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). On King's support it covers between 0.940 and 0.956 in
simulations from the CPB. On the floor support it covers about 0.90 to 0.94,
with most misses on the upper side, because the estimate of alpha is biased
toward zero there (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 does not fall to the cut on the lower side: on King's
support, whose trace runs down to alpha = 1e-6, the lower limit is then
0; on the floor support, whose trace stops at alpha = 0.005, it is that
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
#> 0.4359631 0.6260090
#> attr(,"alpha")
#>
#> 0.5149297
#> attr(,"method")
#> [1] "first-order profile likelihood"
#> attr(,"boundary")
#> [1] FALSE