Module Fourteen: More Complicated Designs – Block Designs, 2n Factorial Designs, etc.

An important goal of experimental design is to increase the precision of the results. Using experimental design techniques, we are able to control the local environment through choice of factors, selection of homogeneous experimental units, random assignment of experiments to treatments, and recording co-variates that may be influential to the results.

In many experiments, we often encounter the limitation of environment such as location, time. For such factors, it is difficult to conduct appropriate local control such as the random assignment of units to treatment, if the experiment units are not homogeneous from time to time or from location to location.

For an example, suppose we conduct an experiment to study the effect of four formulae of fertilizer on a field.

One way to accomplish this experiment is to divide the field into 4xb plots, and randomly assign b plots to each formula. This is the Completely Randomized Design. (CRD).

However, it is common that there may be some characteristics of the filed that make the plots (experiments) non-homogeneous. These may include the irrigation system, the soil type, and so on.

By using CRD, we run into the risk of applying the same or similar formula to the plots with better water irrigation than other formula. The consequence is we will never know if the yield is due to better water irrigation or due to fertilizer.

Using statistical term, the experimental units assigned to each fertilizer formula may be quite non-homogeneous. The experimental error will be much larger. The actual fertilizer effects will be difficult to quantified.

•One simple approach is to take the water irrigation gradient into account when designing the experiment.

When dividing the field into plots, one can split the filed according to the water gradient by splitting the filed into b blocks based on the water gradient so that within each block, the water irrigation is similar. Now, within each block, we split it into 4 plots.

The four fertilizer formulae are then randomly assigned to a plot within each Block.

This design is called Randomized Complete Block Design (RCBD)

1. Divide the experimental units into ‘b’ blocks so that the experimental units within each block are more as similar as possible.

2. Randomly assign one experiment to each treatment of the ‘a’ treatments within each Block.

(NOTE: Step two itself is just a CRD).

3 4 1

2 4 3

2 1 3

2 4 1

Block 1 Block 2 Block 3

2 4 3

4 2 4

3 1 1

1 3 2

A CRD Arrangement for a 4 level treatment with 3 replications. Each cell represents a unit. The number is the assigned treatment level.

A RCBD Arrangement for a 4 level treatment with 3 blocks

If the units within each block are more similar to each other than between blocks, then RCBD will have a great advantage.

Using CRD, the difference between Treatment 2 and 4 may be due to the block difference, not the treatment difference.

What is the difference between Block Designs and Two-Factor design?

It looks that Block is just another factor in the experiment, and that one can consider RCBD a factorial design.

They are different because of the assignment of the experiment units are very different. In addition, when planning a block design, there is a common understanding that there is no interaction effect between Block and Treatment. This means the treatment differences are similar from block to block.

For example, The difficult subject is generally difficult regardless who teaches. Therefore, the average test scores for subjects A,B,C taught by Teacher I are 80, 40, 100. The same subject taught by Teacher II will have a similar pattern. The averages may be 70, 35, 90. But, they should not be 30, 100, 90 for subject A,B,C. This says there is no interaction between teacher and Subject.

In a block design, we assume no interaction between Block and Treatment.

Statistical Model for A RCBD

ij .. i. .. . .. i. . ..

, i = 1,2,...,a; j = 1,2,..., b

~ (0, )

When data are analyzed, each observation can be decomposed as:y = y + (y ) ( ) ( y )

ˆ ˆ =

ij i j ij

ij

j ij j

i

y a b e

e Normal

y y y y y y

a

2 2 2 2ij .. i. .. . .. i. . ..

ˆ ˆ

Sum of Squares Partitions are:

y - y = b (y ) a ( ) + ( y )

SSTO = SSTR + SSB + SSEd

j ij

j ij j

b e

y y y y y y

f: (ab-1) = (a-1) + (b-1) + (a-1)(b-1)

Source Df SS MS F P-value EMS

Block b-1 SSB MSB=SSB/(b-1) MSB/MSE bB

Treatment a-1 SSTR MSTR=SSTR/(a-1) MSTR/MSE b

Error (a-1)(b-1) SSE MSE=SSE/ [(a-1)(b-1)]

Total ab-1 SSTO

The ANOVA Table for RCBD

i.

i. .

The SE of Mean response of a treatment level is

SE(y )

The SE of Mean difference between two treatment levels is:

2SE(y )

Hence, we can also compute confidence intervals for Mean Response

j

MSEb

MSEyb

or Mean Difference.Multiple comparison procedures such as Tukey's pair-wise proceudre can also be applied.

A Case Study – Complete Randomized Block DesignThe self-inductance of coils with iron-oxide cores was measured under different temperature conditions of the measuring bridge. The coil temperature was held constant. Five coils were used for the experiment. The self-inductance of each coil was measured for each of four temperatures (22, 23, 24, 250) for measuring bridge. The temperatures were utilized in a random order for each coil. The data are percentage deviation from a standard.

Row Coil Temp %dev

1 1 22 1.400

2 1 23 1.400

3 1 24 1.375

4 1 25 1.370

5 2 22 0.264

6 2 23 0.235

7 2 24 0.212

8 2 25 0.208

9 3 22 0.478

10 3 23 0.467

Row Coil Temp %dev

11 3 24 0.444

12 3 25 0.440

13 4 22 1.010

14 4 23 0.990

15 4 24 0.968

16 4 25 0.967

17 5 22 0.629

18 5 23 0.620

19 5 24 0.495

20 5 25 0.495

Hands-on ActivityConduct an appropriate analysis for this study.

Conduct a multiple comparison of temperature levels.

Obtain a 95% confidence interval for Temperature = 22.

Note: The treatment can be fixed as well as random effect.

Note: We can have more than one factor in the CRBD design. This is mainly to plan a factorial design within each Block. In the analysis, we need to add one additional source, Block, into the model and the ANOVA table.

The model for a complete Block Design with two fixed effect factors is:

, i = 1,2,...,a; j = 1,2,..., b; k = 1,2,..., r

is the effect of Factor A is the effect of Factor B

is the interaction effect of A and B

is the Blocking effect

ij i j ij k ijk

i

j

ij

k

ij

y a b ab c e

ab

ab

ce

~ (0, )k Normal