A Bayesian Semiparametric Temporally-Stratified Proportional Hazards Model with Spatial Frailties

Abstract

Incorporating temporal and spatial variation could potentially enhance information gathered from survival data. This paper proposes a Bayesian semiparametric model for capturing spatio-temporal heterogeneity within the proportional hazards framework. The spatial correlation is introduced in the form of county-level frailties. The temporal effect is introduced by considering the stratification of the proportional hazards model, where the time-dependent hazards are indirectly modeled using a probability model for related probability distributions. With this aim, an autoregressive dependent tailfree process is introduced. The full Kullback-Leibler support of the proposed process is provided. The approach is illustrated using simulated and data from the Surveillance Epidemiology and End Results database of the National Cancer Institute on patients in Iowa diagnosed with breast cancer.

Figure 1

Simulated data: The posterior mean and 95% point–wise credible interval under the partially specified ARDFP with M = 5 are displayed as dashed lines for each strata. The true density in each case is displayed as a solid lines.

Figure 2

SEER data: Censored histogram of raw survival in months for diagnosis ages 55–75, one diagnosed independent cancer, and “local” stage (left), “regional” (middle), or “distant” (right), with fitted predictive density curves from the best fitting ARDTFP model overlaid for years 1989 (panel (a)), 1990 (panel (b)), 1991 (panel (c)) and 1992 (panel (d)), respectively.

Figure 3

SEER data: Censored histogram of raw survival in months for diagnosis ages 55–75, one diagnosed independent cancer, and “local” stage (left), “regional” (middle), or “distant” (right), with fitted predictive density curves from the best fitting ARDTFP model overlaid for years 1993 (panel (a)), 1994 (panel (b)), 1995 (panel (c)) and 1996 (panel (d)), respectively.

Figure 4

SEER data: Censored histogram of raw survival in months for diagnosis ages 55–75, one diagnosed independent cancer, and “local” stage (left), “regional” (middle), or “distant” (right), with fitted predictive density curves from the best fitting ARDTFP model overlaid for years 1997 (panel (a)) and 1998 (panel (b)), respectively.

Figure 5

SEER data: Predictive densities for age 65, breast cancer only, and disease stages “local” (upper left), “regional” (upper right), and “distant” (lower left) across 200 months.

Figure 6

SEER data: Predictive survival curves of age 65, breast cancer only, and disease stages “local” (upper left), “regional” (upper right), and “distant” (lower left) across 200 months.