Reports the four-parameter fit in terms of the composite \(ac\), which is identified, rather than \(a\) and \(c\) separately, which are not.
Usage
bd_identified_coef(object, level = 0.95)
# S3 method for class 'bd_identified'
print(x, ...)Value
An object of class `"bd_identified"` holding the reparametrised estimates with delta-method standard errors, and the condition numbers of the covariance matrix before and after.
Details
Near the lower tail \(a\) and \(c\) enter the density almost entirely through their product, so the expected information is close to singular in the direction that separates them. Writing \(p = ac\) and \(r = a/c\), the ratio \(r\) carries almost no information while \(p\) is estimated precisely.
Reporting \((ac, b, k)\) is therefore the honest summary of a four-parameter fit that sits on the flat direction: it has finite standard errors and a far better conditioned covariance matrix, and it says exactly what the data determine. The thesis pipeline records a condition number falling from 2750 to 152 on the remission data, with the log-likelihood unchanged, since this is a reparametrisation and not a different model.
For the ED submodel \(a = 1\), so \(ac = c\) and nothing is gained; the function returns the coefficients unchanged with a note.
Examples
# \donttest{
dat <- simulate_bd_data(200, a = 1.5, b = 3, c = 2, k = 0.5, seed = 4)
fit <- fit_betadanish(survival::Surv(time, status) ~ 1, data = dat,
n_starts = 5)
#> Warning: The fitted correlation between a-hat and c-hat is -0.990, so effectively only the product c*a is identified. Individual estimates of a and c should not be interpreted.
#> Warning: 1 starting point(s) reached a degenerate ridge and were discarded. The reported fit is the best admissible optimum. If this is most of the grid, the four-parameter model is a poor choice for these data.
bd_identified_coef(fit)
#> Identified parametrisation
#> a and c enter mainly through their product; ac is reported.
#>
#> estimate se lower upper
#> ac 3.86780 0.76056 2.63078 5.68648
#> b 2.83810 0.66395 1.79430 4.48911
#> k 0.42442 0.24492 0.13696 1.31522
#>
#> a = 130.58, c = 0.02962, ratio a/c = 4408.6
#> covariance condition number: 2.0997e+10 -> 1300.5
# }