Period-two hazards after selective depletion

A share s of a cohort is at high risk of reconviction, h a period, and the rest at low risk, l. If a programme changes who survives the first period without reconviction, the period-two hazard among the survivors is a weighted average of h and l with weight the share of high-risk survivors, and it rises with that share. An arm that keeps more of its high-risk people out of reconviction in period one therefore shows the higher period-two hazard even if it has no effect on anyone in period two: a period-by-period hazard ratio is not a causal contrast; a risk difference at a fixed follow-up is. See proof("hzdsel").

Usage

hzdsel(s, h, l, surh, surl)

Arguments

s: The high-risk share at the start, strictly between 0 and 1.
h: The period hazard of the high-risk type: at most 1.
l: The period hazard of the low-risk type: from 0, below h.
surh: Period-one survival of the high-risk type in the control and treated arms: two numbers above 0 and at most 1, the control arm first, or named control and treated in either order (with surl named too). A list of single numbers or a one-row data frame is accepted.
surl: Period-one survival of the low-risk type, the same way; named inputs are matched to surh by name.

Value

A named numeric vector: share0, share1 (the high-risk share among the survivors of each arm), haz0, haz1 (the period-two hazard in each arm) and ratio (haz1 / haz0).

References

Hernan, M. A. (2010). The hazards of hazard ratios. Epidemiology, 21(1), 13-15. doi:10.1097/EDE.0b013e3181c1ea43

Examples

hzdsel(s = 0.5, h = 0.5, l = 0.1, surh = c(0.5, 0.8), surl = c(0.9, 0.9))
#>    share0    share1      haz0      haz1     ratio 
#> 0.3571429 0.4705882 0.2428571 0.2882353 1.1868512