Competing-risks regression

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1 Competing-risks regression Roberto G. Gutierrez Director of Statistics StataCorp LP Stata Conference Boston 2010 R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

2 Outline 1. Overview 2. The problem of competing risks 3. Cause-specific hazards 4. Cumulative incidence functions 5. The Cox regression approach 6. The Fine and Gray approach (stcrreg) 7. Concluding remarks R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

3 Overview Competing risks are events that occur instead of the failure event of interest, and we cannot treat these as censored When you have competing events, you want to focus on cause-specific hazards rather than standard hazards When you have competing events, you want to focus on the cumulative incidence function (CIF) rather than the survival function Cox regression is fine for cause-specific hazards, but for CIFs you need to go through a lot of work Competing-risks regression by the method of Fine and Gray (1999) is a useful alternative Implemented in the stcrreg command, new to Stata 11 R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

4 The problem of competing risks Definition A competing-risk event is an event that impedes what a researcher actually wants to see For example, if a researcher is interested in recurrence of breast cancer, cancer occurring in another location would be a competing event In general you cannot treat competing events as censored because 1. The competing events might be dependent, and you usually can t test this 2. You are unwilling to apply your results to a counter-factual world where the competing event doesn t exist R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

5 The problem of competing risks Competing events as censored What confuses the matter is that, often, it is okay to think of competing events as censorings. But this is merely a computational device That is, the software will often let you treat competing events as censored, but your interpretation of the results should never do so Even more confusing is that some software, such as that for Kaplan-Meier curves, is not appropriate if you treat competing events as censored R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

6 Cause-specific hazards Breast cancer data Consider data on 423 women treated for breast cancer 175 women were given an experimental new drug, and the other 248 women the standard therapy Time in months until disease progression occurred either at an old tumor site (local relapse) or at a new site (distant relapse) was recorded; local relapse is the event of interest If no event occurred after within 60 months, observations were censored Age and race also recorded R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

7 Cause-specific hazards Breast cancer data. use bc_compete, clear (Breast cancer with competing risks). describe Contains data from bc_compete.dta obs: 423 Breast cancer with competing risks vars: 5 13 Jul :27 size: 11,844 (99.9% of memory free) storage display value variable name type format label variable label age float %9.0g Age at start of the trial drug float %9.0g 1 treatment; 0 control race float %9.0g race 1 white; 2 black; 3 other time float %9.0g time in months status float %9.0g 0 censored; 1 local relapse; 2 distant relapse Sorted by: R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

8 Cause-specific hazards Computing-cause specific hazards A cause-specific hazard is the instantaneous risk of failure from a specific cause given that failure (from any cause) has yet to happen In our example, we have two cause-specific hazards: one for local relapse and one for distant relapse You can estimate, graph, and test a cause-specific hazard with standard survival software, if you treat the other event as censored You need only to modify your interpretation, specifically, you need to consider both hazards jointly and not attempt to disentangle them R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

9 Cause-specific hazards Cause-specific hazard for local relapse. stset time, failure(status = 1) (output omitted ). sts graph, hazard by(drug) kernel(gauss) Smoothed hazard estimates analysis time drug = 0 drug = 1 R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

10 Cause-specific hazards Cause-specific hazard for distant relapse. stset time, failure(status = 2) (output omitted ). sts graph, hazard by(drug) kernel(gauss) Smoothed hazard estimates analysis time drug = 0 drug = 1 R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

11 Cause-specific hazards Cox regressions for the effect of drug Because we believe in proportionality of cause-specific hazards we can use stcox to test for differences. quietly stset time, failure(status = 1). stcox drug Cox regression -- Breslow method for ties No. of subjects = 423 Number of obs = 423 No. of failures = 135 Time at risk = LR chi2(1) = 3.85 Log likelihood = Prob > chi2 = _t Haz. Ratio Std. Err. z P> z [95% Conf. Interval] drug R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

12 Cause-specific hazards Cox regressions for the effect of drug. quietly stset time, failure(status = 2). stcox drug Cox regression -- Breslow method for ties No. of subjects = 423 Number of obs = 423 No. of failures = 78 Time at risk = LR chi2(1) = 6.75 Log likelihood = Prob > chi2 = _t Haz. Ratio Std. Err. z P> z [95% Conf. Interval] drug Interpretation: Drug treatment somewhat decreases risk of local relapse while at the same time increasing the risk of distant relapse R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

13 Cumulative incidence functions Definition Despite the ominous name, a CIF is just the probability that you observe a specific type of event before a given time In our analysis, we have two CIFs; one for local relapse and one for distant relapse For example, the CIF for local relapse at 5 months is just the probability of a local relapse before 5 months CIFs begin at zero at time zero and increase to an upper limit equal to the eventual probability that the event will take place, but this is not equal to one because of competing events Mathematically, the CIF for local relapse is a function of both cause-specific hazards R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

14 Cumulative incidence functions Kaplan-Meier curves Kaplan-Meier curves are not appropriate for competing risks. quietly stset time, failure(status = 1). sts graph, survival by(drug) Kaplan Meier survival estimates analysis time drug = 0 drug = 1 R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

15 Cumulative incidence functions Kaplan-Meier curves The previous Kaplan-Meier curve attempts to answer the question What is the probability of no local relapse before (say) 5 months? In a competing-risks setting, a Kaplan-Meier curve is inadequate for three reasons First, it fails to acknowledge that local relapse may never occur. In reality, the probability of local relapse after time zero is not equal to 1 Second, the Kaplan-Meier calculation does not take into account dependence between competing events R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

16 Cumulative incidence functions Kaplan-Meier curves Third, with competing risks it is better to reverse the temporal ordering of the question It makes better sense to ask What is the probability of local relapse within 5 months? than to ask What is the probability that nothing happens for the first five months, but when something does happen I want it to be a local and not a distant relapse? In summary, using Kaplan-Meier demands too much of your data; it requires independent risks and a world where the competing event doesn t occur As such, you should use the cumulative incidence function (CIF) instead R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

17 Cumulative incidence functions Nonparametric estimation You can estimate CIFs nonparametrically using stcompet, by Coviello and Boggess (2004) and available from the SSC. quietly stset time, failure(status = 1). stcompet cif = ci, compet1(2) by(drug). gen cif_local_drug0 = cif if status == 1 & drug == 0 (333 missing values generated). gen cif_local_drug1 = cif if status == 1 & drug == 1 (378 missing values generated). twoway line cif_local_* _t, connect(step step) sort > ytitle(cumulative Incidence) title(cif of local relapse) R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

18 Cumulative incidence functions Nonparametric estimation CIF of local relapse Cumulative Incidence _t cif_local_drug0 cif_local_drug1 R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

19 The Cox regression approach Modeling Nonparametric estimation is flexible, but it cannot adjust for external covariates such as age and race Previously we applied Cox regression on drug to both cause-specific hazards These we could then extend to adjust for age and race, at the cost of the proportionality assumption We could then use the resulting cause-specific hazards to derive estimates of the CIFs R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

20 The Cox regression approach Tedious, but possible To calculate a CIF from Cox regression, we need to 1. Predict the baseline hazard contributions from both Cox regressions on drug, that for local relapse and that for distant relapse 2. Transform the baseline hazard contributions (which assume drug == 0) to those for drug == 1 where appropriate 3. Use the hazard contributions to calculate a product limit estimator of event-free survival for both levels of drug 4. Calculate the estimated CIF manually for both drug == 0 and drug == 1; see page 209 of [ST] for details 5. Plot the results The point: Assessing covariate effects on the CIF using Cox regression is a lot of work R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

21 The Cox regression approach Comparative CIF curves from Cox regression CIF of local relapse, Cox regression Cumulative Incidence _t cif_local_drug0 cif_local_drug1 R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

22 The Fine and Gray approach (stcrreg) Modeling the subhazard An easier way to do CIF covariate analysis is with competing risks regression, according to the model of Fine and Gray (1999) They posit a model for the hazard of the subdistribution for the failure event of interest, known as the subhazard Unlike cause-specific hazards, there is a one-to-one correspondence between subhazards and CIFs for respective event types; that is the CIF for local relapse is a function of only the subhazard for local relapse Covariates affect the subhazard proportionally, similar to Cox regression You do this in Stata 11 using stcrreg. stcurve after stcrreg will plot comparative CIFs for you R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

23 The Fine and Gray approach (stcrreg) Using stcrreg. quietly stset time, failure(status = 1). stcrreg drug, compete(status = 2) Competing-risks regression No. of obs = 423 No. of subjects = 423 Failure event : status == 1 No. failed = 135 Competing event: status == 2 No. competing = 78 No. censored = 210 Wald chi2(1) = 4.68 Log pseudolikelihood = Prob > chi2 = Robust _t SHR Std. Err. z P> z [95% Conf. Interval] drug R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

24 The Fine and Gray approach (stcrreg) stcurve after stcrreg. stcurve, cif at1(drug=0) at2(drug=1) title("cif of local relapse, stcrreg") CIF of local relapse, stcrreg Cumulative Incidence analysis time drug=0 drug=1 R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

25 Concluding remarks Competing risks are events that prevent an event of interest from occurring If you have competing risks, you want to look at cause-specific hazards instead of standard hazards If you have competing risks, you want to look at CIFs instead of survival functions CIF analysis with Cox regression is possible, but difficult stcrreg followed by stcurve is the easier way to go Keep in mind, however, that easier does not mean correct. There are model assumptions to be made for either of the two approaches R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

26 References Coviello, V. and M. Boggess Cumulative incidence estimation in the presence of competing risks. The Stata Journal. 4: Fine, J. and R. Gray A proportional hazards model for the subdistribution of a competing risk. Journal of the American Statistical Association. 94: R. Gutierrez (StataCorp) Competing-risks regression July 15-16, / 26

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