Intra-block estimates

The analysis of incomplete block designs that was described in Chapter 6.6 was based on the following model for the k'th replicate of the j'th treatment in block i.



yijk  =  µ


 + 
(explained
by blocks
)
βi


 + 
(explained by
treatments
)
γj


 + 

(unexplained)
εijk

where β1 = 0 and γ1 = 0. The treatment parameters j} can be estimated by fitting the above model by least squares to the experimental data. These estimates are called intra-block estimates.

This is impossible if the treatments are varied at block level since the treatments would then be confounded with blocks, but can be done for all other experiments with block.

Inter-block estimates

The intra-block estimates are based on differences within blocks. These differences are independent of the block totals (and block means) but if the blocks and treatments are not orthogonal, the block totals can also hold some information about differences between the treatments.

If the block effects are treated as random variables (with mean zero), the block totals satisfy a linear regression model with the treatment effects as parameters and these parameters can be estimated from the block totals by least squares. These least squares estimates are called inter-block estimates.

The block totals can be used to find a second set of estimates of the treatment effects that are independent of the intra-block estimates.


Balanced incomplete block design

A chemical engineer believes that the reaction time for a chemical process depends on the type of catalyst used and conducts an experiment using different batches of raw material. Four catalysts were tested but each batch of raw material was only large enough for three runs of the experiment so a balanced incomplete block design was used with four blocks of size three. The reaction times are shown in the table below.

        Treatment (catalyst)      
Block
(batch of raw material)
A B C D
1 73 73 75
2 74 75 75
3 67 68 72
4 71 72 75

Using the model


yij  =  µ

 + 
(blocks)
βi

 + 
(treatments)
γj

 + 
(unexplained)
εij

where γA = 0, the block totals are:

Block 1
B1  =  3µ   +   0 × γB   +   1 × γC  +   1 × γD  +  (3β1  +  ε1A +  ε1C +  ε1D)
Block 2
B2  =  3µ   +   1 × γB   +   1 × γC  +   0 × γD  +  (3β2  +  ε2A +  ε2C +  ε2D)
Block 3
B3  =  3µ   +   1 × γB   +   1 × γC  +   1 × γD  +  (3β3  +  ε3A +  ε3C +  ε3D)
Block 4
B4  =  3µ   +   1 × γB   +   0 × γC  +   1 × γD  +  (3β4  +  ε4A +  ε4C +  ε4D)

If the block effects i} are assumed to be normally distributed with mean 0 and variance σB2, and the errors at unit level {εij} have variance σ2, then

var(3βi  +  εiA +  εiC +  εiD)   =   9 σB2  +   3 σ2

The four block totals therefore satisfy the model

Bi  =  µ*   +   xi × γB   +  zi × γC  +  wi × γD  +  εi*

where

and the {εi*} can be treated as independent 'errors' with equal variances in the regression model.

The block totals, {Bi} therefore satisfy a multiple linear regression model with 'explanatory variables' xizi and wi . From them, least squares estimates can be found from this model.

Numerical example

The intra-block and inter-block estimates are shown below.

Select Estimated means from the pop-up menu to display the estimated mean responses for the four treatments.