A new treatment for a disease is being tested, to see...

A new treatment for a disease is being tested, to see whether it is better than the standard treatment. The existing treatment is effective on $50 \%$ of patients. It is believed initially that there is a $2 / 3$ chance that the new treatment is effective on $60 \%$ of patients, and a $1 / 3$ chance that the new treatment is effective on $50 \%$ of patients. In a pilot study, the new treatment is given to 20 random patients, and is effective for 15 of them. (a) Given this information, what is the probability that the new treatment is better than the standard treatment? (b) A second study is done later, giving the new treatment to 20 new random patients. Given the results of the first study, what is the PMF for how many of the new patients the new treatment is effective on? (Letting $p$ be the answer to (a), your answer can be left in terms of $p .)$

A new treatment for a disease is being tested, to see whether it is better than the standard treatment. The existing treatment is effective on $50 \%$ of patients. It is believed initially that there is a $2 / 3$ chance that the new treatment is effective on $60 \%$ of patients, and a $1 / 3$ chance that the new treatment is effective on $50 \%$ of patients. In a pilot study, the new treatment is given to 20 random patients, and is effective for 15 of them.
(a) Given this information, what is the probability that the new treatment is better than the standard treatment?
(b) A second study is done later, giving the new treatment to 20 new random patients. Given the results of the first study, what is the PMF for how many of the new patients the new treatment is effective on? (Letting $p$ be the answer to (a), your answer can be left in terms of $p .)$

Key Concepts

Predictive Distribution

The predictive distribution is used to forecast future observations by averaging over the uncertainty in the parameter estimates. In the context of a new study, it combines the posterior distribution with a binomial model to provide probabilities for the number of successes in additional trials.

Binomial Distribution

This distribution describes the number of successes in a fixed number of independent Bernoulli trials (e.g., patient outcomes). It is frequently used to model data in clinical studies where each treatment response is a success or failure, and it plays a critical role in defining the likelihood function in such Bayesian analyses.

Posterior Distribution

The posterior distribution is obtained by combining the prior distribution with the likelihood function using Bayes’ theorem. It represents the updated beliefs about the parameter values after incorporating the observed data, such as the probability that the new treatment has a higher effectiveness rate than the standard treatment.

Prior Distribution

The prior distribution represents one’s initial beliefs about the parameters before any data is observed. In a testing scenario for a new treatment, it lays out the probabilities for different levels of effectiveness, which is then updated as data is collected.

Bayesian Inference

This concept involves updating prior beliefs with new data using Bayes' theorem. In problems like this, one starts with a prior probability distribution over possible hypotheses or parameters (such as different effectiveness rates), and then incorporates evidence from sample data to form an updated (posterior) distribution, which reflects what is known after seeing the data.

Likelihood Function

The likelihood function measures the probability of the observed data under each hypothetical parameter value. For instance, when outcomes are modeled as successes or failures, the likelihood is often based on the binomial distribution, which is essential for comparing different hypotheses about the treatment’s effectiveness.

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