Confidence Intervals for Differences in Matched Pairs

Introduction

Key Concepts

Understanding Matched Pairs

Matched pairs, also known as paired samples or dependent samples, consist of two related observations. These pairs are typically formed by selecting subjects that share a specific characteristic or by measuring the same subjects under two different conditions. The primary objective is to control for variability between subjects, thereby isolating the effect of the treatment or intervention.

Why Use Matched Pairs?

Matched pairs are employed to reduce the impact of confounding variables. By pairing similar subjects, researchers can focus on the differences attributable to the treatment rather than external factors. This approach increases the statistical power of the test, allowing for more accurate and reliable conclusions.

Confidence Interval Basics

A confidence interval (CI) provides a range of values within which the true population parameter is expected to lie with a certain level of confidence, typically 95%. For matched pairs, the CI estimates the mean difference between the paired observations. The formula for a confidence interval for matched pairs is: $$ \bar{d} \pm t^* \left( \frac{s_d}{\sqrt{n}} \right) $$ where:

• $$\bar{d}$$: The sample mean of the differences
• $$t^*$$: The critical value from the t-distribution
• $$s_d$$: The sample standard deviation of the differences
• $$n$$: The number of pairs

Step-by-Step Calculation

To construct a confidence interval for the difference in matched pairs, follow these steps:

1. Calculate the differences: For each pair, subtract one observation from the other to find the difference.
2. Find the sample mean difference ($$\bar{d}$$): Sum all the differences and divide by the number of pairs.
3. Determine the sample standard deviation of differences ($$s_d$$): Use the standard deviation formula on the list of differences.
4. Select the confidence level: Commonly 95%, which corresponds to a $$t^*$$ value from the t-distribution table with $$n-1$$ degrees of freedom.
5. Compute the margin of error: Multiply $$t^*$$ by $$\frac{s_d}{\sqrt{n}}$$.
6. Construct the interval: Add and subtract the margin of error from $$\bar{d}$$ to get the lower and upper bounds.

Assumptions

• Differences are independent and identically distributed.
• The population of differences is approximately normally distributed, especially important for small sample sizes.

Example

Suppose a researcher wants to determine if a new teaching method affects student performance. She measures the scores of 10 students before and after applying the method.

• Differences ($$d_i$$): After − Before
• Sample mean difference ($$\bar{d}$$): 5 points
• Sample standard deviation ($$s_d$$): 2 points
• Number of pairs ($$n$$): 10

Interpreting the Confidence Interval

The confidence interval provides a range where the true mean difference is likely to fall. In our example, we are 95% confident that the new teaching method increases student scores by between 3.57 and 6.43 points. This interval does not contain zero, indicating a statistically significant improvement.

Comparing Independent and Matched Pairs Confidence Intervals

While both types of confidence intervals aim to estimate population parameters, they differ in methodology and assumptions. Matched pairs control for internal variability by pairing related subjects, making them more powerful in detecting differences when the pairing is effective. In contrast, independent samples do not have this control and may require larger sample sizes to achieve similar power.

Common Mistakes

• Ignoring the pairing and treating the samples as independent.
• Using the wrong critical value (e.g., z instead of t for small samples).
• Miscalculating the standard deviation of differences.
• Assuming normality without verifying, especially in small samples.

When to Use Confidence Intervals for Matched Pairs

• Before-and-after studies, such as testing a new drug's effectiveness.
• Comparing twins in biometric studies.
• Evaluating pre-test and post-test scores in educational research.

Advantages

• Controls for subject variability, enhancing the test's sensitivity.
• Generally requires smaller sample sizes compared to independent samples.
• Provides a clearer inference about the treatment effect.

Limitations

• Requires appropriate pairing, which may not always be feasible.
• Sensitivity to outliers in the differences.
• Assumes the differences are normally distributed, which may not hold in all cases.

Comparison Table

Summary and Key Takeaways