e-Statistics

Test of Effect

A statistical model for randomized block design becomes

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

Data of randomized block design consists of:

the column of the measurement values $X_{ijl}$ 's;
the column of the treatment levels $i = 1,\ldots,k$ ;
the column of the blocks $j = 1,\ldots,b$ .

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 $\varepsilon_{ijl}$ are iid normally distributed random variables with mean 0 and common variance $\sigma^2$ . For the respective effects of treatment and block, AOV tables are calculated and interaction plots visualize the effect if any.

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

-distribution with

degree of freedom. Thus, we reject

with significance level $\alpha$ if $F > F_{\alpha,k-1,kbn-k-b+1}$ . Or, equivalently we can compute the

-value

, and reject

if $p^* < \alpha$ . The analysis of variance table for treatment effects is summarized as follows.

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

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

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

-distribution with

degree of freedom. Thus, we reject

with significance level $\alpha$ if $F > F_{\alpha,b-1,kbn-k-b+1}$ . Or, equivalently we can compute the

-value

and reject

if $p^* < \alpha$ . The analysis of variance for block effects becomes

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