Prognosis of cancer survivors: estimation based on differential equations.

We present a method for estimating several prognosis parameters for cancer survivors. The method utilizes the fact that these parameters solve differential equations driven by cumulative hazards. By expressing the parameters as solutions to differential equations, we develop generic estimators that are easy to implement with standard statistical software. We explicitly describe the estimators for prognosis parameters that are often employed in practice, but also for parameters that, to our knowledge, have not been used to evaluate prognosis. We then apply these parameters to assess the prognosis of five common cancers in Norway.

Distinguishing Immunologic and Behavioral Effects of Vaccination.

The interpretation of vaccine efficacy estimands is subtle, even in randomized trials designed to quantify the immunologic effects of vaccination. In this article, we introduce terminology to distinguish between different vaccine efficacy estimands and clarify their interpretations. This allows us to explicitly consider the immunologic and behavioral effects of vaccination, and establish that policy-relevant estimands can differ substantially from those commonly reported in vaccine trials. We further show that a conventional vaccine trial allows the identification and estimation of different vaccine estimands under plausible conditions if one additional post-treatment variable is measured. Specifically, we utilize a "belief variable" that indicates the treatment an individual believed they had received. The belief variable is similar to "blinding assessment" variables that are occasionally collected in placebo-controlled trials in other fields. We illustrate the relations between the different estimands, and their practical relevance, in numerical examples based on an influenza vaccine trial.

Causal inference with recurrent and competing events.

Many research questions concern treatment effects on outcomes that can recur several times in the same individual. For example, medical researchers are interested in treatment effects on hospitalizations in heart failure patients and sports injuries in athletes. Competing events, such as death, complicate causal inference in studies of recurrent events because once a competing event occurs, an individual cannot have more recurrent events. Several statistical estimands have been studied in recurrent event settings, with and without competing events. However, the causal interpretations of these estimands, and the conditions that are required to identify these estimands from observed data, have yet to be formalized. Here we use a formal framework for causal inference to formulate several causal estimands in recurrent event settings, with and without competing events. When competing events exist, we clarify when commonly used classical statistical estimands can be interpreted as causal quantities from the causal mediation literature, such as (controlled) direct effects and total effects. Furthermore, we show that recent results on interventionist mediation estimands allow us to define new causal estimands with recurrent and competing events that may be of particular clinical relevance in many subject matter settings. We use causal directed acyclic graphs and single world intervention graphs to illustrate how to reason about identification conditions for the various causal estimands based on subject matter knowledge. Furthermore, using results on counting processes, we show that our causal estimands and their identification conditions, which are articulated in discrete time, converge to classical continuous time counterparts in the limit of fine discretizations of time. We propose estimators and establish their consistency for the various identifying functionals. Finally, we use the proposed estimators to compute the effect of blood pressure lowering treatment on the recurrence of acute kidney injury using data from the Systolic Blood Pressure Intervention Trial.

Perspective on 'Harm' in Personalized Medicine.

Avoiding harm is an uncontroversial aim of personalized medicine and other epidemiologic initiatives. However, the precise mathematical translation of "harm" is disputable. Here we use a formal causal language to study common, but distinct, definitions of "harm". We clarify that commitment to a definition of harm has important practical and philosophical implications for decision making. We relate our practical and philosophical considerations to ideas from medical ethics and legal practice.

Identification of Vaccine Effects When Exposure Status Is Unknown.

Results from randomized controlled trials (RCTs) help determine vaccination strategies and related public health policies. However, defining and identifying estimands that can guide policies in infectious disease settings is difficult, even in an RCT. The effects of vaccination critically depend on characteristics of the population of interest, such as the prevalence of infection, the number of vaccinated, and social behaviors. To mitigate the dependence on such characteristics, estimands, and study designs, that require conditioning or intervening on exposure to the infectious agent have been advocated. But a fundamental problem for both RCTs and observational studies is that exposure status is often unavailable or difficult to measure, which has made it impossible to apply existing methodology to study vaccine effects that account for exposure status. In this study, we present new results on this type of vaccine effects. Under plausible conditions, we show that point identification of certain relative effects is possible even when the exposure status is unknown. Furthermore, we derive sharp bounds on the corresponding absolute effects. We apply these results to estimate the effects of the ChAdOx1 nCoV-19 vaccine on SARS-CoV-2 disease (COVID-19) conditional on postvaccine exposure to the virus, using data from a large RCT.

Longitudinal incremental propensity score interventions for limited resource settings.

Many real-life treatments are of limited supply and cannot be provided to all individuals in the population. For example, patients on the liver transplant waiting list usually cannot be assigned a liver transplant immediately at the time they reach highest priority because a suitable organ is not immediately available. In settings with limited supply, investigators are often interested in the effects of treatment strategies in which a limited proportion of patients receive an organ at a given time, that is, treatment regimes satisfying resource constraints. Here, we describe an estimand that allows us to define causal effects of treatment strategies that satisfy resource constraints: incremental propensity score interventions (IPSIs) for limited resources. IPSIs flexibly constrain time-varying resource utilization through proportional scaling of patients' natural propensities for treatment, thereby preserving existing propensity rank ordering compared to the status quo. We derive a simple class of inverse-probability-weighted estimators, and we apply one such estimator to evaluate the effect of restricting or expanding utilization of "increased risk" liver organs to treat patients with end-stage liver disease.

Translating questions to estimands in randomized clinical trials with intercurrent events.

Intercurrent (post-treatment) events occur frequently in randomized trials, and investigators often express interest in treatment effects that suitably take account of these events. Contrasts that naively condition on intercurrent events do not have a straight-forward causal interpretation, and the practical relevance of other commonly used approaches is debated. In this work, we discuss how to formulate and choose an estimand, beyond the marginal intention-to-treat effect, from the point of view of a decision maker and drug developer. In particular, we argue that careful articulation of a practically useful research question should either reflect decision making at this point in time or future drug development. Indeed, a substantially interesting estimand is simply a formalization of the (plain English) description of a research question. A common feature of estimands that are practically useful is that they correspond to possibly hypothetical but well-defined interventions in identifiable (sub)populations. To illustrate our points, we consider five examples that were recently used to motivate consideration of principal stratum estimands in clinical trials. In all of these examples, we propose alternative causal estimands, such as conditional effects, sequential regime effects, and separable effects, that correspond to explicit research questions of substantial interest.

Separating Algorithms From Questions and Causal Inference With Unmeasured Exposures: An Application to Birth Cohort Studies of Early Body Mass Index Rebound.

Observational studies reporting on adjusted associations between childhood body mass index (BMI; weight (kg)/height (m)2) rebound and subsequent cardiometabolic outcomes have often not paid explicit attention to causal inference, including definition of a target causal effect and assumptions for unbiased estimation of that effect. Using data from 649 children in a Boston, Massachusetts-area cohort recruited in 1999-2002, we considered effects of stochastic interventions on a chosen subset of modifiable yet unmeasured exposures expected to be associated with early (

Estimating the effect of increasing utilization of living donor liver transplantation using observational data.

There has been a recent increase in enthusiasm for expansion of living donor liver transplantation (LDLT) programmes. Using all adults initially placed on the waiting list in the United States, we estimated the risk of overall mortality under national strategies which differed in their utilization of LDLT. We used a generalization of inverse probability weighting which can estimate the effect of interventions in the setting of finite resources. From 2005 to 2015, 93 812 eligible individuals were added to the waitlist: 51 322 received deceased donor grafts while 1970 underwent LDLT. Individuals who underwent LDLT had more favourable prognostic factors, including lower mean MELD score at transplant (14.6 vs. 20.5). The 1-year, 5-year and 10-year cumulative incidence of death under the current level of LDLT utilization were 18.0% (95% CI: 17.8, 18.3%), 41.2% (95% CI: 40.8, 41.5%) and 57.4% (95% CI: 56.9, 57.9%) compared to 17.9% (95% CI: 17.7, 18.2%), 40.6% (95% CI: 40.2, 40.9%) and 56.4% (95% CI: 55.8, 56.9%) under a strategy which doubles LDLT utilization. Expansion of LDLT utilization would have a measurable, modest effect on the risk of mortality for the entire cohort of individuals who begin on the transplant waiting list.

How to detect and reduce potential sources of biases in studies of SARS-CoV-2 and COVID-19.

In response to the coronavirus disease (COVID-19) pandemic, public health scientists have produced a large and rapidly expanding body of literature that aims to answer critical questions, such as the proportion of the population in a geographic area that has been infected; the transmissibility of the virus and factors associated with high infectiousness or susceptibility to infection; which groups are the most at risk of infection, morbidity and mortality; and the degree to which antibodies confer protection to re-infection. Observational studies are subject to a number of different biases, including confounding, selection bias, and measurement error, that may threaten their validity or influence the interpretation of their results. To assist in the critical evaluation of a vast body of literature and contribute to future study design, we outline and propose solutions to biases that can occur across different categories of observational studies of COVID-19. We consider potential biases that could occur in five categories of studies: (1) cross-sectional seroprevalence, (2) longitudinal seroprotection, (3) risk factor studies to inform interventions, (4) studies to estimate the secondary attack rate, and (5) studies that use secondary attack rates to make inferences about infectiousness and susceptibility.

A generalized theory of separable effects in competing event settings.

In competing event settings, a counterfactual contrast of cause-specific cumulative incidences quantifies the total causal effect of a treatment on the event of interest. However, effects of treatment on the competing event may indirectly contribute to this total effect, complicating its interpretation. We previously proposed the separable effects to define direct and indirect effects of the treatment on the event of interest. This definition was given in a simple setting, where the treatment was decomposed into two components acting along two separate causal pathways. Here we generalize the notion of separable effects, allowing for interpretation, identification and estimation in a wide variety of settings. We propose and discuss a definition of separable effects that is applicable to general time-varying structures, where the separable effects can still be meaningfully interpreted as effects of modified treatments, even when they cannot be regarded as direct and indirect effects. For these settings we derive weaker conditions for identification of separable effects in studies where decomposed, or otherwise modified, treatments are not yet available; in particular, these conditions allow for time-varying common causes of the event of interest, the competing events and loss to follow-up. We also propose semi-parametric weighted estimators that are straightforward to implement. We stress that unlike previous definitions of direct and indirect effects, the separable effects can be subject to empirical scrutiny in future studies.

A Graphical Description of Partial Exchangeability.

Partial exchangeability is sufficient for the identification of some causal effects of interest. Here we review the use of common graphical tools and the sufficient component cause model in the context of partial exchangeability. We illustrate the utility of single world intervention graphs (SWIGs) in depicting partial exchangeability and provide an illustrative example of when partial exchangeability might hold in the absence of complete exchangeability.

The Effect of Prenatal Treatments on Offspring Events in the Presence of Competing Events: An Application to a Randomized Trial of Fertility Therapies.

When studying the effect of a prenatal treatment on events in the offspring, failure to produce a live birth is a competing event for events in the offspring. A common approach to handle this competing event is reporting both the treatment-specific probabilities of live births and of the event of interest among live births. However, when the treatment affects the competing event, the latter probability cannot be interpreted as the causal effect among live births. Here we provide guidance for researchers interested in the effects of prenatal treatments on events in the offspring in the presence of the competing event "no live birth." We review the total effect of treatment on a composite event and the total effect of treatment on the event of interest. These causal effects are helpful for decision making but are agnostic about the pathways through which treatment affects the event of interest. Therefore, based on recent work, we also review three causal effects that explicitly consider the pathways through which treatment may affect the event of interest in the presence of competing events: the direct effect of treatment on the event of interest under an intervention to eliminate the competing event, the separable direct and indirect effects of treatment on the event of interest, and the effect of treatment in the principal stratum of those who would have had a live birth irrespective of treatment choice. As an illustrative example, we use a randomized trial of fertility treatments and risk of neonatal complications.

A causal framework for classical statistical estimands in failure-time settings with competing events.

In failure-time settings, a competing event is any event that makes it impossible for the event of interest to occur. For example, cardiovascular disease death is a competing event for prostate cancer death because an individual cannot die of prostate cancer once he has died of cardiovascular disease. Various statistical estimands have been defined as possible targets of inference in the classical competing risks literature. Many reviews have described these statistical estimands and their estimating procedures with recommendations about their use. However, this previous work has not used a formal framework for characterizing causal effects and their identifying conditions, which makes it difficult to interpret effect estimates and assess recommendations regarding analytic choices. Here we use a counterfactual framework to explicitly define each of these classical estimands. We clarify that, depending on whether competing events are defined as censoring events, contrasts of risks can define a total effect of the treatment on the event of interest or a direct effect of the treatment on the event of interest not mediated by the competing event. In contrast, regardless of whether competing events are defined as censoring events, counterfactual hazard contrasts cannot generally be interpreted as causal effects. We illustrate how identifying assumptions for all of these counterfactual estimands can be represented in causal diagrams, in which competing events are depicted as time-varying covariates. We present an application of these ideas to data from a randomized trial designed to estimate the effect of estrogen therapy on prostate cancer mortality.

Time-dependent mediators in survival analysis: Modeling direct and indirect effects with the additive hazards model.

We discuss causal mediation analyses for survival data and propose a new approach based on the additive hazards model. The emphasis is on a dynamic point of view, that is, understanding how the direct and indirect effects develop over time. Hence, importantly, we allow for a time varying mediator. To define direct and indirect effects in such a longitudinal survival setting we take an interventional approach (Didelez, 2018) where treatment is separated into one aspect affecting the mediator and a different aspect affecting survival. In general, this leads to a version of the nonparametric g-formula (Robins, 1986). In the present paper, we demonstrate that combining the g-formula with the additive hazards model and a sequential linear model for the mediator process results in simple and interpretable expressions for direct and indirect effects in terms of relative survival as well as cumulative hazards. Our results generalize and formalize the method of dynamic path analysis (Fosen, Ferkingstad, Borgan, & Aalen, 2006; Strohmaier et al., 2015). An application to data from a clinical trial on blood pressure medication is given.

On null hypotheses in survival analysis.

The conventional nonparametric tests in survival analysis, such as the log-rank test, assess the null hypothesis that the hazards are equal at all times. However, hazards are hard to interpret causally, and other null hypotheses are more relevant in many scenarios with survival outcomes. To allow for a wider range of null hypotheses, we present a generic approach to define test statistics. This approach utilizes the fact that a wide range of common parameters in survival analysis can be expressed as solutions of differential equations. Thereby, we can test hypotheses based on survival parameters that solve differential equations driven by cumulative hazards, and it is easy to implement the tests on a computer. We present simulations, suggesting that our tests perform well for several hypotheses in a range of scenarios. As an illustration, we apply our tests to evaluate the effect of adjuvant chemotherapies in patients with colon cancer, using data from a randomized controlled trial.

On the collapsibility of measures of effect in the counterfactual causal framework.

The relationship between collapsibility and confounding has been subject to an extensive and ongoing discussion in the methodological literature. We discuss two subtly different definitions of collapsibility, and show that by considering causal effect measures based on counterfactual variables (rather than measures of association based on observed variables) it is possible to separate out the component of non-collapsibility which is due to the mathematical properties of the effect measure, from the components that are due to structural bias such as confounding. We provide new weights such that the causal risk ratio is collapsible over arbitrary baseline covariates. In the absence of confounding, these weights may be used for standardization of the risk ratio.

Effect heterogeneity and variable selection for standardizing causal effects to a target population.

The participants in randomized trials and other studies used for causal inference are often not representative of the populations seen by clinical decision-makers. To account for differences between populations, researchers may consider standardizing results to a target population. We discuss several different types of homogeneity conditions that are relevant for standardization: Homogeneity of effect measures, homogeneity of counterfactual outcome state transition parameters, and homogeneity of counterfactual distributions. Each of these conditions can be used to show that a particular standardization procedure will result in an unbiased estimate of the effect in the target population, given assumptions about the relevant scientific context. We compare and contrast the homogeneity conditions, in particular their implications for selection of covariates for standardization and their implications for how to compute the standardized causal effect in the target population. While some of the recently developed counterfactual approaches to generalizability rely upon homogeneity conditions that avoid many of the problems associated with traditional approaches, they often require adjustment for a large (and possibly unfeasible) set of covariates.

The additive hazard estimator is consistent for continuous-time marginal structural models.

Marginal structural models (MSMs) allow for causal analysis of longitudinal data. The standard MSM is based on discrete time models, but the continuous-time MSM is a conceptually appealing alternative for survival analysis. In applied analyses, it is often assumed that the theoretical treatment weights are known, but these weights are usually unknown and must be estimated from the data. Here we provide a sufficient condition for continuous-time MSM to be consistent even when the weights are estimated, and we show how additive hazard models can be used to estimate such weights. Our results suggest that continuous-time weights perform better than IPTW when the underlying process is continuous. Furthermore, we may wish to transform effect estimates of hazards to other scales that are easier to interpret causally. We show that a general transformation strategy can be used on weighted cumulative hazard estimates to obtain a range of other parameters in survival analysis, and explain how this strategy can be applied on data using our R packages ahw and transform.hazards.