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McNemar's test, developed in Exercise 54, can also be used when individuals are paired (matched) to yield \(n\) pairs and then one member of each pair is given treatment 1 and the other is given treatment 2 . Then \(X_{1}\) is the number of pairs in which both treatments were successful, and similarly for \(X_{2}, X_{3}\), and \(X_{4}\). The test statistic for testing equal efficacy of the two treatments is given by \(\left(X_{2}-X_{3}\right) / \sqrt{\left(X_{2}+X_{3}\right)}\), which has approximately a standard normal distribution when \(H_{0}\) is true. Use this to test whether the drug ergotamine is effective in the treatment of migraine headaches. \begin{tabular}{cc|cc} & & Ergotamine \\ \hline & & \(\boldsymbol{S}\) & \(\boldsymbol{F}\) \\ \hline Placebo & \(\boldsymbol{S}\) & 44 & 34 \\ & \(\boldsymbol{F}\) & 46 & 30 \\ \hline \end{tabular} The data is fictitious, but the conclusion agrees with that in the article "Controlled Clinical Trial of Ergotamine Tartrate" (British Med. J., 1970: 325-327).

Step by step solution

01

Set up the contingency table

From the given data, we know: \( X_1 = 44 \), \( X_2 = 34 \), \( X_3 = 46 \), \( X_4 = 30 \). Each represents the count of pairs with certain outcomes from both the treatments (Ergotamine and Placebo).

02

Express the McNemar's test statistic

The test statistic for McNemar's test is computed using the formula: \[ T = \frac{(X_2 - X_3)}{\sqrt{(X_2 + X_3)}} \]. This formula checks for the difference in discordant pairs.

03

Compute the test statistic value

Substitute the values \( X_2 = 34 \) and \( X_3 = 46 \) into the test statistic formula: \[ T = \frac{(34 - 46)}{\sqrt{(34 + 46)}} = \frac{-12}{\sqrt{80}} \]. Calculate \( T \):\[ T = \frac{-12}{8.94} \approx -1.34 \].

04

Determine the significance level and critical value

Choose a typical significance level \( \alpha = 0.05 \). For a two-tailed test with \( \alpha = 0.05 \), the critical value from the standard normal distribution is approximately \( \pm 1.96 \).

05

Compare test statistic to critical value

The computed test statistic \( T \approx -1.34 \) lies between \(-1.96\) and \(1.96\). Hence, it does not fall into the critical region that suggests rejecting the null hypothesis.

06

Draw a conclusion

Since the test statistic \( T \) does not exceed the critical values, we fail to reject the null hypothesis. This implies there is insufficient evidence to claim that Ergotamine and Placebo are significantly different in their effectiveness.

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Key Concepts

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

Paired Sample Analysis

Paired sample analysis is a statistical method used when observations can be matched in pairs. Each pair consists of two observations, each from a different treatment or condition applied to the same subject or a closely matched subject. This technique helps to control for variability in the data by ensuring that each pair is as similar as possible, except for the treatment in question. In our example, the two treatments are ergotamine and a placebo.

Key aspects of paired sample analysis include:

• A focus on the differences within each pair rather than differences between groups.
• The assumption that pairs are homogeneous in characteristics other than the treatment.
• A reduction in the variability of data, as intra-pair differences tend to be less than inter-group differences.

These elements make paired sample analysis powerful for clinical trials, where it is vital to attribute effects seen to the treatment itself and not to other external differences.

Contingency Table

A contingency table is a type of table often used in statistics to display the frequency distribution of variables. It helps summarize the relationship between categorical variables by showing the number of observations that fall into each category.

In our clinical trial, a 2x2 contingency table was used:

• The rows represent the outcomes of one treatment (Placebo - success or failure).
• The columns represent the outcomes of the other treatment (Ergotamine - success or failure).

This layout enables easy visualization of the successful and unsuccessful outcomes for each treatment, allowing for the calculation of the differences in efficacy between the two treatments. Contingency tables are crucial in analytical studies for comparing different categories and are commonly used in McNemar's test to help discern patterns among paired samples.

Test Statistic

The test statistic is a crucial component of the hypothesis testing process. It helps determine whether to reject the null hypothesis. For McNemar's test, the test statistic formula is designed to evaluate changes in paired categorical data.

The test statistic is calculated as:\[ T = \frac{(X_2 - X_3)}{\sqrt{(X_2 + X_3)}} \]where:

• \(X_2\) is the count of pairs where the control condition was successful, and the treatment was not.
• \(X_3\) is the count of pairs where treatment was successful, and the control condition was not.

This statistic indicates if the discordant pairs (where the two treatments have different outcomes) suggest any significant improvement in one treatment over the other. By comparing the test statistic to the critical value from a standard normal distribution table, we decide if the observed difference is statistically significant or just due to random chance.

Clinical Trial Interpretation

Clinical trial interpretation involves analyzing and understanding the results of a trial. It requires a thorough examination of statistical data to determine if a treatment is effective.

In the McNemar's test example, the interpretation process went as follows:

• The test statistic calculated was approximately \(T \approx -1.34\).
• A significance level \(\alpha = 0.05\) was chosen, establishing a critical value of \(\pm 1.96\).
• Since the test statistic lies between \(-1.96\) and \(1.96\), no evidence supports rejecting the null hypothesis.

This result indicates insufficient evidence to prove a significant difference in effectiveness between ergotamine and a placebo for treating migraines. Clinical trial interpretation helps in making informed decisions in medical practices based on statistical analyses, ensuring treatments are both safe and effective for patients.