Question

What three statistical tests are available for testing for a difference in location for two populations when the data are paired? What assumptions are required for each of these tests?

Short answer

Answer: The three statistical tests for testing a difference in location for two paired populations are the paired sample t-test, Wilcoxon signed-rank test, and sign test. The paired sample t-test assumes paired data, normality, independence, and interval or ratio scale data. The Wilcoxon signed-rank test assumes paired data, independence, and at least ordinal scale data. The sign test assumes paired data, independence, and randomness in the occurrence of positive and negative differences.

Step-by-step solution

Introduction to Paired Data

Paired data refers to a set of observations where each observation in one population is linked to a specific observation in the other population. This typically occurs in situations where the same subjects are measured before and after a treatment or when two related subjects are compared. To test for differences in location, or central tendency, between paired populations, we use the following statistical tests.

Test 1: Paired Sample T-Test

The paired sample t-test, also known as the dependent sample t-test, is used to determine whether the mean difference between two sets of paired observations is significantly different from zero. Assumptions for Paired Sample T-Test: 1. Paired data: The data must consist of pairs of measurements. 2. Normality: The differences between pairs should be approximately normally distributed. 3. Independence: Each pair of observations should be independent of one another. 4. Scale: The data should be interval or ratio scale.

Test 2: Wilcoxon Signed-Rank Test

The Wilcoxon signed-rank test is a non-parametric alternative to the paired sample t-test. It is used to test whether the median difference between two sets of paired observations is significantly different from zero. Assumptions for Wilcoxon Signed-Rank Test: 1. Paired data: The data must consist of pairs of measurements. 2. Independence: Each pair of observations should be independent of one another. 3. Ordinal scale: The data should be at least ordinal scale, which means that the differences between pairs can be meaningfully ranked.

Test 3: Sign Test

The sign test is another non-parametric test used for paired data. It tests whether the median difference between two sets of paired observations is significantly different from zero, but it only considers the sign (positive or negative) of the differences, not their magnitude. Assumptions for Sign Test: 1. Paired data: The data must consist of pairs of measurements. 2. Independence: Each pair of observations should be independent of one another. 3. Randomness: The positive and negative differences are assumed to occur randomly. In summary, the three statistical tests for testing a difference in location for two paired populations are the paired sample t-test, Wilcoxon signed-rank test, and sign test. Each test has its own set of assumptions that must be met in order to obtain valid results.

Key concepts

Paired Sample T-Test

When working with paired data, the Paired Sample T-Test serves as a powerful method to analyze the difference between two population means. This test is essential when the same group is tested before and after a particular intervention, or when comparing twins or matched individuals.

The procedure revolves around the mean of the differences between paired observations. If the mean difference significantly veers away from zero, it indicates a probable effect due to the intervention or a significant difference within the matched pairs. To ensure the reliability of the test, certain assumptions must be met:

• Paired data: Observations need to be connected in a meaningful way, such as pre-test and post-test scores.
• Normality: The differences between paired observations should follow a normal distribution.
• Independence: Each pair should provide unique information without being influenced by other pairs.
• Scale: The data must be on an interval or ratio scale, allowing for meaningful arithmetic operations.

Failure to meet these assumptions might warrant the use of a non-parametric test instead, such as the Wilcoxon Signed-Rank Test or the Sign Test.

Wilcoxon Signed-Rank Test

The Wilcoxon Signed-Rank Test steps in as a non-parametric counterpart to the Paired Sample T-Test, particularly beneficial when the normality assumption is not tenable. This test is invaluable for ordinal data or for instances where the distribution of differences is skewed.

Unlike its parametric counterpart that uses means, the Wilcoxon test concentrates on the median of differences. It involves ranking the absolute differences between pairs, assigning signs based on the direction of the difference, and computing a test statistic that reflects the sum of these signed ranks.

The assumptions for this test are:

• Paired data: The observations must be in pairs, just like in the t-test.
• Independence: The pairs of observations must not influence each other.
• Ordinal scale: The differences between observations should be rank-able, fitting at least an ordinal scale.

This test is more robust than the T-Test in terms of distribution requirements and is especially useful when dealing with non-numeric rating scales like surveys or questionnaires.

Sign Test

As one of the simplest non-parametric tests for paired data, the Sign Test holds its ground by merely considering the direction of the differences between paired observations, ignoring their magnitudes altogether. This attribute makes it highly useful when the data's scale is nominal or when ordinal rankings are not feasible.

The Sign Test primarily involves counting the number of positive differences versus negative ones. A significant imbalance in these counts can suggest a median difference that is not equal to zero. This method is resilient to outliers and skewed data because it is only concerned with the signs rather than the spread or central tendency of the observations.

Key assumptions of the Sign Test include:

• Paired data: Observations must be interconnected in pairs.
• Independence: Each pairing should be isolated from influences of others.
• Randomness: The occurrences of positive and negative differences should seem random and not systematic.

This test is often preferred when the data is at its crudest form and has minimal information about order or magnitude, but paired comparisons are still possible.