STAT-C 3213 (EXPERIMENTAL DESIGN)

Introduction

Experiments in which only a single factor varies while all others are kept constant are called Single-Factor experiments. In such experiments, the treatments consist solely of different levels of single variable factor. All other factors are applied uniformly to all plots at a single prescribe level.

For example, most crop variety trials are single-factor experiments in which the single variables factor is variety and the factor levels (i.e., treatments) are the different varieties. Only the variety planted differs from one experimental plot to one another and all management factors, such fertilizers, insect control, and water management, are applied uniformly to all plots. Other examples of single-factor experiments are:

·   Fertilizer trials where several rates of a single fertilizer elements are tested.

·   Insecticide trials where several insecticides are tested

·   Plant-population trials where several plant densities are tested.

There are two groups of experimental design that are applicable to a single-factor experiment. One group is the family of Complete Block Designs, which is suited for experiments with smaller number of treatments and is characterized by blocks, each of which contains at least one complete set of treatments. The other group is the family of Incomplete block designs, which is suited for experiments with large number of treatments and is characterized by blocks, each of which contains only a fraction of the treatments to be tested.

We describe three complete block designs (complete randomized, randomized complete block, and Latin square designs) and two incomplete block designs (lattice and group balanced block designs). For each design, we illustrate procedures for randomization, plot layout, and analysis of variance with actual experiments.

LESSON 1. Completely Randomized Design (CRD)

A Completely Randomized Design (CRD) is one where treatments are assigned completely at random so that each experimental unit has the same chance of receiving any one treatment. For the CRD, any difference among experimental units receiving the same treatment is considered as experimental error. Hence, CRD is only appropriate for experiments with homogenous experimental units, such as laboratory experiments, where environmental effects are relatively easy to control. For field experiments, where there are generally large variations among experimental plots, is such environmental factors as soil. CRD is rarely used.

Lesson 1.1 CRD: Randomization and Layout

 The step-by-step procedures for randomization and layout for CRD are given here for a field experiment with four Treatments A, B, C, D each replicated 5 times.

STEP 1. Determine the total number of experimental plots (n) as the product of the number of treatments (t) and the number of replications (r), that is n = rt.

For example, n = (5)(4) = 20

SETP 2. Assign a plot number to each experimental plot in any convenient manner; for example. Consecutively from 1 to n. For our example, the plot numbers 1,…..,20 are assigned to the 20 experimental plots. (see figure below)

STEP 3. Assign the treatments to the experimental plot by any of the following randomization schemes:

A. TABLE OF RANDOM NUMBERS

Step A1.

Locate a starting point in a table of random numbers by closing your eyes and pointing a finger to any position.

Step A2.

Using the starting point obtained in Step A1, read downward vertically to obtain n=20 distinct three-digit random numbers. Three-digit numbers are preferred because they are less likely to include ties than one-or two-digit numbers.

For our example, starting at the intersection of the sixth row and the twelfth column, the 20 distinct three-digit random numbers are as shown here together with their corresponding sequence of appearance. (See sample Below)

STEP A3.

Rank the n random numbers obtained in step A2 in ascending or descending order. For our example, the 20 random numbers are ranked from the smallest to the largest, as shown in the following:

STEP A4.

                 Divide the ranks in STEP A3 into t groups, each consisting of r numbers, according to the sequence in which the random number appeared. For our example, the ranks are divided into four groups, each consisting of five numbers, as follows:

STEP A5.

                 Assign the t treatments to the n experimental plots, by using the group number in STEP A4 as the treatment number and the corresponding ranks in each group as the plot number in which the corresponding treatment is to be assigned. For our example, the first group is assigned to treatment A and plots numbered 17, 2, 15, 7, and 19 are assigned to receive this treatment; the second group is assigned to treatment B with plots numbered 13, 4, 18,1, and 8; the third group is assigned to treatment C with plots numbered 16, 14, 6, 9, and 12; and the fourth group to treatment D with plots numbered 10, 20, 3, 11, and 5. The final Layout is shown below

B. DRAWING CARDS.

STEP B1.

From the deck of ordinary playing cards, draw n card, one at a time, mixing the remaining cards after every draw. This procedure cannot be used when the total experimental units exceed 52 because there are only 52 cards in a pack.

For our example, the 20 cards are ranked form smallest to the largest:

STEP B3.

 Assign the t treatments to the n plots by using the rank obtained in STEP B2 as the plot number. Follow the procedure in Steps A4 and A5. For our example, the four treatments are assigned to 20 experimental plots as follows:

C. DRAW BY LOT.

STEP C1.

Prepare n identical pieces of paper and divide them into t groups, each group with r pieces of paper. Label each paper of the same group with same letter (or number) corresponding to a treatment. Uniformly fold each of the n labeled pieces of paper, mix them thoroughly, and place them in a container. For our example, there should be a piece of paper, five each with treatments A, B, C and D appearing on them 

STEP C2.

Draw one piece of paper at a time, without replacement and with constant shaking of the container after each draw to mix its content. For our example, the label and corresponding sequence in which each piece of paper is drawn may be as follows:

STEP C3.

Assign the treatments to plots based on the corresponding treatment label and sequence, drawn in STEP C2. For our example, treatment A would be assigned to plots numbered 3, 6,12,13 and 20. Treatment B to plots numbered, 2, 4, 9, 14 and 15. Treatment C to plots numbered 5, 8, 16, 18, and 19; and Treatment D to plots numbered 1, 7, 10, 11, and 17.

LESSON 2. Completely Randomized Design (CRD): Analysis of Variance

There are two sources of variation among the n observations obtained from a CRD trial. One is the treatment variation, the other is experimental error. The relative size of the two is used to indicate whether the observed difference among treatments is real or due to chance. The treatment difference is said to be real if treatment variation is sufficiently larger than the experimental error.

A major advantage of the CRD is the simplicity in the computation of its Analysis of Variance, especially when the number of replications is not uniform for all treatments. For most of the other designs, the analysis of variance becomes complicated when the loss of data in some plots results in unequal replications among treatments tested. 

Analysis of Variance: Equal Replication

The steps involved in the Analysis of Variance for data from a CRD experiment with an equal number of replications are given below. We use data from an experiment on chemical control of brown and stem borers in rice.

STEP 1. Group the data by treatments and calculate the Treatment Totals (T) and Grand Total (G).

STEP 2. Construct an outline of the Analysis of Variance as follows:

STEP 3. Using t to represent the number of treatments and r, the number of replications, determine the degree of freedom (df), for each source of variation.

STEP 4. Using Xi to represent the measurement of the ith plot, T as the total of the ith treatment, and n as the total number of experimental plots (n=rt). Calculate the correction factor and the various sums of squares (SS).

STEP 5. Calculate the Mean Square (MS) for each source of variation by dividing each SS by its corresponding d.f:

STEP 6. Calculate the F value for testing significance the treatment difference as:

Note that the F value should be computed only when the error d.f is six or more.

STEP 7. Obtain the Tabular F values. To obtain Tabular F values, locate your f1 = treatment df = (t-1) and f2 = error df = t (r-1). And identify your level of significance (i.e., 1% or 5%)

STEP 8. Enter all the values computed in steps 3-7 in the outline of the analysis of variance constructed in Step 2.

STEP 9. Compare the computed F value of STEP 6 with the tabular F values of STEP 7, and decide on the significance of the difference among treatments using the following rules:

1. If the computed F value is larger than the tabular F value at the 1% level of significance, the treatment difference is said to be highly significant. Such a result is generally indicated by placing two asterisks on the computed F value in the Analysis of Variance.

2. If the computed F value is larger than the tabular F value at the 5% level of significance but smaller than or equal to the tabular value at the 1% level of significance, the treatment difference is said to be significant. Such a result is indicated by placing one asterisk on the computed F value in the Analysis of Variance.

3. If the computed F value is smaller than or equal to the tabular F value at the 5% level of significance, the treatment difference is said to be nonsignificant. Such result is indicated by placing ns on the computed F value in the Analysis of Variance.

Note that a nonsignificant F test in the analysis of variance indicates the failure of the experiment to detect any difference among treatments It does not, in any way, prove that all treatments are the same, because the failure to detect treatment difference, based on the nonsignificant F test, could be the result of either a very small or nil treatment difference or a very large experimental error, or both. Thus, whenever the F test is nonsignificant, the researcher should examine the size of the experimental error and the numerical difference among treatment means. If both values are large, the trial may be repeated and efforts made to reduce the experimental error so that the difference among treatments, if any, can be detected. On the other hand, if both values are small, the difference among treatments is probably too small to be of any economic value and, thus no additional trials are needed.

STEP 10. Compute the Grand Mean and the coefficient of variation cv as follows:

The cv indicates the degree of precision with which the treatments are compared and is good index of reliability of the experiment. It expresses the experimental error as percentage of the mean; thus, the higher the cv value, the lower the reliability of the experiment. The cv value is generally placed below the Analysis of Variance Table.

The cv varies greatly with the type of experiment, the crop grown, and the character measured. An experienced researcher, however can make a reasonably good judgement on the acceptability of a particular cv value of a given type experiment.

Analysis of Variance: Unequal Replication

Because the computational procedure for the CRD is not overly complicated when the number of replications differs among treatments, the CRD is commonly used for studies where the experimental material makes it difficult to use an equal number of replications for all treatments. 

Some examples of these cases are:

• Animal feeding experiments where the number of animals for each breed is not the same

• Experiments for comparing body length of different species of insect caught in an insect trap

• Experiments that are originally set up with an equal number of replications but

The steps involved in the Analysis of Variance for data from a CRD experiment with an unequal number of replications are given below.

STEP 1. Group the data by treatments and calculate the Treatment Totals (T) and Grand Total (G). And Construct an outline of the Analysis of Variance as follows:

STEP 2. Using t to represent the number of treatments and n for the total number of observations, determine the degree of freedom for each source of variation, as follows:

Total d.f = n -1

Treatment d.f = t – 1

Error d.f = Total d.f – Treatment d.f

STEP 3. With the treatment totals (T) and the Grand Total (G), compute the correction factor and the various sums of squares, as follows:

STEP 4. Follow steps 5 to 10 stipulated in the Analysis of Variance for Equal Replication.

Grain Yield of Rice Resulting from Use of different Foliar and Granular Insecticides for the control of brown planthoppers and stem borers, from a CRD Experiment with 4 (r) Replications and 7 (t) Treatments (see data below)

Download Data here

STEP 1. Group the data by treatments and calculate the treatment totals (T) and Grand Total (G)

STEP 2. Analysis of Variance Table

STEP 3. Determine Degrees of Freedom

Total d.f = (r)(t) -1 = (4)(7) -1 = 27

Treatment d.f = t – 1 = 7 – 1 = 6

Error d.f = t (r – 1) = 7 (4 – 1 ) = 21

*Error df = Total d.f – Treatment d.f = 27 – 6 = 21

STEP 4. Solve for the C.F, Total SS, Treatment SS and Error SS.

STEP 5. Calculate the Mean Square (MS)

STEP 6. Calculate the F value

STEP 7. Identify the Tabular F values

STEP 8. Enter all the values computed in the Analysis of Variance Table.

STEP 9. Compare the computed F value with the Tabular F values.

STEP 10. Compute Grand mean and Coefficient of variation (cv)