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.

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.


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:

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


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.


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:

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


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

\[ SSTO = SSA + SSB(A) + SSE \]

where:

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


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 \(SSTO\) \(abn -1\)

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


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:

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


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:

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


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:

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


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.


24.4.3 Random Factor Effects

If factors are random:

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


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 \(\sigma^2\) \(\sigma^2\)

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) \(\frac{MSB(A)}{MSE}\) \(\frac{MSB(A)}{MSE}\)

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.