Cox model has been the commonly used method in past analyses of association between obesity and the risk estimates of cancer in situations where the subjects have also died (or could die) of noncancer events (competing events). The Cox model does not address the presence of competing events convincingly. The competing risk approach accommodates the fact that individuals who died of other causes (competing events) will never die of cancer and thus provides more realistic estimates. This study uses the competing risk approach to study the association of obesity and cancer mortality and compare the analysis results with those based on the traditional Cox model. It was seen that while the cause-specific hazard rate of cancer is significantly higher for obese population compared to normal weight population, the difference is not significant using competing risk approach. We demonstrated that higher cause-specific hazard rate does not necessarily imply higher incidence rate and in situations involving competing events we recommend using competing risk approach in addition to the Cox regression model.
Over the past two decades obesity has established itself as a global health issue. Some of the staggering figures listed by the World Health Organization (WHO) such as “nearly doubled since 1980” and “fifth leading risk for global deaths” seem to indicate catastrophic consequences of obesity at global level (
Besides the association of obesity and all-cause mortality, the association of obesity with specific cause of death such as cardiovascular diseases and cancer has also garnered attention [
The association between obesity and cancer has been controversial. While in general most of the studies provide evidence in the favor of increased risk of cancer for obese patients as compared to normal weight patients [
An alternative approach to the traditional Cox model is to take the competing events into consideration and treat them differently from the censored events. One of the popular competing risk models is the Fine and Gray model which was specifically developed to analyze the association between covariates and risk estimates in the presence of competing events [
The purpose of this paper is to analyze the association between obesity and cancer in presence of competing events using Cox regression as well as the Fine and Gray model and compare the two results. The publicly available Framingham dataset will be used for the analysis. The event of interest is death due to cancer and the competing events are death due to cardiovascular disease as well as death due to other causes.
In this section we will compare the model settings for the Cox model and the Fine and Gray model. We assume that event 1 is the event of interest and event 2 is a competing event. We provide a brief definition of some of the concepts in survival analysis that will be used in the rest of the paper. The definitions are intended to be evocative rather than pedagogical.
The above definitions can be found in the introductory textbook on survival analysis by Klein and Moeschberger [
The standard Cox regression model, also known as the proportional hazard model, is a semiparametric model that essentially models the effect of covariates on the risk (cause-specific hazard) associated with event the of interest as follows:
Any coefficient from
The use of Cox regression models in estimation of cumulative incidence proportion has been considered cumbersome and difficult to interpret [
When the traditional Cox model is used to analyze competing events, each event is analyzed separately and the competing events are considered as censored. When competing events and censored events are independent of the event of interest, the estimated total risk
The Fine and Gray regression model is based on the idea of subdistribution hazard. The subdistribution hazard is defined as the probability of observing the event of interest in a subject under the assumption that the subject is alive at time
Comparison of cause-specific hazard and subdistribution hazard. “Risk set” values in bold font represent the risk set for cause-specific hazard and values in light face represent risk set for subdistribution hazard. “Num. of events” values in bold font represent the number for event of interest and values in light face represent number of competing event. “Hazard” values in bold font represent cause-specific hazard and values in light face represent subdistribution hazard.
Time |
|
|
|
|
|
|
3 |
|
| |||||
Risk set |
|
|
|
10 |
|
9 |
|
8 |
|
8 | ||||
Num. of events |
|
|
|
2 |
|
0 |
|
1 |
|
1 | ||||
Hazard |
|
|
|
|
|
|
|
|
|
|
At time 0, we assume that 10 subjects are being followed where the two possible outcomes are either event of interest or competing event (censored events have been excluded for tabular convenience). At time 1, let us assume 1 subject experiences event of interest and two subjects experience competing event. The cause-specific and subdistribution hazards are both 0.1. However, at time 2, only the subjects that have not experienced any events
Owing to the difference in risk sets, the risk estimates based on Fine and Gray model cannot exceed the risk estimates based on Cox model. Table
Fine and Gray model essentially links the subdistribution hazard to the cumulative incidence function as follows:
Past studies have shown association of age, smoking, and gender with cancer [
Flowchart of analysis.
A dataset of “Framingham Heart Study” is used for analysis. The Framingham Heart Study is a long term prospective study of the etiology of cardiovascular disease among a population of free living subjects in the community of Framingham, Massachusetts. The original cohort of the study consisted of 5209 men and women from Framingham, Massachusetts. Extensive examinations were carried out every other year on this cohort since 1948. The dataset consists of follow-up information of 4526 individuals (2005 men and 2521 women) who were alive at Exam 4 of the study and all the variables were measured at Exam 4.
Table
Summary of variables in the Framingham dataset.
Variables | Percentage | Median follow-up time (yrs) | ||
---|---|---|---|---|
Male |
|
| ||
Female |
|
| ||
Current smokers |
|
| ||
Current Nonsmokers |
|
| ||
Underweight (BMI < 18.5) |
|
| ||
Normal weight (18.5 ≤ BMI < 25) |
|
| ||
Overweight (25 ≤ BMI < 30) |
|
| ||
Obese (BMI ≥ 30) |
|
| ||
Cancer death |
|
| ||
CVD death |
|
| ||
Other cause of death |
|
| ||
Censored |
|
| ||
|
||||
Summary statistics for age (years) | ||||
|
||||
Min | Median | Max | Mean | Standard deviation |
|
||||
34 | 49 | 69 | 49.82 | 8.49 |
|
||||
|
Deaths due to cancer are treated as events of interest and deaths due to other causes are the competing events. Figure
Cause-specific (left based on the Cox model) and subdistribution (right based on the Fine and Gray model) hazard estimated for deaths due to cancer as event of interest.
The three BMI categories (obese, underweight, and overweight) were included in the model and based on the results from stepwise regression “age,” “smoking,” “gender,” and “age × gender” were included the model. The results based on the Cox regression model and the Fine and Gray model are summarized in Table
Summary of analysis for cancer as event of interest.
Cause-specific hazard modelling (Cox model) with cancer as event of interest | ||||
---|---|---|---|---|
Hazard | Std. error |
|
| |
Underweight |
|
|
|
|
Overweight |
|
|
|
|
Obese |
|
|
|
|
Age |
|
|
|
|
Smoking |
|
|
|
|
Gender |
|
|
|
|
Age × gender |
|
|
|
|
|
||||
Fine and Gray modelling with cancer as event of interest | ||||
Hazard | Std. error |
|
| |
|
||||
Underweight |
|
|
|
|
Overweight |
|
|
|
|
Obese |
|
|
|
|
Age |
|
|
|
|
Smoking |
|
|
|
|
Gender |
|
|
|
|
Age × gender |
|
|
|
|
After adjusting for the effect due to age, smoking status, gender, and the interaction between age and gender, the results from Cox regression show that the cause-specific hazard rate of cancer for “obese” group is approximately 4 times higher than “normal” group (Table
Our work essentially compared the association between obesity and risk of cancer using the Cox model for cause-specific analysis and the Fine and Gray model of the competing risk approach. Our findings show that while the cause-specific hazard rate estimated with Cox model for cancer in obese population is significantly higher than that of normal population, the incidence rate of cancer estimated with the Fine and Gray model in obese population is not significantly higher than that of normal population. The analysis thus demonstrates that significantly higher cause-specific hazard rate in a hypothetical world without competing events does not necessarily imply a significantly higher incidence rate in a real world with competing events. These findings are in agreement with the previous studies [
Which approach should be used for research? From implementation aspect, Cox model is easy to use and is flexible in accommodating time-dependent covariates and had been widely used; the Fine and Gray model is not as flexible as the Cox model. For example, it does not address time-dependent covariates and it also does not allow stratified model [
The authors declare no conflict of interests regarding the publication of this paper.