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Keywords:

  • Aalen model;
  • causal effect;
  • confounding;
  • odds of concordance;
  • survival data

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Causal odds-of-concordance measure
  5. Estimation and large sample properties
  6. Numerical results
  7. Closing remarks
  8. Acknowledgements
  9. References
  10. Appendix
  11. 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

  1. Top of page
  2. Abstract
  3. Introduction
  4. Causal odds-of-concordance measure
  5. Estimation and large sample properties
  6. Numerical results
  7. Closing remarks
  8. Acknowledgements
  9. References
  10. Appendix
  11. 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

equation

where T0 and T1 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

  1. Top of page
  2. Abstract
  3. Introduction
  4. Causal odds-of-concordance measure
  5. Estimation and large sample properties
  6. Numerical results
  7. Closing remarks
  8. Acknowledgements
  9. References
  10. Appendix
  11. 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)

equation

where β0(t) is the baseline hazard function, βG(t) is the excess hazard due to treatment and βX(t) denotes the effect of the additional covariates. The odds of concordance is given by

  • display math(1)

where T0 and T1 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

  • display math(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 inline image.

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

equation

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 inline image denote the potential outcome that would have been observed had the treatment been g. The causal odds-of-concordance measure is then defined as

  • display math(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 inline image. 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, inline image, that would have been observed under an intervention, setting the exposure to g. In our setting the G-computation formula reads

equation

where FX 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

  • display math(4)

where

equation

with inline image, and inline image.

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

equation

where

equation

with

equation

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

Estimation and large sample properties

  1. Top of page
  2. Abstract
  3. Introduction
  4. Causal odds-of-concordance measure
  5. Estimation and large sample properties
  6. Numerical results
  7. Closing remarks
  8. Acknowledgements
  9. References
  10. Appendix
  11. Appendix

Let C denote the censoring time and assume that T and C are conditionally independent given (G,X). We observe the first time either failure or censoring occurs, U=min(T,C), and an indicator of whether it is censoring or failure that occurs δ=I(TC). The data consist of n independent replicates of inline image and we observe failure times only in the interval [0,τ] where τ<∞ is the endpoint of the study. These quantities translate into the counting process framework of Andersen et al. (1993) as the counting process inline image and at risk process inline image for the ith individual. With this notation, the estimator of inline image is obtained by well-established methods (Martinussen & Scheike, 2006) as

  • display math(5)

where Z(t) is the generalized inverse of Z(t) with the latter being the n×(p+2)-matrix with ith row inline image. A natural estimator of inline imageis given by

equation

where

equation

The causal OC-measure can now be estimated by

equation

Furthermore, the choice of v may be linked to a quantile q ∈ [0,1] by solving inline image with

equation

We now give the asymptotic properties of the causal OC estimator, the key to this being the establishment of the properties of inline image that are derived using empirical process theory. In appendix A, we show that inline image converges weakly to a zero mean Gaussian process and that

  • display math(6)

where inline image are zero-mean i.i.d. terms specifically given as

equation

with inline image given in Martinussen & Scheike (2006), and inline image given in appendix A. This result may then be used to show that that inline image converges weakly to a zero mean Gaussian process the covariance function of which may be consistently estimated by inline image. The proof of this along with an expression of inline image is given in appendix B.

Numerical results

  1. Top of page
  2. Abstract
  3. Introduction
  4. Causal odds-of-concordance measure
  5. Estimation and large sample properties
  6. Numerical results
  7. Closing remarks
  8. Acknowledgements
  9. References
  10. Appendix
  11. Appendix

Simulation study

A small simulation study was conducted to assess the small-sample behaviour of the proposed estimator under various degrees of confounding and effect sizes. For this purpose event times Ti are simulated according to the hazard

equation

where the Xis are generated from a uniform distribution on (−0.5,0.5) and the Bernoulli variables Gi are generated using the conditional probabilities

equation

We considered the parameter configurations β0=1, βG=1,2, βX=0,1, βGX=0,5, thus varying the effect sizes of both G and X as well as the magnitude of confounding. For each of these scenarios censoring times were generated according to an exponential distribution with an intensity ensuring approximately 20% censoring. Furthermore, we considered the sample sizes N=200,500 and 1000.

For each of the above configurations 1000 data sets were generated. For each data set the logarithm of the proposed estimator inline image was calculated for v solving inline image along with its corresponding estimated standard error. Table 1 summarizes the results.

Table 1. Summary results of the simulation study. Bias corresponds to sample median of the estimated log (OCv)sminus the true value. SE corresponds to the sample standard error and SEE to the median of the estimated standard errors. CP denotes the actual coverage probability of the nominal 95% confidence intervals
NβGβXβGXlog(OCv)biasSESEECP
2001050.690.010.240.230.94
5000.010.150.150.95
10000.000.100.100.95
2002051.100.040.250.240.96
5000.010.160.160.95
10000.010.110.110.94
2001000.690.000.190.180.95
5000.000.120.120.96
10000.000.080.080.96
2002001.100.010.200.190.94
5000.010.120.120.95
10000.000.090.090.95
2001100.700.010.190.180.94
5000.010.120.120.95
10000.000.080.080.96
2002101.110.020.200.190.94
5000.020.120.120.95
10000.000.090.090.94
2001150.700.020.240.220.94
5000.000.150.140.94
10000.010.110.100.94
2002151.110.010.250.240.94
5000.000.150.150.96
10000.010.110.110.95

From Table 1 we conclude that the suggested estimator behaves well for all sample sizes and configurations. The bias is negligible and the estimated standard error is close to the actual one in all scenarios. All coverage probabilities are acceptably close to 95%.

Application to acute myocardial infarction data

In this subsection we consider survival of patients after an acute myocardial infarction event (AMI). The study was carried out at the University Clinical Centre in Ljubljana, where 1040 patients were followed for up to 14 years. The end point was death from any cause, as gathering cause-specific death information was proved impossible to carry out. This data set is also considered in Stare et al. (2005), and contains several variables recorded at the time of admission. We concentrate on the effect of aspirin (1 = yes, 0 = no). There is missing information on aspirin for 20 patients. For some reason aspirin was more likely to be given to younger patients in this cohort. Specifically, the median ages of treated and un-treated individuals were 59 and 66 years, respectively. Since age is a highly significant predictor, it is of interest to study the effect of aspirin while considering age as a confounder. We summarize the effect of aspirin-using the causal OC defined in (3). Figure 1, right display, shows the estimated OCv as a function of v (full curve) and also 95% point wise confidence limits (dashed curves). From this it is seen that the causal OC is estimated to be significantly below 1 with a slight increase over the period of follow-up. For instance at 10 years of follow-up the causal OC is estimated to 0.66 with 95% confidence limits ranging from 0.55 to 0.80. Thus, in a comparison between a treated and an untreated individual in a randomized trial, the aspirin-treated individual has a 34% (20–45%) lower risk of dying first within a 10-year period after acute myocardial infarction.

image

Figure 1. AMI data. Left display: estimated causal survival curves for ‘no aspirin’ treatment (dotted curve) and for ‘aspirin’ treatment (dashed curve). Right display: estimated causal OCv as a function of v (full curve) with 95% pointwise confidence limits (dashed curves).

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Application to the TRACE study

The TRACE study group (Jensen et al., 1997) investigated the prognostic importance of various risk factors on mortality for approximately 6600 patients with myocardial infarction. We consider a random subsample 1000 of these patients that have previously been analysed in Martinussen & Scheike (2006, 2007); the data are available in the R-package timereg (Martinussen & Scheike, 2006). In these analyses ventricular fibrillation was recorded for 71 patients and was identified to be a very important risk factor. Figure 2 displays the estimated cumulative regression coefficient for this variable fitted in the additive Aalen where we also adjust for gender, diabetes status (present/absent), clinical heart failure status (present/absent) and age at enrolment centred around 70 years. A very strong effect is seen, but it vanishes after approximately one month. Specifically, Martinussen & Scheike (2007), using a change-point model, estimated a change point of 33 days. In the study design individuals were followed for approximately six years after which their follow-up was terminated according to a uniformly distributed censoring time in the age-span six to eight years. To compare with the results reported in Martinussen & Pipper (2012), we considered v=1, 3.5 and 6 years of follow-up. The causal OC is estimated as inline image (2.7–5.7), inline image (2.2–3.6) and inline image (2.0–3.0). The corresponding OC's reported in Martinussen & Pipper (2012) were 4.1, 2.9 and 2.6, respectively. Hence, similar results are obtained, indicating that the magnitude of confounding is minor. Some more details on this are given in the next section.

image

Figure 2. TRACE data: cumulative effect of ventricular fibrilation with 95% pointwise confidence limits.

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Closing remarks

  1. Top of page
  2. Abstract
  3. Introduction
  4. Causal odds-of-concordance measure
  5. Estimation and large sample properties
  6. Numerical results
  7. Closing remarks
  8. Acknowledgements
  9. References
  10. Appendix
  11. 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

equation

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

equation

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,

equation

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Causal odds-of-concordance measure
  5. Estimation and large sample properties
  6. Numerical results
  7. Closing remarks
  8. Acknowledgements
  9. References
  10. Appendix
  11. Appendix
  • Aalen, O. O. (1980).A model for non-parametric regression analysis of counting processes. In Lecture notes in statistics-2: mathematical statistics and probability theory (eds W. Klonecki, A. Kozek & J. Rosinski), 125. Springer-Verlag, New York.
  • Andersen, P. K., Borgan, Ø., Gill, R. D. & Keiding, N. (1993). Statistical models based on counting processes. Springer, New York.
  • Chen, L., Lin, D. Y. & Zeng, D. (2010). Attributable fraction functions for censored event times. Biometrika 97, 713726.
  • Cox, D. R. (1972). Regression models and life-tables (with discussion). J. Roy. Statist. Soc. Ser. B 34, 406424.
  • Hernán, M., Brumback, B. & Robins, J. M. (2000). Marginal structural models to estimate the causal effect of zidovudine on the survival of HIV-positive men. Epidemiology 11, 561570.
  • Jensen, G. V., Torp-Pedersen, C., Hildebrandt, P., Kober, L., Nielsen, F. E., Melchior, T., Joen, T. & Andersen, P. K. (1997). Does in-hospital ventricular fibrillation affect prognosis after myocardial infarction? Eur. Heart J. 18, 919924.
  • Martinussen, T. & Scheike, T. H. (2006). Dynamic regression models for survival data. Springer, New York.
  • Martinussen, T. & Scheike, T. H. (2007). Aalen additive hazards change-point model. Biometrika 94, 861772.
  • Martinussen, T. & Pipper, C. B. (2012). Estimation of odds of concordance based on the Aalen additive model. Lifetime Data Anal. DOI 10.1007/510985-012-9234-4.
  • Martinussen, T., Scheike, T. H. & Skovgaard, I. M. (2002). Efficient estimation of fixed and time-varying covariate effects in multiplicative intensity models. Scand. J. Statist. 29, 5774.
  • Martinussen, T. & Vansteelandt, S. (2012). A note on collapsibility and confounding bias in Cox and Aalen regression models. Technical report, Department of Biostatistics, University of Copenhagen.
  • Murphy, S. A. & Sen, P. K. (1991). Time-dependent coefficients in a Cox-type regression model. Stochastic Process. Appl. 39, 153180.
  • Pearl, J. (2000). Causality: models, reasoning, and inference. Cambridge University Press, Cambridge.
  • Robins, J. (1986). A new approach to causal inference in mortality studies with sustained exposure periods – application to control of the healthy worker survivor effect. Math. Model. 7, 13931512.
  • Schemper, M., Wakounig, S. & Heinze, G. (2009). Estimation of average hazard ratios by weighted Cox regression. Statist. Med. 28, 24732489.
  • Stare, J., Henderson, R. & Pohar, M. (2005). An individual measure of relative survival. Appl. Statist. 54, 115126.
  • Tian, L., Zucker, D. M. & Wei, L. J. (2005). On the Cox model with time-varying regression coefficients. J. Amer. Statist. Assoc. 100, 172183.
  • van der Vaart, A. W. (1998). Asymptotic statistics. Cambridge University Press, New York.
  • van der Vaart, A. W. & Wellner, J. A. (1996). Weak convergence and empirical processes. Springer, New York.
  • Zucker, D. M. & Karr, A. F. (1990). Nonparametric survival analysis with time-dependent effects: a penalized partial likelihood approach. Ann. Statist. 18, 329353.

Appendix

  1. Top of page
  2. Abstract
  3. Introduction
  4. Causal odds-of-concordance measure
  5. Estimation and large sample properties
  6. Numerical results
  7. Closing remarks
  8. Acknowledgements
  9. References
  10. Appendix
  11. Appendix
equation

To prove week convergence we impose the following regularity condition

  1. inline image.

Let Pn and P denote the empirical and true distribution and in analogy with Van der Vaart (1998) define

equation

Then

  • display math(7)

According to example 19.7 in van der Vaart (1998)ft belongs to a P-Donsker class under assumption (a) and thus weak convergence of the first term on the right-hand side of (7) is ensured. For the second term on the right-hand side of (7) note that due to the weak convergence of inline image (Martinussen & Scheike, 2006)

  • display math(8)

uniformly over bounded sets of x and t.

This in turn ensures that

equation

Consequently weak convergence of the second term on the right-hand side follows from the weak convergence of inline image.

For the third term of the right-hand side of (7) we may again use (8) to see that

equation

According to example 19.8 in van der Vaart (1998) inline image belongs to a Glivenko–Cantelli class and thus

equation

Consequently by Slutsky's Theorem (example 1.4.7 in van der Vaart & Wellner, 1996)

equation

and we conclude that inline image converges weakly. Also note that combining the above and using the i.i.d. decomposition of inline image (Martinussen & Scheike, 2006) we have the following i.i.d. decomposition

  • display math(9)

Similarly, with

equation

one may show that inline image converges weakly and has the i.i.d. decomposition

  • display math(10)

By standard arguments we further note that

equation

which ensures weak convergence of inline image. From (9) and (10) it also follows that

equation

where

equation

Finally, we note that

equation

from which the weak convergence of inline image and (6) follows.

Appendix

  1. Top of page
  2. Abstract
  3. Introduction
  4. Causal odds-of-concordance measure
  5. Estimation and large sample properties
  6. Numerical results
  7. Closing remarks
  8. Acknowledgements
  9. References
  10. Appendix
  11. Appendix
equation

First note that the Delta method (van der Vaart & Wellner, 1996, ch. 3.9) ensures weak convergence towards a zero mean Gaussian process as both inline image and inline image converge weakly to zero mean Gaussian processes. From the Delta method and the Law of Large Numbers it also follows that the asymptotic covariance function is given as the limit in probability of

equation

where

equation

with

equation

The quantity inline image is now obtained from inline image by plugging in relevant estimates. Their consistency ensure that a consistent estimator of the asymptotic covariance function is given by

equation