Build a measurement-error covariance targeting a given reliability

Returns a Sigma_u (via build_sigma_u()) whose surrogate-measurement error variances are scaled so the mean per-surrogate reliability equals reliability. Surrogate reliability is the intraclass-style ratio var_x / (var_x + var_U_j); the relative surrogate qualities in ratios are preserved (so the surrogates stay unequal for the per-surrogate ladder), and a single multiplicative scale is solved by stats::uniroot() to hit the target average. Use it to sweep measurement quality in a simulation, e.g. reliability in c(0.5, 0.65, 0.8).

Usage

sigma_u_for_reliability(reliability, var_x, ratios = c(2, 2.5, 3), rho = 0.4)

Arguments

reliability

Target mean per-surrogate reliability in (0, 1).

var_x

Variance of the latent exposure X (e.g. x_sd^2 passed to generate_aft_data()).

ratios

Relative marginal error variances across the surrogates; their scale is solved for, only their ratios matter. Default c(2.0, 2.5, 3.0) matches build_sigma_u().

rho

Common pairwise error correlation, passed to build_sigma_u().

Value

A J x J covariance matrix suitable for the Sigma_u argument of generate_aft_data().

Examples

S <- sigma_u_for_reliability(0.65, var_x = 75^2)
# mean per-surrogate reliability is ~0.65:
mean(75^2 / (75^2 + diag(S)))
#> [1] 0.65