Let's break down the question step by step and provide a clear, **rendered table** to help you understand a **three-stage nested design** with **expected mean squares** and **variance component estimates**. --- ### **Problem Setup** - **Three-stage nested design:** - **A**: Fixed factor (e.g., Treatment) - **B(A)**: Random factor nested within A (e.g., Batch within Treatment) - **C(B)**: Random factor nested within B (e.g., Sample within Batch) - The model (for observation \( y_{ijkl} \)) is: \[ y_{ijkl} = \mu + \alpha_i + \beta_{j(i)} + \gamma_{k(j(i))} + \epsilon_{l(k(j(i)))} \] - \( \alpha_i \): Fixed effect of level \(i\) of A - \( \beta_{j(i)} \): Random effect of level \(j\) of B within A - \( \gamma_{k(j(i))} \): Random effect of level \(k\) of C within B - \( \epsilon_{l(k(j(i)))} \): Random error --- ### **Balance and Replication** Suppose: - \(a\) levels of A - \(b\) levels of B within each A - \(c\) levels of C within each B - \(n\) replicates within each C --- ## **Expected Mean Squares (EMS) Table** | Source | Degrees of Freedom | Expected Mean Square | |--------------------------|-------------------|---------------------------------------------| | A | \(a-1\) | \(\sigma^2_e + n\sigma^2_\gamma + nc\sigma^2_\beta + ncb \cdot \text{(A effect)}\) | | B (A) | \(a(b-1)\) | \(\sigma^2_e + n\sigma^2_\gamma + nc\sigma^2_\beta \) | | C (B) | \(ab(c-1)\) | \(\sigma^2_e + n\sigma^2_\gamma \) | | Error | \(abc(n-1)\) | \(\sigma^2_e\) | Where: - \(\sigma^2_\beta\): Variance component for B(A) - \(\sigma^2_\gamma\): Variance component for C(B) - \(\sigma^2_e\): Error variance --- ## **Variance Component Estimates** You estimate variance components by equating observed mean squares to their expectations (solving for each variance): ### **Formulas:** \[ \begin{align*} \hat{\sigma}^2_e &= MS_{Error} \\ \hat{\sigma}^2_\gamma &= \frac{MS_{C(B)} - MS_{Error}}{n} \\ \hat{\sigma}^2_\beta &= \frac{MS_{B(A)} - MS_{C(B)}}{nc} \\ \end{align*} \] - For A (fixed effect), you do **not** estimate a variance component (since it's fixed). --- ## **Summary Table (Rendered)** | Source | D.F. | Expected Mean Square | Variance Estimate Formula | |----------|--------------|-------------------------------------------|--------------------------------------------| | A | \(a-1\) | \(\sigma^2_e + n\sigma^2_\gamma + nc\sigma^2_\beta + ncb \cdot (\text{A effect})\) | N/A (Fixed) | | B(A) | \(a(b-1)\) | \(\sigma^2_e + n\sigma^2_\gamma + nc\sigma^2_\beta\) | \(\hat{\sigma}^2_\beta = \frac{MS_{B(A)} - MS_{C(B)}}{nc}\) | | C(B) | \(ab(c-1)\) | \(\sigma^2_e + n\sigma^2_\gamma\) | \(\hat{\sigma}^2_\gamma = \frac{MS_{C(B)} - MS_{Error}}{n}\) | | Error | \(abc(n-1)\) | \(\sigma^2_e\) | \(\hat{\sigma}^2_e = MS_{Error}\) | --- ### **Restricted Model** The "restricted form" means that the random effects have mean zero and are independent. --- ### **Summary** 1. **Write the model** with fixed/random effects. 2. **Build the EMS table** as above. 3. **Use the mean squares** from your ANOVA table to estimate variance components with the given formulas. If you need an **example with numbers** or a **step-by-step calculation**, just ask!