Paired Data

When to Use Paired Analysis

Use a paired t-test when:

• The same subjects are measured twice (before/after)
• Subjects are matched in pairs based on similar characteristics
• Each observation in one group has a natural pairing with an observation in the other group

Setting Up the Paired t-Test

1. Calculate differences: $d_i = x_{\text{after}} - x_{\text{before}}$ (or $d_i = x_{1i} - x_{2i}$)

2. State hypotheses about the mean difference $\mu_d$:

   • $H_0: \mu_d = 0$ (no difference on average)
   • $H_a: \mu_d \neq 0$ (or $>$ or $<$)

3. Compute: $\bar{d} = \frac{\sum d_i}{n}, \quad s_d = \sqrt{\frac{\sum(d_i - \bar{d})^2}{n-1}}$

4. Test statistic: $t = \frac{\bar{d} - 0}{s_d / \sqrt{n}}, \quad df = n - 1$

Conditions for Paired t-Test

1. Random: Pairs are randomly selected or treatments randomly assigned within pairs
2. 10%: Number of pairs $< 10\%$ of all possible pairs
3. Normal: The differences are approximately Normal (check with graph of differences)

Paired t-Confidence Interval

$\bar{d} \pm t^* \frac{s_d}{\sqrt{n}}$

Paired vs. Two-Sample: How to Decide

| Feature | Paired | Two-Sample | |---------|--------|------------| | Data structure | Natural pairing exists | Independent groups | | Examples | Before/after, twins, left/right | Men vs. women, drug vs. placebo (different people) | | Analyze | Differences | Separate groups | | Advantage | Controls for subject variability | Simpler design |

Why Pairing Helps

Pairing reduces variability by controlling for individual differences. Each subject serves as their own control, so person-to-person variation is removed.

Example: Testing a new study method

• Paired design: Same students take two tests (before and after method)
• Two-sample design: Different students use different methods
• The paired design is more powerful because it eliminates student-to-student variability

Common Mistakes

1. Using a two-sample test when data is paired
2. Forgetting to check Normality of the differences (not the original data)
3. Computing differences inconsistently (always subtract in the same direction)

AP Tip: If the data has a natural pairing, you MUST use a paired t-test. Using a two-sample t-test on paired data is incorrect and will cost points.