Simultaneous bands for event time percentiles in Cox models with an extension to recurrent events

Survival analysis, or time-to-event analysis, is a subject within biostatistics that analyzes time until an event of interest occurs. A main method in survival analysis is Cox Proportional Hazards Regression (Cox PH). Cox PH models are used to describe a relationship between variables and the hazard of having an event. Cox PH models can then be used to characterize the relationship between the variables and time-to-event percentiles. Describing survival time quantiles as a function of co-variates can give healthcare practitioners a helpful application in precision medicine.

The goal in precision medicine is to treat each patient, or a subgroup of patients, based on their individual characteristics instead of treating everyone in the same manner. This could be done through the use of genetic testing or by identifying subgroups of patients who would benefit from certain treatments. The median overall survival time, which quantifies the amount of time after which 50% of the patients have had the event or clinical endpoint, is an overall assessment metric that shows little regard to key patient characteristics. A way to apply survival analysis to precision medicine is to assess the median survival time across a range of values for an important coefficient and add confidence bands to have a range of values that likely contain the true median.

The existing simultaneous band construction from Burr and Doss (1993) works only under settings that are unlikely in a real data scenario, such as a constant hazard or a coefficient that does not does not affect the event hazard in a clinically meaningful way. This dissertation proposes a simultaneous band that addresses the problems of the existing method and improves the overall coverage probabilities of confidence bands. The methods introduced, the Variance Stabilizing Transformation Band (VST) and the Corrected Variance Stabilizing Transformation Band (CVST), provide better coverage than that which is currently in the literature. These two bands are created using the Bartlett Approximation which allowed for a transformation of the asymptotic variance that stabilizes the variance to be constant. The CVST band also estimates a correction factor to account for the variability not attributed to the estimated percentile process. At 0% censoring, the CVST maintained near 95% coverage whereas the Burr and Doss method had coverage proportions as low as 70% in the simulation study.

In survival analysis, it is also possible for a subject to have more than one event; analyzing these type of data is called recurrent event analysis. For example, a patient with kidney disease can be hospitalized multiple times for complications related to their kidney disease. New asymptotic results were developed with recurrent events, which allowed for the creation of confidence bands. These bands are slightly conservative with coverage probabilities as high as 99% observed coverage in simulations for the 95% band.

These methods were used to create confidence bands for median time between hospital visits as a function of age for patients with end-stage renal disease (ESRD). The bands showed that the length of time between hospital admissions is shorter for young patients with ESRD than it is for older patients.