On the choice of time scales in competing risks predictions

Summary

In the standard analysis of competing risks data, proportional hazards models are fit to the cause-specific hazard functions for all causes on the same time scale. These regression analyses are the foundation for predictions of cause-specific cumulative incidence functions based on combining the estimated cause-specific hazard functions. However, in predictions arising from disease registries, where only subjects with disease enter the database, disease-related mortality may be more naturally modeled on the time since diagnosis time scale while death from other causes may be more naturally modeled on the age time scale. The single time scale methodology may be biased if an incorrect time scale is employed for one of the causes and an alternative methodology is not available. We propose inferences for the cumulative incidence function in which regression models for the cause-specific hazard functions may be specified on different time scales. Using the disease registry data, the analysis of other cause mortality on the age scale requires left truncating the event time at the age of disease diagnosis, complicating the analysis. In addition, standard Martingale theory is not applicable when combining regression models on different time scales. We establish that the covariate conditional predictions are consistent and asymptotically normal using empirical process techniques and propose consistent variance estimators for constructing confidence intervals. Simulation studies show that the proposed two time scales method performs well, outperforming the single time-scale predictions when the time scale is misspecified. The methods are illustrated with stage III colon cancer data obtained from the Surveillance, Epidemiology, and End Results program of National Cancer Institute.

1 Introduction

It is well recognized that in the presence of competing risks, standard survival analysis methods for estimation of cause-specific failure probabilities may not be valid, with the Kaplan–Meier estimator (Kaplan and Meier, 1958) and predicted failure probabilities from the proportional hazards regression model (Cox, 1972) corresponding instead to pseudo-survival functions derived from the cause-specific hazard functions (Prentice and others, 1978). As an example, we consider individuals diagnosed with colon cancer in the Surveillance, Epidemiology, and End Results (SEER) database, for whom estimation of the risk of death from colon cancer and from other causes is of public health interest. In the analysis of colon cancer, naively ignoring the occurrence of a competing event of death from colon cancer results in biased estimates of the probability of colon cancer mortality for patients for whom death from other causes is possible. To estimate the cumulative incidence of a particular event type requires synthesizing the cause-specific hazard for the event of interest with those for the competing events. Nonparametric and semiparametric approaches to estimation via the cause-specific hazards have been studied (Aalen and Johansen, 1978; Lin, 1997; Cheng and others, 1998; Shen and Cheng, 1999; Scheike and Zhang, 2003), with a definitive treatment of the theoretical issues provided by martingale arguments (Andersen and others, 1993). The current colon cancer risk estimates provided by the National Cancer Institute utilize proportional hazards models for the cause-specific hazards for colon cancer and for other causes which adjust for patient-specific risk factors (Lee and others, 2012).

In chronic disease registries like the SEER database, subjects are enrolled in the database at the time of disease diagnosis. A primary objective may be to understand the impact of risk factors on subsequent mortality from disease and other causes and to obtain individualized estimates of the probabilities of these outcomes. In the regression analyses of cause-specific hazard functions, it seems natural to model death from disease on the time since diagnosis time scale and death from other causes, which may occur both before and after disease diagnosis, on the age time scale. This choice of the time scale is clearly described in Korn and others (1997), Thiebaut and Benichou (2004), and Pencina and others (2007), with only a single cause of failure. An incorrect choice of the time scale generally results in misspecification of the proportional hazards model for the cause-specific hazard function. An exception is when the true time scale is age and the baseline hazard function on the age time scale follows a Gompertz distribution (Gompertz, 1825). In this case, the proportional hazards model holds on both time scales. The existing methods for prediction of the cumulative incidence function require that analyses of the cause-specific hazards are conducted on the same time scale, which may lead to model misspecification. As an example, if the cause-specific hazard for other cause mortality is incorrectly modeled on the time since diagnosis time scale, cumulative incidence predictions for both cancer mortality and other cause mortality may be biased.

One may consider accounting for the nonproportionality with an interaction term. There may be applications where nonproportionality on the time since diagnosis scale may be adequately modeled using parametric models, potentially including interactions of time since diagnosis with other time-independent covariates, where the baseline hazard function is completely unspecified on the time since diagnosis scale. This approach is potentially less parsimonious than a model specified on the age scale. Moreover, in many applications, including our SEER analysis, the age effect is much stronger than the time since diagnosis effect on other cause mortality, and it is preferable to model the age effect nonparametrically. In other population-based applications, there may be strong interactions between risk factors and age which cannot be easily modeled or interpreted on the time since diagnosis scale. As an example, one might consider racial differences in the US life tables (Arias, 2014). Such data show that whites have lower risk of mortality than African Americans prior to age 88, but that the hazards cross after age 88. This phenomenon can be explained on the age scale using a frailty argument but is much less amenable to interpretation and modeling on the time since diagnosis scale.

In this article, we generalize prediction methods (Andersen and others, 1993) to allow the use of different time scales for different causes. In Section 2, we demonstrate how the fitted models may be combined to obtain predictions for the cumulative incidence functions on the time scale of interest. In applications like the SEER colon cancer population, those predictions are most naturally calculated on the time since diagnosis time scale for disease-related death and on the age time scale for death from other causes. The registry data structure creates complications, since the analysis of the cause-specific hazards of death from other causes is subject not only to right censoring by disease-related death but also left truncation by the age of disease diagnosis. That is, only subjects diagnosed with disease enter the registry database. Fitting the proportional cause-specific hazards model for death from other causes must account for such truncation, as detailed in Section 2. In addition, standard martingale theory results (Andersen and others, 1993) for the predicted cumulative incidences are only valid when the cause-specific hazard models for all causes are specified on the same time scale. In supplementary materials available at Biostatistics online, we establish the consistency and asymptotic normality of those predictions using an alternative approach based on empirical processes, and present variance estimators for the construction of confidence intervals.

The methods perform well in the simulations presented in Section 3, exhibiting superior performance to analyses which specify the same time scale for all causes (Cheng and others, 1998) when the assumption is violated and modest efficiency losses when the baseline hazard on the age time scale has a form of a Gompertz hazard. In Section 4, a thorough analysis of the colon cancer data illustrates the practical utility of the two time scales methodology versus the single time scale methodology in disease registries where the choice of time scale may be unclear a priori. Practical remarks related to the choice of time scales in the competing risks setting are provided in Section 5.

2 Cumulative incidence prediction with two time scales

Let be the time to failure since diagnosis and be the cause of failure. In this article, we consider only two causes of failure, , where the cause of interest is cause 1 and other competing events are combined into cause 2. Define , where is the censoring time. Let be a vector of time-independent covariates. We assume that is independent of () conditional on . Let , where is the indicator function. The observed data consist of .

In the analysis of competing risks, one may be interested in predicting the probability of experiencing a particular type of failure by time in the presence of other competing risks. The probability is called the cumulative incidence function defined by . The cause-specific hazard function for cause conditional on covariates is defined by , which implies that the cumulative incidence function can be expressed as a function of all cause-specific hazard functions as , where is the overall survival function and is the cumulative cause-specific hazard function for cause .

It is common to assume that the proportional hazards model (Cox, 1972) holds for each of the cause-specific hazard functions on a single time scale . That is, , where is a regression parameter vector for cause and is an unspecified baseline hazard function for cause . Under the cause-specific proportional hazards model, the cumulative incidence function for cause conditional on covariates is expressed by , where and . Following Andersen and others (1993), Cheng and others (1998) showed that the cumulative incidence function can be consistently estimated by plugging in the corresponding estimators as

$$\begin{array}{ll}{\widehat{F}}_k(t;z)\hfill & ={\displaystyle {\int }_0^t\widehat{S}}(u;z)\exp ({\widehat{\beta }}_k^{T}z)\text{d}{\widehat{\Lambda }}_{0k}(u)\hfill \\ \hfill & ={\displaystyle \sum _{i=1}^n\widehat{S}}(X_i;z)\exp ({\widehat{\beta }}_k^{T}z)\frac{I({\delta }_i=k)I(X_i\le t)}{{\displaystyle \sum _{j=1}^{n}I}(X_i\le X_j)\exp ({\widehat{\beta }}_k^T{Z}_j)},\hfill \end{array}$$ (2.1)

where is the partial likelihood estimator for , is the Breslow (1974) estimator for , and . Martingale theory yields the large sample properties of the estimator (2.1), with inferences based on asymptotic normality and martingale-based variance estimators.

In cancer studies with competing causes of death, medical investigators may be interested in predicting both cancer and other cause mortality by a particular time after diagnosis for a patient diagnosed with cancer at age with specific covariates . In such cases, death from cancer and death from other causes are typically modeled on the time since diagnosis time scale with age at diagnosis included as a covariate in the models for death from cancer and from other causes along with other covariates . In this analysis, the failure time random variable refers to the time to event on the time since diagnosis time scale. If the cause-specific proportional hazards model holds for death from cancer and from other causes on the time since diagnosis time scale, the cancer and other cause mortality by time for the patient can be estimated by methods of Cheng and others (1998) given in (2.2) with time since diagnosis as the time scale. This strategy is frequently employed in practice (Lee and others, 2012).

However, as described in Korn and others (1997), Thiebaut and Benichou (2004), Pencina and others (2007), and Cho and others (2013), it seems more natural to model death from other causes on the age time scale rather than the time since diagnosis time scale. This can be applied to the analysis of competing risks allowing different time scales for different causes, that is, cause 1 on the time since diagnosis time scale and cause 2 on the age time scale. When the true time scale for cause 2 is age, an incorrect choice of the time scale leads to misspecification of the proportional hazards model. For example, the proportionality assumption holds for cause 1 on the time since diagnosis time scale, whereas the assumption holds for cause 2 on the age time scale but not on the time since diagnosis time scale. In this case, the methods for predicting the cumulative incidence functions on the same time scale (i.e., time since diagnosis time scale) for causes 1 and 2 may result in a bias in the estimates of the cumulative incidence function. An exception is when the proportional hazards model for cause 2 holds on the age time scale with the Gompertz baseline hazard function, in which case, the proportional hazards model also holds on the time since diagnosis time scale. In this scenario, both time scales are valid for cause 2 and yield consistent predictions.

To develop the two time-scale predictions, additional notation is needed. Let be the age at which a subject is diagnosed with disease. Then the event time on the age time scale is given by , which is the age at which a subject experiences an event of cause 1 or 2. The potentially censored time on the age time scale is . In the database, there may be time-dependent covariates, in addition to time-independent covariates. We partition the time-independent covariates at the age of diagnosis into , consisting of time-independent covariates such as race and gender, and , consisting of time-dependent covariates measured at age of diagnosis (i.e., entry of study) such as comorbidity scores. Thus, . The observed data on the age time scale are . Note that the observed event times for cause 2 are left-truncated by age at diagnosis since we can observe only subjects diagnosed with disease. Thus, when we fit a model for cause 2 on the age time scale, estimation must account for such left truncation. On the other hand, when fitting the cause-specific hazard model for cause 2 on the time since diagnosis time scale, the truncation can be ignored without affecting the validity of the analysis as all subjects in the registry are observed from time 0 on this time scale.

If we choose age as the time scale for cause 2, it is appropriate to assume the model for cause 2 proportional on the age time scale. We still adopt a cause-specific proportional hazards model for cause 1 on the time since diagnosis time scale, given by , where is time since diagnosis. The implied model for cause 1 on the age time scale is

$${\lambda }_1^{\text{age}}(a;A,Z_1,Z_2(A))=\{\begin{array}{cc}{h}_1(a,Z_1,\{Z_2(s),0<s<a\}),& 0<a<A,\\ {\lambda }_{01}(a-A)\exp ({\beta }_{11}A+{\beta }_{12}{Z}_1+{\beta }_{13}{Z}_2(A)),& a\ge A,\end{array}$$

where is age. Let the cause-specific proportional hazards model for cause on the age time scale be

$${\lambda }_2^{\text{age}}(a;A,Z_1,Z_2(A))=\{\begin{array}{cc}{h}_2(a,Z_1,\{Z_2(s),0<s<a\}),& 0<a<A,\\ {\lambda }_{02}(a)\exp ({\beta }_{21}A+{\beta }_{22}{Z}_1+{\beta }_{23}{Z}_2(A)),& a\ge A.\end{array}$$

On the time since diagnosis time scale, it is .

In the equations above, and are unspecified nonnegative functions which may depend on the time-independent covariates and on the history of time-dependent covariates up to age prior to the time of diagnosis, . Note that is theoretically defined prior to age at diagnosis. However, we observe only subjects whose age of diagnosis are known, i.e., . Individuals that died from non-cancer causes prior to being diagnosed with cancer (i.e., ) would not be included in the registry. Conceptually, the models for and for specify the risks of cancer and other cause mortality prior to diagnosis while the models for specify the risks of cancer and other cause mortality after diagnosis. The functions and are not identifiable from data without making additional assumptions about the joint distribution of and , because data before the time of diagnosis are not observed. Inference should be made conditionally on the left truncation of . We leave and completely unspecified for , as these models are not needed for predictions conditioned on diagnosis at time . Hence, we make no assumptions about and because we do not have the relevant data and because they could differ from and for many different reasons. The models for and for include as a covariate, capturing that the risks of cancer and other cause mortality are conditional on having been diagnosed at time . The standard left-truncated right-censored methodology (Andersen and others, 1993) for fitting the cause-specific proportional hazards model is valid conditionally on this model specification. Note that technically may not be proportional on the age time scale on the whole range of age axis from to since we do not specify the model before the time of diagnosis. However, for , the cause-specific hazards for cause are proportional on the age time scale.

With cause 1 on the time since diagnosis time scale and cause 2 on the age time scale, the cause-specific proportional hazards models we use for prediction are

$$\begin{array}{l}{\lambda }_1^{\text{time}}(t;A,Z_1,Z_2(A))={\lambda }_{01}(t)\exp ({\beta }_{11}A+{\beta }_{12}{Z}_1+{\beta }_{13}{Z}_2(A)),\\ {\lambda }_2^{\text{age}}(a;A,Z_1,Z_2(A))={\lambda }_{02}(a)\exp ({\beta }_{21}A+{\beta }_{22}{Z}_1+{\beta }_{23}{Z}_2(A)),a\ge A.\end{array}$$ (2.2)

Our interest is to make a valid prediction based on two time scales since the time of diagnosis. Thus, we assume that the prediction models on two time scales depend only on the value of covariates measured at the time of diagnosis and not on the value of covariates measured after the time of diagnosis. That is, for time-dependent covariates, only their values at the time of diagnosis enter the models. The values of the covariates after diagnosis are not relevant to predictions.

Using that , one can translate one time scale into the other time scale. The overall survival function can be expressed on either time scale as on the time since diagnosis time scale and on the age time scale, where and . The cumulative incidence function for cause 1 on the time since diagnosis time scale can be defined using the overall survival function and the cause-specific hazard function for cause 1 on the time since diagnosis time scale . The cumulative incidence function for cause 2 on the age time scale can be defined using the overall survival function and the cause-specific hazard function for cause 2 on the age time scale . Given , the cumulative incidence functions for causes 1 and 2 on two time scales can be expressed as

$$\begin{array}{ll}{F}_1(t;a_0,z_0)\hfill & =\mathrm{Pr}(T\le t,ϵ=1\mid a_0,z_0)\hfill \\ \hfill & ={\displaystyle {\int }_0^t\exp }\left\{-{\Lambda }_1^{\text{time}}(u;a_0,z_0)-{\Lambda }_2^{\text{age}}(a_0+u;a_0,z_0)\right\}{\lambda }_1^{\text{time}}(u;a_0,z_0)\text{d}u,\hfill \\ F_2(a_0+t;a_0,z_0)\hfill & =\mathrm{Pr}(a_0<T^{\text{age}}\le a_0+t,ϵ=2\mid a_0,z_0)\hfill \\ \hfill & ={\displaystyle {\int }_{a_0}^{a_0+t}\exp }\left\{-{\Lambda }_1^{\text{time}}(s-a_0;a_0,z_0)-{\Lambda }_2^{\text{age}}(s;a_0,z_0)\right\}{\lambda }_2^{\text{age}}(s;a_0,z_0)\text{d}s,\hfill \end{array}$$

where .

Let be the partial likelihood estimator for on the time since diagnosis time scale, treating the failure times with as censored. Let be the partial likelihood estimator for on the age time scale, treating the failure times with as censored and accounting for left truncation by age at diagnosis . Let and be the Breslow (1974) estimators for on the time since diagnosis time scale and for on the age time scale accounting for left truncation by age at diagnosis , respectively, defined as

$$\begin{array}{ll}{\widehat{\Lambda }}_{01}(t)\hfill & ={\displaystyle \sum _{i=1}^n\frac{I({\delta }_i=1)I(X_i\le t)}{{\displaystyle \sum _{j=1}^{n}I}(X_i\le X_j)\exp ({\widehat{\beta }}_{11}{A}_j+{\widehat{\beta }}_{12}{Z}_{1j}+{\widehat{\beta }}_{13}{Z}_{2j}(A_j))}},\hfill \\ {\widehat{\Lambda }}_{02}(a_0+t)\hfill & ={\displaystyle \sum _{i=1}^n\frac{I({\delta }_i=2)I(a_0\le X_i^{\text{age}}\le a_0+t)}{{\displaystyle \sum _{j=1}^{n}I}(A_j\le X_i^{\text{age}}\le X_j^{\text{age}})\exp ({\widehat{\beta }}_{21}{A}_j+{\widehat{\beta }}_{22}{Z}_{1j}+{\widehat{\beta }}_{23}{Z}_{2j}(A_j))}}.\hfill \end{array}$$

Under the models (2.2), the cumulative hazard functions and overall survival functions on two time scales can be estimated by , , , and . The cumulative incidence functions for causes 1 and 2 on two time scales can be estimated by

$$\begin{array}{ll}{\widehat{F}}_1(t;a_0,z_0)\hfill & ={\displaystyle {\int }_0^t\exp }\left\{-{\widehat{\Lambda }}_1^{\text{time}}(u;a_0,z_0)-{\widehat{\Lambda }}_2^{\text{age}}(a_0+u;a_0,z_0)\right\}d{\widehat{\Lambda }}_1^{\text{time}}(u;a_0,z_0)\hfill \\ \hfill & ={\displaystyle \sum _{i=1}^n{\widehat{S}}_1}(X_i;a_0,z_0)\frac{\exp ({\widehat{\beta }}_{11}{a}_0+{\widehat{\beta }}_{12}{z}_{01}+{\widehat{\beta }}_{13}{z}_{02}(a_0))I({\delta }_i=1)I(X_i\le t)}{{\displaystyle \sum _{j=1}^{n}I}(X_i\le X_j)\exp ({\widehat{\beta }}_{11}{A}_j+{\widehat{\beta }}_{12}{Z}_{1j}+{\widehat{\beta }}_{13}{Z}_{2j}(A_j))},\hfill \\ {\widehat{F}}_2(a_0+t;a_0,z_0)\hfill & ={\displaystyle {\int }_{a_0}^{a_0+t}\exp }\left\{-{\widehat{\Lambda }}_1^{\text{time}}(s-a_0;a_0,z_0)-{\widehat{\Lambda }}_2^{\text{age}}(s;a_0,z_0)\right\}d{\widehat{\Lambda }}_2^{\text{age}}(s;a_0,z_0)\hfill \\ \hfill & ={\displaystyle \sum _{i=1}^n{\widehat{S}}_2}(X_i^{\text{age}};a_0,z_0)\frac{\exp ({\widehat{\beta }}_{21}{a}_0+{\widehat{\beta }}_{22}{z}_{01}+{\widehat{\beta }}_{23}{z}_{02}(a_0))I({\delta }_i=2)I(a_0\le X_i^{\text{age}}\le a_0+t)}{{\displaystyle \sum _{j=1}^{n}I}(A_j\le X_i^{\text{age}}\le X_j^{\text{age}})\exp ({\widehat{\beta }}_{21}{A}_j+{\widehat{\beta }}_{22}{Z}_{1j}+{\widehat{\beta }}_{23}{Z}_{2j}(A_j))}.\hfill \end{array}$$

Cheng and others (1998) derived the asymptotic properties of the predicted cumulative incidence using the standard martingale theory (Andersen and others, 1993). This approach is only valid when the cause-specific hazard models are specified on the same time scale. A similar derivation for the proposed estimator employing two different time scales will not be valid. The problem is that the Martingale theory holds for each cause with respect to different filtrations, defined with respect to the chosen time scales. We establish the theoretical properties on two time scales using empirical processes. In Appendix A of supplementary materials available at Biostatistics online, we show that and are asymptotically equivalent to (A.4) and (A.7) which converge to normal distributions with means 0 and respective variances and , where . The variances can be consistently estimated by and in (A.5) and (A.8).

Pointwise confidence intervals for can be constructed based on a transformation of to ensure confidence intervals for are bounded between 0 and 1 and improve the coverage probability. Denote , where is a known function with nonzero continuous derivative and is a weight function which converges to a nonnegative bounded function. By the functional delta method, the process is asymptotically equivalent to , where . We use and , as in Cheng and others (1998). Pointwise confidence intervals for are given by

$$\exp \left[-\exp \left\{\log \{-\log ({\widehat{F}}_k(u;a_0,z_0)\}\pm z_{\alpha /2}\frac{n^{-1/2}{\widehat{V}}_k{(u;a_0,z_0)}^{1/2}}{{\widehat{F}}_k(u;a_0,z_0)\log \{{\widehat{F}}_k(u;a_0,z_0)\}}\right\}\right],$$ (2.3)

where is an upper percentile of the standard normal distribution.

3 Simulation studies

3.1 Generating data

Simulation studies were conducted to evaluate the performance of the proposed estimator. We generated four covariates: age at diagnosis , a time-independent covariate , and two time-dependent covariates measured at the time of diagnosis, to be included in the model for cause 1 and to be included in the model for cause 2. Age at diagnosis was generated as the absolute value of a normal random variable with mean 50 and standard deviation 25. A time-independent covariate was generated from a Bernoulli distribution with success probability . Time-dependent covariates and were generated from the normal distribution with mean 5 and standard deviation 2 and with mean increasing with age and standard deviation 2, respectively, and then rounded to an integer. To generate data conditional on age at diagnosis , we treat and age at diagnosis as latent variables and specify a joint model for these variables by generating as described above and based on models formulated for causes 1 and 2 on the age time scale. We do this because the occurrence of an event prior to age at diagnosis truncates the diagnosis with disease. It is easier to assume a simulation model for the untruncated data (, ), and then subset the generated data to only subjects with .

We considered two scenarios. In the first scenario, the cause-specific hazards model for cause was proportional on both the time since diagnosis and age time scales. We assumed the cause-specific hazard functions on the age time scale for causes and as

$${\lambda }_1^{\text{age}}(a;A,Z_1,Z_2(A))=\{\begin{array}{cc}0,& 0<a<A\\ {\alpha }_1\exp ({\beta }_{11}A+{\beta }_{12}{Z}_1+{\beta }_{13}{Z}_2(A)),& a\ge A,\end{array}$$

$${\lambda }_2^{\text{age}}(a;A,Z_3(A))=\{\begin{array}{cc}\exp ({\alpha }_2+{\theta }_{2}a),& 0<a<A\\ \exp ({\alpha }_2+{\theta }_{2}a)\exp ({\beta }_{21}A+{\beta }_{22}{Z}_3(A)),& a\ge A.\end{array}$$

The true parameter values are = (0.02, 0.1, 0.01, 0.01, -9.5, 0.075, 0.095, 0.01). Note that the baseline hazard function for cause is assumed to be that of a Gompertz distribution. Under this scenario, we had 24% failures from cause 1 and 76% failures from cause 2 in the absence of censoring. In the second scenario, the model for cause was proportional on the age time scale but not on the time since diagnosis time scale. We assumed the same model for cause as scenario 1 and the model for cause on the age time scale as

$${\lambda }_2^{\text{age}}(a;A,Z_3(A))=\{\begin{array}{cc}{\alpha }_{21}I(a<{\gamma }_2)+{\alpha }_{22}I(a\ge {\gamma }_2),& 0<a<A\\ \{{\alpha }_{21}I(a<{\gamma }_2)+{\alpha }_{22}I(a\ge {\gamma }_2)\}\exp ({\beta }_{21}A+{\beta }_{22}{Z}_3(A)),& a\ge A.\end{array}$$

The true parameter values are ()=(0.03, 0.01, 0.1, 0.01, 0.01, 0.01). The values of () are (0.05, 0.5, 40). Under this scenario, we had 32% failures from cause 1 and 68% failures from cause 2 in the absence of censoring. Additional details of data generation can be found in Appendix B of the supplementary materials available at Biostatistics online.

In each scenario, we compared the predicted cumulative incidences on two time scales with predictions from Cheng and others (1998), which used the same time scale (i.e., time since diagnosis time scale) for events of both types under the cause-specific proportional hazards model. The objectives are to demonstrate the advantages of the proposed two time scales estimator when the single time scale model is misspecified and to assess the loss of efficiency compared with the single time scale approach on the time since diagnosis time scale when the cause-specific proportional hazards assumption holds for both causes 1 and 2 on the time since diagnosis time scale. We conducted 1000 simulations with sample size 200 and 600.

3.2 Simulation results

Table 1 shows the true values of and , biases of and , empirical variances (Emp.Var), averages of the variance estimates (E()), and empirical coverage probabilities (CP) for 95% confidence intervals given in (2.3) from two time scales and from the time since diagnosis time scale. When the cause-specific hazards model for cause 2 was proportional on the time since diagnosis time scale and hence the use of Cheng and others (1998) was appropriate, we observed some loss of efficiency due to left truncation on the age time scale. The increased variances with the two time scales methodology can be explained by the fact that when we estimate the hazard function for cause 2 on the age time scale, due to left truncation we have fewer subjects at risk for each event of type 2 than when we work on the time since diagnosis time scale. The empirical coverage probabilities for the proposed estimator on two time scales were slightly lower than the nominal level for the sample size of 200 and improved to the nominal level when the sample size increased to 600.

TABLE 1.

Simulation results when the model for cause 2 is proportional on both the age and time since diagnosis time scales; true values of (True), empirical variances (Emp.Var), averages of the variance estimates , and empirical coverage probabilities (CP).

					Method
					Two time scales				Time since diagnosis time scale
Sample size	% censored	Cause	Time	True	Bias	Emp.Var		CP	Bias	Emp.Var		CP
200	10%		5	0.140	0.0059	0.00162	0.00161	0.939	0.0060	0.00161	0.00160	0.933
		1	8	0.209	0.0086	0.00272	0.00270	0.932	0.0091	0.00262	0.00265	0.943
			10	0.249	0.0084	0.00351	0.00342	0.936	0.0090	0.00334	0.00333	0.946
			5	0.068	0.0050	0.00261	0.00247	0.775	0.0006	0.00031	0.00033	0.959
		2	8	0.113	0.0066	0.00331	0.00314	0.911	0.0010	0.00069	0.00073	0.954
			10	0.145	0.0061	0.00357	0.00338	0.917	0.0009	0.00101	0.00109	0.953
200	35%		5	0.140	0.0081	0.00231	0.00232	0.937	0.0083	0.00227	0.00231	0.942
		1	8	0.209	0.0104	0.00442	0.00445	0.940	0.0111	0.00425	0.00443	0.949
			10	0.249	0.0126	0.00601	0.00615	0.935	0.0134	0.00578	0.00613	0.944
			5	0.068	0.0070	0.00389	0.00351	0.616	0.0006	0.00040	0.00040	0.954
		2	8	0.113	0.0086	0.00486	0.00468	0.877	0.0008	0.00103	0.00107	0.947
			10	0.145	0.0075	0.00558	0.00542	0.907	0.0037	0.00177	0.00185	0.943
600	10%		5	0.140	0.0016	0.00052	0.00054	0.950	0.0017	0.00052	0.00053	0.953
		1	8	0.209	0.0019	0.00093	0.00091	0.940	0.0021	0.00091	0.00089	0.944
			10	0.249	0.0018	0.00116	0.00114	0.954	0.0021	0.00113	0.00110	0.946
			5	0.068	0.0025	0.00088	0.00088	0.948	0.0003	0.00010	0.00010	0.948
		2	8	0.113	0.0025	0.00115	0.00111	0.945	0.0005	0.00023	0.00024	0.945
			10	0.145	0.0028	0.00121	0.00118	0.943	0.0006	0.00035	0.00035	0.947
600	35%		5	0.140	0.0029	0.00077	0.00079	0.939	0.0029	0.00076	0.00078	0.945
		1	8	0.209	0.0031	0.00149	0.00152	0.948	0.0033	0.00145	0.00148	0.948
			10	0.249	0.0033	0.00214	0.00210	0.949	0.0035	0.00207	0.00205	0.950
			5	0.068	0.0010	0.00127	0.00136	0.945	0.0003	0.00014	0.00013	0.937
		2	8	0.113	0.0021	0.00161	0.00173	0.948	0.0003	0.00035	0.00034	0.940
			10	0.145	0.0024	0.00180	0.00192	0.949	0.0014	0.00054	0.00059	0.956

As shown in Table 2, when the cause-specific hazards model for cause 2 was not proportional on the time since diagnosis time scale, the estimates of the cumulative incidence functions for causes 1 and 2 on two time scales had small biases that diminished with increasing samples sizes. The empirical coverage probabilities were close to the nominal level. The estimates from the time since diagnosis time scale had very large systematic biases for cause , hence significant biases for cause , and yielded very poor empirical coverage probabilities. For both approaches, the variance estimates agreed with their empirical variances. As the sample size increases, the empirical coverage probabilities for the estimates from the time since diagnosis time scale decline dramatically, due to the fact that the systematic biases persist and the variance estimates decrease. Thus, when the cause-specific proportional hazards assumption does not hold for cause on the time since diagnosis time scale, the methods of Cheng and others (1998) may lead to biased results, which become worse as the sample size increases. Additional simulations with or 0.25 showed that the results were similar to Table 2 (see Tables S1 and S2 of supplementary materials available at Biostatistics online), indicating that the performance of the two time scales method is insensitive to the size of the hazard change at age and evidencing poor performance of the single time scale method, but with reduced bias owing to the smaller jump in the cause-specific hazard. Simulations under different covariate distributions evidence similar performance; see Appendix C of the supplementary materials available at Biostatistics online.

TABLE 2.

Simulation results when the model for cause 2 is proportional on the age time scale but not on the time since diagnosis time scale; true values of (True), empirical variances (Emp.Var), averages of the variance estimates , and empirical coverage probabilities (CP)

					Method
					Two time scales				Time since diagnosis time scale
Sample size	% censored	Cause	Time	True	Bias	Emp.Var		CP	Bias	Emp.Var		CP
200	10%		5	0.178	0.0024	0.00190	0.00194	0.945	0.0237	0.00147	0.00148	0.887
		1	8	0.245	0.0013	0.00313	0.00311	0.947	0.0459	0.00207	0.00206	0.817
			10	0.278	0.0019	0.00371	0.00372	0.945	0.0619	0.00229	0.00229	0.746
			5	0.264	0.0073	0.00403	0.00420	0.958	0.1668	0.00174	0.00217	0.031
		2	8	0.362	0.0102	0.00443	0.00440	0.940	0.2055	0.00196	0.00246	0.008
			10	0.411	0.0119	0.00437	0.00428	0.939	0.2168	0.00205	0.00253	0.009
200	35%		5	0.178	0.0018	0.00231	0.00226	0.944	0.0228	0.00176	0.00171	0.888
		1	8	0.245	0.0016	0.00386	0.00370	0.940	0.0446	0.00253	0.00244	0.836
			10	0.278	0.0021	0.00474	0.00450	0.946	0.0599	0.00288	0.00274	0.777
			5	0.264	0.0086	0.00492	0.00557	0.961	0.1607	0.00228	0.00239	0.077
		2	8	0.362	0.0106	0.00510	0.00572	0.960	0.1978	0.00250	0.00287	0.034
			10	0.411	0.0122	0.00510	0.00553	0.952	0.2080	0.00268	0.00304	0.039
600	10%		5	0.178	0.0003	0.00068	0.00064	0.943	0.0216	0.00051	0.00049	0.816
		1	8	0.245	0.0008	0.00103	0.00103	0.949	0.0441	0.00069	0.00068	0.610
			10	0.278	0.0009	0.00123	0.00124	0.952	0.0598	0.00077	0.00076	0.451
			5	0.264	0.0045	0.00147	0.00143	0.942	0.1689	0.00062	0.00072	0.000
		2	8	0.362	0.0055	0.00155	0.00148	0.943	0.2091	0.00073	0.00081	0.000
			10	0.411	0.0054	0.00152	0.00143	0.936	0.2207	0.00078	0.00083	0.000
600	35%		5	0.178	0.0003	0.00070	0.00075	0.954	0.0218	0.00054	0.00056	0.864
		1	8	0.245	0.0005	0.00114	0.00122	0.957	0.0439	0.00077	0.00080	0.677
			10	0.278	0.0001	0.00141	0.00150	0.956	0.0586	0.00088	0.00090	0.515
			5	0.264	0.0058	0.00183	0.00189	0.947	0.1632	0.00068	0.00079	0.000
		2	8	0.362	0.0066	0.00190	0.00193	0.954	0.2015	0.00083	0.00094	0.000
			10	0.411	0.0074	0.00185	0.00185	0.936	0.2126	0.00090	0.00099	0.000

4 Case study: seer colon cancer database

We illustrate our inference procedure using stage III colon cancer data obtained from the November 2008 submission of the SEER population-based cancer registry program of National Cancer Institute. We use cases 66 years and older diagnosed between the years 1994 and 2005 with colon cancer and comorbidity scores for stage III colon cancer who had surgery from the SEER 13 registries (except Alaska) and California. Covariates included site, substage, grade, marital status, race/ethnicity, sex, age at diagnosis, comorbidity scores (derived from SEER cases linked to Medicare data; Klabunde and others, 2000), year of diagnosis (1994–2005), and an interaction between age at diagnosis and comorbidity scores. We excluded patients with a prior diagnosis of cancer or who were diagnosed either at autopsy or by death certificate only. Descriptive statistics for the data used in the analysis are given in Table S3 of supplementary material available at Biostatistics online. Of the 14,657 patients in stage III with surgery performed, 5685 patients died from colon cancer, 3123 other causes, and 5849 (39.91%) were censored. A maximum follow-up was 10 years. Details of parameterizations for age at diagnosis and comorbidity scores can be found in Appendix D of the supplementary materials available at Biostatistics online.

Table 3 shows the regression parameter estimates and standard errors under the cause-specific proportional hazards model on the time-on-study scale for death from colon cancer and on both the time-on-study and age scales for death from other causes, respectively. We included only significant predictors for death from colon cancer and from other causes in the model at the significance level of 0.05. Site, substage, grade, marital status, race/ethnicity, sex, age at diagnosis, and year of diagnosis were significant for death from colon cancer while comorbidity scores (-value ) and the interaction between age at diagnosis and comorbidity scores (-value ) were not significant for death from colon cancer. Grade, marital status, race/ethnicity, sex, age at diagnosis, comorbidity scores, year of diagnosis, and the interaction between age at diagnosis and comorbidity scores were significant predictors of death from other causes on both the time-on-study and age scales. The regression parameter estimates and standard errors for death from other causes on the age scale were similar to those on the time-on-study scale except age at diagnosis, where the regression coefficient of age at diagnosis for death from other causes is 0.097 in the time-on-study model and 0.018 in the age-scale model. This is easily understandable using that .

TABLE 3.

Regression parameter estimates and standard errors under the cause-specific proportional hazards model with time-on-study and age as the time scale

	time-on-study scale						age scale
	Death from colon cancer			Death from other causes			Death from other causes
		se()	-value		se()	-value		se()	-value
Site
Proximal	0	—	—	—	—	—	—	—	—
Distal	0.084	0.029	0.003	—	—	—	—	—	—
Substage
Stage IIIA	0	—	—	—	—	—	—	—	—
Stage IIIB	0.833	0.068	0.001	—	—	—	—	—	—
Stage IIIC	1.416	0.069	0.001	—	—	—	—	—	—
Grade
Grade I/II	0	—	—	0	—	—	0	—	—
Grade III/IV	0.261	0.029	0.001	0.120	0.039	0.002	0.119	0.040	0.003
Marital Status
Married	0	—	—	0	—	—	0	—	—
Single	0.129	0.030	0.001	0.302	0.041	0.001	0.294	0.041	0.001
Race/ethnicity
Non-Hispanic white	0	—	—	0	—	—	0	—	—
Hispanic	0.114	0.060	0.058	0.041	0.089	0.647	0.026	0.089	0.773
Non-Hispanic black	0.232	0.051	0.001	0.219	0.068	0.001	0.236	0.068	0.001
AI/AN Non-Hispanic	0.447	0.251	0.074	0.501	0.318	0.116	0.447	0.319	0.161
API Non-Hispanic	0.207	0.059	0.001	0.380	0.088	0.001	0.376	0.088	0.001
Sex
Male	0	—	—	0	—	—	0	—	—
Female	0.108	0.029	0.001	0.358	0.040	0.001	0.355	0.040	0.001
Age at diagnosis	0.036	0.002	0.001	0.097	0.004	0.001	0.018	0.009	0.035
Comorbidity	—	—	—	4.118	0.368	0.001	3.765	0.358	0.001
Year of diagnosis	0.024	0.004	0.001	0.023	0.006	0.001	0.029	0.006	0.001
Age at diagnosiscomorbidity	—	—	—	0.040	0.005	0.001	0.035	0.005	0.001

For comparison, we estimated cumulative incidences for an individual with specific covariates using two time scales and methods of Cheng and others (1998). Figure 1 shows the cumulative incidence estimates with pointwise 95% confidence interval for a married non-Hispanic white man aged 66 or 76 years diagnosed in 2005 with substage IIIB, grade III/IV, proximal colon, and comorbidity scores of 0.4 or 1.4. For a man diagnosed at 66 years old with a comorbidity score of 1.4, the cumulative incidence estimates for death from cancer from the two time scales are slightly lower than those from the time-on-study scale while the cumulative incidence estimates for death from other causes from the two time scales are higher than those from the time-on-study scale (Figure 1(a)). For a man diagnosed at 76 years old with a comorbidity score of 0.4, the cumulative incidence estimates for death from cancer and from other causes from the two time scales are similar to those from the time-on-study scale as shown in Figure 1(b). In Figure 1(a), the two approaches, seem to provide different estimates for a young patient with high comorbidity scores. However, the 95% confidence intervals for the two approaches overlap, which suggests that the difference in prediction between the two time scales and time-on-study scale is within the limit of sampling variation.

Fig. 1.

Estimated cumulative incidence probability with 95% confidence interval (CI) for a married non-Hispanic white man aged 66 or 76 diagnosed in 2005 with substage IIIB, grade III/IV, proximal colon, and comorbidity scores of 0.4 or 1.4.

Model accuracy assessed using calibration plots does not evidence substantive differences between the fitted models based on the two time scales approach and the standard prediction methods. In addition, the assumption of proportional hazards was assessed using Schoenfeld residual plots and goodness-of-fit tests, with deviations from proportionality being relatively modest on both time-on-study and age scales. Details may be found in Appendices E and F of the supplementary materials available at Biostatistics online.

Korn and others (1997) formally established that the Cox model on the time-on-study scale adjusting for age at entry of study as a covariate yields regression parameters for other covariates which are identical to those from the Cox model with age as the time scale if the baseline hazard function of the age-scale model is that of a Gompertz distribution (i.e., for some and ). The difference between the two models occurs with a nonparametric effect of the time scale as well as in other non-Gompertz parametric models. We computed the Breslow estimates of the baseline cumulative hazard function for death from other causes using age as the time scale in Figure 2, which are in rough agreement with a Gompertz distribution. In order to more formally evaluate the Gompertz distribution, we fitted a parametric proportional hazards model for death from other causes assuming a Gompertz form for the baseline hazard function. The estimator was obtained by maximizing the left-truncated right-censored likelihood function with age as the time scale given by , where , , , and and are the observed entry and failure/censoring times for the th patient on the age scale. We used the same covariates in the model for death from other causes on the age scale (see Table 3). We plotted the maximum likelihood estimates of the Gompertz baseline cumulative hazard function in Figure 2. The Breslow estimates agree well with the Gompertz estimates of the baseline cumulative hazard function. This explains why we have similar results from the two approaches. This agrees with demographic studies showing that the Gompertz often makes a good fit to total mortality, at least in an age range of 30–90 years, and with a one-year factor of around 1.1, which is close to in the age-scale model.

Fig. 2.

The Breslow estimates and Gompertz estimates of the baseline cumulative hazard function for death from other causes on the age scale.

5 Discussion

In cancer studies including the SEER registry data, it is more natural to model death from cancer on the time-on-study scale and death from other causes on the age scale. When the true time scale for cause 2 is age, an incorrect choice of the time scale may lead to biased estimation, as evidence in the numerical studies, while our proposed methods on two time scales provide valid estimates. When the baseline hazard function for death from other causes on the age scale follows a Gompertz distribution (Gompertz, 1825), the two approaches may provide similar results, as shown in analyses of the SEER colon cancer registry data in Section 4. This suggests that power might be gained from an analysis based on the time-on-study scale for both causes of death when the Gompertz assumption for the baseline hazard function for death from other causes holds. However, even if the Gompertz distribution provides a good overall fit, there may be some specific combinations of covariates for which the hazard of death from other causes deviates from a Gompertz distribution, producing different estimates when fitting equivalent models using the time-on-study scale for both causes of death when compared with using two time scales. Since age is the natural time scale for death from other causes it is generally a safer choice to use the two time scales approach. However, the tradeoff in using models on different time scales is that the time-on-study model makes more efficient use of the data than the two time scales approach because of left truncation on the age scale.

Another benefit of fitting the cause-specific hazards model for other causes on the age scale is that it permits estimation of the “net” residual life expectancy without cancer across a wide age range that is not possible when fitting the cause-specific hazards model for other causes on the time since diagnosis scale. That is, if one assumes death from cancer and death from other causes are independent (Tsiatis, 1978), then the residual distribution of the latent failure time corresponding to other cause mortality may be obtained from the fitted cause-specific hazards model on the age scale. This “net” distribution may be interpreted as the distribution of time to non-cancer mortality in a hypothetical population where cancer mortality has been eliminated, which is of interest to patients and physicians. Hence, it is useful to report this quantity, with the important caveats regarding the reliance of its interpretation on strong and unverifiable assumptions. Estimation of this “net” distribution is much more limited when modeling on the time since diagnosis scale, owing to the fact that the baseline hazard on the time since diagnosis scale is defined over a much shorter time interval (roughly 10 years in the SEER databases for years with consistent up to-date staging) than is the baseline hazard on the age scale (66–90 years of age in the SEER databases linked to Medicare).

In the cancer registry data, subjects are enrolled in the database at the time of disease diagnosis. On the age scale, the baseline hazard may not be stable at the minimum age that the analysis starts, since the risk set can rise and fall. This differs from time since diagnosis where the risk set is always largest at time 0 and gets progressively smaller the further subjects are from diagnosis. Thus, one has to be careful to only make predictions at time points where the risk sets are of sufficient size to provide stable estimates of the baseline hazard. The proposed methods are most applicable to large population-based registry analyses.

Supplementary material

Supplementary material is available online at http://biostatistics.oxfordjournals.org

Funding

This study was supported by 2016 Research Grant from Kangwon National University.