Absolutely! Let’s break down and **explain step by step** how to answer this question about a **three-stage nested design** in ANOVA, with some rendered tables for clarity. --- ### **1. Understanding the Model** - **Three-stage nested design**: A ⊃ B ⊃ C (C is nested in B, which is nested in A) - **A is fixed**: Levels of A are chosen deliberately (not random) - **B and C are random** - **Balanced**: Same number of observations in each group #### **Model Equation** \[ Y_{ijkl} = \mu + \alpha_i + \beta_{j(i)} + \gamma_{k(ij)} + \epsilon_{l(ijk)} \] Where: - \(i = 1, ..., a\) (levels of A) - \(j = 1, ..., b\) (levels of B within A) - \(k = 1, ..., c\) (levels of C within B and A) - \(l = 1, ..., n\) (replicates within C, B, A) - \(\alpha_i\) = fixed effect of A - \(\beta_{j(i)}\) = random effect of B within A, \(\sim N(0, \sigma^2_B)\) - \(\gamma_{k(ij)}\) = random effect of C within B and A, \(\sim N(0, \sigma^2_C)\) - \(\epsilon_{l(ijk)}\) = random error, \(\sim N(0, \sigma^2)\) --- ### **2. Expected Mean Squares Table** | Source | df | Mean Square (MS) | Expected Mean Square (EMS) | |----------------|----------------------|--------------------------|------------------------------------------------------------| | A | \(a-1\) | \(MS_A\) | \(\sigma^2 + n\sigma^2_C + nc\sigma^2_B + ncb \text{(fixed effect)}\) | | B(A) | \(a(b-1)\) | \(MS_{B(A)}\) | \(\sigma^2 + n\sigma^2_C + nc\sigma^2_B\) | | C(B(A)) | \(ab(c-1)\) | \(MS_{C(B(A))}\) | \(\sigma^2 + n\sigma^2_C\) | | Error | \(abc(n-1)\) | \(MS_{Error}\) | \(\sigma^2\) | **Note:** For a fixed effect, the EMS for A does not include a variance component for A. --- ### **3. **Formulas for Estimating Variance Components** Let’s solve for each variance component in terms of the observed mean squares: #### **a. Error Variance (\(\sigma^2\)):** \[ \hat{\sigma}^2 = MS_{Error} \] #### **b. Variance due to C (\(\sigma^2_C\)):** \[ \hat{\sigma}^2_C = \frac{MS_{C(B(A))} - MS_{Error}}{n} \] #### **c. Variance due to B (\(\sigma^2_B\)):** \[ \hat{\sigma}^2_B = \frac{MS_{B(A)} - MS_{C(B(A))}}{nc} \] --- ### **4. Table with Formulas** | Component | Formula | |-------------------|----------------------------------------------------------| | \(\sigma^2\) | \(MS_{Error}\) | | \(\sigma^2_C\) | \(\frac{MS_{C(B(A))} - MS_{Error}}{n}\) | | \(\sigma^2_B\) | \(\frac{MS_{B(A)} - MS_{C(B(A))}}{nc}\) | --- ### **5. Summary Table** | Source | df | Mean Square | Expected Mean Square (EMS) | Variance Component Estimate | |----------------|-------------|-----------------|----------------------------------|--------------------------------------------| | A (fixed) | \(a-1\) | \(MS_A\) | \(\sigma^2 + n\sigma^2_C + nc\sigma^2_B + ncb\;[\text{effect of A}]\) | N/A (Fixed) | | B(A) (random) | \(a(b-1)\) | \(MS_{B(A)}\) | \(\sigma^2 + n\sigma^2_C + nc\sigma^2_B\) | \(\frac{MS_{B(A)} - MS_{C(B(A))}}{nc}\) | | C(B(A)) (random) | \(ab(c-1)\)| \(MS_{C(B(A))}\)| \(\sigma^2 + n\sigma^2_C\) | \(\frac{MS_{C(B(A))} - MS_{Error}}{n}\) | | Error | \(abc(n-1)\)| \(MS_{Error}\) | \(\sigma^2\) | \(MS_{Error}\) | --- ## **Summary** - **Model**: \(Y_{ijkl} = \mu + \alpha_i + \beta_{j(i)} + \gamma_{k(ij)} + \epsilon_{l(ijk)}\) - **EMS Table**: Shows how each level’s mean square is built from the variance components. - **Variance components**: Estimated by equating observed mean squares to their expectations and solving. --- If you want a **numerical example** or a **diagram**, let me know!