EXPERIMENTAL DESIGNS USED IN RICE RESEARCH
Learning Objective
At the end of this lesson, the participants should be able to •
identify the advantages and disadvantages of using each of the following designs: -
completely randomized design randomized complete block design latin square design split plot design strip plot design
•
layout an experiment using any of the above designs; and
•
understand and learn what proper blocking is.
Introduction
Completely Randomized and Randomized Complete Block Designs are the most common designs. The Completely Randomized Design is used when the experimental units are nearly homogenous. On the other hand, when the variability variability to be controlled is unidirectional, say fertility gradient, then the Randomized Complete Block Design is used. Other designs arise in consequence of various features of the treatments, constraints of land or other resources resources or concentration of interest. For example: 1. Variability to be controlled is in two directions: Latin Squares 2. Number of treatments exceeds convenient block size: Incomplete Block Designs 3. Treatment structure is according to a combination of two or more factors (factorial experiment) but certain factors must go on larger units or additional treatments must be incorporated into an existing experiment exp eriment or (more rarely) rarely)
Experimental Designs Used in Rice Research
97
4. higher precision is wanted on some types of comparison than on others: Split-plot and Strip-plot Designs 5. A factorial factorial experiment but the total number of treatment treatment combinations exceeds practicable size of experiment: Fractional Factorial Designs
Completely Randomized Design
This is the simplest type type of design. The treatments are assigned assigned completely at random random so that each experimental unit has the the same chance of receiving each of the treatments. treatments. In addition the units should be processed in random order at all subsequent stages of the experiment where this order is likely to affect results (Cochran and Cox, 1957). For the CRD, any difference among experimental units receiving the same treatment is considered an experimental error. error. Hence, the CRD is only appropriate for experiments with homogenous experimental units. Advantages of the Design
This design has several conveniences: 1. Flexible. Any number of treatments and replicates replicates may be used. The number of replications may vary from treatment to treatment in order to place more emphasis on treatments of special interest. 2. Easy to analyze even if the number of replicates is not the same from treatment to treatment. 3. The method of analysis remains simple even if results of some units are missing or rejected. In addition, the relative relative loss of information due to missing data is smaller than in other designs. Disadvantages of the Design
The disadvantage of a Completely Randomized Design is that it is usually suited only for small numbers of treatments and for homogenous experimental materials. When large numbers of treatments are included, a relatively large amount of experimental material must be used. This generally increases the variation variation among treatment responses responses and thus makes the experimental error large. This error may be reduced with the use of a different different design, unless the units are highly homogenous or the experimenter has no information by which to arrange or handle the units in more homogenous groups.
98
Completely Randomized Design
CRD is seldom used for field experiments because experience has shown other designs to be more suitable. Layout of the Design
The term layout refers to the assignment of experimental treatments on the experimental site whether it be over space, time or type type of material. The whole of the experimental area or material is partitioned into a number of experimental units, say, N. A random selection of r1 experimental units is made and one of the t treatments is applied to these units. A random random selection selection of r2 of the remaining N-r1 experimental units is made and one of the remaining t-1 treatments is applied to these particular units. units. This procedure procedure continues until all treatments have been applied. When each treatment is replicated an equal number of times, r1=r2=...=rv=r and Σri=rt=N experimental units. Unless practical limitations dictate otherwise, such as scarcity of units, or unless some treatments are more variable or are of greater interest than others, equal replication for each treatment is recommended. For example, if an experiment with five treatments replicated four times is to be conducted in a Completely Randomized Design, the layout may be as follows:
T4
T5
T3
T2
T3
T1
T4
T5
T5
T2
T3
T1
T2
T5
T1
T3
T4
T2
T4
T1
Randomized Complete Block Design
The Randomized Complete Block Design is characterized by blocks of equal sizes, each containing a complete set of all treatments. Each block constitutes a replicate. replicate. At all stages of the experiment the objective is to keep the experimental errors within
Experimental Designs Used in Rice Research
99
each block as small as practicable. Thus, during the course of the experiment, a uniform technique should be employed in all experimental units within within a group. Any changes changes in technique or conditions that may affect the results should be made between groups. In RCB design, the assumption is that there is no treatment × block interaction. That is, on the average, treatments have the same effect in each block. Blocking Technique
The primary purpose of blocking is to reduce experimental error by eliminating the contribution of known sources of variation among experimental unit. This is done by grouping the experimental units into blocks such that variability within each block is minimized and variability variability among blocks is maximized. Because only the variability within a block becomes part of the experimental error, blocking is most effective when the experimental area has a predictable predictable pattern of variability. variability. With a predictable pattern, pattern, plot shape and block orientation can be chosen so that much of the variation is accounted for by the difference among blocks, and experimental plots within the same block are kept as uniform as possible. There are two important decisions that have to be made in arriving at an appropriate and effective blocking technique. These are: •
The selection of the source of variability to be used as the basis for blocking.
•
The selection of the block shape and orientation.
An ideal source of variation to use as the basis for blocking is one that is large and highly predictable. Examples are: •
Soil heterogeneity, in a fertilizer or variety trial where yield data is the primary character of interest.
•
Direction of insect migration, in an insecticide trial where insect infestation is the primary character of interest.
•
Slope of the field, in a study of plant reaction to water stress.
After identifying the specific source of variability to be used as the basis for blocking, the size and shape of the the blocks must be selected selected to maximize variability among blocks. The guidelines for this decision are: 1. When the gradient is unidirectional (i.e., there is only one gradient), use long and narrow blocks. Furthermore, orient these blocks so their their length is perpendicular to the direction of the gradient.
100
Randomized Complete Block Design
2. When the fertility gradient occurs in two directions with one gradient much stronger than the other, ignore the weaker gradient and follow the preceding guideline for the case of the unidirectional gradient. 3. When the fertility gradient occurs in two directions with both gradients equally strong and perpendicular to each other, choose o ne of these alternatives: •
Use blocks that are square as possible.
•
Use long and narrow blocks with their length perpendicular to the direction of one gradient and use the covariance technique to take care of the other gradient.
•
Use latin square design with two-way blocking, one for each gradient.
4. When the pattern of variability is not predictable, blocks should be as square as possible. Advantages of the Design
The advantages of this design are: 1. Accuracy. This design has been known to be more accurate than the Completely Randomized Design for most types of experimental work provided the blocking has been correctly applied, that is, units within block are more similar than those between blocks. 2. Flexibility. There is no restriction restriction on the number of treatments treatments and replications. At least two replicates are required to obtain tests tests of significance. significance. In addition, some some treatments may be included more than once to increase precision of their measurements with little complication to the analysis. 3. Ease of analysis. The statistical statistical analysis analysis is simple and easy to perform. perform. Moreover, the error of any treatment comparison may be isolated and any number of treatments may be omitted from the analysis without complicating it. These features may be useful when certain treatment differences turn out to be very large, when some treatments produce failure or when the experimental errors for the various comparisons are heterogeneous. Disadvantages of the Design
The chief disadvantage of the Randomized Complete Block Design is that it is not suitable for large numbers of treatments or for cases in which the complete blocks contain
Experimental Designs Used in Rice Research
101
considerable variability. variability. Another small disadvantage is that some resources of the experiment are used to estimate block effects. effects. If there are are no differences between blocks these resources are wasted. Layout of the Design
When the experimental units have been blocked, the treatments are assigned at random to the units within each group. group. A new randomization is made for each block group. Using the same example as in the Completely Randomized Design, the layout may look as follows:
Block I
Block II
Block III Block IV
T2
T5
T1
T2
T3
T1
T4
T5
T5
T4
T3
T1
T4
T2
T5
T3
T1
T3
T2
T4
Latin Square Design
The Latin Square Design has the capability to block an area in two directions. The name was derived from an ancient puzzle that is concerned with the number of different ways Latin letters can be arranged in a square matrix so that each letter appears once and only once in each row and each column. The effect of the double blocking is to eliminate from the experimental error differences among rows and differences differences among columns before comparing the treatments. Thus the Latin Square provides more opportunity than Randomized Blocks for the reduction of error by skillful planning.
102
Latin Square Design
Advantages of the Design
The advantages of a Latin Square design are: 1. With a two-way blocking, the Latin Square controls more of the variation than the Completely Randomized or Randomized Complete Block Designs. The two-way elimination of variation often results in a small error mean square. 2. The analysis is simple; it is only slightly more complicated than that for the Randomized Complete Block Design. Disadvantages of the Design
The major disadvantage of the Latin Square Design is that the number of treatments must equal the number of replications. If the number of treatments becomes very large, large, the number of replications required becomes impractical. impractical. Like the RCB design, but to a larger extent, some resources resources are used to estimate row and column effects. These are wasted if the the material is homogenous. In small squares this is a serious serious restriction restriction because not enough degrees of freedom for estimating the experimental error unless squares themselves are replicated. Layout of the Design
Examples of Latin Squares are shown below: 3×3
4×4
A B C B C A C A B
A B C D E
5×5 B C D A E C D A E E B A C D B
A B C D
E D B C A
1 B C A D D B C A
D C A B
A B C D E F
A B C D
2 B C C D D A A B
6×6 B C D E F D C A D E F B A F E C C A B F E B A D
F E A B D C
D A B C
A B C D
3 B C D A A D C B
A B C D E F G
7×7 B C D E C D E F D E F G E F G A F G A B G A B C A B C D
D C B A
A B C D
4 B C A D D A C B
F G A B C D E
G A B C D E F
D C B A
Experimental Designs Used in Rice Research
103
A B C D E F G H
8×8 B C D E F G C D E F G H D E F G H A E F G H A B F G H A B C G H A B C D H A B C D E A B C D E F
H A B C D E F G
9×9 A B C D E F G H I B C D E F G H I A C D E F G H I A B D E F G H I A B C E F G H I A B C D F G H I A B C D E G H I A B C D E F H I A B C D E F G I A B C D E F G H
10 × 10 A B C D E F G H I J B C D E F G H I J A C D E F G H I J A B D E F G H I J A B C E F G H I J A B C D F G H I J A B C D E G H I J A B C D E F H I J A B C D E F G I J A B C D E F G H J A B C D E F G H I
11 × 11 A B C D E F G H I J K B C D E F G H I J K A C D E F G H I J K A B D E F G H I J K A B C E F G H I J K A B C D F G H I J K A B C D E G H I J K A B C D E F H I J K A B C D E F G I J K A B C D E F G H J K A B C D E F G H I K A B C D E F G H I J
The method of randomization for this plan is as follows: 1. 3 × 3. Arrange all columns and rows at random. 2. 4 × 4. Select at random one of the four squares. Arrange at random all columns and rows. 3. 5 × 5 and higher squares. independently at random.
Arrange all rows, rows, columns and treatments treatments
Factorial Experiments
Experiments so far presented involved single-factor experiments, that is, only one factor is being tested. Strictly speaking, results results of single factor experiments are valid only under the condition or particular level they were were tested. If one wishes wishes to broaden the scope of the applicability of the results, then a factorial experiment should be considered.
104 Factorial Experiments
That is, testing testing two or more factors simultaneously. simultaneously. experiments are:
The advantages of factorial
•
They provide information on the interaction among various factors tested.
•
They are economical by comparison to several single factor experiments.
•
They broaden the applicability of main effect conclusions.
Interaction between Factors
Two factors are said to interact if the effect of one factor changes as the level of the other factor changes. changes. Consider the following figure: figure: Grain Yield
Grain Yield V1
V1
V2
V2
Nitrogen Rate Figure a
Nitrogen Rate Figure b
Figure a shows an interaction between variety and nitrogen. That is, nitrogen nitrogen does not affect the yield yield of the two varieties varieties in the same manner. In this case, while nitrogen fertilization increases the yield of the first variety, it decreases the yield of the second. Figure b still shows interaction interaction between variety and nitrogen. Here, nitrogen increases increases yield of both V1 and V2 but the rate of increase is much greater for V1 than for V2. Factorial Experiments in Complete Block Designs
The complete block (CRD, RCB and Latin Square) designs are not only appropriate for single-factor experiments but for factorial experiments as well.
Experimental Designs Used in Rice Research
105
Suppose an experiment involving 5 N-rates and 4 Varieties is to be conducted in a RCB design in 3 replications. replications. The layout may look as follows:
V2 60 V4 90 V3 30 V1 30 V4 60
V4 30 V2 30 V4 120 V2 120 V1 90
V3 90 V1 60 V2 0 V4 0 V3 60
V1 120 V3 0 V3 120 V2 90 V1 0
V4 60 V3 0 V2 30 V1 30 V4 0
Rep I
V2 120 V1 120 V3 120 V2 90 V2 60
V3 90 V4 120 V2 30 V3 60 V4 90
Rep II
V1 60 V2 0 V3 120 V1 90 V3 30
V3 60 V2 120 V2 0 V4 30 V1 60
V1 120 V1 0 V4 30 V2 60 V4 90
V4 120 V3 30 V3 120 V3 0 V2 30
V2 90 V1 90 V4 0 V1 30 V3 90
Rep III
Factor A: Nitrogen (0, 30, 60, 90, 120) rate (kg/ha) Factor B: Variety (V1, V2, V3, V4)
Split plot Design
The split-plot designs are frequently used for factorial experiments in which the nature of the experimental material or the operations involved make it difficult to handle all factor combinations in the same manner. The basic split-plot split-plot design involves assigning the levels of one factor to main plots arranged in a completely random, randomized complete block or latin square design. design. The levels of the second second factor are assigned to subplots subplots within each main plot. The design usually sacrifices precision precision in estimating the average effects of the treatments treatments assigned to main plots. It often improves the precision for comparing the average effects of treatments assigned to subplots and, when interaction exist, for comparing the effects of subplot treatments for a given main plot treatment. This arises from the fact that experimental error for main plots is usually larger than the experimental error used to compare subplot treatments. Usually, the error term for subplot treatments is smaller than would be obtained if all treatment combinations were arranged in a randomized complete block design. Advantages of the Design
The advantages of the split-plot design are: 1. Experimental units which are large by necessity or design may be utilized to compare subsidiary treatments. 2. Increased precision on the test of subplot main effect and interaction effect.
106 Split Plot Design
Disadvantages of the Design
The disadvantages of the split-plot design are: 1. The main plot treatments are measured with less precision than they are in a randomized complete block design. 2. When missing data occur, the increase in complexity of the analysis for the split-plot design is greater than for the randomized complete block design. Layout of the Design
Suppose an experiment involving 5 N-rates and 4 Varieties is to be conducted in a Split plot design in 3 replications. The layout of the design may look as follows: V2
V3
V1
V4
V4
V2
30 V3
V4
V2
V1
V3
V4
60 V2
V1
V4
V3
V4
V3
V4
V1
V2
V1
V2
V3
V2
V3
V2
V1
V4
V3
V4
V1
V2
V1
V2
V1
V3
V2
V1
V4
V3
V2
120 V3
V2
V1
V4
V4
V1
V3
V2
90
120 V3
V4 60
30
0 V2
V3
90
90
V1
V1 0
0 V4
V3
V4
V1
120
60
30
Rep I
Rep II
Rep III
Mainplot factor: Nitrogen (0, 30, 60, 90, 120) rate (kg/ha) Subplot factor: factor: Variety (V1, V2, V3, V4)
Strip-Plot Design
If in a split-plot design the subplot treatments are not separately randomized for each mainplot, but are randomly allocated to strips of subplots cutting across each replication, we have what is called the strip-plot, criss-cross, criss-cross, or split-block design. design. The first is to be preferred, the levels of each factor forming strips at right-angles across each block. There are two types of mainplot corresponding to the two factors, the levels of each factor being separately randomized in each block. This is a valid design and has a use when it is convenient to apply both factors to large large areas. However, corresponding to the three types of unit (2 mainplots and subplot) there are three error terms, one for each main effect and one for the interaction.
Experimental Designs Used in Rice Research
107
Advantages of the Design
The advantages of the strip-plot design are: 1. The subplots may be kept relatively small, even though the mainplots for both factors must be relatively large experimental units. 2. More precise information is obtained for the interaction. Disadvantages of the Design
The disadvantages of the design are: 1. Precision in estimating the main effects is less than if the experiment were conducted in a randomized complete block design. 2. The analysis of the design is more complex than the ordinary randomized complete block design. Layout of the Design
Suppose we wish to conduct a two-factor experiment involving 5 varieties and 4 nitrogen rates arranged in a strip-plot design with 4 replications. The layout of the design may look as follows: N1
N3
N2
N4
N3
N4
N1
N2
N3
N2
N1
N4
N2
N1
V2
V4 V4
V1
V5
V5
V3 V3
V2
V1
V4
V2 V2
V3
V4
V1
V5 V5
V5
V3
V3 Rep I
V1 V1 Rep II
V4 Rep III
V2 Rep IV
108 Strip Plot Design
N4
N3
References:
Box, G.E.P., Hunter, W.G. & Hunter, J.S. (1978). Statistics for Experiments, Experiments, John Wiley & Sons, Inc. Cochran, W.G. & Cox, G. M. (1950). (1950). Experimental Design, John Wiley & Sons, Inc. Federrer, W.T. (1955). Experimental Design, McMillan. Gomez, K.A. & Gomez, A.A. (1984). Statistical Procedures for Agricultural Agricultural Research, John Wiley & Sons, Inc.. John, P.W.M. (1971). Statistical Design Design and Analysis Analysis of Experiments, The McMillan McMillan Company. Little, T.M. & Hills, F.J. (1978). Agricultural Experimentation - Design and Analysis, Analysis, John Wiley & Sons. Montgomery, D.C. D.C. (1984). Design and Analysis Analysis of Experiments, 2nd Ed., John Wiley & Sons, Inc. Ostle, B. & Mensing, R.W. (1975). Statistics in Research, 3rd. ed., The Iowa Iowa State University Press. Rayner, A.A. (1967). A First Course in Biometry Biometry for Agriculture Agriculture Students, University of Natal Press.
Experimental Designs Used in Rice Research
109
110
