20.5 Nested Designs

Let \(\mu_{ij}\) be the mean response when factor A is at the i-th level and factor B is at the j-th level.
If the factors are crossed, the jth level of B is the same for all levels of A.
If factor B is nested within A, the j-th level of B when A is at level 1 has nothing in common with the j-th level of B when A is at level 2.

Factors that can’t be manipulated are designated as classification factors, as opposed to experimental factors (i.e., you assign to the experimental units).

20.5.1 Two-Factor Nested Designs

  • Consider B is nested within A.
  • both factors are fixed
  • All treatment means are equally important.

Mean responses

\[ \mu_{i.} = \sum_j \mu_{ij}/b \]

Main effect factor A

\[ \alpha_i = \mu_{i.} - \mu_{..} \]

where \(\mu_{..} = \frac{\mu_{ij}}{ab} = \frac{\sum_i \mu_{i.}}{a}\) and \(\sum_i \alpha_i = 0\)

Individual effects of B is denoted as \(\beta_{j(i)}\) where \(j(i)\) indicates the j-th level of factor B is nested within the it-h level of factor A

\[ \beta_{j(i)} = \mu_{ij} - \mu_{i.} \\ = \mu_{ij} - \alpha_i - \mu_{..} \\ \sum_j \beta_{j(i)}=0 , i = 1,...,a \]

\(\beta_{j(i)}\) is the specific effect of the jth level of factor B nested within the ith level of factor A. Hence,

\[ \mu_{ij} \equiv \mu_{..} + \alpha_i + \beta_{j(i)} \equiv \mu_{..} + (\mu_{i.} - \mu_{..}) + (\mu_{ij} - \mu_{i.}) \]

Model

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

where

  • \(Y_{ijk}\) response for the kth treatment when factor A is at the i-th level and factor B is at hte jth level (i = 1,..,a; j = 1,..,b; k = 1,..n)
  • \(\mu_{..}\) constant
  • \(\alpha_i\) constants subject to restriction \(\sum_i \alpha_i = 0\)
  • \(\beta_{j(i)}\) constants subject to restriction \(\sum_j \beta_{j(i)} = 0\) for all i
  • \(\epsilon_{ijk} \sim iid N(0,\sigma^2)\)

\[ E(Y_{ijk}) = \mu_{..} + \alpha_i + \beta_{j(i)} \\ var(Y_{ijk}) = \sigma^2 \]

there is no interaction term in a nested model

ANOVA for Two-Factor Nested Designs

Least Squares and MLE estimates

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

residual \(e_{ijk} = Y_{ijk} - \bar{Y}_{ijk}\)

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

ANOVA Table

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 SSB(A) a(b-1) MSB(A) \(\sigma^2 + n \frac{\ | | | | um \sum e ta_{i)}^ 2}{a(b-1)}\)
Error SSE ab(n-1) MSE \(\sigma^2\)
Total SSTO abn -1

Tests For Factor Effects

\[ H_0: \text{ All } \alpha_i =0 \\ H_a: \text{ not all } \alpha_i = 0 \]

\(F = \frac{MSA}{MSE} \sim f_{(1-\alpha;a-1,(n-1)ab)}\) reject if \(F > f\)

\[ H_0: \text{ All } \beta_{j(i)} =0 \\ H_a: \text{ not all } \beta_{j(i)} = 0 \]

\(F = \frac{MSB(A)}{MSE} \sim f_{(1-\alpha;a(b-1),(n-1)ab)}\) reject \(F>f\)

Testing Factor Effect Contrasts

\(L = \sum c_i \mu_i\) where \(\sum c_i =0\)

\[ \hat{L} = \sum c_i \bar{Y}_{i..} \\ \hat{L} \pm t_{(1-\alpha/2;df)}s(\hat{L}) \]

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

Testing Treatment Means

\(L = \sum c_i \mu_{.j}\) estimated by \(\hat{L} = \sum c_i \bar{Y}_{ij}\) with confidence limits:

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

where

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

Unbalanced Nested Two-Factor Designs

If there are different number of levels of factor B for different levels of factor A, then the design is called unbalanced

The model

\[ Y_{ijk} = \mu_{..} + \alpha_i + \beta_{j(i)} + \epsilon_{ijk} \\ i = 1,2;j =1,..,b_i;k=1,..,n_{ij} \\ b_1 = 3, b_2= 2, n_{11} = n_{13} =2, n_{12}=1,n_{21} = n_{22} = 2\\ \sum_{i=1}^2 \alpha_i =0 \\ \sum_{j=1}^3 \beta_{j(1)} = 0 \\ \sum_{j=1}^2 \beta_{j(2)}=0 \]

where \(\alpha_1,\beta_{1(1)}, \beta_{2(1)}, \beta_{1(2)}\) are parameters. And constraints: \(\alpha_2 = - \alpha_1, \beta_{3(1)}= - \beta_{1(1)}-\beta_{2(1)}, \beta_{2(2)}=-\beta_{1(2)}\)

4 indicator variables

\[\begin{equation} X_1 = \begin{cases} 1&\text{if obs from school 1}\\ -1&\text{if obs from school 2}\\ \end{cases} \end{equation}\]

\[\begin{equation} X_2 = \begin{cases} 1&\text{if obs from instructor 1 in school 1}\\ -1&\text{if obs from instructor 3 in school 1}\\ 0&\text{otherwise}\\ \end{cases} \end{equation}\]

\[\begin{equation} X_3 = \begin{cases} 1&\text{if obs from instructor 2 in school 1}\\ -1&\text{if obs from instructor 3 in school 1}\\ 0&\text{otherwise}\\ \end{cases} \end{equation}\]

\[\begin{equation} X_4 = \begin{cases} 1&\text{if obs from instructor 1 in school 1}\\ -1&\text{if obs from instructor 2 in school 1}\\ 0&\text{otherwise}\\ \end{cases} \end{equation}\]

Regression Full Model

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

Random Factor Effects

If

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

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

Test Statstics

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 the precision of treatment comparisons by reducing variability is to use regression models to adjust for differences among experimental units (also known as analysis of covariance).