Question

Below are diastolic blood pressures \((\mathrm{mm} \mathrm{Hg})\) of ten patients before and after treatment for high blood pressure. Test the hypothesis that the treatment has no effect on blood pressure using a Wilcoxon signed-rank test, (a) using the exact significance level and (b) using a normal approximation. Discuss briefly. \(\begin{array}{llrrrrrrrrr}\text { Before } & 94 & 105 & 101 & 106 & 118 & 107 & 96 & 102 & 114 & 95 \\ \text { After } & 96 & 96 & 95 & 103 & 105 & 111 & 86 & 90 & 107 & 84\end{array}\)

Short answer

Exact test: T = 52 > critical value 8, do not reject null. Normal approximation: Z = 2.85, reject null hypothesis.

Step-by-step solution

Calculate the Differences

First, we need to find the difference between the before and after values for each patient. \[\text{Differences} = \text{Before} - \text{After}\]The differences are: \[0, 9, 6, 3, 13, -4, 10, 12, 7, 11\]

Assign Ranks to Absolute Differences

Next, we calculate the absolute differences and rank them in ascending order, ignoring the zero difference.The absolute differences are: \[9, 6, 3, 13, 4, 10, 12, 7, 11\]Rank these values: \[3 (3), 4 (6), 5 (7), 6 (9), 7 (10), 8 (11), 9 (12), 10 (13)\]Assign these ranks back to their respective differences.

Apply Sign Ranks

The ranks need their original signs. For differences we get: \[-4: 3.5, 3: 2, 6: 4.5, 10: 6.5, 12: 8.5, 7: 5.5, 11: 7.5, 9: 5, 13: 9, \] and the zero value gets no rank because it contributes nothing to the test statistic.

Calculate Test Statistic T

Now, we calculate the test statistic using the sum of the positive signed ranks:\[T = 3.5 + 2 + 4.5 + 6.5 + 8.5 + 5.5 + 7.5 + 5 + 9 = 52\]

Determine Critical Value from Exact Table

For a Wilcoxon signed-rank test with 9 pairs of data and no ties, consult an exact significance table for a two-tailed test with a significance level of 0.05. The critical value for \(n=9\) at \(\alpha=0.05\) is 8.

Compare T with Critical Value

Since \(T = 52\) is greater than the critical value of 8, we do not reject the null hypothesis based on the exact significance level.

Use Normal Approximation

We use the normal approximation for larger samples (here \(n>25\)). The test statistic is approximated to a Z-score. Calculate the expected value (mean) and standard deviation:\[\mu_T = \frac{n(n + 1)}{4}, \sigma_T = \sqrt{\frac{n(n + 1)(2n + 1)}{24}}\]For \(n = 9\), this gives \(\mu_T = 22.5\) and \(\sigma_T \approx 10.34\).

Calculate Z-score

Calculate Z-score:\[Z = \frac{T - \mu_T}{\sigma_T} = \frac{52 - 22.5}{10.34} \approx 2.85\]Using the standard normal distribution table, \(Z = 2.85\) corresponds to a p-value that is less than 0.05.

Conclusion

Based on normal approximation, the Z-score suggests rejecting the null hypothesis, concluding that the treatment affects blood pressure.

Key concepts

Hypothesis Testing

Hypothesis testing is a fundamental part of statistical science. It enables us to make decisions based on data. Basically, you pose an assumption, called the null hypothesis, which you try to challenge with your data. In the context of the Wilcoxon Signed-Rank Test, the null hypothesis typically states that there is no effect or difference in the data before and after treatment.

When you perform hypothesis testing, there are two outcomes:

• Reject the null hypothesis, meaning there's an effect or difference.
• Fail to reject the null hypothesis, implying no effect or difference.

After collecting and processing the data, you calculate a test statistic, which helps determine the probability of observing your data, assuming the null hypothesis is true. If this probability (the p-value) is very low, you may reject the null hypothesis.

Statistical Significance

Statistical significance relates to the likelihood that a result is not due to chance. When performing the Wilcoxon Signed-Rank Test, you look for statistical significance to decide whether the change between two conditions (such as before and after treatment) is meaningful.

A result is statistically significant if the p-value is below a predetermined threshold, usually 0.05. This means there is less than a 5% chance that the observed differences are due to random chance. Essentially, statistical significance helps us understand if we are likely to see similar results in other studies or contexts. It's a crucial step to ensure the reliability of our conclusions.

Normal Approximation

Normal approximation is a method used when dealing with larger datasets and applies to tests like the Wilcoxon Signed-Rank Test. For small sample sizes, exact tests are usually performed. However, when the sample size is larger (often considered above 25), calculating exact test values can become complex.

Normal approximation simplifies this by using the normal distribution to approximate the distribution of test statistics. This involves calculating the mean and standard deviation of the test statistic under the null hypothesis. With these values, a Z-score is calculated. This score tells you where your data lies on this distribution. You then compare the Z-score to critical values to decide whether the result is statistically significant.

Critical Value

The critical value is a point or threshold on a statistical distribution. It helps you decide if you should reject the null hypothesis in favor of the alternative hypothesis. In hypothesis tests like the Wilcoxon Signed-Rank Test, critical values are determined by the chosen significance level (such as 0.05) and the sample size.

When using tables, the critical value corresponds to the point beyond which the probability of finding a test statistic under the null hypothesis is less than the significance level. If your test statistic is greater than this critical value, it suggests that your results are unlikely under the null hypothesis. In turn, leading you to potentially reject it.

Understanding critical values is an essential part of making valid statistical decisions. They serve as the benchmark against which your test statistics are measured to interpret your data correctly.