Survival times, in days, of 72 guinea pigs injected with virulent tubercle bacilli, taken from the principal regimen of the study of Bjerkedal (1960). A complete sample: every animal was observed to death, so `status` is 1 throughout.
Format
A data frame with 72 rows and 2 columns:
- time
Survival time in days
- status
Event indicator, 1 for all observations (no censoring)
Source
Bjerkedal, T. (1960). Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli. *American Journal of Epidemiology*, 72(1), 130-148. doi:10.1093/oxfordjournals.aje.a120129
Details
These data are the reference case for a well-identified four-parameter fit. The scaled total-time-on-test transform is unimodal and the upper tail is determinate rather than heavy, and the maximum-likelihood fit of the full Beta-Danish model attains a finite interior optimum: every estimate lies strictly inside the parameter space, with \(\hat b = 3.64\) (Wald standard error 1.20), so that \((\hat b - 1)/\mathrm{SE} = 2.20\) places \(b\) more than two standard errors clear of the \(b = 1\) identifiability ridge.
By contrast, on `remission` and `carbon_fibres` the four-parameter fit sits on the flat \((a, c)\) direction of the likelihood with \(\hat a < 1\) and is only weakly identified. Use this dataset when you want to see the parent model behaving well; see the Identifiability section of [fit_betadanish()] for what to watch for elsewhere.
Examples
data(guinea_pig)
summary(guinea_pig$time)
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 12.00 54.75 70.00 99.82 112.75 376.00
# \donttest{
fit_full <- fit_betadanish(survival::Surv(time, status) ~ 1,
data = guinea_pig)
#> Warning: The times look recorded on a grid of 1. The point-density likelihood treats them as exact, which understates the standard errors. Consider grouped = TRUE.
#> Warning: 2 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.
fit_sub <- fit_betadanish(survival::Surv(time, status) ~ 1,
data = guinea_pig, submodel = TRUE)
#> Warning: The times look recorded on a grid of 1. The point-density likelihood treats them as exact, which understates the standard errors. Consider grouped = TRUE.
compare_models(fit_full, fit_sub)
#> Likelihood Ratio Test (a = 1 vs a != 1)
#>
#> Model LogLik Chisq Df Pr(>Chisq)
#> 1 Submodel (3-param) -390.0731 NA NA NA
#> 2 Full Model (4-param) -390.0592 0.0277516 1 0.8676942
# }