The exact limit of the two-sample Cox estimate on a patient set

For a set of patients of an ahr_population(), the limit $\beta^*$ of the log hazard ratio estimated by a Cox fit of treatment alone on that set (1:1 randomisation, analysis at the population's calendar time with each patient's uniform entry window, administrative censoring only): the root of the limiting score (Struthers and Kalbfleisch), with the information, the model-based and robust (Lin-Wei) variances per patient, the event probabilities, the type table and $\omega$ (the H* share of events), and the set's AHR, CDE and its margHR at infinity (no censoring). Computed by quadrature; nothing is drawn.

Usage

ahr_limit(
  pop,
  set,
  b = NULL,
  h = NULL,
  psi = NULL,
  horizon = NULL,
  variance = TRUE,
  b_R = NULL,
  h_R = NULL
)

Arguments

pop: An ahr_population().
set: A set name: "all", "nonregion", "region", "region_Gstar" (region outside H*), "Hstar", "Hstar_region", "Hstar_nonregion"; or a logical vector of length pop$N.
b, h: Hazard ratios outside and inside H*: the treated log hazard is $\theta_0 + \log b + \log(h/b) 1\{H^*\}$. Both NULL (and psi NULL): the DGM's own per-patient log hazard ratios (loghr_po).
psi: NULL, or a per-patient log hazard ratio vector of length pop$N (overrides b, h, b_R, h_R).
horizon: Calendar time of the analysis; NULL (default) is pop$x; Inf gives the uncensored limit.
variance: Logical: compute the robust variance (default TRUE).
b_R, h_R: Hazard ratios inside the region, outside and inside H*: with b and h they give the four-cell treated log hazard $\theta_0 + \log b$ (outside H*, outside the region), $\log h$ (H*, outside), $\log b_R$ (outside H*, region), $\log h_R$ (H*, region). NULL (default) means b_R = b and h_R = h, the two-cell model. Used only with b and h. When they differ from b, h, a type group whose cells carry different hazard ratios has psi = NA in types.

Value

A list: beta (the limit) and hr (margHR at the horizon), i1 (information per patient), var_model, var_robust, var_robust_fixed (allocation fixed at n/2) (variances per patient, so divide by the set's size), sandwich, d1, d0, d (event probabilities by arm and at 1:1, i.e. expected events per patient), types (per type group: patients, log-HR, events, event share, own and pooled imbalance moments), omega, ahr, cde, band_share, omega0, marghr_inf, n_pop, share ($n_S/N$), horizon, and the node arrays and grid used by ahr_design().

Details

A standard set at given (b, h) with $0.2 \le b, h \le 30$ uses the population's cached grid and Laplace sums (the route of ahr_design()); any other call (a logical set, psi, or the DGM's own effects) builds the set's own grid.

Limits

Weibull AFT DGMs from inject_mrct_structure() only; entry uniform on $(a_i, a_i + R)$ for each entry group; administrative censoring only (the random censoring model of the DGM, e.g. cens_adjust, is not in the layer); correlations exact for nested sets. Each distinct delay is one quadrature group with its own breaks, so a delay vector with many distinct values is slow. Derivations: the technical report (docs/manuscript/techreport/analytic-layer.qmd, Chapter 6, of the mrct-consistency repository) and the theory notes (docs/theory/AHR_mrct_dgm.qmd, docs/theory/AHR_complete.qmd).

References

Struthers, C. A. and Kalbfleisch, J. D. (1986). Misspecified proportional hazards models. Biometrika 73, 363-369.

Examples

if (FALSE) { # \dontrun{
pop <- ahr_population(dgm, 40, 11)
e <- ahr_limit(pop, "region", b = 0.55, h = 2)
c(margHR40 = e$hr, AHR = e$ahr, CDE = e$cde, margHR_inf = e$marghr_inf)
} # }