Let's break down **Question 14.7** step by step: ### **Three-Stage Nested Design:** Suppose you have three factors: - \(A\): Fixed (levels: \(a\)) - \(B\) within \(A\): Random (levels: \(b\)) - \(C\) within \(B\): Random (levels: \(c\)) - \(n\): Number of observations per combination #### **Model:** \[ Y_{ijkl} = \mu + \alpha_i + \beta_{j(i)} + \gamma_{k(ij)} + \epsilon_{l(ijk)} \] where: - \(\alpha_i\): Effect of fixed factor \(A\) - \(\beta_{j(i)}\): Random effect of \(B\) nested within \(A\), \( \sim N(0, \sigma^2_B)\) - \(\gamma_{k(ij)}\): Random effect of \(C\) nested within \(B\) and \(A\), \( \sim N(0, \sigma^2_C)\) - \(\epsilon_{l(ijk)}\): Error, \( \sim N(0, \sigma^2)\) All random effects are independent and have mean 0. --- ### **Expected Mean Squares (EMS):** 1. **Total Variance:** \[ \text{Var}(Y_{ijkl}) = \sigma^2 + \sigma^2_C + \sigma^2_B \] 2. **EMS for each source:** #### **For Error (Within C):** \[ \text{EMS}_E = \sigma^2 \] #### **For C within B within A:** \[ \text{EMS}_C = \sigma^2 + n \sigma^2_C \] #### **For B within A:** \[ \text{EMS}_B = \sigma^2 + n \sigma^2_C + nc \sigma^2_B \] #### **For A:** \[ \text{EMS}_A = \sigma^2 + n \sigma^2_C + nc \sigma^2_B + nbc \frac{\sum (\alpha_i - \bar\alpha)^2}{a-1} \] But since \(A\) is fixed, the last term represents the variation due to the fixed effect, not a variance component. --- ### **Formulas for Estimating Variance Components:** From the EMS, **solve for variance components**: 1. **Estimate \(\sigma^2\):** \[ \hat{\sigma}^2 = \text{MS}_E \] 2. **Estimate \(\sigma^2_C\):** \[ \hat{\sigma}^2_C = \frac{\text{MS}_C - \text{MS}_E}{n} \] 3. **Estimate \(\sigma^2_B\):** \[ \hat{\sigma}^2_B = \frac{\text{MS}_B - \text{MS}_C}{nc} \] --- ### **Summary Table:** | Source | df | EMS | Variance Estimator | |----------------|---------------------|-------------------------------------------------------------------|----------------------------------------------| | A (fixed) | \(a-1\) | \(\sigma^2 + n\sigma^2_C + nc\sigma^2_B + nbc \text{(A effect)}\) | — (fixed) | | B(A) | \(a(b-1)\) | \(\sigma^2 + n\sigma^2_C + nc\sigma^2_B\) | \(\frac{\text{MS}_B - \text{MS}_C}{nc}\) | | C(B(A)) | \(ab(c-1)\) | \(\sigma^2 + n\sigma^2_C\) | \(\frac{\text{MS}_C - \text{MS}_E}{n}\) | | Error | \(abc(n-1)\) | \(\sigma^2\) | \(\text{MS}_E\) | --- ### **Key Points:** - **A** is fixed: only estimate its effect, not a variance component. - **B** and **C** are random: estimate their variance components from the expected mean squares. - **Assume restricted (REML) form:** above formulas are typical ANOVA estimators; for REML, use unbiased estimators based on these differences. --- **If you need the full derivation or a worked example with specific numbers, let me know!**