Extending the Peters-Belson approach for assessing disparities to right censored time-to-event outcomes

Abstract

The Peters-Belson (PB) method was developed for quantifying and testing disparities between groups in an outcome by using linear regression to compute group-specific observed and expected outcomes. It has since been extended to generalized linear models for binary and other outcomes and to analyses with probability-based sample weighting. In this work, we extend the PB approach to right-censored survival analysis, including stratification if needed. The extension uses the theory and methods of expected survival on the basis of Cox regression in a reference population. Within the PB framework, among the groups to be compared, one group is chosen as the reference group, and outcomes in that group are modeled as a function of available predictors. By using this fitted model's estimated parameters, and the predictor values for a comparator group, the comparator group's expected outcomes are then calculated and compared, formally with testing and informally with graphics, with their observed outcomes. We derive the extension, show how we applied it in a study of incontinence in nursing home elderly, and discuss issues in implementing it. We used the 'survival' package in the R system to do computations.

Keywords: Cox proportional hazards regression; Ederer estimate; log rank test; population survival.

Figure 1

Observed and expected time without Dual Incontinence (DI) among Black non-Hispanic nursing home residents age 65+ years and free of DI at admission. (Solid black line and dotted black lines: Actual survival curve [Kaplan-Meier] for Blacks with 95% pointwise confidence limits. Dashed line: Expected survival curve for Blacks based on the Cox regression in Whites.)

Figure 2

Observed and expected time without Dual Incontinence (DI) among Hispanic nursing home residents age 65+ years and free of DI at admission. (Solid black line and dotted black lines: Actual survival curve [Kaplan-Meier] for Hispanics with 95% pointwise confidence limits. Dashed line: Expected survival curve for Hispanics based on the Cox regression in Whites.)

Figure 3

Observed and expected time without Dual Incontinence (DI) among Asian nursing home residents age 65+ years and free of DI at admission. (Solid black line and dotted black lines: Actual survival curve [Kaplan-Meier] for Asians with 95% pointwise confidence limits. Dashed line: Expected survival curve for Asians based on the Cox regression in Whites.)

Figure 4

Observed and expected time without Dual Incontinence (DI) among Black non-Hispanic (a,d), Hispanic (b,e), and Asian (c,f) nursing home residents age 65+ years and free of DI at admission. (Solid black line and dotted black lines: Actual survival curve [Kaplan-Meier] with 95% pointwise confidence limits. Other lines: Expected survival curve calculated via Ederer’s estimate (red) or Hakulinen’s estimate (blue, green) based on the Cox regression in Whites; see Sections 3.4 and 4.1 for calculation details.)