How to transfer distribution of probability parameter from 5 years into 1 year

Asked: 29 Feb 2020 | 828

Dear Dr and Prof, I'm trying to find probability parameters for my model of chronic hepatitis B. However, most study I found come from observational studies, with different duration of follow-up . In my model, I need the probability and the parameters for beta distribution (alpha, beta). And I don't know how to transfer information in the literature to be used in my model. I would like to pick up an example to illustrate. I want to estimate the annual probability of transiting from Immune tolerant (IT) state into immune clearance (IC) in Chronic Hepatitis B disease. I looked at literature and found some observational studies provide information about this, for example: during a median follow-up of 8.2 years, 20 out of 102 patients transfered from IT to IC. So the probability during 8.2 years was 0.1961 and this probability followed the beta distribution, with alpha=20, beta=102-20=82. Base on those information, I can calculate the annual probability of transiting form IT to IC is 0.0262. But how about the distribution parameters (alpha, beta) for 1 year? I faced the similar situation a lot when I try to find the transition probabilities. So, I really hope that you could guide me what is the proper way to do this. I'm looking for hearing from you and thank you so much for your help.

Modelling

Expert Replies:

Yot Teerawattananon

Senior Researcher | Thailand | Replied: 16 Mar 2020 at 14:24

It's an interesting question and replies from Mark and Alec. If I may, I would like to propose another approach which may or may not be similar to what Mark suggested. I would suggest a very simple solution as follow:
i) annual rate of IT to IC = -(LN(1-0.196))/8.2 = 0.027
ii) annual prob of IT to IC = 1-EXP(-0.027) = 0.026
iii) making a simple markov with 102 samples at time 0, then I can make the following prediction of samples with IT and IC stage
IT IC
0 102.000 0.000
1 99.321 2.679
2 96.712 5.288
3 94.172 7.828
4 91.699 10.301
5 89.290 12.710
6 86.945 15.055
7 84.661 17.339
8 82.438 19.562

From this table, you can have alpha and beta for beta distribution based on a big assumption of constant rate of progression.

The weak point of this data is that you may need to take into account of mortality in the cohort.

Alec Morton

Professor | United Kingdom | Replied: 10 Mar 2020 at 00:23

Dear Dr Ha
I am sorry for the delay in replying. I interpreted your question a bit differently from Mark. My interpretation was as follows: you have survival data based on a 8.2 year cutoff and calculate mean 8.2 year survival and the beta hyperdistributions based on the absolute number of deaths and survivors. You then takes the 8.2th root of mean survival to get the annual rate. But your question is what is the hyperdistribution of that annual rate?
I consulted some friends who are mathematical statisticians and they tell me that the hyperdistribution you should use is a generalised beta distribution and suggest the following paper
Mcdonald, J. B. and Y. X. Xu (1995). "A Generalization of the Beta-Distribution with Applications." Journal of Econometrics 66(1-2): 133-152.
Hoping this is helpful,
Alec

Mark Jit

Professor | United Kingdom | Replied: 02 Mar 2020 at 15:41

Dear Dr. Ha,

I assume you are using a continuous time Markov model. There is no easy way to do this since you are converting an 8.2 year cumulative incidence into an instantaneous flow and then back into a 1 year cumulative incidence. The best thing is probably to fit your transition probability between IT to IC using a likelihood-based method. You can find a simple explanation of it here, or in many textbooks on statistical inference:
https://towardsdatascience.com/probability-concepts-explained-maximum-likelihood-estimation-c7b4342fdbb1

I think you can use a binomial likelihood since the underlying data you are fitting to is 20 out of 102 patients transferred by 8.2 years.

Mark