One-way ANOVA

Definition

One-way ANOVA

One-way ANOVA is used to analyze the significance of differences of means between groups. Let an $i$-th response of the $j$-th group be $X_{ij}={\mu }_j+{ϵ}_{ij}$ where ${ϵ}_{ij}\sim N(0,{\sigma }^2),\forall i=1,2,\dots ,n_j,\forall j=1,2,\dots ,c$

We want to test the null hypothesis $H_0:{\mu }_1={\mu }_2=\cdots ={\mu }_c$ i.e. there is no treatment effect. $F=\frac{\sum _{j=1}^c{n}_j({\bar{X}}_{.j}-{\bar{X}}_{..}{)}^2/(c-1)}{\sum _{j=1}^c\sum _{i=1}^{n_j}(X_{ij}-{\bar{X}}_{.j}{)}^2/(N-c)}=\frac{SSB/(c-1)}{SSW/(N-c)}\sim F(c-1,N-c)$ where $N=\sum _{j=1}^c{n}_j$ is the total number of observations, ${\bar{X}}_{.j}$ represents the mean of the $j$-th group and ${\bar{X}}_{..}$ represents the overall mean, $\frac{1}{{\sigma }^2}\sum _{j=1}^c{n}_j({\bar{X}}_{.j}-{\bar{X}}_{..}{)}^2\sim {\chi }^2(c-1)$ of the numerator indicates between-group variance (SSB) and $\frac{1}{{\sigma }^2}\sum _{j=1}^c\sum _{i=1}^{n_j}(X_{ij}-{\bar{X}}_{.j}{)}^2\sim {\chi }^2(N-c)$ of the denominator indicates within-group variance (SSW)

The Likelihood Ratio Test rejects $H_0$ if $F>F_{\alpha }(c-1,N-c)$

One-way ANOVA with a Regression Model

$Y={\beta }_0+{\beta }_1{D}_1+{\beta }_2{D}_2+\cdots +{\beta }_{c-1}{D}_{c-1}+ϵ$ where $D_i$ are dummy variables represent each category, and $c$ is the number of categories

In the setting, a corner-point constraint is used.

The null hypothesis $H_0:{\beta }_1={\beta }_2=\cdots ={\beta }_{c-1}=0$ i.e. there is no treatment effect, yields same test statistic as the null hypothesis of the one-way ANOVA with equal replication $H_0:{\mu }_1={\mu }_2=\cdots ={\mu }_c$

Deviance Test for One-way ANOVA

The null hypothesis $H_0:D_1=D_2=\cdots =D_{c-1}=0$ i.e. there is no treatment effect, can be tested with the Deviance. If ${\sigma }^2$ is known, the test statistic is defined as $\Delta D=D_0-D_1=\frac{1}{{\sigma }^2}\left(\frac{1}{n_c}\sum _{j=1}^c{y}_{j.}^2-\frac{1}{N}{Y}_{..}^2\right)\sim {\chi }^2(c-1)$ And reject the $H_0$ if $\Delta D>{\chi }_{\alpha }^2(c-1)$

If ${\sigma }^2$ is unknown, the test statistic is defined as $F=\frac{D_0-D_1}{(N-1)-(N-c)}/\frac{D_1}{N-c}\sim F(c-1,N-c)$ And reject the $H_0$ if $F>F_{\alpha }(c-1,N-c)$