# A Case Study Using the Beta-Danish Distribution

# Introduction

This vignette demonstrates a typical survival analysis workflow using the BetaDanish package.

    library(BetaDanish)
    library(survival)
    #> Warning: package 'survival' was built under R version 4.5.3
    #>
    #> Attaching package: 'survival'
    #> The following objects are masked from 'package:BetaDanish':
    #>
    #>     leukemia, transplant
    data('remission', package = 'BetaDanish')
    head(remission)
    #>   time status
    #> 1 0.08      1
    #> 2 2.09      1
    #> 3 3.48      1
    #> 4 4.87      1
    #> 5 6.94      1
    #> 6 8.66      1

# Fitting the Beta-Danish model

    fit <- fit_betadanish(Surv(time, status) ~ 1, data = remission, n_starts = 1)
    summary(fit)
    #>
    #> Call:
    #> fit_betadanish(formula = Surv(time, status) ~ 1, data = remission,
    #>     n_starts = 1)
    #>
    #> Beta-Danish Distribution Fit
    #> Model: Full 4-Parameter Model
    #>
    #>    Estimate Std. Error Lower 95% Upper 95% z value Pr(>|z|)
    #> a  0.686576   0.795132 -0.871883  2.245035  0.8635 0.387877
    #> b  4.078205   1.478893  1.179575  6.976834  2.7576 0.005823 **
    #> c  2.196505   2.576019 -2.852492  7.245502  0.8527 0.393840
    #> k  0.082973   0.076769 -0.067495  0.233441  1.0808 0.279782
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #> ---
    #> Log-Likelihood: -409.9137
    #> AIC: 827.8274  | BIC: 839.2356

# Three-parameter submodel

    fit_sub <- fit_betadanish(Surv(time, status) ~ 1, data = remission, submodel = TRUE, n_starts = 1)
    compare_models(fit, fit_sub)
    #> Likelihood Ratio Test (a = 1 vs a != 1)
    #>
    #>                  Model    LogLik      Chisq Df Pr(>Chisq)
    #> 1   Submodel (3-param) -409.9541         NA NA         NA
    #> 2 Full Model (4-param) -409.9137 0.08081129  1   0.776201

# Diagnostic plots

    plot(fit, type = 'survival')

    plot(fit, type = 'hazard')

# Interpretation

The fitted model can be used to estimate survival probabilities, hazard behavior, and overall model fit. Users should compare the Beta-Danish model with alternative lifetime distributions and inspect diagnostic plots before drawing final conclusions.