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

for level and block .

Here (i) denotes the overall average, (ii) is called -th treatment effect (or factor effect), and (iii) is -th block effect. Furthermore, it is assumed that are iid normally distributed random variables with mean 0 and common variance .

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

which is known as the hypothesis test for treatment effect. To proceed the statistical analysis of treatment effects, the total sum of squares within blocks must be formulated by

Under the null hypothesis above, the test statistic

has the -distribution with degree of freedom. Thus, we reject with significance level if . Or, equivalently we can compute the -value , and reject if . The analysis of variance table for treatment effects is summarized as follows.

 Source Degree of freedom Mean square F-statistic Treatment Error Total within blocks

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

By rejecting we are also justifying the appropriateness of the model for randomized block design. Here we need to introduce the total sum of squares within treatments by

Under the null hypothesis above, the test statistic

has the -distribution with degree of freedom. Thus, we reject with significance level if . Or, equivalently we can compute the -value and reject if . The analysis of variance for block effects becomes

 Source Degree of freedom Mean square F-statistic Block Error Total within treatments

The AOV table is calculated as follows.

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