PHLT 8500 Week 10 Survival Model Brief Example

Counting the Years Until Diabetes, Not Only Whether It Came: A Survival Brief for the Private-Well Cohort That Keeps Participants Who Moved Away and Treats Death as a Competing Event

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Walden University

PHLT 8500: Advanced Biostatistics

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Month Day, Year

Counting the Years Until Diabetes, Not Only Whether It Came: A Survival Brief for the Private-Well Cohort That Keeps Participants Who Moved Away and Treats Death as a Competing Event

Clock, Event, Exits

The Week 5 logistic model asked whether participants developed diabetes by the six-year visit and set aside the 192 who missed it. A survival analysis uses everyone. Every one of the 1,612 enrollees has a clock that starts at enrollment and stops at the first of four things: a diabetes diagnosis or qualifying A1c, the event of interest; moving out of the county and leaving the study; death; or the end of follow-up at eight years. Over a median of 7.1 years, 214 participants developed diabetes, 96 moved away, 41 died without diabetes and the rest reached the end of follow-up without an event.

Censored, Not Discarded

A participant who moved away after four years without diabetes is not a missing case. We know she was diabetes-free for four years; we do not know what happened afterward. Survival methods use that partial information by keeping her in the risk set for four years and then removing her, a process called censoring (Clark et al., 2003). The approach assumes that people who leave are, at the time they leave, at similar risk to those who stay, given the covariates. Moving for a job is unlikely to violate this; moving to be nearer a hospital might, and the brief notes it as a limitation.

Curves Before the Model

Kaplan-Meier curves show the cumulative probability of remaining diabetes-free over time. Split by well arsenic at the federal limit, the curves separate gradually from the second year. By eight years, the estimated cumulative incidence is 13.0% among participants with wells below 10 micrograms per liter and 17.6% among those at or above it, with a log-rank test p = .02. The curves do not cross, a first visual sign that the proportional hazards assumption is reasonable.

Hazard Ratios, Read Narrowly

A doubling of well arsenic carries a 12% higher hazard of diabetes at any point during follow-up, among participants still at risk and with the same covariates. A hazard ratio compares instantaneous rates, not cumulative risks, and should not be described as "12% more likely to develop diabetes." For a reader, the clearer statement is the pair of adjusted cumulative incidences: about 13% versus 17% over eight years. With 214 events and 10 parameters, the model has about 21 events per parameter, adequate by the standards examined by Vittinghoff and McCulloch (2007).

Table 1

Cox Proportional Hazards Model for Incident Diabetes (Composite Figures, n = 1,612, 214 events)

Note. Composite figures. Separate models, each with the standard covariate set.

Proportionality Checked

The Cox model assumes the hazard ratio is constant over time. A test based on scaled Schoenfeld residuals (Grambsch & Therneau, 1994) gives p = .48 for arsenic, so there is no evidence that its hazard ratio changes over follow-up. Age shows a mild violation, p = .03, with its hazard ratio declining slightly over time. Refitting with age as the time scale, instead of time since enrollment, handles this and leaves the arsenic hazard ratio at 1.12.

Death as a Competing Event

Forty-one participants died without diabetes, and a person who has died can no longer develop it. Treating deaths as ordinary censoring is appropriate for estimating the cause-specific hazard ratio, which describes the rate of diabetes among those alive and at risk, but it overstates cumulative incidence if read as the probability of diabetes in the presence of death. Austin et al. (2016) explain the difference between the cause-specific and subdistribution approaches. A Fine-Gray model gives a subdistribution hazard ratio of 1.11 per doubling, nearly identical, because deaths were few relative to diabetes events. The brief reports the cause-specific estimate for the etiologic question and the cumulative incidence accounting for death for the public health question.

What Survival Analysis Adds

Compared with the Week 5 logistic model, the survival model uses all participants, including those who left early, uses the timing of each case and separates death from the outcome. Its estimate, a 12% higher hazard per doubling, is close to the Week 5 odds ratio of 1.14, which is reassuring: the association is not an artifact of who completed the six-year visit.

References

Austin, P. C., Lee, D. S., & Fine, J. P. (2016). Introduction to the analysis of survival data in the presence of competing risks. Circulation, 133(6), 601-609. https://doi.org/10.1161/CIRCULATIONAHA.115.017719

Clark, T. G., Bradburn, M. J., Love, S. B., & Altman, D. G. (2003). Survival analysis part I: Basic concepts and first analyses. British Journal of Cancer, 89(2), 232-238. https://doi.org/10.1038/sj.bjc.6601118

Grambsch, P. M., & Therneau, T. M. (1994). Proportional hazards tests and diagnostics based on weighted residuals. Biometrika, 81(3), 515-526. https://doi.org/10.1093/biomet/81.3.515

Vittinghoff, E., & McCulloch, C. E. (2007). Relaxing the rule of ten events per variable in logistic and Cox regression. American Journal of Epidemiology, 165(6), 710-718. https://doi.org/10.1093/aje/kwk052