# Fitting Weibull PH model to Weibull mixture cure model

I generated a mixture cure model with Weibull distributed survival times (Weibull mixture cure model) . This is my baseline survival function conditional on the Bernoulli distributed covariate $X_i$ : $S_0(t_i \vert x_i) = exp (-(e^{\gamma_0 + \gamma_1x_i}) t^{\alpha}$) where $e^{\gamma_0 + \gamma_1x_i}$ is the scale parameter and $\alpha$ is the shape parameter.

I generated the data with about 30% censoring, P(x=0) = 0.2 (probability of being cured when x = 0) and P(x=1)=0.3 (probability of being cured when x = 1), $\alpha = 1.5$, $\gamma_0 = 0.5$ and $\gamma_1 = 0.0$.

Aim:

After generating the data, my aim is fit the true model (the Weibull mixture cure model which I did using nlm function in R to minimize the negative log likelihood), the Cox PH model and the Weibull PH model to the data. I recorded p-values under $\gamma_1 = 0.0$ and did this 1000 times to find the empirical Type I error (to validate the Wald test statistic).

Null hypothesis : $\gamma_1 = 0$ Alternative hypothesis : $\gamma_1 = 1$

It worked fine for the true model and the estimates, the empirical Type I error is 0.040 which is within 95% confidence interval for 0.05 significance level (0.0365, 0.0635).

It did not work fine for the Cox PH model and I am assuming that the Cox PH assumptions were violated.

I was however expecting the Weibull PH model to somehow have empirical Type I error within the 95% significance level for me to compare the estimates with the true model. Under $\gamma_1 = 0.0$, I received 0.915 of empirical type I error for the Weibull PH model (way too high) . Under the $\gamma_1 = 1.0$, I received empirical power which is around 0.032. Is this expected? At the end of the day, I want to see if Weibull PH would still be reasonable model when it ignores cure proportions.

I tried to find literature that specifically compares mixture cure model with semiparametric/parametric PH models through simulations but could find none.

Has anyone done such simulations before to compare two survival models please enlighten me?

Thank you so much.