sum_{j=1}^n \frac{\partial U_j}{\partial \psi}(\hat{\psi})$$ is zero. A modified standard error calculation can be made as follows. Let $$O_i$$ be the observed data recorded for subject $$i$$. Applying Hajek projections to the above estimating equation (van der Vaart, 1998), we then find that \[\sqrt{n}(\hat{\psi}-\psi)=\frac{1}{\sqrt{n}}\sum_{i=1}^n E\left\{\frac{\partial W_i}{\partial \psi}(\psi)\right\}^{-1}W_i(\psi)+o_p(1), \] where \[W_i(\psi)\equiv E\left\{U_i(\psi)\frac{\partial U_j}{\partial \psi}(\psi)|O_i\right\}+E\left\{U_j(\psi)\frac{\partial U_i}{\partial \psi}(\psi)|O_i\right\}\!. \] It follows that the asymptotic variance of $$\hat{\psi}$$ can be consistently estimated as the sample variance of \[\tilde{W}_i(\psi)\equiv U_i(\hat{\psi})\frac{1}{n}\sum_{j=1}^n \frac{\partial U_j}{\partial \psi}(\hat{\psi}) +\frac{\partial U_i}{\partial \psi}(\hat{\psi})\frac{1}{n}\sum_{j=1}^n U_j(\hat{\psi}),\] divided by \[\left[\left\{\frac{1}{n}\sum_{i=1}^n \frac{\partial U_i}{\partial \psi}(\hat{\psi})\right\}^2 +\frac{1}{n}\sum_{i=1}^n U_i(\hat{\psi})\frac{1}{n}\sum_{i=1}^n \frac{\partial^2 U_i}{\partial \psi^2}(\hat{\psi}) \right]^2.\] This reduces to the usual sandwich variance estimator (as previously derived) when $$\sum_{i=1}^n U_i(\hat{\psi})=0$$, but not otherwise. 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BiostatisticsOxford University Press

Published: Sep 27, 2017

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