### Abstract

- Top of page
- Abstract
- Introduction
- Causal odds-of-concordance measure
- Estimation and large sample properties
- Numerical results
- Closing remarks
- Acknowledgements
- References
- Appendix
- Appendix

A simple summary of a treatment effect is attractive, which is part of the explanation of the success of the Cox model when analysing time-to-event data since the relative risk measure is such a convenient summary measure. In practice, however, the Cox model may fail to give a reasonable fit, very often because of time-changing treatment effect. The Aalen additive hazards model may be a good alternative as time-changing effects are easily modelled within this model, but results are then evidently more complicated to communicate. In such situations, the odds of concordance measure (OC) is a convenient way of communicating results, and recently Martinussen & Pipper (2012) showed how a variant of the OC measure may be estimated based on the Aalen additive hazards model. In this study, we propose an estimator that should be preferred in observational studies as it always estimates the causal effect on the chosen scale, only assuming that there are no un-measured confounders. The resulting estimator is shown to be consistent and asymptotically normal, and an estimator of its limiting variance is provided. Two real applications are provided.

### Introduction

- Top of page
- Abstract
- Introduction
- Causal odds-of-concordance measure
- Estimation and large sample properties
- Numerical results
- Closing remarks
- Acknowledgements
- References
- Appendix
- Appendix

Cox regression is a popular method to analyse right-censored time-to-event data. Its proportional hazards assumption gives indeed a convenient way of reporting the effect of explanatory variables in terms of relative risks. However, this assumption often fails in practice, usually because of time-changing covariate effects. Some methods to handle this within the Cox model do exist, but most of these are *ad hoc* such as the one proposed by Cox (1972) where one investigates a specific deviation from the Cox model. Non-parametric estimation of a potential time-varying effect has also been suggested, see Zucker & Karr (1990), Murphy & Sen (1991), Martinussen *et al.* (2002) and Tian *et al.* (2005), but they all require some kind of smoothing. A useful alternative in such a case is the Aalen additive hazards model (Aalen, 1980) since time-varying covariate effects are easily estimated using this model without needing any kind of smoothing. Reporting of results will inevitably be more complicated in this setting, however, and it is desirable to have a simple quantity that summarizes the effect of a given covariate in a convenient way under such scenarios. An appealing summary measure of a given treatment effect is the odds-of-concordance, OC, defined as

where *T*^{0} and *T*^{1} denote independent random variables describing the lifetime corresponding to treatment 0 and 1, respectively.

In a recent study, Martinussen & Pipper (2012) showed how to estimate a variant of this measure focussing on the time span where data are actually observed. In contrast to previously suggested estimators of the OC such as the weighted Cox regression (Schemper *et al.*, 2009) this estimator was provided with large sample properties, and an estimator of its limiting variance was also given justifying the computation of confidence intervals. Although their suggestion allows for adjustment of confounders the proposed estimators may not estimate the causal treatment effect on this scale. Hence, with a non-randomized treatment there is a need for an alternative estimator. The key to this is to formulate the OC in terms of potential outcomes (Robins, 1986) and then use the results of Martinussen & Vansteelandt (2012). They derived the causal hazard function using the G-computation formula (Robins, 1986). In the present study, we study the corresponding causal OC measure and show how to estimate this quantity. We show that the proposed estimator is consistent and asymptotically normal, and provide a consistent estimator of the limiting variance making inference possible. The proof of the asymptotic properties relies on empirical process theory. Small sample properties are investigated in numerical studies, and the methods are applied to two real examples.

### Causal odds-of-concordance measure

- Top of page
- Abstract
- Introduction
- Causal odds-of-concordance measure
- Estimation and large sample properties
- Numerical results
- Closing remarks
- Acknowledgements
- References
- Appendix
- Appendix

Let *T* denote the possibly right-censored failure time and suppose that we have recorded the covariates *G*,*X*, where *G* is a binary indicator of treatment and *X* is a *p*-dimensional vector of additional covariates. With this setup we assume the Aalen additive hazards model (Aalen, 1980)

where *β*_{0}(*t*) is the baseline hazard function, *β*_{G}(*t*) is the excess hazard due to treatment and *β*_{X}(*t*) denotes the effect of the additional covariates. The odds of concordance is given by

- (1)

where *T*^{0} and *T*^{1} denote independent random variables describing the lifetime corresponding to treatment 0 and 1, respectively, and with the same value, *x*, of the back ground covariates. Martinussen & Pipper (2012) considered the following modification of the OC(*x*)-measure

- (2)

where *v* can be chosen according to substance matter or in a more automated manner such as for instance *v*=*h*^{−1}(*q*) with .

Note that the latter approach links (2) to a constant *q* ∈ [0,1] that does not depend on the cohort specifics and thus allows for comparison between different cohorts. In either case (2) consitutes a restricted version of the OC for which the choice of *v* or *q* determines the time frame in which the exposure effect is assessed. Specifically, (2) may be rewritten to

from which it is clear that (2) should be interpreted as the odds of an exposed individual having an event before an unexposed individual given that at least one of them has an event before *v*. When it comes to estimation of OC(*x*,*v*) this corresponds to censor all lifetimes after time point *v*, which was also suggested by Schemper *et al.* (2009).

By construction (2) is a measure of the exposure effect for a given covariate configuration, *x*. We will now derive its causal counterpart that does not depend on a specific value of the back ground covariate *x*. To this end we use the notion of potential outcomes of Robins (1986). Let denote the potential outcome that would have been observed had the treatment been *g*. The causal odds-of-concordance measure is then defined as

- (3)

where *v* denotes some time-point that may again be chosen according to substance matter or by for instance *v*=*h*^{−1}(*q*) with . Note that this quantity does not depend on back ground covariates as it corresponds to the population-averaged causal exposure effect using the odds-of-concordance as measure. For more on causal effect and potential outcomes we refer to Pearl (2000).

The population-averaged causal exposure effect may be calculated using the G-computation formula (Robins, 1986; Pearl, 2000) for the distribution, , that would have been observed under an intervention, setting the exposure to *g*. In our setting the G-computation formula reads

where *F*_{X} denotes the marginal distribution function of *X*. Martinussen & Vansteelandt (2012) used this to show that the causal hazard function, under the assumed additive hazards model, is given by

- (4)

where

with , and .

The distribution of is thus given by the hazard function in (4). Hence

where

with

It is thus seen that the OC_{v}-measure is a function of . We now turn to the estimation of the OC_{v}-measure based on data where we have independent replicates from the above model, and where we allow for right-censoring.

### Closing remarks

- Top of page
- Abstract
- Introduction
- Causal odds-of-concordance measure
- Estimation and large sample properties
- Numerical results
- Closing remarks
- Acknowledgements
- References
- Appendix
- Appendix

It is also possible to derive an expression for the causal OC function assuming an underlying Cox model, see Martinussen & Vansteelandt (2012) for an expression of the causal hazards function under the Cox model. We have, however, chosen to work with the Aalen additive hazards model as it often gives a good fit in practice since it is capable of capturing time-changing effects. To this end note that the interpretation of the suggested OC measure is not affected by the actual model used for its calculation.

In Martinussen & Pipper (2012), the following overall OC measure was suggested

that corresponds to comparing the survival of an untreated and a treated individual with the same covariate configuration randomly chosen from the study population, whereas the causal OC measure given in (3) corresponds to comparing the survival of an untreated and a treated individual in a randomized trial. Both measures rely on the no-unmeasured confounders assumption. If there is no confounding, the two measures coincide.

We have used G-computation to estimate the causal OC. A possible alternative route is to use inverse probability weighted estimators for marginal structural models (Hernán, *et al.*, 2000) to arrive at estimators of the causal hazard function. An advantage of G-computation is that it gives more stable estimators by avoiding inverse probability weights, and that the models on which it relies – in this case the Aalen additive hazards model – can be readily validated using techniques described, for instance in Martinussen & Scheike (2006).

We have considered the situation with a binary exposure variable *G*, but one may extend the theory to cover also the case where *G* is continuous. The measure should then be modified to

considering the case where the exposure level is changed from *g* to *g*+1. It is easy to calculate this quantity under the proposed model, for instance,