Latin square design

Prepared By: Group 3

*

*

*A class of experimental designs that allow for two

sources of blocking.

*Can be constructed for any number of treatments,

but there is a cost. If there are t treatments, then t2

experimental units will be required.

*If one of the blocking factors is left out of the

design, we are left with a design that could have

been obtained as a randomized block design.

*Analysis of a Latin square is very similar to that of a

RBD, only one more source of variation in the model.

*Two restrictions on randomization.

*The major feature of this design is its capacity to

simultaneously handle two known sources among

experimental units.

*The two directional blocking in a LS Design, commonly

referred to as row-blocking and column-blocking.

*It is accomplished by ensuring that every treatment

occurs only once in each row-block and once in each

column block.

*This procedure makes it possible to estimate the

variation among row-blocks as well as column blocks and

to remove them from experimental error.

*

*Field trials in which the experimental area has two fertility gradients

running perpendicular to each other, or has a unidirectional fertility

gradient but also has residual effects from previous trials.

* Insecticide field trials where the insect migration has a predictable

direction that is perpendicular to the dominant fertility gradients of

the experimental field

*Greenhouse trials in which the experimental pots are arranged in

straight line perpendicular to the glass or screen walls, such that the

difference among rows and the distance from the glass wall are

expected to be the two major sources of variability among the

experimental pots.

*Laboratory trials with replication over time, such that the difference

among experimental units conducted at the same time and among

those conducted over time constitute the two known sources of

variability.

*

•A researcher wishes to perform a yield experiment under field conditions, but she/he knows or suspects that there are two fertility trends running perpendicular to each other across the study plots.

•An animal scientists wishes to study weight gain in piglets but knows that both litter membership and initial weights significantly affect the response.

•In a greenhouse, researchers know that there is variation in response due to both light differences across the building and temperature differences along the building.

•An agricultural engineer wishing to test the wear of different makes of tractor tire, knows that the trial and the location of the tire on the (four wheel drive, equal tire size) tractor will significantly affect wear.

*

*Advantages:

•Allows for control of two extraneous sources of variation.

•Analysis is quite simple.

*Disadvantages:

•Requires t2 experimental units to study t treatments.

•Best suited for t in range: 5 t 10.

•The effect of each treatment on the response must be

approximately the same across rows and columns.

•Implementation problems.

•Missing data causes major analysis problems.

*

*Step 1: Select a sample LS plan with five

treatments from Appendix K.

Example:

A B C D E

B A E C D

C D A E B

D E B A C

E C D B A

*Step 2: Randomized the row arrangement of the plan selected

in step 1, following one of the randomization schemes.

*For this experiment, the table-of-random-numbers method is

applied.

*Select five three-digit random numbers; for example: 628, 846,

475, 902 and 452.

*Rank the selected random numbers from lowest to highest:

Random No. Sequence Rank

628 1 3

846 2 4

475 3 2

902 4 5

452 5 1

*Step 3: Randomize the column arrangement, using

the same procedure used for row arrangement in

step 2. For example, the five random numbers

selected and their ranks are:

Random No. Sequence Rank

792 1 4

032 2 1

947 3 5

293 4 3

196 5 2

*

*There are four sources of variation in a LS design, two more

than that for the CRD and one more than that for the RCB

design. The sources of variation are row column, treatment

and experimental error.

*To illustrate the computation procedure for the analysis of

variance of a LS design, we use data on grain field of three

promising maize hybrids (A,B and D) and of a check (C) from an

advanced yield trial 4x4 Latin square design.

*