Hospices Civils de Lyon, Service de Biostatistique, Lyon, France

Université de Lyon, Lyon, France

Université Lyon 1, Villeurbanne, France

CNRS, UMR 5558, Laboratoire de Biométrie et Biologie Evolutive, Equipe Biostatistique-Santé, Villeurbanne, France

Institut de Veille Sanitaire, Département des Maladies Chroniques et des Traumatismes, Saint-Maurice, France

INSERM ERI3 « Cancers & Populations », Caen, France

Registre Bourguignon des Cancers Digestifs, Inserm U866, CHU Dijon, Dijon, France

Laboratoire d'Enseignement et de Recherche sur le Traitement de l'Information Médicale, EA 3283, Aix-Marseille Université, Faculté de Médecine, Marseille, France

Abstract

Background

In medical research, one common competing risks situation is the study of different types of events, such as disease recurrence and death. We focused on that situation but considered death under two aspects: "expected death" and "excess death", the latter could be directly or indirectly associated with the disease.

Methods

The excess hazard method allows estimating an excess mortality hazard using the population (expected) mortality hazard. We propose models combining the competing risks approach and the excess hazard method. These models are based on a joint modelling of each event-specific hazard, including the event-free excess death hazard. The proposed models are parsimonious, allow time-dependent hazard ratios, and facilitate comparisons between event-specific hazards and between covariate effects on different events. In a simulation study, we assessed the performance of the estimators and showed their good properties with different drop-out censoring rates and different sample sizes.

Results

We analyzed a population-based dataset on French colon cancer patients who have undergone curative surgery. Considering three competing events (local recurrence, distant metastasis, and death), we showed that the recurrence-free excess mortality hazard reached zero six months after treatment. Covariates sex, age, and cancer stage had the same effects on local recurrence and distant metastasis but a different effect on excess mortality.

Conclusions

The proposed models consider the excess mortality within the framework of competing risks. Moreover, the joint estimation of the parameters allow (i) direct comparisons between covariate effects, and (ii) fitting models with common parameters to obtain more parsimonious models and more efficient parameter estimators.

Background

Analysis of failure time data is one of the major fields of statistics, death (whatever its cause) being the event of interest. However, in some other situations, several types of events are considered and the occurrence of one type prevents the occurrence of the others, creating a context of competing risks. Initially, in that context, a subject would fail because of only one of several different event types (for example, different causes of death). However, a patient may undergo successively several event types and be considered in a situation of competing risks. For example, in a study of the efficacy of a treatment on a chronic disease, it may be interesting to analyse time to recurrence. However, a patient may present recurrence and then die or die before recurrence. This example may be analysed within the competing risks framework limiting the analysis to the first occurring event, this event being sufficient to indicate treatment failure

Our work was motivated by a dataset from a French population-based study on colon cancer patients who have undergone curative surgery. Although surgery remains the primary treatment, the incidence of recurrence after surgery increases during the first five years reaching 12.8% for local recurrence and 25.6% for distant metastasis

Our objective was therefore to estimate both the excess mortality hazard and the recurrence-event hazard. Moreover, we were interested in (i) checking whether a given covariate may have the same impact on the different events, in particular on two related types of events such as "local recurrence" and "distant metastasis", and (ii) testing whether the excess mortality and recurrence-hazard are proportional or not; i.e., whether their ratio is constant over time or time-dependent in order to model it in a flexible way when appropriate. To achieve these objectives, among the many developments of the competing risk theory over the last 30 years

The paper is organized as follows. In the next section, we present our motivating example of competing risks in a prognostic study of colon cancer. Section Methods introduces the excess hazard model in case of a single event, the background of the competing risk methodology, then presents our competing risk models proposed to estimate jointly the excess mortality and the recurrent-specific hazards. In section Results, we present the analysis strategy of the motivating example and show the results. We conclude this article with a discussion of the findings and an outline of further developments.

Motivating example

FRANCIM network is an association that joins all validated French Cancer Registries. The original dataset that stems from a "High-Resolution study" of nine French cancer registries consists of 1016 incident cases of colon cancer (caecum to rectosigmoid junction; C18 and C19 according to the International Classification of Diseases for Oncology, 3^{rd }revision) diagnosed in 1995 and treated with curative intent (surgery or endoscopic resection). Observations with unknown cancer stage at diagnosis were excluded from the analysis (n = 45). Moreover, 35 patients with synchronous distant metastasis at diagnosis (i.e., Stage IV) were also excluded because the occurrence of metastasis was one of the events under study. Thus, the analysis concerned 936 incident cases of colon cancer.

Three events were of interest: local recurrence, distant metastasis, and death. The delay from diagnosis to the first observed event was calculated in each case and patients' follow-up was restricted to the first seven years after diagnosis, time at which patients still at risk were censored. The mean age at diagnosis was 71 years (range: 21 to 100). The patients were 455 women (49%) and 481 men (51%). The cancer stages at diagnosis were 256 stage I (27%), 289 stage II (42%), and 291 stage III (31%). During the study period, there were 60 local recurrences, 143 distant metastases, and 206 deaths.

The study population presented a persistent mortality after curative intent treatment. Within such a context, an analysis of excess death with no recurrence (local or distant) should provide new insights into the course of the disease and the impact of the treatment.

Methods

The excess mortality hazard model

In the classical additive form, the overall mortality hazard function, _{
O
}, is split into an excess hazard function, _{
+
}, and a population (or expected) hazard function _{
P
}

where **x **a vector of covariates, and **z **a vector of population characteristics

The population hazard function _{
P
}(**z**) in (1) is assumed to be known and is usually quantified on the basis of a vector **z **of population characteristics (generally age, sex, and possibly place of residence, etc.) and may be obtained from national statistics institutes. In previous works, the excess hazard function was modelled by proportional hazard (PH) models _{+}(**x**) = _{0 }(**βx**) with _{0}(

Background of the competing risks

Competing risk data in a sample of N patients give rise to a right censored sample (_{i},δ_{i},j_{i}
**x**
_{
i
}), _{
i
}is the time to the first event, _{i }
_{i }
_{i }
_{i }
_{
i
}), and **x**
_{
i
}a vector of covariates.

Assuming a random censoring mechanism, the full likelihood function

where _{j}
**x**) is the event _{i }
_{i}

When death is among the

New competing risk models in excess mortality analysis

The proposed models

The new model combining the excess mortality hazard model and the competing risk model may be written:

where, in case of _{1 }= 0 and _{1k
}= 0,

To simplify the interpretation of model (3), we considered _{1}(**x **equal to 0. In model (3), the log of the baseline hazard of event 1, log(_{1}(_{1}(_{
j
}(**x **= 0) = _{1}(_{j}
_{1}(_{3}) which represents the baseline of the "event-free excess death hazard". The model (3) assumes PH effects of covariates **x **on the event-specific hazards. For one unit increase of a given covariate _{
k
}, the effect is split into a _{k}
_{jk}
_{k}
_{k }
_{jk}
_{0}: _{jk }
_{k }

However, the assumption of a common pattern for event-specific hazards through _{1}
_{j}

The new flexible model may be written:

where, in case of _{1}(_{1k
}= 0,

In the flexible model (4), the log of the baseline hazard of event 1, log(_{1}(_{j}
_{j }
_{j}
_{j}

Estimation procedure

In both models (3) and (4), the maximum likelihood estimates are obtained using the trick of data duplication. A detailed description of data duplication and coding can be found in references

Performance of the estimators

Simulation studies were conducted to assess the performance of the estimators obtained from model (4) in the case of three competing events (of whom death) and different sample sizes and censoring rates. Data generation, simulation design, and results are detailed in Additional file

**Data generation and design of simulation studies**. Detailed presentation of the way data are generated and simulation are carried out (see also references

Click here for file

Briefly, the times to the events were supposed to depend on three independent prognostic factors. Different rates of drop-out censoring (0%, 15%, and 30%) and different sample sizes (N = 400 and N = 1000) were considered. The relative biases (RBs) were close to zero (range: -0.047 to 0.05) whatever the sample size and the drop-out censoring rate (Additional file

Results

Analysis strategy

In this analysis, our objectives were: (i) estimate the baseline hazards, and their ratios, for local recurrence, distant metastasis, and recurrence-free excess death; (ii) to estimate the effects of sex, age at diagnosis, and stage at diagnosis associated with each event-specific hazard; and (iii) to test whether the effects of covariates are common to the two related events "local recurrence" and "distant metastasis" or not.

The strategy consisted of three steps. The first step was to determine the pattern of the time-dependent HRs between event-specific hazards. Thus, a Cox PH model with constant HRs between event-specific hazard functions was used on duplicated data; i.e., with a dummy variable denoting each type of event introduced as covariate

Results of the analysis

At the first step of our strategy, it was obvious to use a constant hazard ratio of the baseline hazards of distant metastasis event to that of local recurrence event, while a cubic regression spline with one knot at one year was used to model the time-dependent HR of the death event to that of local recurrence event.

At the second step, the AIC approach selected a regression cubic spline with one knot at 1 year to model the baseline hazard relative to the local recurrence event. As shown in Figure _{P}

Event-specific baseline hazard functions for local recurrence, distant metastasis, and excess death in 936 French colon cancer patients

**Event-specific baseline hazard functions for local recurrence, distant metastasis, and excess death in 936 French colon cancer patients**.

Event-specific baseline hazard functions for local recurrence, distant metastasis, and overall death in 936 French colon cancer patients

**Event-specific baseline hazard functions for local recurrence, distant metastasis, and overall death in 936 French colon cancer patients**.

Then, in the third step, the tests of common covariate effects on the events "local recurrence" and "distant metastasis" were not significant (p = 0.58, p = 0.94, and p = 0.07 for sex, age, and stage, respectively). The estimated hazard ratios are shown in Table

Results of the analysis of 936 French colon cancer patients: Adjusted Hazard Ratios for covariates sex, age and stage of cancer at diagnosis associated with the events local recurrence, distant metastasis and excess death, with the 95% confidence interval.

**Type of event and Covariate**

**Adjusted Hazard Ratio with 95% confidence interval**

**p-value for a 2-tailed Wald test**

Local recurrence

Man

1

Woman^{#}

0.71 [ 0.53; 0.94 ]

0.02

Age^{#}

1.00 [ 0.99; 1.01 ]

0.55

Stage I

1

Stage II^{#}

3.50 [ 2.11; 5.80 ]

< 0.01

Stage III^{#}

7.36 [ 4.50; 12.05 ]

< 0.01

Distant metastasis

Man

1

Woman^{#}

0.71 [ 0.53; 0.94 ]

0.02

Age^{#}

1.00 [ 0.99; 1.01 ]

0.55

Stage I

1

Stage II^{#}

3.50 [ 2.11; 5.80 ]

< 0.01

Stage III^{#}

7.36 [ 4.50; 12.05 ]

< 0.01

Excess death

Man

1

Woman

0.77 [ 0.42; 1.42 ]

0.74

Age

1.12 [ 1.09; 1.16 ]

< 0.01

Stage I

1

Stage II

1.48 [ 0.73; 2.98 ]

0.05

Stage III

2.76 [ 1.39; 5.48 ]

0.02

^{# }The same effect was estimated on local recurrence event and distant metastasis event.

Discussion

To our knowledge, model (3) and its flexible refinement model (4) proposed here are the first to consider competing risks within the framework of the excess hazard regression model. These new models make it possible to estimate (i) the hazard function for each type of pre-specified event, including the recurrence-free excess death hazard function, (ii) changes over time in their ratio, and (iii) the effect of covariates on the hazard of each event, including the excess death event. Furthermore, the joint estimation of all parameters allows comparisons between covariate effects associated with different types of events in a single analysis. Analysis of the population-based dataset on French colon cancer patients using model (4) underlines the importance to model in a flexible way the ratio of the baseline hazards of the events and permits new insights into the benefit of surgery.

A joint modelling of the hazards allows fitting models with common parameters; this results in more parsimonious models and more efficient parameter estimators

In our new model (3), we assumed a common pattern for all event-specific hazards, which may be dubious in most cases. Whenever the assumption of a common pattern does not hold, a simple approach could be to analyse the competing risk data stratified on the type of event. While this approach assigns different baseline hazard functions for each type of event, it does not allow comparisons between all types of events in a single analysis. In our new flexible model (4), the introduction of a time-dependent log HRs _{k}
_{k }
_{k}

The present work focuses on modelling the event-specific hazard which is not directly interpretable as the marginal hazard function. Indeed, interpreting the event-specific hazard as the marginal function would be equivalent to assuming independence between competing events, whereas this assumption cannot be met

Regression splines have been widely used in classical survival studies

The data on colon cancer patients showed an important excess death that occurred just after surgery but decreased thereafter to become null a few months later. We have shown that if the expected mortality hazard is not taken into account, the overall mortality hazard will be more important and never reach zero. Local recurrence and distant metastasis hazards reached peaks nearly one year after diagnosis and then decreased slowly, confirming the importance of keeping patients under close medical supervision

In this work, we limited the analysis to the first occurring event but other recurrences, as well as death after a recurrence, may be observed. An interesting future work, based on the idea of our new model, would be to study all times to events as multivariate failure-time data, including an unmeasured "frailty" term to take into account the correlation between times to events, such as that between the time to distant metastasis and the time to excess death

Conclusions

The new models proposed in this paper allow considering competing risks within the framework of excess hazard regression model. They make it possible to estimate in a flexible way the hazard function for each type of pre-specified events, including the recurrence-free excess death hazard function. A joint estimation of all parameters allows direct comparison between covariate effects and may provide more parsimonious models and more efficient parameter estimators.

List of abbreviations

PH: proportional hazard; LR: Likelihood Ratio; HR: Hazard Ratio; AIC: Akaike Information Criterion; df: degree of freedom;

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

AB and RG conceived the study, performed the statistical analysis and drafted the manuscript. LR, GL and VJ conceived the study and helped drafting the manuscript. All authors read and approved the final manuscript.

Acknowledgements

The authors thank the French network of cancer registries FRANCIM and the following French cancer registries for their contribution to the high-resolution study on colorectal cancer: Bas-Rhin General Cancer Registry (M. Velten), Calvados Digestive Cancer Registry (G. Launoy), Côte d'Or Digestive Cancer Registry (A.-M. Bouvier), Doubs General Cancer Registry (A. Danzon), Hérault General Cancer Regisrty (B. Trétarre), Isère General Cancer Registry (M. Colonna), Manche General Cancer Registry (S. Bara), Somme General Cancer Registry (N. Bourdon-Raverdy), Tarn General Cancer Registry (P. Grosclaude).

The authors are also very grateful to Jean Iwaz, PhD, Hospices Civils de Lyon, for revising the manuscript.

Pre-publication history

The pre-publication history for this paper can be accessed here: