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Fizer (NOT Pfizer) is a pharmaceutical company which has developed a new vaccine for COVID-1 order to bring their vaccine to market, the FDA requires them to demonstrate the efficacy of th vaccine through clinical trials. With this aim, they conduct the following experiment. N = n + m participants are recruited randomized clinical trial.

Fizer (NOT Pfizer) is a pharmaceutical company which has developed a new vaccine for COVID-1 order to bring their vaccine to market, the FDA requires them to demonstrate the efficacy of th vaccine through clinical trials. With this aim, they conduct the following experiment. N = n + m participants are recruited randomized clinical trial.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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## Clinical Trial Design for COVID-19 Vaccine Efficacy **Background:** Fizer (not Pfizer) is developing a new COVID-19 vaccine. To gain FDA approval, they must prove the vaccine's efficacy through clinical trials. **Experimental Setup:** - Participants: A total of \( N = n + m \) subjects are recruited for a randomized trial. - **Experimental Group:** Consists of \( n \) subjects receiving the real vaccine. - **Control Group:** Comprises \( m \) subjects receiving a placebo. A survey checks how many participants contract COVID-19 post-trial. **Statistical Variables:** - For the experiment group: \( X_1, X_2, \ldots, X_n \) are independent and identically distributed variables represented as \( X_i \sim \text{Ber}(\theta_X) \). Here, each variable is 1 if the subject contracts COVID-19. - For the control group: \( Y_1, Y_2, \ldots, Y_m \) are similarly distributed. \( Y_j \sim \text{Ber}(\theta_Y) \). **Analysis Questions:** 1. **Interpretation of \(\theta_X\) and \(\theta_Y\):** - Discuss what \(\theta_X\) and \(\theta_Y\) signify. Explain the interpretation of the difference \(\theta_Y - \theta_X\). 2. **Maximum Likelihood Estimator:** - Define \(\theta = \theta_Y - \theta_X\). Compute the maximum likelihood estimator \(\hat{\theta}_{\text{MLE}}\) for \(\theta\) and find \(\text{Var}(\hat{\theta}_{\text{MLE}})\). 3. **Sampling Distribution:** - Use the central limit theorem to derive the sampling distribution for \(\hat{\theta}_{\text{MLE}}\). 4. **Confidence Intervals:** - Explain various methods of constructing a confidence interval, e.g., two-sided, left-sided, right-sided. If advising the FDA, specify the type of confidence interval desired to assess vaccine efficacy and its interpretation. Each analysis step builds a deeper understanding of vaccine efficacy evaluation through statistical methods.
Transcribed Image Text:## Clinical Trial Design for COVID-19 Vaccine Efficacy **Background:** Fizer (not Pfizer) is developing a new COVID-19 vaccine. To gain FDA approval, they must prove the vaccine's efficacy through clinical trials. **Experimental Setup:** - Participants: A total of \( N = n + m \) subjects are recruited for a randomized trial. - **Experimental Group:** Consists of \( n \) subjects receiving the real vaccine. - **Control Group:** Comprises \( m \) subjects receiving a placebo. A survey checks how many participants contract COVID-19 post-trial. **Statistical Variables:** - For the experiment group: \( X_1, X_2, \ldots, X_n \) are independent and identically distributed variables represented as \( X_i \sim \text{Ber}(\theta_X) \). Here, each variable is 1 if the subject contracts COVID-19. - For the control group: \( Y_1, Y_2, \ldots, Y_m \) are similarly distributed. \( Y_j \sim \text{Ber}(\theta_Y) \). **Analysis Questions:** 1. **Interpretation of \(\theta_X\) and \(\theta_Y\):** - Discuss what \(\theta_X\) and \(\theta_Y\) signify. Explain the interpretation of the difference \(\theta_Y - \theta_X\). 2. **Maximum Likelihood Estimator:** - Define \(\theta = \theta_Y - \theta_X\). Compute the maximum likelihood estimator \(\hat{\theta}_{\text{MLE}}\) for \(\theta\) and find \(\text{Var}(\hat{\theta}_{\text{MLE}})\). 3. **Sampling Distribution:** - Use the central limit theorem to derive the sampling distribution for \(\hat{\theta}_{\text{MLE}}\). 4. **Confidence Intervals:** - Explain various methods of constructing a confidence interval, e.g., two-sided, left-sided, right-sided. If advising the FDA, specify the type of confidence interval desired to assess vaccine efficacy and its interpretation. Each analysis step builds a deeper understanding of vaccine efficacy evaluation through statistical methods.
**5.** Using your answer from parts (3) and (4), derive the expression for the confidence interval at level \( (1-\alpha) \), i.e., find \( l_{\alpha} \) and \( u_{\alpha} \) such that \[ CI(\theta; \alpha) = [l_{\alpha}, u_{\alpha}]. \] --- **6.** The code below contains a snippet for constructing a confidence interval. Complete the code for \( l_{\alpha} \) and \( u_{\alpha} \) so that the output of the cell is your confidence interval from part (5) at level \( \alpha = 0.05 \). What do you conclude from your answer?
Transcribed Image Text:**5.** Using your answer from parts (3) and (4), derive the expression for the confidence interval at level \( (1-\alpha) \), i.e., find \( l_{\alpha} \) and \( u_{\alpha} \) such that \[ CI(\theta; \alpha) = [l_{\alpha}, u_{\alpha}]. \] --- **6.** The code below contains a snippet for constructing a confidence interval. Complete the code for \( l_{\alpha} \) and \( u_{\alpha} \) so that the output of the cell is your confidence interval from part (5) at level \( \alpha = 0.05 \). What do you conclude from your answer?
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