在双向方差分析中交互作用的零假设是什么？

假设我们有两个因子（A和B），每个因子有两个级别（A1，A2和B1，B2）和一个响应变量（y）。

在执行类型的双向ANOVA时：

y~A+B+A*B

我们正在测试三个原假设：

1. 因子A的均值没有差异
2. 因子B的均值没有差异
3. 因子A和B之间没有相互作用

写下后，很容易提出前两个假设（对于1来说是）$H_0:\; \mu_{A1}=\mu_{A2}$

但是假设3应该如何表述呢？

编辑：以及如何将其制定为两个以上级别的情况？

谢谢。

我认为将假设及其相应的检验清晰分开很重要。对于以下内容，我假设受试者之间采用均衡的CRF- 设计（相等的像元大小，柯克表示法：完全随机因子设计）。$pq$

是观察我在治疗 Ĵ因子的甲和治疗 ķ因子的乙与 1 ≤ 我≤ Ñ， 1 ≤ Ĵ ≤ p和 1 ≤ ķ ≤ q。该模型是 ÿ 我Ĵ ķ = μ Ĵ ķ + ε 我（Ĵ ķ ）$Y_{ijk}$$i$$j$$A$$k$$B$$1 \leq i \leq n$$1 \leq j \leq p$$1 \leq k \leq q$$Y_{ijk} = \mu_{jk} + \epsilon_{i(jk)}, \quad \epsilon_{i(jk)} \sim N(0, \sigma_{\epsilon}^2)$

设计： 乙1个... 乙ķ ... 乙q 阿1 μ 11 ... μ 1 ķ ... μ 1个q μ 1 ... ... ... ... ... ... ... 甲Ĵ μ Ĵ 1 ... μ Ĵ ķ ... μ Ĵ q μ Ĵ 。... ... ... ... ... ... ... 甲p μ p 1 ... μ$\begin{array}{r|ccccc|l} ~ & B 1 & \ldots & B k & \ldots & B q & ~\\\hline A 1 & \mu_{11} & \ldots & \mu_{1k} & \ldots & \mu_{1q} & \mu_{1.}\\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots\\ A j & \mu_{j1} & \ldots & \mu_{jk} & \ldots & \mu_{jq} & \mu_{j.}\\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots\\ A p & \mu_{p1} & \ldots & \mu_{pk} & \ldots & \mu_{pq} & \mu_{p.}\\\hline ~ & \mu_{.1} & \ldots & \mu_{.k} & \ldots & \mu_{.q} & \mu \end{array}$

是在小区中的预期值 Ĵ ķ， ε 我（Ĵ ķ ）与人的测量关联的误差我在该小区中。的（）表示法表示的索引 Ĵ ķ是固定的任何给定的人我因为该人仅在一个条件下观察。效果的一些定义：$\mu_{jk}$$jk$$\epsilon_{i(jk)}$$i$$()$$jk$$i$

（用于治疗的平均预期值Ĵ因子的甲）$\mu_{j.} = \frac{1}{q} \sum_{k=1}^{q} \mu_{jk}$$j$$A$

（用于治疗的平均预期值ķ因子的乙）$\mu_{.k} = \frac{1}{p} \sum_{j=1}^{p} \mu_{jk}$$k$$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:

1. $H_{0_{I}}: \sum_{j}\sum_{k} (\alpha \beta)^{2}_{jk} = 0$
   (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.)

2. $H_{0_{I}}: \alpha_{j}^{(k)} - \alpha_{j}^{(k')} = 0 \quad \forall \, j \, \wedge \, \forall \, k, k' \quad (k \neq k')$
   (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.)

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

4. $H_{0_{I}}$: 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.

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:

$H_0: C = 0\quad\text{vs.}\quad H_A: C \neq 0.$