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Aerosolized Vaccine for Measles. An aerosolized vaccine for measles was developed in Mexico and has been used on more than 4 million children since 1980 . Aerosolized vaccines have the advantages of being able to be administered by people without clinical training and do not cause injectionassociated infections. Despite these advantages, data about efficacy of the aerosolized vaccines against measles compared to subcutaneous injection of the vaccine have been inconsistent. Because of this, a large randomized controlled study was conducted using children in India. The primary outcome was an immune response to measles measured 91 days after the treatments. Among the 785 children receiving the subcutaneous injection, 743 developed an immune response, while among the 775 children receiving the aerosolized vaccine, 662 developed an immune response. 3 a. Compute the proportion of subjects experiencing the primary outcome for both the aerosol and injection groups. b. Can we safely use the large-sample confidence interval for comparing the proportion of children who developed an immune response to measles in the aerosol and injection groups? Explain. c. Give a \(95 \%\) confidence interval for the difference between the proportion of children in the aerosol and injection groups who experienced the primary outcome. d. The study described is an example of a noninferiority clinical trial intended to show that the effect of a new treatment, the aerosolized vaccine, is not worse than that of the standard treatment by more than a specified margin. 4 Specifically, is the percentage of children who developed an immune response for the aerosol treatment more than \(5 \%\) below the percentage for the subcutaneous injected vaccine? The 5percentage-point difference was based on previous studies and the fact that with a bigger difference, the aerosolized vaccine would not provide the levels of protection necessary to achieve herd immunity. Using your answer in part (c), do you feel the investigators demonstrated the noninferiority of the aerosolized vaccine? Explain.

Step by step solution

01

Calculate Proportion for the Injection Group

To find the proportion of children who developed an immune response in the injection group, divide the number of children with a response by the total number of children in that group. So, for the subcutaneous injection:\[P_{\text{injection}} = \frac{743}{785} \approx 0.9465\]

02

Calculate Proportion for the Aerosol Group

Similarly, compute the proportion for the aerosol group by dividing the number of children who developed an immune response by the total number in that group:\[P_{\text{aerosol}} = \frac{662}{775} \approx 0.8542\]

03

Assess Large-Sample Confidence Interval Applicability

Determine if a large-sample confidence interval is appropriate by checking if the sample sizes and proportions satisfy conditions for normal approximation. Both sample sizes should be large enough, and the estimated proportions should allow for expected counts greater than 5 for both successful and unsuccessful outcomes (\(n \times p \) and \( n \times (1-p) \)). Both proportions meet these criteria, indicating that the large-sample confidence interval is applicable.

04

Calculate the Standard Error for Difference in Proportions

Calculate the standard error for the difference between the two proportions:\[SE_{\text{diff}} = \sqrt{ \frac{P_{\text{injection}} (1-P_{\text{injection}})}{785} + \frac{P_{\text{aerosol}} (1-P_{\text{aerosol}})}{775} }\]Using the proportions calculated:\[SE_{\text{diff}} = \sqrt{ \frac{0.9465 \times 0.0535}{785} + \frac{0.8542 \times 0.1458}{775} } \approx 0.015\]

05

Compute the 95% Confidence Interval

Using the standard error and a Z-score of 1.96 for a 95% confidence interval, calculate the interval:\[CI = (P_{\text{aerosol}} - P_{\text{injection}}) \pm 1.96 \times SE_{\text{diff}}\]\[CI = (0.8542 - 0.9465) \pm 1.96 \times 0.015 \approx (-0.107, -0.071)\]

06

Evaluate Noninferiority Based on Confidence Interval

Determine whether the aerosolized vaccine demonstrates noninferiority by seeing if the confidence interval's upper bound is above the specified 5% noninferiority margin. Since the upper confidence limit is -0.071, which is more than 5% below the injection group proportion, the aerosolized vaccine does not meet the noninferiority criteria with a margin of -0.05, as the entire confidence interval lies below it.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Proportion Calculation

Proportion calculation helps in understanding the part of a group exhibiting a certain outcome. Here, we are calculating the proportion of children who developed an immune response to the measles vaccine. For the subcutaneous injection group, we find the proportion by dividing the number of children with an immune response by the total number of children in that group. This can be represented by the formula: \(P_{\text{injection}} = \frac{743}{785} \).

The calculation gives us \(P_{\text{injection}} \approx 0.9465\), meaning about 94.65% of the children in the injection group had an immune response.

• Similarly, for the aerosol group, the proportion is calculated as \(P_{\text{aerosol}} = \frac{662}{775} \)
• This results in \(P_{\text{aerosol}} \approx 0.8542\), indicating that about 85.42% of children in the aerosol group developed an immune response.

Understanding proportions is crucial as it gives a clear picture of how effective each treatment was in producing the desired immune response.

Confidence Interval

A confidence interval gives us an estimated range that is likely to contain the true difference in proportions, providing a measure of precision. To determine if aerosol vaccines are noninferior, we construct a 95% confidence interval for the difference between the group proportions.

This interval is calculated using the formula: \[CI = (P_{\text{aerosol}} - P_{\text{injection}}) \pm 1.96 \times SE_{\text{diff}}\]Where \(SE_{\text{diff}}\) is the standard error of the difference in proportions.

• The difference between the aerosol and injection groups’ proportions is \(0.8542 - 0.9465 = -0.0923\)
• After substituting into the confidence interval formula, we find the interval is approximately (-0.107, -0.071).

This negative interval suggests that the aerosol vaccination might be less effective than the subcutaneous injection. Confidence intervals in trials like these help researchers determine the reliability and variability of their results.

Standard Error

The standard error (SE) measures the variability in the sample proportion difference. It tells us how much the proportions vary by chance. The formula to compute the standard error for two proportions is:

\[SE_{\text{diff}} = \sqrt{ \frac{P_{\text{injection}} (1-P_{\text{injection}})}{785} + \frac{P_{\text{aerosol}} (1-P_{\text{aerosol}})}{775} }\]

• The SE gives insight into how far the sample’s estimates are from the true population value.
• In this scenario, the SE is roughly 0.015, indicating that the difference in sample proportions occasionally varies by this amount purely by random sampling variation.

The smaller the standard error, the more precise the estimate is considered to be, offering a clearer picture of the difference between treatment effects.

Randomized Controlled Study

Randomized Controlled Studies (RCS) are a fundamental type of experiment used in clinical research. They help establish causal relationships by randomly assigning participants to different treatment groups, in this case, aerosolized versus subcutaneous vaccine.

• Randomization reduces bias and ensures that differences in outcomes between groups are likely due to the treatment rather than confounding factors.
• The method provides a robust framework to compare vaccine efficacy, allowing researchers to draw reliable conclusions about the effects of the treatments being compared.

By utilizing RCS, the study effectively assesses the immune response induced by different vaccine administration methods, thus contributing valuable insights into vaccine efficacy.