A Matched Pairs Confidence Interval (or Paired t-interval) is used when data consists of two measurements taken from the same individual or from two individuals who are closely linked (e.g., twins or siblings).
The parameter of interest is the population mean difference, denoted as , which represents the average difference between the two measurements across the entire population.
The process involves calculating the difference () for every pair in the sample and then performing inference on this single list of differences.
The point estimate for the interval is the sample mean difference, , which is the average of all individual differences calculated from the sample pairs.
The fundamental logic of matched pairs is the reduction of variability. By comparing an individual to themselves (or a similar counterpart), we eliminate 'between-subject' variation, making it easier to detect a true treatment effect.
Because we are analyzing a single set of differences, the sampling distribution of the mean difference follows a t-distribution with degrees of freedom, where is the number of pairs.
The standard error of the mean difference, , accounts for the spread of the differences in the sample and the sample size.
Step 1: Calculate Differences: Subtract the two values for each pair. Ensure the direction of subtraction (e.g., Before - After) is consistent for all pairs.
Step 2: Verify Conditions: Check for Randomness (random sample or assignment), Independence (10% rule if sampling without replacement), and Normality (the population of differences is normal or ).
Step 3: Calculate Statistics: Find the sample mean difference and the sample standard deviation of the differences .
Step 4: Determine Critical Value: Use a t-table or calculator to find based on the desired confidence level and .
Step 5: Construct the Interval: Apply the formula:
Formula:
| Feature | Matched Pairs (Paired t) | Two-Sample (Independent t) |
|---|---|---|
| Data Source | One group measured twice or linked pairs | Two separate, unrelated groups |
| Variable | Mean of differences () | Difference of means () |
| Degrees of Freedom | (where is number of pairs) | Complex (approx. smaller ) |
| Focus | Within-pair change | Between-group comparison |
Incorrect Sample Size: A common error is using as the sample size. In matched pairs, is the number of pairs, not the total number of observations.
Normality Check: Students often check the normality of the two original groups separately. You must check the normality of the differences specifically.
Independence Confusion: While the pairs themselves must be independent of other pairs, the two observations within a pair are intentionally dependent.