24.4 Nested Designs

A nested design occurs when one factor is entirely contained within another. This differs from a crossed design, where all levels of one factor are present across all levels of another factor.

• Crossed Design: If Factor B is crossed with Factor A, then each level of Factor B appears at every level of Factor A.
• Nested Design: If Factor B is nested within Factor A, then each level of Factor B is unique to a particular level of Factor A.

Thus, if Factor B is nested within Factor A:

• Level 1 of B within A = 1 has nothing in common with

• Level 1 of B within A = 2.

Types of Factors

• Classification Factors: Factors that cannot be manipulated (e.g., geographical regions, subjects).
• Experimental Factors: Factors that are randomly assigned in an experiment.

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24.4.1 Two-Factor Nested Design

We consider a nested two-factor model where:

• Factor A has $a$ levels.

• Factor B is nested within Factor A, with $b$ levels per level of A.

• Both factors are fixed.

• All treatment means are equally important.

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The mean response at level $i$ of Factor A:

$$\mu_{i.} = \frac{1}{b} \sum_j \mu_{ij}$$

The main effect of Factor A:

$$\alpha_i = \mu_{i.} - \mu_{..}$$

where:

$$\mu_{..} = \frac{1}{ab} \sum_i \sum_j \mu_{ij} = \frac{1}{a} \sum_i \mu_{i.}$$

with the constraint:

$$\sum_i \alpha_i = 0$$

The nested effect of Factor B within A is denoted as $\beta_{j(i)}$, where:

$$\begin{aligned} \beta_{j(i)} &= \mu_{ij} - \mu_{i.} \\ &= \mu_{ij} - \alpha_i - \mu_{..} \end{aligned}$$

with the restriction:

$$\sum_j \beta_{j(i)} = 0, \quad \forall i = 1, \dots, a$$

Since $\beta_{j(i)}$ is the specific effect of the $j$-th level of factor $B$ nested within the $i$-th level of factor $A$, the full model can be written as:

$$\mu_{ij} = \mu_{..} + \alpha_i + \beta_{j(i)}$$

or equivalently:

$$\mu_{ij} = \mu_{..} + (\mu_{i.} - \mu_{..}) + (\mu_{ij} - \mu_{i.})$$

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The statistical model for a two-factor nested design is:

$$Y_{ijk} = \mu_{..} + \alpha_i + \beta_{j(i)} + \epsilon_{ijk}$$

where:

• $Y_{ijk}$ = response for the $k$-th observation when:

  • Factor A is at level $i$.

  • Factor B (nested within A) is at level $j$.

• $\mu_{..}$ = overall mean.

• $\alpha_i$ = main effect of Factor A (subject to: $\sum_i \alpha_i = 0$).

• $\beta_{j(i)}$ = nested effect of Factor B within A (subject to: $\sum_j \beta_{j(i)} = 0$ for all $i$).

• $\epsilon_{ijk} \sim iid N(0, \sigma^2)$ = random error.

Thus, the expected value and variance are:

$$\begin{aligned} E(Y_{ijk}) &= \mu_{..} + \alpha_i + \beta_{j(i)} \\ var(Y_{ijk}) &= \sigma^2 \end{aligned}$$

Note: There is no interaction term in a nested model, because Factor B levels are unique within each level of A.

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The least squares and maximum likelihood estimates:

Parameter	Estimator
$\mu_{..}$	$\bar{Y}_{...}$
$\alpha_i$	$\bar{Y}_{i..} - \bar{Y}_{...}$
$\beta_{j(i)}$	$\bar{Y}_{ij.} - \bar{Y}_{i..}$
$\hat{Y}_{ijk}$	$\bar{Y}_{ij.}$

The residual error:

$$e_{ijk} = Y_{ijk} - \bar{Y}_{ij.}$$

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The total sum of squares (SSTO) is partitioned as:

$$SSTO = SSA + SSB(A) + SSE$$

where:

$$\begin{aligned} \sum_i \sum_j \sum_k (Y_{ijk} - \bar{Y}_{...})^2 &= bn \sum_i (\bar{Y}_{i..} - \bar{Y}_{...})^2 + n \sum_i \sum_j (\bar{Y}_{ij.} - \bar{Y}_{i..})^2 \\ &+ \sum_i \sum_j \sum_k (Y_{ijk} - \bar{Y}_{ij.})^2 \end{aligned}$$

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24.4.1.1 ANOVA Table for Nested Designs

Source of Variation	SS	df	MS	E(MS)
Factor A	$SSA$	$a-1$	$MSA$	$\sigma^2 + bn \frac{\sum \alpha_i^2}{a-1}$
Factor B (A)	$SSB(A)$	$a(b-1)$	$MSB(A)$	$\sigma^2 + n \frac{\sum \beta_{j(i)}^2}{a(b-1)}$
Error	$SSE$	$ab(n-1)$	$MSE$	$\sigma^2$
Total	$SSTO$	$abn -1$

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24.4.1.2 Tests For Factor Effects

• Factor A:

  $$F = \frac{MSA}{MSB(A)} \sim F_{(a-1, a(b-1))}$$

  Reject $H_0$ if $F > f_{(1-\alpha; a-1, a(b-1))}$.

• Factor B within A:

  $$F = \frac{MSB(A)}{MSE} \sim F_{(a(b-1), ab(n-1))}$$

  Reject $H_0$ if $F > f_{(1-\alpha; a(b-1), ab(n-1))}$.

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24.4.1.3 Testing Factor Effect Contrasts

A contrast is a linear combination of factor level means:

$$L = \sum c_i \mu_i, \quad \text{where} \quad \sum c_i = 0$$

The estimated contrast:

$$\hat{L} = \sum c_i \bar{Y}_{i..}$$

The confidence interval for $L$:

$$\hat{L} \pm t_{(1-\alpha/2; df)} s(\hat{L})$$

where:

$$s^2(\hat{L}) = \sum c_i^2 s^2(\bar{Y}_{i..}), \quad \text{where} \quad s^2(\bar{Y}_{i..}) = \frac{MSE}{bn}, \quad df = ab(n-1)$$

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24.4.1.4 Testing Treatment Means

For treatment means, a similar approach applies:

$$L = \sum c_i \mu_{.j}, \quad \hat{L} = \sum c_i \bar{Y}_{ij}$$

The confidence limits for $L$:

$$\hat{L} \pm t_{(1-\alpha/2; (n-1)ab)} s(\hat{L})$$

where:

$$s^2(\hat{L}) = \frac{MSE}{n} \sum c_i^2$$

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24.4.2 Unbalanced Nested Two-Factor Designs

When Factor B has different levels for different levels of Factor A, the design is unbalanced.

$$\begin{aligned} Y_{ijk} &= \mu_{..} + \alpha_i + \beta_{j(i)} + \epsilon_{ijk} \\ \sum_{i=1}^2 \alpha_i &= 0, \quad \sum_{j=1}^3 \beta_{j(1)} = 0, \quad \sum_{j=1}^2 \beta_{j(2)} = 0 \end{aligned}$$

where:

• Factor A: $i = 1, 2$.

• Factor B (nested in A): $j = 1, \dots, b_i$.

• Observations: $k = 1, \dots, n_{ij}$.

Example case:

• $b_1 = 3, b_2 = 2$ (Factor B has different levels for A).

• $n_{11} = n_{13} = 2, n_{12} = 1, n_{21} = n_{22} = 2$.

• Parameters: $\alpha_1, \beta_{1(1)}, \beta_{2(1)}, \beta_{1(2)}$.

Constraints:

$$\alpha_2 = -\alpha_1, \quad \beta_{3(1)} = -\beta_{1(1)} - \beta_{2(1)}, \quad \beta_{2(2)} = -\beta_{1(2)}$$

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The unbalanced design can be modeled using indicator variables:

1. Factor A (School Level): $$X_1 = \begin{cases} 1 & \text{if observation from school 1} \\ -1 & \text{if observation from school 2} \end{cases}$$

2. Factor B (Instructor within School 1): $$X_2 = \begin{cases} 1 & \text{if observation from instructor 1 in school 1} \\ -1 & \text{if observation from instructor 3 in school 1} \\ 0 & \text{otherwise} \end{cases}$$

3. Factor B (Instructor within School 1): $$X_3 = \begin{cases} 1 & \text{if observation from instructor 2 in school 1} \\ -1 & \text{if observation from instructor 3 in school 1} \\ 0 & \text{otherwise} \end{cases}$$

4. Factor B (Instructor within School 1): $$X_4 = \begin{cases} 1 & \text{if observation from instructor 1 in school 1} \\ -1 & \text{if observation from instructor 2 in school 1} \\ 0 & \text{otherwise} \end{cases}$$

Using these indicator variables, the full regression model is:

$$Y_{ijk} = \mu_{..} + \alpha_1 X_{ijk1} + \beta_{1(1)} X_{ijk2} + \beta_{2(1)} X_{ijk3} + \beta_{1(2)} X_{ijk4} + \epsilon_{ijk}$$

where $X_1, X_2, X_3, X_4$ represent different factor effects.

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24.4.3 Random Factor Effects

If factors are random:

$$\begin{aligned} \alpha_1 &\sim iid N(0, \sigma^2_\alpha) \\ \beta_{j(i)} &\sim iid N(0, \sigma^2_\beta) \end{aligned}$$

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Expected Mean Squares for Random Effects

Mean Square	Expected Mean Squares (A Fixed, B Random)	Expected Mean Squares (A Random, B Random)
MSA	$\sigma^2 + n \sigma^2_\beta + bn \frac{\sum \alpha_i^2}{a-1}$	$\sigma^2 + bn \sigma^2_{\alpha} + n \sigma^2_\beta$
MSB(A)	$\sigma^2 + n \sigma^2_\beta$	$\sigma^2 + n \sigma^2_\beta$
MSE	$\sigma^2$	$\sigma^2$

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F-Tests for Factor Effects

Factor	F-Test (A Fixed, B Random)	F-Test (A Random, B Random)
Factor A	$\frac{MSA}{MSB(A)}$	$\frac{MSA}{MSB(A)}$
Factor B(A)	$\frac{MSB(A)}{MSE}$	$\frac{MSB(A)}{MSE}$

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Another way to increase precision in treatment comparisons is by adjusting for covariates using regression models. This is called Analysis of Covariance (ANCOVA).

Why use ANCOVA?

• Reduces variability by accounting for covariate effects.

• Increases statistical power by removing nuisance variation.

• Combines ANOVA and regression for more precise comparisons.

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