Generalized Randomized Block Design and Experimental Error

Addelman, Sidney. (1969). The Generalized Randomized Block Design. American Statistician, 23, 35-36. Gates, Charles. E. (1999).What really is experimental error in block designs. American Statistician 49, 362-363.

Summary By Lu Wang

Design

Analysis

The lack of emphasis on design

determines the manner in which the experiment is carried out;

reflected in the mathematical model;

determines the appropriate techniques of analysis.

The experimental design:

Two-way classification: GRBD vs. RBDS

TrtBlock 1 2 ... t

1

2

...

b

r units r units r units

r units

r units

r units

r units

r units

r units

...

...

...

... ... ... ...

GRBD with equal block sizes

5

btr experimental units

b blocks, tr units per block

t treatments, r experimental units per treatment

btr

tr tr tr

r r r r r r

......

... ... ...t t t

RBDS

6

bt

t t

1 1 1 1

bt experimental units

b blocks, t units per block

one unit per treatment

r observational units per experimental unit

t t

r r r r

......

... ... ... ...

...... ......

b

Mathematical Model

Generalized randomized block design with equal block sizes

Y=Treatment Block Treatment*Block(R) Plot(Treatment*Block) (R)

The Generalized Randomized Block Design SIDNEY ADDELMAN

State University of New York at Buffalo

In the literature on the design and analysis of experi- ments far more emphasis has been placed on analysis than on design. The lack of emphasis on design is a curious phenomenon because the experimental design determines the manner in which the experiment is car- ried out, which is reflected in the mathematical model, which in turn determines the appropriate techniques of analysis.

The experimental design consists of the experimental plan and an appropriate randomization procedure for allocating the treatments to the experimental units. It is important that the experimental design be pre- sented along with the mathematical model so that one can see whether or not the model reflects the actual procedure for performing the experiment.

Although most users of experimental statistics are familiar with the randomized block, the Latin square and the split plot designs, very few have heard of the generalized randomized block design. This is a direct result of the lack of emphasis of the design aspects of experimentation in the recent literature. Although this design was mentioned in a few of the writings of the early and mid 1950's [1], [2], [3], very little attention has been given it since. The analysis of the generalized randomized block design with either equal or unequal block sizes is usually described in the literature as the analysis for the two-way classification with an interac- tion and either equal or unequal but proportional num- bers of observations per classification. Some of the better known references are [4], [5], [6] and more recently [7].

The two-way classification is almost always intro- duced by either a two-way table with single or multiple observations in each cell or by a statement such as ""Assume that we have a two-way classification with interaction and equal cell frequencies". There then follows the mathematical model

Yijk = t' + /i + ri + (/T) ii + CEjk (1) i = 1,2... b; j = 1,2.... t; k = 1,2....

with the terms defined in the usual manner. The analy- sis techniques for the two-way classification are then presented for situations where the interaction effect is or is not significant, aiid when one or both of the factors are fixed or random.

Unfortunately, there are two different experimental designs that will produce the same two-way table with multiple observations, only one of which results in the model given by (1). The design that results in model (1) is the generalized randomized block with equal block sizes obtained as follows: There are btr experimental units that are divided into b blocks of tr units in such a way that the units in different blocks are as hetero- geneous as possible andl the units in the same block

as homogeneous as possible. The t treatments are ran- domly allocated to the tr units in each block in such a manner that each treatment is allocated to exactly r units, where a different random allocation is used for each block. The second design is the randomized block with sampling which consists of bt experimental units that are divided into b blocks of t homogeneous units. The t treatments are randomly allocated to the units in a block so that every treatment is allocated to one unit per block, a different random allocation being used for each block. A sample of r observational units is then randomly selected from within each experimental unit. Although the two-way table with r observations per cell is an appropriate summary table for this design, just as it was for the generalized randomized block design, the inodel (1) is not appropriate for this design. The appropriate model does not contain a block-treat- ment interactioni term but rather two error terms. The first of these is called experimental error; it is the error associated with the experimental units, and it accounts for the differences in yields that would occur if the same treatment was applied to different experimental units in the same block. The second, called sampling error, is the error associated with the observational units. It accounts for the differences in the yields of observa- tional units within the same experimental unit.

It is evident that the two experimental designs could not possibly lead to the same model, for if they did, the model would not reflect the manner in which the experi- ment was carried out. In order to lessen the chance of using an inappropriate mathematical model, the experi- mental design on which the model is based should always be specified.

The generalized randomized block design with un- equal sized blocks may be obtained in the same manner as was the generalizedl randomized block with equal block sizes except that each block will contain tri experi- mental units, where i = 1,2 . . . b and each treatment is randomly allocated to ri units in the i-th block. The unequal sized blocks are reflected in model (1) by letting k now range from 1 to r -.

An awareness of the generalized randomized block design should lead to its more frequent use. The ran- domized block design. which is the most popular of all experimental designs, requires that the number of blocks be equal to the number of replicates of each treatment, whether or not the experimental units divide naturally into that many blocks of homogeneous units. Thus every situation for which t treatments are to be tested with, say, 6 replicates, utilizing a randomized block design requires that the 6t experimental units be divided into 6 blocks of t units. If the 6t experimental units divide naturally illto two blocks of 3t homogeneous

35

Generalized randomized block design with unequal block sizes

Mathematical Model

It may be obtained in the same manner as was generalized randomized block design with equal block sizes expect that each block will contain tri experimental units, i=1,2,...,b.

– The randomized block design requires that the number of blocks be equal to the number of replicates of each treatment, whether or not the experimental units divide naturally into that many blocks of homogeneous units;

– GRBD allows us to test the block-treatment interaction;

– GRBD gives the advantage of blocking---increased precision.

Why should we use GRBD?

10

GRBD

Conclusion: Use GRBD

Exactly what is the composition of " experimental error " in designs such as the randomized (complete) block design (RBD)?

Experimental Error

Experimental error measures the variance, assumed homogeneous, of the levels of treatment factors applied to the experimental units, and thus is the appropriate error term for testing hypotheses concerning the treatment factor.

A definition of experimental error

Wrong understanding

Experimental error is simply the interaction of blocks with treatments!!!

What is a GRBD?

It is an randomized (complete) block design, RBD, in which more than one experimental unit within blocks is assigned randomly the same treatment.

Example 1

RBD

GRBD

GRBD vs RBDS

Example 2

GRBDS

Y= B T B *T(R) P(B *T) (R) S(P*B*T) (R)

GRBDS with s=1 GRBD

Y= B T B *T(R) P(B *T) (R)

GRBDS with p=1 RBDS

Y= B T B *T(R) S(B*T)(R)

GRBDS with p=s=1 RBD

Y= B T B *T(R)

Conclusion

Experimental error for testing treatments, is comprised of three sources of variability: block by treatment interaction, within block plot-to-plot variability, and within experimental plot sampling variation.

Thank you very much!

Lu Wang

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