Two Useful Reformulations of the Hazard Ratio

Abstract

The hazard ratio is a standard summary for comparing survival curves yet hazard ratios are often difficult for scientists and clinicians to interpret. Insight into the interpretation of hazard ratios is obtained by relating hazard ratios to the maximum difference and an average difference between survival probabilities. These reformulations of the hazard ratio are useful in classroom discussions of survival analysis and when discussing analyses with scientists and clinicians. Large-sample distribution theory is provided for these reformulations of the hazard ratio. Two examples are used to illustrate the ideas.

• Cox model
• Proportional hazards model
• Survival analysis

APPENDIX: TECHNICAL RESULTS

A.1 Derivation of (3)

Let r(a, b) = E {S(T_a)^exp (bτ)} for constants a and b, where T_a has cdf F_a(t) = 1 − S(t)^exp (aτ). The probability integral transformation (Casella and Berger 2002, p. 54) of T_a given by U = F_a(T_a) = 1 − S(T_a)^exp (aτ) has a uniform distribution on (0, 1). Also, (1 − U)^{exp {τ(b − a)}} = S(T_a)^exp (bτ), where 1 − U is uniform on (0, 1) and so If we define A in (2) with G(t) = F_j(t), where j = 0 or 1, then If j = 0, then A = 1/{1 + exp (τ)} − 1/2 = 1/2 − 1/{1 + exp ( − τ)} whereas if j = 1 then A = 1/2 − 1/{1 + exp ( − τ)}. As the same result is obtained for F₀(t) and F₁(t), we can conclude that (3) also applies for F_β(t) = βF₁(t) + (1 − β)F₀(t) for any 0 ⩽ β ⩽ 1.

A.2 Derivation of D′ τ

For − ∞ < τ < ∞, define w(τ) = τ/{1 − exp (τ)}. Note that this function is everywhere differentiable with w′(τ) = {1 + w(τ)exp (τ)}/{1 − exp (τ)}. A direct calculation shows that D(τ) = exp {w(τ)} − exp {w(τ)exp (τ)}. Using the chain rule