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.
APPENDIX: TECHNICAL RESULTS
A.1 Derivation of (3)
Let r(a, b) = E {S(Ta)exp (bτ)} for constants a and b, where Ta has cdf Fa(t) = 1 − S(t)exp (aτ). The probability integral transformation (Casella and Berger Citation2002, p. 54) of Ta given by U = Fa(Ta) = 1 − S(Ta)exp (aτ) has a uniform distribution on (0, 1). Also, (1 − U)exp {τ(b − a)} = S(Ta)exp (bτ), where 1 − U is uniform on (0, 1) and so
If we define A in (2) with G(t) = Fj(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 Fβ(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