The interval inverts the likelihood-ratio test of alpha, re-maximizing the
coefficients at each value. By default the cut is the chi-square one (a
first-order interval); it covers about 0.90 to 0.94 in simulations from the
CPB, with nearly all misses on the upper side, because the estimate of
alpha is biased toward zero (calibrate_alpha() explains the mechanism).
A fit that went through calibrate_alpha() gets the interval calibrated by
parametric bootstrap instead. Either interval is model-based: it assumes the
CPB and independent observations.
Value
A length-2 numeric vector (lower, upper) with attributes alpha (the
point estimate), method (first-order or calibrated) and boundary (TRUE
when the profile has not fallen to the cut by alpha = 0.005, so that the
lower limit is the parameter bound).
Examples
set.seed(7); x <- rnorm(200)
N <- pmax(round(exp(1.5 + 0.4 * x) / 0.5), 1); y <- rbinom(200, N, 0.5)
fit <- cpb(y ~ x, data.frame(y = y, x = x)[y > 0, ], se = "none")
alpha_confint(fit)
#> lower upper
#> 0.4359848 0.6107149
#> attr(,"alpha")
#>
#> 0.4815937
#> attr(,"method")
#> [1] "first-order profile likelihood"
#> attr(,"boundary")
#> [1] FALSE