Exam Pattern: Paired Power Planning in R

This note abstracts the makeup midterm question where the main challenge is choosing the correct R power calculation for a paired design.

Source Question

  • STA305 Makeup W2026, Question 3(a)

Setup

The design is paired: each twin pair contributes one treated unit and one control unit. The meaningful target is the within-pair difference, not the marginal variance of responses under treatment and control separately.

If the clinically meaningful difference is $\delta$ and the standard deviation of the paired differences is $\sigma_D$, then power planning should be done on the differences.

Core Decision Rule

Use a paired power calculation when:

  • the experimental units are matched pairs
  • inference is on within-pair differences
  • the question gives the standard deviation of the differences

Do not use a two-sample calculation based on the marginal standard deviation when the design is paired.

R Template

power.t.test(delta = 2.5, sd = 2,
             sig.level = 0.05,
             power = 0.85,
             type = "paired",
             alternative = "one.sided")

Why This Is the Right Output

  • delta = 2.5 because that is the smallest clinically meaningful paired difference
  • sd = 2 because the question gives the standard deviation of the within-pair differences
  • type = "paired" because treatment is assigned within each pair
  • alternative = "one.sided" because the wording is directional

Common Wrong Choices

  • using sd = 3, the marginal standard deviation rather than the difference SD
  • using type = "two.sample" as if the twin data were independent groups
  • using a two-sided alternative when the scientific claim is one-sided

Exam Checklist

  • Identify whether the design is paired before touching R
  • Match sd to the variance of the paired differences
  • Match type to the design
  • Match alternative to the scientific wording