Calibrate the interval for the CPB dispersion parameter

An opt-in step that stores, in the fit, the parametric-bootstrap distribution of the signed root of the profile likelihood ratio for alpha. Once it is stored, alpha_confint(), confint.cpb(), implied_ceiling() and summary() report the calibrated interval instead of the first-order one. It is the remedy for floor-support fits (support = "floor"). On King's support, the default, the first-order interval covers close to its level, and the calibration is an optional refinement.

Usage

calibrate_alpha(object, B = 199, cores = 1L, seed = NULL, ...)

# S3 method for class 'cpb'
calibrate_alpha(object, B = 199, cores = 1L, seed = NULL, ...)

# S3 method for class 'hurdle_cpb'
calibrate_alpha(object, B = 199, cores = 1L, seed = NULL, ...)

# Default S3 method
calibrate_alpha(object, B = 199, cores = 1L, seed = NULL, ...)

Arguments

object: A pooled cpb() fit, or a hurdle_cpb() fit whose intensity is a pooled CPB.
B: Number of bootstrap replicates (default 199).
cores: Worker processes for the refits.
seed: Optional seed for the simulated responses; the caller's random number state is restored afterwards, as in stats::simulate().
...: Unused.

Value

object, with the component alpha.cal: a list holding the signed roots r (NA for a replicate that failed to fit), B, B_ok and seed.

Details

The first-order interval cuts the profile log-likelihood qchisq(level, 1) / 2 below its maximum on both sides. On King's support that cut is close to calibrated, because the probabilities change continuously with the parameters. The simulations from the CPB cover nine designs (n from 200 to 1,000, true alpha 0.3, 0.5 and 0.7). There the first-order 95% interval covers between 0.940 and 0.956, 0.947 on average, and more of its misses fall above the upper limit than below the lower one. With B = 39 replicates, the calibrated interval covers 0.948 on average in the same simulations. It is about 12% longer, and its misses are more nearly balanced.

On the floor support the cut is not calibrated. The support 0, ..., floor(lambda / (1 - alpha)) moves with the parameters, the log-ceiling coefficients behave like endpoint parameters, and alpha-hat inherits a bias toward zero from them. Across the same nine designs the first-order 95% interval covers between 0.90 and 0.94, and most of its misses have the true alpha above the upper limit.

The calibration simulates B responses from the fitted model (the fitted rates, alpha-hat, the same truncation and offset), refits each one, and records the signed root r* = sign(alpha* - alpha-hat) sqrt(2 (l* - l*_p(alpha-hat))). The interval is then {alpha : q_lo <= r(alpha) <= q_hi}, with q_lo and q_hi the order statistics of the r* at ranks (B + 1)(1 - level)/2 from each end (the 5th smallest and 5th largest of 199 at the 95% level), so the upper limit sits q_lo^2 / 2 and the lower limit q_hi^2 / 2 below the maximum of the profile. On the floor support, in the same simulations, the calibrated 95% interval covers 0.92 to 0.96, with its misses balanced between the two sides. On the floor support, the first-order interval falls short only when the design has many distinct covariate patterns (the endpoint effect needs many distinct ceilings): with an intercept only, or a covariate taking ten or fewer values, it covers at or above its level, and the calibration, which stays close to the nominal level there too (0.95 with a binary covariate), is not needed.

The responses are all drawn first, on the calling process, and the refits use no random numbers, so the result depends on seed but not on cores. With B = 199 the two cuts carry simulation error (a standard deviation of about 0.2 in q_lo and q_hi), which moves a limit by a fraction of its distance from the estimate; B = 999 reduces it and is needed for a 99% interval. Each replicate costs about a third of the original fit.

The interval is model-based. It assumes the CPB and independent observations, and it is not cluster-robust: when the fit was given cluster, a message says so. Under misspecification alpha has no fixed target (on data with a soft upper tail its estimate rises with the sample size), and no interval computed under the fitted CPB repairs that.

Calibration is refused, with the reason, where it has nothing to work with: a fit whose ceiling reaches max.support (the guard, not the data, bounds alpha), weights that are not whole numbers, and a fit saved by version 0.1.0. Whole-number weights are treated as frequency weights: a row of weight w contributes w independent simulated responses.

Examples

# \donttest{
set.seed(7); x <- rnorm(100)
d <- data.frame(y = rcpb(100, exp(1.5 + 0.4 * x), 0.5, truncated = TRUE), x = x)
fit <- cpb(y ~ x, d, se = "none")
alpha_confint(fit)                       # first-order
#>     lower     upper 
#> 0.3866204 0.6670207 
#> attr(,"alpha")
#>           
#> 0.4955812 
#> attr(,"method")
#> [1] "first-order profile likelihood"
#> attr(,"boundary")
#> [1] FALSE
## 19 replicates keep the example short and can serve a 90% interval; use 199 or more
fit <- calibrate_alpha(fit, B = 19, seed = 1)
alpha_confint(fit, level = 0.90)         # calibrated
#>     lower     upper 
#> 0.4185265 0.6310002 
#> attr(,"alpha")
#>           
#> 0.4955812 
#> attr(,"method")
#> [1] "profile likelihood, signed root calibrated by parametric bootstrap (19 of 19 replicates)"
#> attr(,"boundary")
#> [1] FALSE
# }