# Bayesian inference MCMC model for interval censored data

I have interval censored data for incubation period and I suppose that the exact time Ti included in each interval of the data [L;U] follows a lognormal distribution (mu,sigma2). I would like to use a bayesian approach MCMC model to estimate first mu and sigma2 and then the different Ti for the different patients. How I could do that ?

Thank you for your help

## 1 Answer

Since you can write down the survival function (or CDF) of the log-normal distribution, then just have to specify this into the likelihood function. The contribution of the ith censored observation to the likelihood function is:

$$p_i = \Phi\left(\frac{\log(U_i)-\mu}{\sigma}\right) - \Phi\left(\frac{\log(L_i)-\mu}{\sigma}\right).$$

Once you specify this, just run your MCMC sampler to obtain a posterior sample. In this case, a Metropolis-Hastings should work well.

Using the posterior samples you can approximate the predictive survival function as follows:

$$PS(t) = \frac{1}{N} \sum_j \left[1-\Phi\left(\frac{\log(t)-\mu_j}{\sigma_j}\right)\right],$$

where $(\mu_j,\sigma_j)$ represent the $N$ posterior samples of these parameters.

The point values $T_i$ cannot be estimated, but you don't need to do this since you can get inference about the parameters without them.

• Thank you very much ! I will get the parameters with that, but in fact I would like also to estimate Ti as I would like to do a logistic regression afterthat with the incubation period (Ti) as covariate. Do you have any ideas ? – Georges Harendt Sep 24 '14 at 14:06
• @Victor I am not sure I understand your second question (perhaps you should post it as a separate question), but my impression is that you could use some sort of data augmentation. – Kant Sep 24 '14 at 14:21
• Thank Karl for all these information. I will probably post this second question as a separate question ! – Georges Harendt Sep 25 '14 at 2:12