 Twoway ANOVA definition
 Twoway ANOVA table
 Tukey test for nonadditivity
 Halfnormal distribution
 ANOVA without replication
In this post, we will generalize over oneway ANOVA to twoway and multiway ANOVA, and discuss a few solutions to test for interaction terms when there is no replication.
Twoway ANOVA definition
Cellmeans version of the model: \(y_{ijk} = \mu_{ij} + \epsilon_{ijk}, \epsilon_{ijk} \sim N(0, \sigma^2)\)
Factorlevel version: \(y_{ijk} = \mu + \alpha_i + \beta_j + (\alpha\beta)_{ij} + \epsilon_{ijk}\)
where (these constraints provide the desired model identifiability and degree of freedoms)
\(\sum_{i=1}^{a} \alpha_i =0\),
\(\sum_{j=1}^{b} \beta_j =0\),
\(\sum_{i=1}^a (\alpha\beta)_{ij} = \sum_{j=1}^b (\alpha\beta)_{ij} = 0\).
Analysis of variance based on parition of total sum of squared deviations from grand mean.
\(\sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n} (y_{ijk}  \bar{y}_{...})^2 =\) \(\sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n} (y_{ijk}  \bar{y}_{ij.})^2 + \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n} ( \bar{y}_{ij.}  \bar{y}_{...} )^2\)
The above equation can be seen as 1) $SS_{residual}$, or withingroup variance; and 2) $SS_{treatment}$, or betweengroup variance. The Betweengroup variance can be further partitioned as:
\(SS_{treatment} = \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n} ( [\bar{y}_{i..}  \bar{y}_{...}] + [\bar{y}_{.j.}  \bar{y}_{...}] + [\bar{y}_{ij.}  \bar{y}_{i..}  \bar{y}_{.j.} + \bar{y}_{...} )^2\) \(= SSA + SSB + SSAB\)
that is, the main effect os A/estimate of $\alpha_i$ (SSA), the main effect os B/estimate of $\beta_j$ (SSB), and the A.B interaction estimate $(\alpha\beta)_{ij}$ (SSAB).
Twoway ANOVA table
Source  df  SS  MS  E(MS)  F 

A  a1  SSA  MSA  $\sigma^2 + \frac{bn\sum_i(\mu_{i..}\mu{…})}{(a1)}$  $F_A = \frac{MSA}{MSE}$ 
B  b1  SSB  MSB  $\sigma^2 + \frac{an\sum_j(\mu_{.j.}\mu{…})}{(b1)}$  $F_B = \frac{MSB}{MSE}$ 
AB  (a1)(b1)  SSAB  MSAB  $\sigma^2 + \frac{n\sum_i\sum_j(\mu_{ij.}\mu{…})}{(a1)(b1)}$  $F_{AB} = \frac{MSAB}{MSE}$ 
Error  ab(n1)  SSE  MSE  $\sigma^2$  
Total  abn1 
E(MS) is the expected mean, or noncentrality parameter for the Ftest.
When there are no replications, assuming interaction terms in twoway ANOVA will eat the error sum of squares, hence is statistically inhibitive. To fix this, need to treat interaction term as error term and assume no interaction effects (pool interaction/error).
If assumed wrong model (i.e. use a noninteractive model when there is interaction effects), it would tend to yield “conservative” tests for A,B main effects since expectation of denominator of Ftest is potentially greater than $\sigma^2$.
Tukey test for nonadditivity
In essense Tukey test for nonadditivity is testing for a multiplicative interaction term when there are no replications, where $(\alpha\beta)_{ij}=D\cdot\alpha_i\cdot\beta_j$, $D$ is some constant. The hypothesis testing is $H_1: D\ne0$ vs $H_0:D=0$.
Below is how a twoway ANOVA table will look like for testing interactions without replications:
Source  df  SS  MS  F 

A  a1  SSA  MSA  $F_A = \frac{MSA}{MS_{Rem}}$ 
B  b1  SSB  MSB  $F_B = \frac{MSB}{MS_{Rem}}$ 
AB  (a1)(b1)  SSAB  MSAB  [Not doable due to 0df for error term, use Tukey partition] 
Nonaddiviity  1  SSAB*  MSAB*=SSAB*  $F_{AB} = \frac{MSAB^*}{MS_{Rem}} \sim F_{1, abab}$ 
Remainder  ab  a b  SSRem  $MS_{Rem}$  
Total  ab1 
Halfnormal distribution
Daniel, 1959 Technometrics
If $Z \sim N(0,1)$, then $Z$ follows a “halfnormal” distribution, often denoted as $\chi$ (square root of $\chi^2$).
Suppose we break down all effects into orthonormal 1df. contrasts. [same lengthimportant so that contrasts have same variance]. The following procedure is used to test interaction terms using Halfnormal distribution:
 Take absolute values of each $\hat{L}=c^T \cdot \bar{Y}$, (i.e. contrasts ) order from smallest to largest.
As a side note, using the Halfnormal here is because without replication, we cannot estimate the standard error $SE^2(\hat{L}) = MSE \sum_{i=1}{r} \frac{c_i^2}{n_i}$, where MSE cannot be computed.

Plot $contrasts$ (i.e. $\hat{L}$) against expected order statistics from $\chi$.

Carry at test of largest contrasts.

Remove largest, repeat process 23.
ANOVA without replication
Three strategies:
 Assume no interaction effects and treat interaction as error terms
 Tukey test of nonadditivity; parametrize certain type of interactions (i.e. multiplicative combinations of main effects)
 Halfnormal plot