### Competing risks and cumulative incidence

15 June 2018

Papers: The importance of the ECG in patients with acute heart failure

Programs: PaulBrownPhD/comprisk

### Introduction

In this study, the secondary endpoint was time from hospital discharge to first heart failure related readmission. For the analysis, if the proportional hazards assumption is reasonable, we would use standard Cox regression to adjust for factors deemed clinically important (e.g. age, sex blood pressure etc.) The proportional hazards assumption was confirmed using a formal statistical test and a visual inspection of the log-cumulative hazard plot [see AFT blog post]. The important issue here is that time to readmission is not independent of mortality which leads to censored survival times. There have been review papers [ref] calling for more appropriate analyses of such 'competing risks' and thus we can expect non-statistical reviewers to be conversant of the issue i.e. reviewers would certainly ask about censoring of readmission by death. There is a lack of consistency in how death is handled in this context [ref]. Because time to readmission starts at discharge, we lose the in-hospital deaths. However, mortality accounted for an appreciable number of the censoring events observed in this study (>30%).There are two approaches. The cause-specific hazard is obtained by including the competing risk as censored observations (see first proc phreg below). Alternatively, it is a simple matter in proc phreg to implement Fine & Gray's model for the cumulative incidence function (CIF) using 'eventcode' (see the second proc phreg below, or see this SAS example). This is a new feature in SAS 9.4. As Ying et al. note (see weblink below): "An increasingly common practice of assessing the probability of a failure in competing risks analysis is to estimate the cumulative incidence function, which is the probability subdistribution function of failure from a specific cause".

The only difference in the second phreg proc are the values listed in 'status' (which indicates censoring), i.e. status=2 (the competing risk) is not included and the event of interest appears in the eventcode statement. It is useful then, when deriving time-to-event endpoints, to include a status variable indicating 0=censored, 1=readmission and 2=patients who die before readmission. For the full analysis program see tableS1.sas at the GitLab link above.

There is a SAS macro for estimating cumulative incidence (CI) based on Gooley et al. (see Gooley weblink below). CI is variably referred to as the cause-specific risk and the crude incidence curve. A key remark in Gooley et al. is the following explanation of the probability calculation: "The contribution to the estimate of the probability of failure from the cause of interest due to failures that occur after patients are censored is increased over the contribution from previous failures. The increase is equal to the potential contribution from the censored patient(s) that is redistributed among patients known to be at risk of failure beyond the time that censoring occurred [so far, so good]. Note that if a patient fails from a competing risk, the potential contribution to the estimate for this patient becomes zero, as failure from the event of interest is no longer possible. Hence, patients who fail from a competing risk are treated differently from patients who are censored due to a lack of follow-up." I have placed the full SAS macro "CumIncid (full macro).sas" in GitLab; in addition, a concise version which I edited down to the essential code "CumIncid (simplified).sas" is made available.

As an aside: we should ideally account for multiple readmissions within patients (see MTRE blog post). This can be handled by a shared frailty model or Andersen-Gill etc. In this case, the competing risk of death could be treated as just 'another event in the event process' i.e. the final event [ref]. This is crude, however, and a joint frailty model should be considered (see the SAS code at the bottom of Liu & Huang).

### Appendix SAS procs for survival data

__proc lifetest__: nonparametric, Kaplan-Meier estimates, log-rank and Wilcoxon tests (use the latter if proportional hazards assumption is not reasonable); in strata statement after forward slash specify e.g. 'test=logrank' (then obtain separate estimates of S(t) for each strata); as Dave Collett p332 noted: "there can be a difference between the results of the log-rank test obtained using the strata statement alone and that obtained using the test statement alone. This is due to the different ways in which tied survival times are handled by the two statements"

__proc lifereg__: parametric, specify baseline hazard or accelerated failure time model (see AFT blog post); yields acceleration factor

__proc phreg__: semi-parametric, Cox's proportional hazards model, allows adjustment for covariates; yields hazard ratio; Wald test, likelihood score test (if there are no tied survival times this test in Cox regression is identical to the log-rank test)

### Notable sources:

Analyzing Survival Data with Competing Risks Using SAS SoftwareYing, Using the PHREG Procedure to Analyze Competing-Risks Data

Gooley, Estimation of failure probabilities in the presence of competing risks

Fine & Gray, A Proportional Hazards Model for the Subdistribution of a Competing Risk