91Ó°ÊÓ

Clopidogrel and Aspirin A large study was conducted to test the effectiveness of clopidogrel in combination with aspirin in warding off heart attacks and strokes. \({ }^{14}\) The trial involved more than 15.500 people 45 years of age or older from 32 countries, including the United States, who had been diagnosed with cardiovascular disease or had multiple risk factors. The subjects were randomly assigned to one of two groups. After two years, there was no difference in the risk of heart attack, stroke, or dying from heart disease between those who took clopidogrel and low-dose aspirin daily and those who took low-dose aspirin plus a dummy pill. The two-drug combination actually increased the risk of dying \((5.4 \%\) versus \(3.8 \%\) ) or dying specifically from cardiovascular disease ( \(3.9 \%\) versus \(2.2 \%\) ). a. The subjects were randomly assigned to one of the two groups. Explain how you could use the random number table to make these assignments. b. No sample sizes were given in the article: however, let us assume that the sample sizes for each group were \(n_{1}=7720\) and \(n_{2}=7780 .\) Determine whether the risk of dying was significantly different for the two groups. c. What do the results of the study mean in terms of practical significance?

Step by step solution

01

Understanding random number tables

Random number tables are essentially lists of digits that are chosen randomly and have no pattern. They are widely used in various fields, including statistical research, to generate random assignments or to make random selections.

02

Assigning subjects to groups using a random number table

To use a random number table to assign subjects into two groups, follow these steps: 1. Number the subjects from 1 to 15,500. 2. Choose a starting point in the random number table and read the numbers sequentially. 3. Define a rule for assigning subjects to each group. For example, assign the subject to Group 1 (clopidogrel and aspirin) if the random number is even and Group 2 (aspirin plus a dummy pill) if the random number is odd. 4. Continue this process until all subjects are assigned to one of the two groups. ##Part B: Hypothesis test for the difference in proportions##

03

Set up the hypotheses

The null hypothesis (H0) is that there is no difference in the risk of dying between the two groups, while the alternative hypothesis (H1) is that there is a significant difference in the risk of dying between these groups: \(H_0: p_1 = p_2\) \(H_1: p_1 \neq p_2\)

04

Calculate the sample proportions and pooled proportion

Calculate the sample proportions for each group (\(\hat{p}_1\) and \(\hat{p}_2\)) and the pooled proportion (\(\hat{p}\)): \( \hat{p}_1 = \frac{7720 \times 5.4\%}{7720} = 0.054\) \( \hat{p}_2 = \frac{7780 \times 3.8\%}{7780} = 0.038\) \( \hat{p} = \frac{7720 \times 0.054 + 7780 \times 0.038}{7720 + 7780} = \frac{416.88+295.64}{15500} ≈ 0.0459\)

05

Calculate the test statistic and p-value

Compute the test statistic (z) and the p-value for the hypothesis test: \( z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}} = \frac{0.054 - 0.038}{\sqrt{0.0459(1-0.0459)(\frac{1}{7720} + \frac{1}{7780})}} ≈ 5.224\) Use a z-table or software to find the p-value corresponding to the test statistic. In this case: \( p \approx 0.0000\) Since the p-value is less than the typical significance level (\(α = 0.05\)), we reject the null hypothesis, which means there is a significant difference in the risk of dying between the two groups. ##Part C: Practical significance## The results of the study indicate that, statistically, there is a significant difference in the risk of dying between the two groups. However, practical significance must take into consideration the magnitude of this difference, the relevance of the findings in real-world situations, and the potential impact on healthcare decisions. It is crucial to discuss these aspects with healthcare experts and decision-makers, who can better assess whether the observed differences are practically significant in a medical context.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Random Assignment

Random assignment is a crucial technique in experimental studies that ensures each participant has an equal chance of being placed into any group. This unbiased method helps eliminate potential selection bias, ensuring the only variable affecting the results is the treatment itself. This is vital for obtaining reliable and valid data.

To achieve random assignment in a study, researchers often use tools like a random number table. In such tables, digits are listed without any pattern, allowing for unbiased decisions. By numbering each participant and using the table, one can randomly assign them to groups. For instance, if we assign even-numbered participants to one group and odd-numbered to another, we ensure each group is balanced, minimizing bias.

This procedure not only reinforces the validity of an experiment but also enhances the credibility of its findings. Using a random number table is one of the simplest and most effective ways to support random assignment.

Sample Proportions

Sample proportions are statistical measures that reflect the ratio of outcomes in a sample compared to the whole. When calculating sample proportions, researchers can understand the part of the sample that exhibits a particular trait or outcome.

In the case study, the sample proportions of people dying from each group were calculated. These proportions are derived by dividing the number of favorable outcomes by the total sample size. For instance, the proportion for the group taking clopidogrel and aspirin was derived by dividing the number of fatalities by the total number of participants in that group.

Understanding sample proportions is key to comparing different experimental groups. It highlights differences or similarities in outcomes, aiding in the assessment of the effectiveness of treatments or interventions. Such comparisons are especially crucial in large-scale studies where subtle changes can have significant implications.

Statistical Significance

Statistical significance is a cornerstone concept in hypothesis testing, used to determine whether observed differences in data are likely to be due to chance or a true effect. When statisticians report a result as statistically significant, it signifies that the observed outcome is unlikely to have occurred by random chance.

Typically, a p-value is used to measure significance. In hypothesis testing, if the p-value falls below a certain threshold (often 0.05), it suggests that the data provide strong evidence against the null hypothesis. In our case, the p-value indicated a statistically significant difference in mortality between the two groups.

While statistical significance provides confidence about the presence of a difference, it does not convey the size or importance of that difference in practical terms. It merely indicates that the difference is not attributable to random variation alone. Understanding this can guide researchers in making informed decisions about their findings.

Practical Significance

Practical significance moves beyond mere statistics to consider the real-world implications of a study's findings. It evaluates the magnitude and relevance of an effect, determining if it is large or impactful enough to influence decision-making or policy.

In evaluating practical significance, it's crucial to consider not just if a result is statistically significant, but if it is meaningful in a practical sense. Even a small statistical difference can be practically significant if it influences crucial aspects of treatment, risk management, or overall patient care.

For instance, while statistical analysis might show a significant difference in mortality rates due to a drug, if the difference is small, stakeholders need to weigh the benefits against risks, costs, and other factors. Practical significance thus bridges the gap between statistical analysis and pragmatic decision-making, grounding statistical findings in the context of real-life applications.