Competing risks framework
Notation and assumptions
We will introduce the notation and the treatment effect estimands for RHC data example, where there are two causes of “discharge”, due to improvement and due to death. That is the number of competing events \(K=2\).
Let \(T\) be the lag time from the admission to an ICU and consideration for RHC procedure to the discharge or death in a hospital, whichever comes first. And let \(E\) be an indicator of the type of the event, with \(E=1\) if \(T\) corresponds to discharge, and \(E=2\) if \(T\) corresponds to death. If neither discharge nor death is observed during the follow-up, \(T\) is right-censored by the time to the last visit, and in this case \(E=0\). In our length-of-stay analysis of RHC data, there is no censoring, and for all the patients (except for 1 person) the date of the end-of-stay in an ICU is known.
Let \(A\) be the treatment assignment random variable with \(A=1\) for those who had RHC procedure and \(A=0\) for those who did not have, and \(C\) be a vector of covariates.
Let \((T^a, E^a)\), \(a=0,1\), be potential outcomes that would be observed if a patient would receive treatment \(a\). Our causal assumptions are:
no unmeasured confounding: treatment assignment \(A\) is independent of potential outcomes given \(C\). That is, given \(C\), \(A\) and \((T^0, E^0)\) are independent, and \(A\) and \((T^1, E^1)\) are independent;
positivity: \(0<P(A=a\mid C)<1\) for \(a=0,1\);
SUTVA (Stable Unit Treatment Value Assumption): the outcome of every patient does not depend on the treatment of others (noninterference), and the outcome does not depend on the way a treatment was assigned (consistency).
If there is right censoring in the data, our methods assume that given \(A\), the time to the last visit (censoring time) is independent of the outcome \((T,E)\).
Treatment effect estimands
In order to introduce treatment effect estimands, we need first to define single-world cause-specific hazards of transitioning to states “1. discharge” or “2. in-hospital death” (Figure 1) in a counterfactual world corresponding to treatment \(a\): \[ h_k^a(t)=\lim_{\Delta t \to 0} \frac{1}{\Delta t} P(t\leq T^a < t+\Delta t, E^a=k \mid T^a \geq t), \quad k=1,2, \] and single-world cause-specific cumulative incidence functions (CIF): \[ CIF_a(t, k)=E\left[I_{(T^a \leq t, E^a=k)}\right]=P(T^a \leq t, E^a=k), \quad k=1,2. \]
Figure 1: A single-world competing risks model with two events discharge and in-hospital death.
Hazards \(h_k^a(t)\) are usually interpreted as rates of the competing processes, and \(CIF_a(t, k)\) are often called risk functions - see, e.g., Andersen et al. (2012) and Cole et al. (2015). For a more detailed definition of \(CIF_a(t, k)\), we refer to Charpignon et al. (2021) (Methods section).
Hazards ratios (HR), risk differences (RD) and risk ratios (RR)
The hazard ratios for both events are defined by \[ HR_k(t)=\frac{h_k^1(t)}{h_k^0(t)}, \quad k=1,2. \] Notice that here \(HR_k(t)\) can depend on \(t\) because we do not make any modeling assumptions regarding the hazard functions \(h_k^a(t)\).
The risk difference functions are defined by \[ RD(t, k)=E\left[I_{(T^1 \leq t, E^1=k)}\right] - E\left[I_{(T^0 \leq t, E^0=k)}\right]= CIF_1(t, k) - CIF_0(t, k), \quad k=1,2. \] \(RD(t,k)\) are between-treatment-arms differences of probabilities of having an event of type \(k\) by time \(t\). They represent the average treatment effects on different types of events, for the target population. The risk ratios, respectively, are defined by \[ RR(t, k)=\frac{E\left[I_{(T^1 \leq t, E^1=k)}\right]}{E\left[I_{(T^0 \leq t, E^0=k)}\right]}= \frac{CIF_1(t, k)}{CIF_0(t, k)}, \quad k=1,2. \]
Restricted mean time (RMT) gained/lost due to treatment
If we want to quantify the effect of treatment in terms of the number of days/months/years gained (or lost) on average from being treated, we can “translate” \(CIF_a(t, k)\) from probabilities to the time domain by using the following Restricted Mean Time functionals \(RMT_a(\tau, k)\) which are defined by \[ RMT_a(\tau, k) = \int_0^{\tau} CIF_a(t,k)dt, \quad k=1,2, \quad a=0,1, \] where \(\tau\) is some prespecified time point of interest, e.g. 5 years. \(RMT_a(\tau, k)\) is the area under the CIF curve, which can be interpreted as the restricted average post-event time corresponding to event \(k\). If the event is desirable, we would like to see a longer average post-event time. For example, for recovery we would like patients to recover faster which would result in a longer average time in recovery - see, e.g., McCaw et al. (2020). Notice that \(RMT_a(\tau, k)\), \(k=1,2\), are cause-specific functionals that correspond to event \(k\). The average treatment effect based on RMT can be defined as a difference: \[ RMT_1(\tau, k)-RMT_0(\tau, k). \]
Use of the RMT for quantifying treatment effects is becoming more established in the univariate survival analysis, where this effect is usually defined as a difference between the Restricted Mean Survival Times (RMST) which is defined by \[ RMST_a(\tau) = E(T^a I(T^a \leq \tau)), \] that is the expectation of the restricted time-to-event. However, in competing risks, the RMT is still very rarely used. Two recent papers of Calkins et al. (2018) and McCaw et al. (2020) have the real data examples that illustrate the use of the RMT in the competing risks setting.
From the definition of the RMST, it is clear that it does not distinguish between the causes of the event, but only uses the information on the time-to-event. Alternatively, the RMST can be obtained as the area under the survival curve up to \(\tau\). The RMST is interpreteted as the expected restricted sojourn time in state 0 (Figure 1).
The relation between the RMST, used in the univariate survival analysis, and the cause-specific RMT, used in the competing risks analysis, can be seen from the following: \[ RMST_a(\tau) = E(T^a I(T^a \leq \tau))= \int_0^{\tau} \left\{1- CIF_a(t,1) - CIF_a(t,2)\right\} dt =\\ \tau - \int_0^{\tau} CIF_a(t,1)dt - \int_0^{\tau}CIF_a(t,2) dt= \tau- RMT_a(\tau,1) - RMT_a(\tau,2). \]
That is, combined together the cause-specific \(RMT_a(\tau, 1)\) and \(RMT_a(\tau, 2)\) complement the Restricted Mean Survival Time, \(RMST_a(\tau)\).
Estimation
Propensity scores (PS)
When we estimate the ATE from observational data where the treatment was not randomized, we need to account for the treatment assignment selection bias, e.g., through an emulation of baseline randomization. The goal of this emulation is to create similar treatment arms (at least with respect to measured baseline covariates). This can be achieved under the assumption of “no unmeasured confounding”, e.g., by inverse-probability-of-treatment-weighting (IPTW) based on a treatment assignment model, or a propensity score (PS) model. A propensity score for an individual with observed treatment \(A=a\) and a vector of covariates \(C=c\), is defined by: \[ ps=P(A=a \mid C=c). \] Further, we denote the estimates of \(ps\) by \(\widehat{ps}\).
There are many considerations playing a role in:
- choosing a PS model,
- selection of variables to include in this model,
- choosing appropriate balancing weights based on a fitted PS model,
- model diagnostics,
- and finally, taking into account the uncertainty resulting from a treatment assignment modeling step.
The aim of the PS model is to balance covariates between treatment groups, and there were proposed several ways to achieve the balance. For example, Imai and Ratkovic (2014) propose to fit a PS model while “optimizing the covariate balance”, that is they combine the steps of fitting a PS model and obtaining (optimal) balancing weights into one procedure. In contrast to this method, we take a step-by-step approach where any part of the whole process of building PS weights can be replaced with its alternative, e.g., anothe typr of a PS model, or another type of weights. In what follows we address some (but not all) of the listed above considerations:
(A) A fitted PS model can be formally assessed for the goodness-of-fit (GOF) to the observed data using formal statistical tests, but this type of diagnostics is usually not done in practice. The reason for that is that the goal of the PS model is to balance baseline covariates, therefore the main criterion for a good fit of the PS model is usually the degree of achieved balance for important confounders. However, it is recognized that the balance cannot be attained with a poorly fitted PS model, and a flexible family of models for propensity scores is more likely to fit the data well.
causalCmprsk uses logistic regression for the propensity scores model. All the factors provided by the user to the functions get.weights, fit.nonpar and fit.cox are included in the PS model, and no variable selection is implemented. The factors in the PS model can be main effects or interactions. Provided a PS model fits the data well and if there is no problem of non-overlap between the treatment arms, the covariates will be balanced in the weighted population, meaning that the weighted marginal distributions will be similar in two treatment arms (if their main effects were included in the PS model), and the weighted joint distributions will be similar in both treatment arms as well (if their corresponding interactions were included in the PS model).
The problem of non-overlap, though, can be overcome by choosing, e.g., overlap weights (F. Li, Morgan, and Zaslavsky (2018), F. Li, Thomas, and Li (2018), Thomas, Li, and Pencina (2020)).
(B) For theoretical and practical considerations on confounder selection we refer to Ding, VanderWeele, and Robins (2017), Austin and Stuart (2015) and VanderWeele (2019).
Currently, causalCmprsk does not implement any confounder selection procedure, and all the factors provided by the user to the functions get.weights, fit.nonpar and fit.cox in the argument C are included in the PS model.
In general, we just remind that the process of variable selection in the PS model involves not only a covariate-treatment relationship but also how strongly a covariate is related to the studied outcome, since only variables that are related to both the treatment assignment and the outcome are defined as confounders. The list of potential confounders should be screened by a subject matter expert to avoid inclusion of a possible instrumental variable (IV) in the PS model, since this might amplify the residual bias in the presence of unmeasured confounding or increase the variance if all the confounders are measured (Ding, VanderWeele, and Robins (2017)).
(C) It is important to diagnose a fitted PS model before we use it, since it is the basis for our inference on the average treatment effects. The importance of diagnostics of a fitted PS model is recognized by all the authors who use propensity scores, but by “the diagnostics of a PS model” they usually mean different procedures (Granger et al. (2020)).
For example, the standard statistical diagnostic test for the goodness-of-fit (GOF) of a logistic regression model, the Hosmer-Lemeshow test (Hosmer and Lemeshow (1980)), is very rarely used for diagnostics of a PS model (Granger et al. (2020)). Although it is important that the model-based predicted probabilities of treatment assignment are similar to the corresponding empirical probabilities (F. Li, Morgan, and Zaslavsky (2018)), most often a fitted PS model is assessed by its ability to balance baseline covariates which is verified by looking at the absolute standardized differences (ASD) (Austin and Stuart (2015), Granger et al. (2020), F. Li, Morgan, and Zaslavsky (2018), Thomas, Li, and Pencina (2020)). For continuous covariates, it is even more informative to compare (at least visually) the distributions of weighted covariates in two treatment groups, and not only their weighted averages or ASD.
Moreover, we notice that given the same PS model, the attained covariate balance depends on the type of weights we choose. For example, “overlap weights estimated from a logistic model yield exact balance between groups in the means of each covariate included in the model” (F. Li, Morgan, and Zaslavsky (2018)).
In order to diagnose a fitted PS model and check the resulting covariate balance, we recommend to call first get.weights function, which can be done externally, before any estimation of ATEs is done.
Through the analysis of RHC data, we will illustrate how to visually verify whether the PS model fits the data well, check overlap and evaluate the attained balance.
(D) The choice of weights plays a crucial role in the estimation of average treatment effects, since weights define the target population for which we estimate the ATE. “The ATE targets just one patient mix among many possible patient mixes. With heterogeneous treatment effects, different patient mixes lead to different estimands” (Mao, Li, and Greene (2019)).
causalCmprsk has the following types of weights (wtype) which correspond to different target populations and thus to different ATEs:
- “ATE”, defined by \(\frac{1}{\widehat{P}(A=a_i\mid C=c_i)}\), corresponds to the ATE for the overall cohort;
- “stab.ATE”, defined by \(\frac{\widehat{P}(A=a_i)}{\widehat{P}(A=a_i\mid C=c_i)}\), is a stabilized version of ATE weights (Hernán, Brumback, and Robins (2000)), so that “stab.ATE” corresponds to the ATE for the overall cohort;
- “ATT”, defined by \(\frac{\widehat{P}(A=1\mid C=c_i)}{\widehat{P}(A=a_i\mid C=c_i)}\), corresponds to the ATE for the population of treated;
- “ATC”, defined by \(\frac{\widehat{P}(A=0\mid C=c_i)}{\widehat{P}(A=a_i\mid C=c_i)}\), corresponds to the ATE for the population of untreated (i.e. a control group);
- “overlap”, defined by \(\frac{\widehat{P}(A=1\mid C=c_i)\widehat{P}(A=0\mid C=c_i)}{\widehat{P}(A=a_i\mid C=c_i)}\), corresponds to the ATE for the overlap population. Please see Table 1 in F. Li, Morgan, and Zaslavsky (2018).
The important point to remember is that if the treatment has heterogeneous effects for different individuals, the different weighting schemes target different population average effects, and it is important to characterize this target population (L. Thomas, Li, and Pencina (2020)), e.g., to describe covariate distributions of this target population. We will illustrate this in RHC data analysis below.
Cox-based estimation
causalCmprsk implements the Cox-based estimation by assuming the Cox proportional hazards model for the single-world cause-specific hazards \(h^a_k(t)\), \(k=1,2\):
\[
h_k^0(t)=h_{0k}(t); \quad h_k^1(t)=h_{0k}(t) \exp(\beta_k), \quad k=1,2.
\]
This model assumes that the cause-specific hazards are proportional in two counterfactual worlds, a world where everyone is treated and a world where everyone is not treated:
\[
HR_k(t)=\frac{h^1_k(t)}{h^0_k(t)}=\frac{h^0_k(t)\exp(\beta_k)}{h^0_k(t)}=\exp(\beta_k)=HR_k,
\]
and a proportionality factor between them is \(\exp(\beta_k)\). This outcome model implies time-invariant hazard ratios, \(HR_k\).
The balancing weights that aim to adjust for the treatment assignment selection bias, play a role of “case weights” where the contribution of each observation is \(w_i\) instead of 1. For more details on the implemented Cox-based estimation, and for the explicit expressions of the estimating equations and the estimators we refer to Charpignon et al. (2021) (Methods section).
Nonparametric estimation
In nonparametric survival framework, the single-world cause-specific hazards \(h^0_k(t)\) and \(h^1_k(t)\) (\(k=1,2\)) stay unspecified, that is \(h^0_k(t)\) and \(h^1_k(t)\) freely vary and are not tied together by any assumptions. This implies that now \(HR_k(t)\) can be time-dependent. The nonparametric estimates provide a visual way to test the Cox proportional hazards assumption (under our causal assumptions 1,2,3). We illustrate how to do this in RHC data analysis below.
The nonparametric estimation procedure comprises: first, the estimation of all single-world cause-specific cumulative hazard functions, \(H^a_k(t)=\int_0^t h^a_k(s) ds\), \(k=1,2\), \(a=0,1\), and second, using them as “building blocks” to derive the estimators of all other parameters:
- The estimator of the cumulative hazards \(H^a_k(t)\) is a weighted version of the Breslow-type estimator: \[ \widehat{H}^a_k(t)=\sum_{i: e_i=k, a_i=a} \frac{ \widehat{w}_i I(t_i\leq t)} { \sum_{j:a_j=a} \widehat{w}_j I(t_i\leq t_j)}, \quad k=1,2, \quad a=0,1. \]
- By plugging-in the estimated hazards, \(\widehat{H}^a_k(t)\), we obtain the estimator of \(CIF_a(t,k)\): \[ \widehat{CIF}_a(t,k)=\sum_{i: e_i=k, a_i=a, t_i\leq t} \frac{ \widehat{w}_i \exp(-\widehat{H}^a_1(t_i) - \widehat{H}^a_2(t_i))} { \sum_{j:a_j=a} \widehat{w}_j I(t_i\leq t_j)}, \quad k=1,2, \quad a=0,1. \]
- The estimator of \(RMT_a(t,k)\) is \[ \widehat{RMT}_a(\tau, k) = \int_0^{\tau} \widehat{CIF}_a(t,k)dt, \quad k=1,2, \quad a=0,1. \]
- Then, the estimators of the treatment effect measures, i.e. HR, RD, RR and RMT differences, are easily obtained by plugging-in.
Similar to the Cox-based estimation, here too, the balancing weights are used as “case weights”.
Notice that the resulting estimates of \(H^a_k(t)\), \(CIF_a(t,k)\) and of their functionals are \(c\grave{a}dl\grave{a}g\) (right-continuous with left-limits) step functions. The estimates of \(RMT_a(t,k)\) and its differences are not step functions.
Quantifying uncertainty
causalCmprsk uses Bayesian bootstrap to obtain confidence intervals (see, e.g., Lo (1993)), where each bootstrap sample comprises the original cohort with a random weight, assigned to each observation. This weight is generated from \(Exp(1)\) distribution, then truncated at 5, and lastly standardized so that the sum of all weights would equal the sample size.
For each bootstrap sample, we repeat the steps of 1. fitting the propensity scores model and obtaining balancing weights, and 2. estimating all the parameters of interest, that is the absolute measures (i.e., cumulative hazards functions, CIFs and RMTs) and the treatment effect measures (i.e., HR, RD, RR, RMT differences). This bootstrap estimator takes into account the uncertainty that comes from both estimation of weights and estimation of the parameters of interest.
All our parameters (except for \(\beta_k\) under the Cox model) are time-dependent, and their confidence intervals are obtained pointwise for each time point.
In survival data, the Bayesian bootstrap is expected to provide a more stable alternative compared to the simple bootstrap since the risk sets in all bootstrap samples change at the same time points as in the original cohort. The 95% confidence intervals are obtained as 2.5% and 97.5% quantiles from the sample distributions of bootstrap estimates.
