Question

Verzani (2014), page 323 , presented a data set concerning the effect that different dosages of the drug AZT have on patients with HIV. The responses we consider are the p24 antigen levels of HIV patients after their treatment with AZT. Of the \(20 \mathrm{HIV}\) patients in the study, 10 were randomly assign the dosage of \(300 \mathrm{mg}\) of AZT while the other 10 were assigned \(600 \mathrm{mg}\). The hypotheses of interest are \(H_{0}: \Delta=0\) versus \(H_{1}: \Delta \neq 0\) where \(\Delta=\mu_{600}-\mu_{300}\) and \(\mu_{600}\) and \(\mu_{300}\) are the true mean p24 antigen levels under dosages of \(600 \mathrm{mg}\) and \(300 \mathrm{mg}\) of AZT, respectively. The data are given below but are also available in the file aztdoses. rda. \begin{tabular}{|l|llllllllll|} \hline \(300 \mathrm{mg}\) & 284 & 279 & 289 & 292 & 287 & 295 & 285 & 279 & 306 & 298 \\ \hline \(600 \mathrm{mg}\) & 298 & 307 & 297 & 279 & 291 & 335 & 299 & 300 & 306 & 291 \\ \hline \end{tabular} (a) Obtain comparison boxplots of the data. Identify outliers by patient. Comment on the comparison plots. (b) Compute the two-sample \(t\) -test and obtain the \(p\) -value. Are the data significant at the \(5 \%\) level of significance? (c) Obtain a point estimate of \(\Delta\) and a \(95 \%\) confidence interval for it. (d) Conclude in terms of the problem.

Short answer

The answer will depend on the computations for the statistical tests. If the p-value of the t-test is less than 0.05, there is a significant difference between the mean p24 antigen levels of the two dosage groups. The 95% confidence interval for delta will show the range of true values for this difference. The outliers in the boxplots will identify patients who react differently to the treatments.

Step-by-step solution

Create Comparison Boxplots

First, create separate boxplots for each drug dosage group (300mg and 600mg). These boxplots will visually display the distribution of the data, including median and any possible outliers. Outliers would be values that clearly fall far from the bulk of the data.

Perform T-Test

We perform a two-sample t-test with the null hypothesis \(H_0: \Delta = 0\), which states that there is no difference between the means of the two dosage groups, against the alternative hypothesis \(H_1: \Delta \neq 0\), which states that a difference does exist. Calculate the t-statistic and corresponding p-value using the formula for the t-test for independent samples. If the obtained p-value is less than 0.05, we reject the null hypothesis.

Calculate Point Estimate of Delta

To calculate the point estimate of \(\Delta\) (\(\mu_{600} - \mu_{300}\)), you need to subtract the mean p24 antigen level of the 300 mg group from the mean p24 antigen level of the 600 mg group.

Obtain Confidence Interval for Delta

You can compute a 95% confidence interval for \(\Delta\) by using the formula for a confidence interval for the difference between two means. This will give a range for the true value of the difference between the means of the p24 antigen levels under the two dosages.

Conclude in Terms of Problem

Based on the obtained p-value and confidence interval for \(\Delta\), determine whether there is a significant difference between the mean p24 antigen levels of the two dosage groups. If there is a significant difference, it suggests that one of the dosages is more effective than the other.