Question

Recovery Rates Clinical data concerning the effectiveness of two drugs in treating a particular condition (as measured by recovery in 7 days or less) were collected from 10 hospitals. You want to know whether the data present sufficient evidence to indicate a higher recovery rate for one of the two drugs. a. Test using the sign test. Choose your rejection region so that \(\alpha\) is near. \(05 .\) b. Why might it be inappropriate to use the Student's \(t\) -test in analyzing the data? $$ \begin{array}{cccc} \hline &&{\text { Drug A }} \\ \hline & & {\text { Number }} \\ & \text { Number in } & \text { Recovered } & \text { Percentage } \\ \text { Hospital } & \text { Group } & \text { (7 days or less) } & \text { Recovered } \\ \hline 1 & 84 & 63 & 75.0 \\ 2 & 63 & 44 & 69.8 \\ 3 & 56 & 48 & 85.7 \\ 4 & 77 & 57 & 74.0 \\ 5 & 29 & 20 & 69.0 \\ 6 & 48 & 40 & 83.3 \\ 7 & 61 & 42 & 68.9 \\ 8 & 45 & 35 & 77.8 \\ 9 & 79 & 57 & 72.2 \\ 10 & 62 & 48 & 77.4 \\ \hline \end{array} $$ $$ \begin{array}{cccc} \hline && {\text { Drug B }} \\ \hline && {\text { Number }} \\ & \text { Number in } & \text { Recovered } & \text { Percentage } \\ \text { Hospital } & \text { Group } & \text { (7 days or less) } & \text { Recovered } \\ \hline 1 & 96 & 82 & 85.4 \\ 2 & 83 & 69 & 83.1 \\ 3 & 91 & 73 & 80.2 \\ 4 & 47 & 35 & 74.5 \\ 5 & 60 & 42 & 70.0 \\ 6 & 27 & 22 & 81.5 \\ 7 & 69 & 52 & 75.4 \\ 8 & 72 & 57 & 79.2 \\ 9 & 89 & 76 & 85.4 \\ 10 & 46 & 37 & 80.4 \\ \hline \end{array} $$

Short answer

In this step, we will count the number of positive and negative differences in our table: | Sign | Count | |-------|-------| | Positive | 2 | | Negative | 8 | #tag_title# Step 3: Perform the Sign Test Next, we will perform the sign test by calculating the test statistic and comparing it to a critical value from a binomial distribution table, with a desired significance level of α near 0.05. - Test statistic: W = the smaller frequency of positive or negative differences = 2 - Sample size: n = total number of paired data points = 10 We will then compare W to the critical value from the binomial distribution table for the chosen significance level. If W ≤ critical value, we reject the null hypothesis, which is that there is no difference in the recovery rates of the two drugs. In our case, the critical value for a one-tailed binomial test with n = 10 and α = 0.05 is 1. If we have W ≤ 1, we would reject the null hypothesis in favor of the alternative hypothesis, which states that there is a significant difference in recovery rates between the two drugs. Since our W = 2, which is greater than the critical value (1), we fail to reject the null hypothesis. This means there is insufficient evidence to conclude that one drug has a significantly higher recovery rate. #Explanation# Using Student's t-test in this situation might be inappropriate because the t-test is a parametric test that assumes the data is normally distributed and has equal variances. The sign test is more suitable for this case, as it is a non-parametric method that does not make assumptions about the normality and homoscedasticity of the data. This provides a more robust analysis when normality and equal variances cannot be assumed.

Step-by-step solution

Create a Table of Differences between Paired Data Points>\

Step 2: Count the Number of Positive and Negative Differences

Key concepts

Sign Test

The sign test is a nonparametric method used to determine whether two paired data sets (like the recovery rates for two different drugs) differ from each other in a statistically significant way. It is particularly useful when the assumptions of other tests, such as the t-test, are not met. For instance, the sign test does not assume normal distribution or homoscedasticity (equal variances) among comparisons.

To perform the sign test, one simply counts the number of positive and negative differences between the paired scores. A 'positive' or 'negative' sign is given depending on whether the difference is above or below zero. The computed number of positives and negatives are then compared to a reference distribution—typically a binomial distribution—to determine if the observed result is statistically significant, which means it is unlikely to have occurred by chance alone.

When applying it to our exercise, where the recovery rates from two different drugs at 10 hospitals are compared, one would calculate the differences in recovery rates between Drug A and Drug B, designate signs to these differences, and then assess the count of positive and negative signs to make inferences about the drugs' effectiveness.

Recovery Rate Statistics

Recovery rate statistics involve analyzing the proportion of subjects in a study that recover from a certain condition within a specified timeframe. In clinical trials, this manages to be a critical outcome metric as it directly relates to a drug's effectiveness. Statistical methods can then be applied to these rates to deduce the likelihood of recovery upon treatment with a drug.

A key aspect of recovery rate statistics is determining whether observed differences in rates between two treatment groups (or drugs) are due to chance or to a real effect of the treatments. Calculation of percentages, as seen in our exercise, is a straightforward method for representing recovery rates. However, interpreting these raw percentages and comparing them to make meaningful conclusions requires the use of statistical tests, such as the sign test, to account for random variability in the data.

In the current exercise, the primary statistical interest is to compare the recovery rates between Drug A and Drug B to establish which drug can provide a higher recovery rate. The statistical analysis will use these recovery percentages to determine if there is enough evidence to conclude a significant difference in effectiveness.

Paired Data Analysis

Paired data analysis is a statistical approach that is used when the data collected come in pairs, often representing 'before and after' scenarios, or in this case, outcomes from two treatments given over the same subject or under the same conditions. Each pair is considered to be connected or matched, so differences within pairs can give insight into the treatment effect by reducing variability not due to treatment.

This method is particularly useful when dealing with crossover studies in clinical research or matched case-control studies. The focus is on the comparison within the pair, rather than between different individual units. Such a paired design improves the study's efficiency and controls for confounding factors that might vary between subjects but are constant within each subject pair.

In our textbook exercise, the paired data comes from the recovery percentages of patients from the same hospital being treated with two different drugs, Drug A and Drug B. The paired data comparison will shed light on whether there is evidence to suggest a true difference in the effectiveness of these drugs, or if any observed differences might be attributed to chance.