Paired or independent two-sample

• Two-sample (independent) sampling takes a random sample from the population and randomly assigns them into two groups
  • Example: Assigning two groups of patients into trial and placebo groups for a drug safety test
• Paired sampling randomly pairs up individuals in the population (sampling) and assigns each of the individual to a group
  • Example: Testing effects of socioeconomic condition on dietary preferences by comparing identical twins raised in separate adoptive families

• Sometimes the same research question can have either design
  • Example question: Does clear-cutting a forest affect the number of salamanders present?

<aside> ⁉️ Challenge 1 Describe how you can test the above question both with a independent and a paired study design. Which one might be better?

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• Each has its own benefits based on the study design, however since the sampling unit in a paired study design is the pair, two measurements made on each pair is reduced to a single value
  • Estimate the mean of the differences

Paired comparison of means

• Reduces the variation among sampling units that are non-dependent (i.e. the treatment)
  • Increases the power and precision of estimates
• Paired t-test is used to test whether the mean difference of paired measurements equals a specific value (essentially a one-sample t-test of mean difference)
  • $H_0$: the mean difference is $x$.
  • $H_1$: the mean difference is not $x$.
• Assumptions
  • Sampling units (pairs) are randomly selected from the population
  • Paired differences have a normal distribution in the population
    • Individual measurements from the pairs can have any distribution

Antibody production rate before and after implantation of testosterone in 13 red-winged blackbirds

Differences in antibody production rate in blackbirds before and after implantation of testosterone

Two-sample comparison of means

• If the variable $Y$ is normally distributed in both populations (i.e. $Y_1\sim\mathcal{N}(\mu_1,\sigma^2_1)$ and $Y_2\sim\mathcal{N}(\mu_1,\sigma^2_1)$), the sampling distribution for the difference between the sample means is also normal

$$ Y_1\sim\mathcal{N}(\mu_1,\sigma^2_1)\land Y_2\sim\mathcal{N}(\mu_1,\sigma^2_1)\Rightarrow \bar{Y_2}-\bar{Y_1}\sim\mathcal{N}(\mu_3,\sigma^2_3) $$