e-Statistics

Test of Effect

The following data of randomized block design consists of:
  1. the response column;
  2. the treatment column;
  3. the block column.

The statistical model for randomized block design becomes

$\displaystyle Y_{ij} = \mu + \alpha_i + \beta_j + \varepsilon_{ij}$    for level $ i = 1,\ldots,k$ and block $ j = 1,\ldots,b$.

Here (i) $ \mu$ denotes the overall average, (ii) $ \alpha_i$ is called $ i$-th treatment effect (or factor effect), and (iii) $ \beta_j$ is $ j$-th block effect. Furthermore, it is assumed that $ \varepsilon_{ij}$ are iid normally distributed random variables with mean 0 and common variance $ \sigma^2$.

The objective of experiment is typically to determine whether there are ``some treatment effects'' or not. Then the hypothesis testing problem becomes

$\displaystyle H_0:\: \alpha_1 = \cdots = \alpha_k = 0,
$

which is known as the hypothesis test for treatment effect. To proceed the statistical analysis of treatment effects, the total sum $ SS_{\mbox{total:bl}}$ of squares within blocks must be formulated by

$\displaystyle SS_{\mbox{total:bl}} = SS_{\mbox{tr}} + SS_{\mbox{error}}
= \sum_{j=1}^b\sum_{i=1}^k\sum_{l=1}^n
(X_{ijl} - \bar{X}_{\cdot j \cdot})^2 .
$

Under the null hypothesis above, the test statistic

$ \displaystyle F = \frac{MS_{\mbox{tr}}}{MS_{\mbox{error}}}$

has the $ F$-distribution with $ (k-1, kbn-k-b+1)$ degree of freedom. Thus, we reject $ H_0$ with significance level $ \alpha$ if $ F > F_{\alpha,k-1,kbn-k-b+1}$. Or, equivalently we can compute the $ p$-value $ p^*$, and reject $ H_0$ if $ p^* < \alpha$. The analysis of variance table for treatment effects is summarized as follows.

Source Degree of freedom $ SS$ Mean square F-statistic
Treatment $ k-1$ $ SS_{\mbox{tr}}$ $ \displaystyle MS_{\mbox{tr}} = \frac{SS_{\mbox{tr}}}{k-1}$ $ \displaystyle F = \frac{MS_{\mbox{tr}}}{MS_{\mbox{error}}}$
Error $ kbn-k-b+1$ $ SS_{\mbox{error}}$ $ \displaystyle
MS_{\mbox{error}} = \frac{SS_{\mbox{error}}}{n-k-b+1}$  
Total within blocks $ kbn-b$ $ SS_{\mbox{total:bl}}$    

It is important to detect whether there are ``some block effects'' or not. For this we can similarly conduct the hypothesis testing problem

$\displaystyle H_0:\: \beta_1 = \cdots = \beta_k = 0.
$

By rejecting $ H_0$ we are also justifying the appropriateness of the model for randomized block design. Here we need to introduce the total sum $ SS_{\mbox{total:tr}}$ of squares within treatments by

$\displaystyle SS_{\mbox{total:tr}} = SS_{\mbox{bl}} + SS_{\mbox{error}}
= \sum_{i=1}^k\sum_{j=1}^b\sum_{l=1}^n
(X_{ijl} - \bar{X}_{i \cdot\cdot})^2
$

Under the null hypothesis above, the test statistic

$ \displaystyle F = \frac{MS_{\mbox{bl}}}{MS_{\mbox{error}}}$

has the $ F$-distribution with $ (b-1, kbn-k-b+1)$ degree of freedom. Thus, we reject $ H_0$ with significance level $ \alpha$ if $ F > F_{\alpha,b-1,kbn-k-b+1}$. Or, equivalently we can compute the $ p$-value $ p^*$ and reject $ H_0$ if $ p^* < \alpha$. The analysis of variance for block effects becomes

Source Degree of freedom $ SS$ Mean square F-statistic
Block $ b-1$ $ SS_{\mbox{bl}}$ $ \displaystyle MS_{\mbox{bl}} = \frac{SS_{\mbox{bl}}}{b-1}$ $ \displaystyle F = \frac{MS_{\mbox{bl}}}{MS_{\mbox{error}}}$
Error $ kbn-k-b+1$ $ SS_{\mbox{error}}$ $ \displaystyle
MS_{\mbox{error}} = \frac{SS_{\mbox{error}}}{n-k-b+1}$  
Total within treatments $ kbn-k$ $ SS_{\mbox{total:tr}}$    

The AOV table is calculated as follows.

The following interaction plot helps visualize the corresponding effects if any.