Question
Suppose historical data from 50 similar clinical trials, each with 100 patients, is available. In these trials, the average success rate was approximately 70%. Explain how you would use this historical data to assess if the number of positive responses in a trial of 100 patients can be modeled by a binomial distribution. Your explanation must include:
(i) How to estimate the parameters of the binomial distribution from the historical data.
(ii) How to apply a goodness-of-fit test to evaluate the model’s validity.
Hint
To estimate parameters, consider the given average success rate. For goodness-of-fit, think about comparing observed frequencies of outcomes in the historical data to expected frequencies under the binomial model. The Chi-squared test is a suitable method.
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Answer
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Mark Scheme
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Explanation
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Complete Answer
To validate the binomial model using historical data:
(i) Parameter Estimation: Estimate the success probability \(p\) from the historical data. Given the average success rate is approximately 70%, we take \(p = 0.7\). The number of trials per experiment is \(n = 100\).
(ii) Goodness-of-fit Test: Apply a Chi-squared goodness-of-fit test. First, determine the observed frequencies of the number of positive responses from the 50 historical trials. Then, for each possible number of positive responses (0 to 100), calculate the expected frequency based on the binomial distribution with \(n=100\) and \(p=0.7\), and multiply by 50 (number of trials). Calculate the Chi-squared statistic by summing \(\frac{(O_k – E_k)^2}{E_k}\) for each category, where \(O_k\) is the observed frequency and \(E_k\) is the expected frequency. Compare the calculated statistic to the critical value from the Chi-squared distribution with appropriate degrees of freedom, or find the p-value. If the p-value is greater than a chosen significance level (e.g., 0.05), we do not reject the binomial model.
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Answer Explanation