## Test of Effect

The following data of randomized block design consists of:- the response column;
- the treatment column;
- 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

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

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.