Absolutely! Let’s break down and explain **Problem 14.7** step by step. --- ### **A. Background: Three-Stage Nested Design** Suppose your factors are: - **A**: Fixed main factor (e.g., schools) - **B**: Random, nested in A (e.g., classes within schools) - **C**: Random, nested in B (e.g., students within classes) The model for the ith level of A, jth level of B (within A), kth level of C (within B), and lth replication is: \[ Y_{ijkl} = \mu + \alpha_i + \beta_{j(i)} + \gamma_{k(ij)} + \varepsilon_{l(ijk)} \] - \(\mu\): overall mean - \(\alpha_i\): fixed effect of \(A\) (fixed) - \(\beta_{j(i)}\): random effect of \(B\) (nested in \(A\)), \(N(0, \sigma^2_B)\) - \(\gamma_{k(ij)}\): random effect of \(C\) (nested in \(B\)), \(N(0, \sigma^2_C)\) - \(\varepsilon_{l(ijk)}\): random error, \(N(0, \sigma^2)\) --- ### **B. Expected Mean Squares (EMS)** Suppose: - \(a\): number of levels of A - \(b\): number of levels of B within each A - \(c\): number of levels of C within each B - \(n\): number of replications per C #### **ANOVA Structure** | Source | d.f. | Mean Square | |------------------|---------------------|--------------| | A | \(a-1\) | \(MS_A\) | | B(A) | \(a(b-1)\) | \(MS_{B(A)}\)| | C(B(A)) | \(ab(c-1)\) | \(MS_{C(B(A))}\)| | Error | \(abc(n-1)\) | \(MS_E\) | #### **Expected Mean Squares** - **Error:** \(\mathbb{E}[MS_E] = \sigma^2\) - **C(B(A))**: \(\mathbb{E}[MS_{C(B(A))}] = \sigma^2 + n\sigma^2_C\) - **B(A)**: \(\mathbb{E}[MS_{B(A)}] = \sigma^2 + n\sigma^2_C + nc\sigma^2_B\) - **A**: \(\mathbb{E}[MS_A] = \sigma^2 + n\sigma^2_C + nc\sigma^2_B + nbc\frac{\sum_{i=1}^a (\alpha_i - \bar{\alpha})^2}{a-1}\) (since A is fixed, this term is not a variance component, but a function of the fixed effects) --- ### **C. Estimation of Variance Components** We use the differences between mean squares to estimate variance components: 1. **Estimate \(\sigma^2\) (Error variance):** \[ \hat{\sigma}^2 = MS_E \] 2. **Estimate \(\sigma^2_C\):** \[ \hat{\sigma}^2_C = \frac{MS_{C(B(A))} - MS_E}{n} \] 3. **Estimate \(\sigma^2_B\):** \[ \hat{\sigma}^2_B = \frac{MS_{B(A)} - MS_{C(B(A))}}{nc} \] --- ### **D. Notes on the Mixed Model (Restricted Form)** - The fixed factor (A) means its effect is not a variance component, but rather a specific set of fixed values. - The variance components (\(\sigma^2_B, \sigma^2_C, \sigma^2\)) are for the random effects and error. --- ## **Summary Table** | Component | Variance Component | Estimate | |--------------|------------------------|-------------------------------------------| | **Error** | \(\sigma^2\) | \(MS_E\) | | **C(B(A))** | \(\sigma^2_C\) | \(\frac{MS_{C(B(A))} - MS_E}{n}\) | | **B(A)** | \(\sigma^2_B\) | \(\frac{MS_{B(A)} - MS_{C(B(A))}}{nc}\) | --- **If you want to see a step-by-step algebraic derivation of the expected mean squares, just let me know!**