21.5 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 \(j\)-th 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).
21.5.1 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
\[ \begin{aligned} \beta_{j(i)} &= \mu_{ij} - \mu_{i.} \\ &= \mu_{ij} - \alpha_i - \mu_{..} \\ \sum_j \beta_{j(i)}&=0 , i = 1,...,a \end{aligned} \]
\(\beta_{j(i)}\) is the specific effect of the \(j\)-th level of factor \(B\) nested within the \(i\)-th 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 \(k\)-th treatment when factor \(A\) is at the \(i\)-th level and factor \(B\) is at the \(j\)-th 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 \(i\)
- \(\epsilon_{ijk} \sim iid N(0,\sigma^2)\)
\[ \begin{aligned} E(Y_{ijk}) &= \mu_{..} + \alpha_i + \beta_{j(i)} \\ var(Y_{ijk}) &= \sigma^2 \end{aligned} \]
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
\[ \begin{aligned} &H_0: \text{ All } \alpha_i =0 \\ &H_a: \text{ not all } \alpha_i = 0 \end{aligned} \]
\(F = \frac{MSA}{MSE} \sim f_{(1-\alpha;a-1,(n-1)ab)}\) reject if \(F > f\)
\[ \begin{aligned} &H_0: \text{ All } \beta_{j(i)} =0 \\ &H_a: \text{ not all } \beta_{j(i)} = 0 \end{aligned} \]
\(F = \frac{MSB(A)}{MSE} \sim f_{(1-\alpha;a(b-1),(n-1)ab)}\) reject \(F>f\)
Testing Factor Effect Contrasts
\(L = \sum c_i \mu_i\) where \(\sum c_i =0\)
\[ \begin{aligned} \hat{L} &= \sum c_i \bar{Y}_{i..} \\ \hat{L} &\pm t_{(1-\alpha/2;df)}s(\hat{L}) \end{aligned} \]
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
\[ \begin{aligned} Y_{ijk} &= \mu_{..} + \alpha_i + \beta_{j(i)} + \epsilon_{ijk} \\ \sum_{i=1}^2 \alpha_i &=0 \\ \sum_{j=1}^3 \beta_{j(1)} &= 0 \\ \sum_{j=1}^2 \beta_{j(2)}&=0 \end{aligned} \]
where
\(i = 1,2;j =1,..,b_i;k=1,..,n_{ij}\)
\(b_1 = 3, b_2= 2, n_{11} = n_{13} =2, n_{12}=1,n_{21} = n_{22} = 2\)
\(\alpha_1,\beta_{1(1)}, \beta_{2(1)}, \beta_{1(2)}\) 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
\[ \begin{aligned} \alpha_1 &\sim iid N(0,\sigma^2_\alpha) \\ \beta_{j(i)} &\sim iid N(0,\sigma^2_\beta) \end{aligned} \]
| 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 Statistics
| 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).