Evaluating Percentiles of Survival

To the Editor:

Findings from observational studies of time to an event of interest (eg, death, incidence, relapse) are commonly presented in terms of hazard ratios or rate ratios. These are relative measures of risk. A patient or health scientist, however, may also be interested in knowing the absolute risk of having the event of interest after a given time point. Survival percentiles provide this information. For example, if the 50th percentile of survival, or median survival, is 6 years, then the risk of having the event within 6 years is 50%.

We illustrate the use of Laplace regression to directly model percentiles of survival as a function of the exposure, adjusting for potential confounders.¹ As an example, we examined the association between abdominal obesity (as measured by waist circumference) and mortality from cardiovascular disease (CVD) in a cohort of 13,479 men living in Sweden who were 60–79 years of age at the start of follow-up in 1998. During 11 years of follow-up, we documented 1348 deaths from CVD. Detailed information is provided in the eAppendix (https://links.lww.com/EDE/A602).

Percentiles of survival range between 1% and 99%, but in any given study, the number and time of events determines which percentiles one can estimate. In our data, 10% of the cohort died of CVD during 11 years. Here, we focus on the 10th percentile of survival time expressed in months across waist circumference levels. To facilitate a comparison with previous studies, we also present the survival analysis using a Cox proportional hazard model (eAppendix, https://links.lww.com/EDE/A602).

To flexibly model waist circumference as a quantitative variable, we used restricted cubic splines with 3 knots at 85, 96, and 108 cm of waist circumference. The Figure shows the change of both the 10th survival percentile and the mortality rate over the range of waist circumference. The solid line in panel (A) indicates the decreasing time interval by which 10% of men die as waist circumference increases. Overall, 90% of men with 94 cm waist circumference lived longer than 133 months (11 years). While every 10-cm increment in waist circumference reduced time to CVD death by 12 months (95% CI = 2–22 months), which is equivalent to a 20% higher mortality rate (9%–33%). Below the reference value of 94 cm, the change in mortality was slight (Figure).

Figure: A, Tenth percentile differences (differences in time by which 10% of the cohort has died) and (B) rate ratios (ratios of hazards of death), from CVD as a function of waist circumference. Data were fitted using (A) Laplace regression and (B) Cox regression. The estimates were adjusted for baseline age, body mass index, total physical activity, alcohol consumption, smoking status, educational level, marital status, and self-reported health. Dashed lines represent 95% confidence limits. The reference value of waist circumference is 94 cm. Tick marks (A) represent waist circumference of men who died of CVD, and the histogram shows the distribution of waist circumference in the cohort.

We next investigated this dose-response relationship among never-smokers. A flexible restricted cubic spline model suggests a constant change in the survival percentile throughout the range of waist circumference (eAppendix, https://links.lww.com/EDE/A602). The time by which 10% of never-smokers died of CVD decreased by 10 months (1–18), for every 10-cm increment in waist circumference.

Survival at percentiles lower than the 10th provided similar results. Any attempt to model percentiles higher than the 10th, for example, the median survival (50th percentile), would require extrapolating beyond the observed range of survival time.

In the absence of covariates, Laplace regression provides maximum likelihood estimates of survival percentiles similar to the well-known nonparametric Kaplan-Meier method. Unlike Kaplan-Meier, however, Laplace regression allows one to model the effects of continuous exposures on survival time while adjusting for confounders and assessing interactions. Laplace regression is different from accelerated failure time models because regression coefficients are expressed directly in terms of survival percentile differences. This approach can model several percentiles simultaneously and test variations in exposure effects across survival percentiles.

Information on survival percentiles may complement the information provided by relative risks (ie, hazard ratios, rate ratios), facilitating the interpretation and communication of results in both a clinical and a public health context.

Nicola Orsini

Unit of Nutritional Epidemiology

Institute of Environmental Medicine

Karolinska Institutet

Stockholm, Sweden

Unit of Biostatistics

Institute of Environmental Medicine

Karolinska Institutet

Stockholm, Sweden

[email protected]

Alicja Wolk

Unit of Nutritional Epidemiology

Institute of Environmental Medicine

Karolinska Institutet

Stockholm, Sweden

Matteo Bottai

Unit of Biostatistics

Institute of Environmental Medicine

Karolinska Institutet

Stockholm, Sweden