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I am reading a paper on cost-effective analysis and trying to replicate their results. (paper: https://ascopubs.org/doi/full/10.1200/GO.20.00288)

The probabilistic model in the paper assumes that the transition probabilities and utility weights follow the beta distribution. However, the only values given for the transition probability and utility are the base values. The lower limit and upper limit for the parameters are also given, but they can be solved by just multiplying the base values by 0.8 and 1.2, respectively. Is there any way to compute the shape parameters $\alpha$ and $\beta$ of the beta distribution Beta$(\alpha,\beta)$ using just the base values and the lower and upper limits?

What I have researched so far:

According to this book (https://www.herc.ox.ac.uk/downloads/decision-modelling-for-health-economic-evaluation), if the data are represented by a number of events $r$ observed in a sample $n$, we can set $\alpha=r$ and $\beta=n-r.$ The same is said in the appendix (p. 306) of an article (http://www.med.mcgill.ca/epidemiology/courses/EPIB654/Summer2010/EF/example%20PPI.pdf), but again, the only given value on the paper is the base value.

Going back to the paper, they assumed that there are 1000 patients for the study. Using this and from the research, can I set $n$ to be 1000, and $r$ to be 1000*transition probability? Or should I create a health states distribution and set $n$ and $r$ depending on the treatment cycle I am in?

shyguy
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1 Answers1

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The short answer is no. The lower and upper limits of the beta distribution are 0 and 1, so it is not clear what setting, e.g., the base rate to 0.2 and the limits to 0.16 and 0.25 would mean.

For the sake of argument let $p=\frac{\alpha}{\alpha+\beta}$ be your base rate. If you set $p=0.2$ and $\beta = \frac{\alpha}{p} -\alpha$ you could search for a value of $\alpha$ that makes your variable fall between 0.16 and 0.25 with some (high) probability.

However, my guess would be that the stated limits have nothing to do with the probabilistic analysis. They have probably just used $\alpha = 2, 20, 200$ $\beta=8,80,800$ to get the right mean.

Edit: Apparently, there is a non-standard beta distribution. The limits could refer to this, but I doubt it. https://stats.stackexchange.com/a/186467/264076

Still, there is no way to know what parameters they used.

Jonathan
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