# Chapter 8 ANOVA for Unbalanced Two-Factor Experiments

when data are unbalanced, the type 1 ANOVA test for two-way interactions is the same as the test for two-way interactions discussed previously. However, the type 1 ANOVA tests for individual factors are not the tests for main effects discussed previously. Furthermore, the type 1 results for individual factors depend on the order that the factors appear in the type 1 ANOVA table.

The test for main effects is based on LSMEANS (equal weighted), but Type 1 ANOVA test does not compute LSMEANS

Different types of Sums of Squares

Source Type I Type II Type III A SS(A|1) SS(A|1, B) SS(A|1, B, AB) B SS(B|1, A) SS(B|1, A) SS(B|1, A, AB) AB SS(AB|1, A, B) SS(AB|1, A, B) SS(AB|1, A, B) Error SSE SSE SSE C. Total SSTotal ? ? - For balanced data, Type I = Type II = Type III, The ANOVA F tests in the ANOVA table can be used to test for factor main effects and interaction
- For unbalanced data, Type I sums of squares always add to the total sum of squares, Type II and Type III sums of squares do not add to anything special when data are unbalanced.
*The ANOVA F tests in the Type III ANOVA table can be used to test for factor main effects and interaction.*

SAS code

```
;
proc glmclass time temp;
= time temp time*temp / ss1 ss2 ss3;
model y ; run
```

How to fit the reduced model \(A|1,B, AB\)? Code like

`lm(y~b+a:b)`

or`model y = b a*b`

does not fit the reduced model with no`a`

main effect. Note \(\bar \mu_{1.} = \bar \mu_{2.} \Leftrightarrow \mu_{22} = \mu_{11} + \mu_{12} - \mu_{21}\). So we can replace \(\mu_{22}\) by three cell means.Alternative Computation of Sums of Squares: Let \(SS = y'(P_{X_{reduced+term}} - P_{X_{reduced}})y\) represent any Type I, II or III sum of squares. Let \(q = rank(X_{reduced+term}) - rand(X_{reduced})\) be the degrees of freedom associated with \(SS\). Let \(C\) be any \(q\times p\) matrix whose rows are a basis for the row space of \((P_{X_{reduced+term}} - P_{X_{reduced}})X\). Then the ANOVA F statistic \[ \frac{SS/q}{MSE} = \frac{\hat\beta'C'[C(X'X)^-C']^{-1}C\hat\beta/q}{\hat\sigma^2} \] Thus, any \(SS\) can be computed as \(\hat\beta'C'[C(X'X)^-C']^{-1}C\hat\beta\) for an appropriate matrix \(C\).