Day 8 Applied examples
June 10th, 2026
8.3 Some topics that will arise today
Consider the general linear model for a specific case of an experiment with a two-wat factorial in a CRD:
\[y_{ijk} \sim N(\mu_{ijk}, \sigma^2), \\ \mu_{ijk} = \mu + A_i +C_j + AC_{ij},\]
where \(A_i\) is the effect of the \(i\)th level of \(A\), \(C_j\) is the effect of the \(j\)th level of \(C\), and \(AC_{ij}\) is the interaction between the \(i\)th level of \(A\) and the \(j\)th level of \(C\).
Oftentimes, the objective of our study is to make mean comparisons: essentially, we’re interested in the mean difference as much (or more) as the treatment mean estimate. We can consider the mean difference is \(\mu_i - \mu_{i'}\), where \(i \neq i'\). We could find different approaches to comparing said means:
- Evaluate the mean difference \(\mu_i - \mu_{i'}\) and its 95% confidence interval.
- Do a hypothesis test where \(H_0: \mu_i - \mu_{i'} = 0\) and \(H_a: \mu_i - \mu_{i'} \neq 0\)
The objective of our study can also justo be obtaining treatment mean estimates. To describe them, we probably want to have point estimates and 95% confidence intervals.
Considering all the above:
- How to obtain a confidence interval: \(CI_{\hat{\theta}, 95\%} = \hat{\theta} \pm se(\hat{\theta}) \cdot t_{\frac{\alpha}{2}, dfe}\).
- Let’s discuss what that \(dfe\) may be.