The built-in selection bias of hazard ratios formalized using structural causal models

When interested in time-to-event outcomes, ideally, one would like to know the hazard rates of an individual in the worlds with and without exposure. It is then standard practice to fit the observed hazard rates with a (time-invariant) Cox model (Cox 1972) to estimate the ratio of the expected hazard rates in both worlds. A decade ago, Hernán (2010) raised awareness that hazard ratios estimated from a randomized controlled trial (RCT) are unsuitable for causal inference. Firstly, the average hazard ratio could be uninformative as there will typically be time-varying hazard ratios. More importantly, even when period-specific hazard ratios are estimated, these can vary solely due to the loss of randomization over time by conditioning on survivors. The exposure assignment and risk factors become dependent when conditioning on individuals that survived t, i.e. survival time \(T{\ge } t\), even if these risk factors are unrelated to the exposure (Aalen et al. 2015). As a result, effect measures based on hazard rates can suffer from non-collapsibility (Martinussen and Vansteelandt 2013; Aalen et al. 2015; Sjölander et al. 2016; Daniel et al. 2021).

In practice, the ratio of (partly) marginalized hazards, is estimated, that by the non-collapsibility, deviates from the conditional (causal) hazard ratio. This contrast is referred to as the built-in selection bias of hazard ratios as the bias results from conditioning on prior survival (Hernán 2010; Aalen et al. 2015; Sjölander et al. 2016; Stensrud et al. 2018; Young et al. 2020; Martinussen et al. 2020). This bias should not be confused with confounding bias that is absent when using data from an RCT (Didelez and Stensrud 2021). For exposure assignment A, and the potential survival time when the exposure is intervened on to a denoted by \(T^{a}\), the expected observed hazard ratio from an RCT satisfies(De Neve and Gerds 2020; Martinussen et al. 2020). The expected observed hazard ratio thus equals the ratio of hazard rates at time t for the potential outcomes of individuals from different populations; those for which \(T^{a}{\ge } t\) and those for which \(T^{0}{\ge } t\). As indicated before, these populations will typically not be exchangeable in other risk factors, implying that an effect found cannot be (solely) assigned to the exposure. The effect does thus not reflect how the hazard rate of an individual is affected by exposure. Only for cause-effect relations such that (1) is time-invariant, the estimand can be interpreted as \(\frac{\log ({\mathbb {P}}(T^{a}{\ge } t))}{\log ({\mathbb {P}}(T^{0}{\ge } t))}\). It has been recommended to use better interpretable estimands such as contrasts of quantiles, the restricted mean survival or survival probabilities of the potential outcomes respectively (Hernán 2010; Stensrud et al. 2018; Bartlett et al. 2020; Young et al. 2020), or the probabilistic index derived from the latter (De Neve and Gerds 2020). Alternatively, one can avoid interpretation issues by using accelerated failure time models (Hernán et al. 2005; Hernán 2010) or additive hazard models (Aalen et al. 2015; Martinussen et al. 2020).

Nevertheless, particularly in medical sciences, observed hazard ratios are still commonly presented by practitioners. In this paper, we formalize how the expected observed hazard ratio deviates from the causal hazard ratio (as defined in Sect. 2), and thus quantify the built-in selection bias. To do so, we first present a general parameterization of cause-effect relations for time-to-event outcomes using a structural causal model in Sect. 2 and explain the loss of randomization over time by conditioning on survivors. We will limit ourselves to systems where the causal effect is appropriately described by a causal hazard ratio. The quantitative examples in which the hazard under no exposure varies among individuals, i.e. frailty, as presented in the literature (Aalen et al. 2015; Stensrud et al. 2017; Balan and Putter 2020) do fit in our framework, and we will formalize results for these examples. Additionally, we will extend these examples with causal effect heterogeneity, i.e. the causal effect on the hazard rate might vary between individuals (Stensrud et al. 2017). In Sect. 3, we define the causal hazard ratio and explain why this estimand is not identifiable from data. Practioners instead compute the hazard ratio from data, and in Sect. 4 (which comprises the bulk of this paper) we derive what estimand is estimated: the survivor marginalized causal hazard ratio. This estimand describes a combination of the causal effect of interest and the difference in latent frailty- and modifying-features distribution between survivors in the exposed and unexposed universe. We point out exactly how this estimand deviates from the causal hazard ratio in the presence of frailty and effect heterogeneity. To develop understanding of how selection of frailty- and modifying-features affect the value of the estimand, we presented examples for systems in the presence of frailty (Sect. 4.1), effect heterogeneity (Sect. 4.2) or both (Sect. 4.3). In Sect. 5, we shortly discuss the implications of our results for the traditional Cox estimand. Finally, we present some concluding remarks in Sect. 6.

In this paper, probability distributions of factual and counterfactual outcomes are defined in terms of the potential outcome framework (Neyman 1990; Rubin 1974). Let \(T_{i}\) and \(A_{i}\) represent the (factual) stochastic outcome and exposure assignment level of individual i. Let \(T_{i}^{a}\) equal the potential outcome of individual i under an intervention of level a (counterfactual when \(A_{i} \ne a\)). For those more familiar with the do-calculus, \(T^{a}\) is equivalent to \(T \mid do(A{=}a)\) as e.g. derived in (Pearl 2009, Equation 40) and (Bongers et al. 2021, Definition 8.6). Throughout this paper, we will assume causal consistency: if \(A_{i} = a\), then \(T_{i}^{a} = T_{i}^{A_{i}} = T_{i}\), implying that potential outcomes are independent of the assigned exposure levels of other individuals.

The hazard rate of \(T^{a}\) can vary among the individuals in the population of interest. We will parameterize this heterogeneity for hazards of \(T^{0}\) using a random variable \(U_{0i}\) that represents the frailty of individual i (see for example (Aalen et al. 2008, Chapter 6) or Balan and Putter (2020)). There can also be (relative) effect heterogeneity that we parameterize using the random variable \(U_{1i}\), giving rise to an individual-specific hazard ratio. The hazard of the potential outcome \(T_{i}^{a}\) can be parameterized with a function that depends on \(U_{0i}\), \(U_{1i}\) and a. We describe cause-effect relations with a structural causal model (SCM) which is commonly used in the causal graphical literature, see e.g. (Pearl 2009, Chapter 1.4) and (Peters et al. 2018, Chapter 6), to model observations. Instead, we include details on individual effect modifier \(U_{1}\) as well as the latent common cause of the outcomes \(U_{0}\), to describe all the potential outcomes of an individual jointly. A SCM as presented in this paper is therefore a union of the SCM for observations (\(A{=}a\)), and the so-called intervened SCMs for all possible \(do(A{=}a)\). SCMs have been used before to describe hazards of potential outcomes (Hernán et al. 2000, 2001, 2005). Formulation of hazard rates of potential outcomes presented in the literature, e.g. by Aalen et al. (2015) and by Stensrud et al. (2017), naturally fit in this parameterization. However, as mentioned before, the dependence of \(T^{a}\) and \(T^{0}\) beyond shared frailty is typically not specified. The SCM consists of a joint probability distribution of \((N_{A}, U_{0}, U_{1}, N_{T})\) and a collection of structural assignments \((f_{A}, f_{\lambda })\) such that

Note that the data generating mechanism is described by this SCM as \(T_{i}^{A_{i}} = T_{i}\). If in SCM (2),then there exists confounding as the distributions of \((U_{0}, U_{1}, N_{T})\) are not exchangeable between exposed and non-exposed individuals. However, in this work we focus on the distribution of data observed from a properly executed RCT, where by the randomization so that there is no confounding. Note that \(\lambda _{i}^{a}(t)\) equals the hazard of the potential outcome of individual i under exposure a, i.e. and is thus a random variable when we consider an arbitrary individual. In this parameterization, \(U_{0}\) results in heterogeneity of the hazard under no exposure between individuals, and the presence of \(U_{1}\) results in heterogeneity of the effect of the exposure on the hazard between individuals. The SCM could be re-parameterized by including more details, e.g. measured risk factors, so that part of the unmeasured heterogeneity can be explained.

To understand how the so-called built-in selection bias is introduced, realize that \(T_{i}^{a}\) depends on a and the random variables \(U_{0i}\), \(U_{1i}\), \(N_{Ti}\) only, i.e.  \(T_{i}^{a}{:=}~ \min \left\{ t{>}0{:}~ \exp \left( -\int _{0}^{t}f_{\lambda }(s, U_{0i}, U_{1i}, a)ds\right) {\le } N_{Ti}\right\} = g(U_{0i}, U_{1i}, N_{Ti}, a)\), for some function g. In an RCT, , soHowever, this independence might not hold conditionally on survival at time t since \(T_{i}{:=}~ g(U_{0i}, U_{1i}, N_{Ti}, A_{i})\), so that \(A_{i}{\mid }T_{i}{\ge } t\) can inform on \((U_{0i}, U_{1i}, N_{Ti})\) and thus on \(T_{i}^{a}\), thenOn the contrary,as \(T_{i}^{a}{:=}~g(U_{0i}, U_{1i},N_{Ti}, a)\), \(A_{i}{\mid }T^{a}_{i}{\ge } t\) does not inform on \((U_{0i},U_{1i}, N_{Ti})\). In the literature, the dependence in (6) is often implicitly derived by recognizing that \(\{T{\ge } t \}\) is a collider that can thus open a back-door path between A and \(T^{a}\) (Aalen et al. 2015; Sjölander et al. 2016). The bias that results from conditioning on this collider is referred to as the built-in selection bias of the hazard ratio (Hernán 2010). This complicates causal inference which requires that the distribution of potential outcomes can be expressed in terms of the observed distribution.

In SCM (2), we did not restrict the distribution of \(U_{0}\) and \(U_{1}\) and only restricted \(f_{\lambda }\) and \(f_{A}\) to be properly defined hazard and inverse cumulative distribution functions respectively, so that the structural model is very general. It is important to realize that a SCM cannot be validated with data as it describes potential outcomes from different universes. For each individual the outcome can only be observed in one of the universes, and only the fit of the distribution of the outcomes in the factual world can be verified. In this work we focus on settings where the causal effect can be accurately described with a causal hazard ratio, which is defined in the next section. This will be the case when in SCM (2),In the remainder of this manuscript we will restrict ourselves to cause-effect relations that meet this restriction.

If \(f_{\lambda }(t,U_{0i},U_{1i},a) = f_{0}(t,U_{0i})f_{1}(t,U_{1i},a)\) and \(f_{1}(t,U_{1i},0) = 1\), then the individual causal effect is described by \(f_{1}(t,U_{1i},a)\). The latter equals the ratio at time t of the hazard of an individual’s potential outcome when exposed to level a and when not exposed, i.e. \(\frac{\lambda _{i}^{a}(t)}{\lambda _{i}^{0}(t)}\). In the case of homogeneous effects, \(f_{1}(t, U_{1i}, a) = f_{1}(t, a)\) is equal for all individuals. In the case of heterogeneity of effects, \(f_{1}(t, U_{1i}, a)\) is the individual multiplicative causal effect. From a public health perspective, the ratio of the expected hazard rates in the world where everyone is exposed to a and in the world where all individuals are not exposed is of interest. This causal hazard ratio (CHR) of interest can be obtained as the ratio of the marginalized (over \(U_{0}\) and \(U_{1}\)) conditional hazard rates in both worlds as presented in Definition 1.

When the parameterization of the cause-effect relations as SCM (2) would be known, the CHR can be expressed in terms of the distribution of the data generating mechanism as presented in Theorem 1.

For an example, consider the commonly used frailty model where effect heterogeneity is absent, i.e.The CHR equals the multiplicative effect that does not differ among individuals and equals \(f_{1}(t, a)\). By applying Theorem 1, this CHR is indeed derived to equalIt is important to note that \(f_{1}(t,a)\) deviates from the expected observed hazard ratio equal toas we will elaborate on in Sect. 4.

In summary, it became clear that to derive the CHR from data, inference on the distribution of the latent frailty \(U_{0}\) and effect modifier \(U_{1}\) must be made. When their distributions are known, inference can be drawn from observed data, even when the parameters of the distributions are unknown. Software available to estimate frailty parameters are described by Balan and Putter (2020), and such methods could also be adapted to estimate the latent modifier distribution. However, in practice, the distributions of these latent variables are unknown. Even in the case without causal effect heterogeneity, it is impossible to distinguish the presence of frailty from a time-dependent causal effect (Balan and Putter 2020, Section 2.5). More precisely, different combinations of (varying) effect sizes and frailty distributions give rise to the same marginal distribution. The same applies to combinations that also involve effect modifiers. In the case of clustered survival data (e.g. family data (Valberg et al. 2018)), at least theoretically, the shared frailty could be distinguished from violation of proportional hazards (Balan and Putter 2020). Reasoning along the same lines, individual frailty and marginal time-varying effects could only be derived from effect heterogeneity in the case of recurrent events with stationary distributions.

In Theorem 1, the actual CHR (see Definition 1) has been expressed in terms of the distributions of the observed data, and we concluded that these are not identifiable without making untestable assumptions on the distribution of \((U_{0}, U_{1})\). Instead, practitioners often compute the hazard ratio from data, which expectation we refer to as the observed hazard ratio (OHR) and equalsTo be precise, at time t the hazard rate can only be observed for non-censored individuals at that time \((C(t)=0)\). However, in this work we will assume independent censoring, so that \({\mathbb {P}}\left( T \mid T{\ge }t, A{=}a \right) \) is equal to \({\mathbb {P}}\left( T \mid T{\ge }t, A{=}a, C(t){=}0\right) \).

To compare the OHR to the CHR that quantifies the causal effect of interest, the OHR should be expressed in terms of potential outcomes. For data from an RCT, by independence (7) and causal consistency, the OHR equals the survivor marginalized causal hazard ratio (SMCHR), i.e. This SMCHR should not be confused with the ‘marginal causal hazard ratio’ defined by Martinussen et al. (2020) asthat could also be named the cross-world survivor marginalized causal hazard ratio and is not considered in this work.

We will study how the SMCHR (and thus the OHR from an RCT) differs from the CHR over time. By the law of total probability, the SMCHR in (10) equalsAs the integration in the result of Theorem 1 is with respect to the population distribution of \(U_{0}\) and \(U_{1}\), instead of those individuals for which \(T^{a}{\ge } t\) or \(T^{0}{\ge } t\), the SMCHR deviates from the CHR, resulting in the built-in selection bias of the hazard (Hernán 2010; Aalen et al. 2015; Stensrud et al. 2018).

The problem induced for estimation of the CHR thus results from inference on a different estimand; the combined effect of the exposure of interest and the difference in latent frailty (and effect modification) distribution. To formalize how (10) deviates from the CHR that equals \(\frac{{\mathbb {E}}\left[ f_{0}(t,U_{0})f_{1}(t,U_{1},a)\right] }{{\mathbb {E}}\left[ f_{0}(t,U_{0})\right] }\), we focus on hazard functions that satisfy Condition 1 and do thus not have an infinite discontinuity.

The value of the SMCHR at time t is derived in Theorem 2 and can deviate from the CHR.

From the proof presented in Appendix A.2, it becomes clear that the conditional expectations that determine the value of the SMCHR equal weighted means of \(f_{0}(t,u_{0})f_{1}(t,u_{1},a)\) and \(f_{0}(t,u_{0})\) with weights \(\tfrac{{\mathbb {P}}(T^a{\ge } t \mid U_{0} = u_{0},U_{1} = u_{1})}{{\mathbb {P}}(T^{a}{\ge } t)}\) and \(\tfrac{{\mathbb {P}}(T^0{\ge } t \mid U_{0} = u_{0})}{{\mathbb {P}}(T^{a}{\ge } t)}\) respectively.

To develop our understanding of the difference between the SMCHR and the CHR, we will first continue to study the difference due to frailty and heterogeneity separately in the next two subsections. In the remainder of the section we present examples for cause-effect relations with effect heterogeneity in the presence of frailty, both for independent and dependent \(U_{0}\) and \(U_{1}\). All programming codes used in the examples presented in this paper can be found online at https://​github.​com/​RAJP93/​CHR.

We have demonstrated that in the presence of frailty and effect heterogeneity, even when the CHR is time-invariant, the SMCHR varies over time. Then, the proportional hazards assumption will not hold for an observed hazard ratio from an RCT (that is, with independent censoring, equal to the SMCHR as discussed at the start of Sect. 4). Despite the many options to deal with non-proportional hazards (see, e.g. (Thernau and Grambsch 2000, Section 6.5) or Bennett 1983; Hess 1994; Wei and Schaubel 2008), in the majority of epidemiological time-to-event studies, the misspecified traditional Cox’s proportional hazard model is fitted. The logarithm of the Cox estimate can be interpreted as the logarithm of the OHR marginalized over the observed death times (Schemper et al. 2009), i.e. \({\mathbb {E}}[\log (\text {OHR}(T)) \mid C = 0]\) for censoring indicator C. The logarithm of the Cox estimate obtained from an RCT thus equals a time-weighted average of the logarithm of the OHR. In the case of non-proportional hazards, even for independent censoring, the estimate is well-known to be affected by the censoring distribution. It differs from the average log hazard ratio \({\mathbb {E}}\left[ \log (\text {OHR}(T))\right] \) (Xu and O’Quigley 2000; Schemper et al. 2009; Boyd et al. 2012). Therefore, the bias of the Cox estimate, when the estimand is the CHR, will depend on the joint distribution of \((U_0, U_1)\) as well as the censoring distribution. In most cases considered in Sect. 4, the deviation of the OHR from the CHR increased over time. For independent censoring, the probability of censoring increases over time, so the Cox estimate is closer to the OHR at short times. In Fig. 6, this is demonstrated for the gamma-frailty case (\(\text {var}(U_{0}) = 1\), for which the SMCHR was presented in Fig. 3) by presenting empirically obtained \({\mathbb {E}}[\log (\text {OHR}(T)) \mid C = 0]\) (with 1, 000, 000 replications) based on a varying follow-up time and loss to follow-up modelled with an exponential censoring-time distribution with varying means.

A time-varying OHR violates the proportional hazard assumption that can be verified when fitting a Cox model. When the assumption is not rejected in practice, the statistical test used is probably underpowered. In the presence of heterogeneity, only when the actual CHR would be time-varying, the OHR can be approximately constant when the selection effect and the change in CHR roughly cancel out (Stensrud et al. 2018; Stensrud and Hernán 2020). The data cannot be used to distinguish the latter case from the case with a constant CHR but no heterogeneity and, thus, no selection effect. Similarly, as mentioned at the end of Sect. 3, when the OHR would vary over time, we can never conclude whether this is the result of a time-varying causal effect or due to selection. However, the proportional hazard assumption would be violated in both cases, and a standard Cox model is inappropriate.

In this paper, we have formalized how heterogeneity leads to deviation of the SMCHR (see Equation (10)) from the CHR of interest (see Definition 1) due to the selection of both the individual frailty factor \((U_{0})\) and the individual effect modifier \((U_{1})\). This work generalizes frailty examples presented in the literature (Hernán 2010; Aalen et al. 2015; Stensrud et al. 2017), by considering the possibility of multiplicative effect (on the hazard) heterogeneity that also results in non-exchangeability of exposed and unexposed individuals over time. As a result of the individual effect modifier (\(U_{1}\)), the individuals that survive in the exposed groups are expected to benefit more or suffer less from the exposure. At the same time \(U_{0}{\mid }T^{1}{\ge } t\) will have a different distribution than \(U_{0}{\mid }T^{0}{\ge } t\). When the CHR is larger than one, and , the selection effects act in the same direction. On the other hand, when the CHR is smaller than one and , the selection effects can act in opposite directions so that the SMCHR might be closer to the CHR than in the case without effect heterogeneity (see Fig. 3).

For data from an RCT, with independent censoring, the expected observed hazard ratio equals the studied SMCHR so that all results directly relate to this OHR. For observational data, the OHR does not equal the SMCHR due to confounding. However, when all confounders \({\varvec{L}}\) are observed, i.e.  , one can study the conditional (on \({\varvec{L}}\)) OHR that in turn is equal to the conditional SMCHR. The presented theorems are valid while conditioning on \({\varvec{L}}\).

The intuition explained by Hernán (2010) suggests that an appropriate estimate of the SMCHR is expected to underestimate the actual effect size, while the sign of the logarithms of the SMCHR and the CHR are equal. However, we have shown that in the presence of effect heterogeneity, an SMCHR equal to a particular value x can occur both under \(\text {CHR}{>}1\) as well as \(\text {CHR}{<}1\) as summarized in Table 1. Therefore, OHRs from RCTs are not guaranteed to present a lower bound for the causal effect without making untestable assumptions on the \((U_{0}, U_{1})\) distribution. We have derived how the SMCHR will evolve due to the selection of frailty and effect modifiers in Theorem 1. However, in practice, only the evolution of the OHR can be found (assuming sufficient data is available). Even after assuming the absence of confounding (e.g. for an RCT), the CHR is non-identifiable without making (untestable) assumptions on the \((U_{0}, U_{1})\) distribution as discussed at the end of Sect. 3. We can thus not distinguish between a time-varying CHR without selection of \(U_{0}\) and \(U_{1}\) or a time-invariant CHR with selection, see e.g. Stensrud and Hernán (2020). Adjusting for other risk factors can lower the remaining variability of \(U_{0}\) and \(U_{1}\) so that the difference between the conditional OHR and CHR is reduced. Even for an RCT, it may thus help to focus on adjusted hazard ratios despite the absence of confounding. Nevertheless, adjusting for other risk factors will require more data and modelling decisions.

Finally, we want to remark that for cause-effect relations that cannot be described by SCM (2) with \(f_{\lambda }(t,U_{0i},U_{1i},a) = f_{0}(t,U_{0i})f_{1}(t,U_{1i},a)\), the CHR is not the appropriate measure to quantify the causal effect. Then, other causal hazard contrasts can be relevant that may or may not have an observable analog. For example, additive hazard models (when well-specified) do not suffer from the frailty selection as shown by Aalen et al. (2015), but these models will still suffer from latent modifier selection in the presence of effect heterogeneity (\({\mathbb {E}}[U_{1} \mid T^{1}{\ge } t]{>}{\mathbb {E}}[U_{1}]\)) as demonstrated in our companion paper (Post et al. 2024).

We hope that the discussed effect heterogeneity and formalization of the built-in selection bias of the OHR show the need to use more suitable estimands. As suggested by others, contrasts of the survival probabilities, the median, or the restricted mean survival time of potential outcomes are proper measures to quantify causal effects on time-to-event outcomes (Hernán 2010; Stensrud et al. 2018; Bartlett et al. 2020; Young et al. 2020). Modelling and estimating hazard rates can still be helpful for causal inference when the hazards are used to derive one of the appropriate causal estimands (Ryalen et al. 2018).