What is the NULL hypothesis for interaction in a two-way ANOVA? Let's say we have two factors (A and B), each with two levels (A1, A2  and  B1, B2) and a response variable (y).
The when performing a two way ANOVA of the type:
y~A+B+A*B

We are testing three null hypothesis:


*

*There is no difference in the means
of factor A 

*There is no difference
in means of factor B 

*There is no interaction between factors A and B


When written down, the first two hypothesis are easy to formulate (for 1 it is $H_0:\; \mu_{A1}=\mu_{A2}$)
But how should hypothesis 3 be formulated?
edit: and how would it be formulated for the case of more then two levels?
Thanks.
 A: I think it's important to clearly separate the hypothesis and its corresponding test. For the following, I assume a balanced, between-subjects CRF-$pq$ design (equal cell sizes, Kirk's notation: Completely Randomized Factorial design).
$Y_{ijk}$ is observation $i$ in treatment $j$ of factor $A$ and treatment $k$ of factor $B$ with $1 \leq i \leq n$, $1 \leq j \leq p$ and $1 \leq k \leq q$. The model is $Y_{ijk} = \mu_{jk} + \epsilon_{i(jk)}, \quad \epsilon_{i(jk)} \sim N(0, \sigma_{\epsilon}^2)$
Design:
$\begin{array}{r|ccccc|l}
~      & B 1      & \ldots   & B k      & \ldots   & B q      & ~\\\hline
A 1    & \mu_{11} & \ldots   & \mu_{1k} & \ldots   & \mu_{1q} & \mu_{1.}\\
\ldots & \ldots   & \ldots   & \ldots   & \ldots   & \ldots   & \ldots\\
A j    & \mu_{j1} & \ldots   & \mu_{jk} & \ldots   & \mu_{jq} & \mu_{j.}\\
\ldots & \ldots   & \ldots   & \ldots   & \ldots   & \ldots   & \ldots\\
A p    & \mu_{p1} & \ldots   & \mu_{pk} & \ldots   & \mu_{pq} & \mu_{p.}\\\hline
~      & \mu_{.1} & \ldots   & \mu_{.k} & \ldots   & \mu_{.q} & \mu
\end{array}$
$\mu_{jk}$ is the expected value in cell $jk$, $\epsilon_{i(jk)}$ is the error associated with the measurement of person $i$ in that cell. The $()$ notation indicates that the indices $jk$ are fixed for any given person $i$ because that person is observed in only one condition. A few definitions for the effects:
$\mu_{j.} = \frac{1}{q} \sum_{k=1}^{q} \mu_{jk}$ (average expected value for treatment $j$ of factor $A$)
$\mu_{.k} = \frac{1}{p} \sum_{j=1}^{p} \mu_{jk}$ (average expected value for treatment $k$ of factor $B$)
$\alpha_{j} = \mu_{j.} - \mu$ (effect of treatment $j$ of factor $A$, $\sum_{j=1}^{p} \alpha_{j} = 0$)
$\beta_{k} = \mu_{.k} - \mu$ (effect of treatment $k$ of factor $B$, $\sum_{k=1}^{q} \beta_{k} = 0$)
$(\alpha \beta)_{jk} = \mu_{jk} - (\mu + \alpha_{j} + \beta_{k}) = \mu_{jk} - \mu_{j.} - \mu_{.k} + \mu$
(interaction effect for the combination of treatment $j$ of factor $A$ with treatment $k$ of factor $B$, $\sum_{j=1}^{p} (\alpha \beta)_{jk} = 0 \, \wedge \, \sum_{k=1}^{q} (\alpha \beta)_{jk} = 0)$ 
$\alpha_{j}^{(k)} = \mu_{jk} - \mu_{.k}$
(conditional main effect for treatment $j$ of factor $A$ within fixed treatment $k$ of factor $B$, $\sum_{j=1}^{p} \alpha_{j}^{(k)} = 0 \, \wedge \, \frac{1}{q} \sum_{k=1}^{q} \alpha_{j}^{(k)} = \alpha_{j} \quad \forall \, j, k)$
$\beta_{k}^{(j)} = \mu_{jk} - \mu_{j.}$
(conditional main effect for treatment $k$ of factor $B$ within fixed treatment $j$ of factor $A$, $\sum_{k=1}^{q} \beta_{k}^{(j)} = 0 \, \wedge \, \frac{1}{p} \sum_{j=1}^{p} \beta_{k}^{(j)} = \beta_{k} \quad \forall \, j, k)$
With these definitions, the model can also be written as:
$Y_{ijk} = \mu + \alpha_{j} + \beta_{k} + (\alpha \beta)_{jk} + \epsilon_{i(jk)}$
This allows us to express the null hypothesis of no interaction in several equivalent ways:


*

*$H_{0_{I}}: \sum_{j}\sum_{k} (\alpha \beta)^{2}_{jk} = 0$
(all individual interaction terms are $0$, such that $\mu_{jk} = \mu + \alpha_{j} + \beta_{k} \, \forall j, k$. This means that treatment effects of both factors - as defined above - are additive everywhere.)

*$H_{0_{I}}: \alpha_{j}^{(k)} - \alpha_{j}^{(k')} = 0 \quad \forall \, j \, \wedge \, \forall \, k, k' \quad (k \neq k')$
(all conditional main effects for any treatment $j$ of factor $A$ are the same, and therefore equal $\alpha_{j}$. This is essentially Dason's answer.)

*$H_{0_{I}}: \beta_{k}^{(j)} - \beta_{k}^{(j')} = 0 \quad \forall \, j, j' \, \wedge \, \forall \, k \quad (j \neq j')$
(all conditional main effects for any treatment $k$ of factor $B$ are the same, and therefore equal $\beta_{k}$.)

*$H_{0_{I}}$: In a diagramm which shows the expected values $\mu_{jk}$ with the levels of factor $A$ on the $x$-axis and the levels of factor $B$ drawn as separate lines, the $q$ different lines are parallel.
A: An interaction tells us that the levels of factor A have different effects based on what level of factor B you're applying.  So we can test this through a linear contrast.  Let C = (A1B1 - A1B2) - (A2B1 - A2B2)  where A1B1 stands for the mean of the group that received A1 and B1 and so on.  So here we're looking at A1B1 - A1B2 which is the effect that factor B is having when we're applying A1.  If there is no interaction this should be the same as the effect B is having when we apply A2: A2B1 - A2B2.  If those are the same then their difference should be 0 so we could use the tests:
$H_0: C = 0\quad\text{vs.}\quad H_A: C \neq 0.$
