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In a 2018 study reported in The Lancet, Molina et al. reported on a study for treatment of patients with HIV-1. The study was a randomized, controlled, double-blind study that compared the effectiveness of ritonavir-boosted darunavir (rbd), the drug currently used to treat HIV-1, with dorovirine, a newly developed drug. Of the 382 subjects taking ritonavir-boosted darunavir, 306 achieved a positive result. Of the 382 subjects taking dorovirine, 321 achieved a positive outcome. See page 430 for guidance. a. Find the sample percentage of subjects who achieved a positive outcome in each group. b. Perform a hypothesis test to test whether the proportion of patients who achieve a positive outcome with the current treatment (ritonavir-boosted darunavir) is different from the proportion of patients who achieve a positive outcome with the new treatment (dorovirine). Use a significance level of \(0.01\). Based on this study, do you think dorovirine might be a more effective treatment option for HIV-1 than ritonavir-boosted darunavir? Why or why not?

Step by step solution

01

Calculate the Sample Percentage

First, find the sample percentage of subjects who achieved positive outcome in each group. This can be done by dividing the number of positive cases by the total number of subjects and then multiplying the result by 100. \n For ritonavir-boosted darunavir (rbd) group: \(\frac{306}{382} * 100 = 80.1%\)\n For dorovirine group: \(\frac{321}{382} * 100 = 84.0%\)

02

Setup Hypotheses

H0 (Null hypothesis): The proportions of positive outcomes with rbd and dorovirine are the same (i.e., \(p1 - p2 = 0\)) \n H1 (Alternative hypothesis): The proportions of positive outcomes with rbd and dorovirine are different (i.e., \(p1 - p2 ≠ 0\))

03

Calculate the Difference in Proportions

\(\Delta p = p1 - p2 = 0.801 - 0.84 = -0.039\)

04

Calculate the Combined Sample Proportion

\(P = \frac{(X1 + X2)}{(n1 + n2)} = \frac{(306 + 321)}{(382 + 382)} = 0.8205\)

05

Calculate the Standard Error

The Standard Error (SE) can be calculated using the formula: \(SE = \sqrt{P * (1 - P) * [\frac{1}{n1} + \frac{1}{n2}] } = \sqrt{0.8205 * (1 - 0.8205) * [\frac{1}{382} + \frac{1}{382}] } = 0.0215\)

06

Calculate the Z-Score

The Z score can be calculated using the formula: \(Z = \frac{\Delta p - 0}{SE} = \frac{-0.039}{0.0215} = -1.81\)

07

Analyze the Z-score

A Z-score of -1.81 falls within the acceptance range for a two-tailed test at the significance level of 0.01 (Z = ±2.57). Hence, we accept the null hypothesis (H0), making the conclusion that there's no significant difference in effectiveness between doravirine and rbd.

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Key Concepts

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

Sample Percentage Calculation

When analyzing the effectiveness of two treatments, researchers often start by calculating the success rate, or sample percentage, for each group. This is the percentage of participants who achieved the desired outcome. For example, in the provided exercise, to compute the sample percentage for the ritonavir-boosted darunavir (rbd) group, the number of positive outcomes (306) was divided by the total number of subjects (382), and the result was multiplied by 100 to get a percentage:
-For rbd group: \(\frac{306}{382} \times 100 = 80.1\%\)
-For dorovirine group: \(\frac{321}{382} \times 100 = 84.0\%\)
This initial step helps establish a baseline from which the effectiveness of each drug can be compared.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) is a statement that no effect or no difference exists, while the alternative hypothesis (H1) proposes that there is an effect or a difference. In the context of the exercise, the null hypothesis asserts that the proportion of positive outcomes for both HIV-1 treatments is equal:
-H0: \(p1 - p2 = 0\)
Conversely, the alternative hypothesis suggests that the proportions are not equal:
-H1: \(p1 - p2 eq 0\)
It is crucial to clearly define these hypotheses, as they guide the subsequent statistical analysis.

Z-Score Analysis

The Z-score is a statistical measure that describes a value's relationship to the mean of a group of values, expressed in terms of standard deviations from the mean. In hypothesis testing, it is used to determine how many standard deviations an observed value is from the hypothesis' expected value. In the case of the exercise:
-Z = \(\frac{\Delta p - 0}{SE}\)
A calculated Z-score is then compared to critical values from the standard normal distribution, which correspond to the significance level chosen for the test. If the absolute value of the Z-score is higher than the critical value, the null hypothesis is rejected.

Standard Error Calculation

The standard error (SE) measures the variability or uncertainty in a statistic, such as a sample proportion. It is crucial for working out how concentrated the sample distribution is around the population mean. The formula for the Standard Error in comparing two proportions is:
-SE = \(\sqrt{P \times (1 - P) \times [\frac{1}{n1} + \frac{1}{n2}]}\)
Within our exercise, 'P' is the pooled sample proportion and 'n1' and 'n2' are the sample sizes. The smaller the SE, the more certain we can be about the estimate.

Significance Level

The significance level, often denoted by \(\alpha\), is the threshold at which the results of a statistical test are declared significant. This means, if the probability of observing such extreme results, given that the null hypothesis is true, is less than the significance level, the null hypothesis is rejected. Common levels include 0.01, 0.05, and 0.10. In the given problem, an \(\alpha\) of 0.01 indicates we have a 1% risk of rejecting the null hypothesis when it is actually true (Type I error). A more stringent level like 0.01 means we require stronger evidence to reject the null hypothesis.

Randomized Controlled Trials

A randomized controlled trial (RCT) is a scientific study design that randomly allocates participants into an experimental group or a control group to measure the effectiveness of interventions. RCTs are considered the 'gold standard' in determining causality and evaluating new treatments. The study mentioned in the exercise compares the efficacy of two drugs, ritonavir-boosted darunavir and dorovirine, for treating HIV-1. By randomizing, both known and unknown confounding variables are evenly distributed, minimizing bias and improving the reliability of the results.