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, 16 (2), 196-214

Proportional Hazards and Threshold Regression: Their Theoretical and Practical Connections

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Proportional Hazards and Threshold Regression: Their Theoretical and Practical Connections

Mei-Ling Ting Lee et al. Lifetime Data Anal.

Abstract

Proportional hazards (PH) regression is a standard methodology for analyzing survival and time-to-event data. The proportional hazards assumption of PH regression, however, is not always appropriate. In addition, PH regression focuses mainly on hazard ratios and thus does not offer many insights into underlying determinants of survival. These limitations have led statistical researchers to explore alternative methodologies. Threshold regression (TR) is one of these alternative methodologies (see Lee and Whitmore, Stat Sci 21:501-513, 2006, for a review). The connection between PH regression and TR has been examined in previous published work but the investigations have been limited in scope. In this article, we study the connections between these two regression methodologies in greater depth and show that PH regression is, for most purposes, a special case of TR. We show two methods of construction by which TR models can yield PH functions for survival times, one based on altering the TR time scale and the other based on varying the TR boundary. We discuss how to estimate the TR time scale and boundary, with or without the PH assumption. A case demonstration is used to highlight the greater understanding of scientific foundations that TR can offer in comparison to PH regression. Finally, we discuss the potential benefits of positioning PH regression within the first-hitting-time context of TR regression.

Figures

Fig. 1
Fig. 1
A TR demonstration for time to infection in kidney dialysis using two methods of catheter placement. The TR model is the first hitting time of the zero level in a Wiener diffusion process. Process parameters are the natural logarithm of the initial health level ln y0 and the process mean μ. Panel (a) shows Kaplan-Meier plots and fitted survival curves for the TR model. Panel (b) shows TR regression output from a public software. Panel (c) shows the hazard functions for the two methods and hazard ratios at two time points
Fig. 2
Fig. 2
Constructing boundaries in an FHT context that generate proportional hazard functions
Fig. 3
Fig. 3
A boundary crossing for a gamma process which illustrates how the sample path overshoots the boundary at the first hitting time S = s
Fig. 4
Fig. 4
Three nested boundaries in Brownian motion that generate proportional hazard functions for three levels of covariate z
Fig. 5
Fig. 5
Proportional hazard functions generated by Brownian motion reaching the three boundaries in Fig. 4
Fig. 6
Fig. 6
Stylized description of two scenarios for a simulated clinical trial with two arms. Scenario 1 has y0 twice as large on the treatment arm as on the control arm (2 versus 1), with a common value of μ (equal to −1). Scenario 2 has a value for μ that is half as large on the treatment arm as on the control arm (−0.5 versus −1) with a common value of y0 (equal to 1)
Fig. 7
Fig. 7
Plots of the true inverse Gaussian hazard functions for the two study arms (the lefthand panel for each scenario) as well as plots of the log-hazard ratio (the righthand panel for each scenario)

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