After ANOVA

After Anova ... What Then? 181

AFTER ANOVA ... WHAT THEN ?

Learning Objective

At the end of this lesson, the participants should be able to:

• distinguish between planned and unplanned mean comparisons;

• gain skills in partitioning treatment sums of squares; and

• perform trend analysis for quantitative treatment levels.

The result of the Analysis of Variance verifies the existence of some differences among

treatments tested but does not specify which treatment means significantly differ from

one another. To obtain this information, pairwise mean comparison is usually performed

after the ANOVA. However, there are statistical techniques that can extract more

information on the treatment means other than the pairwise mean comparison.

Some of the statistical techniques useful for further analysis after ANOVA are:

• Pairwise comparison

- Planned

- Unplanned

• Partitioning of sums of squares (PSS)

- Group comparison

- Factorial comparison

- Orthogonal Polynomials

• Regression analysis (when response variable and factors are quantitative)

182 Pairwise Comparison

Pairwise Comparison

Pair comparison is the simplest and most commonly used comparison in agricultural

research. There are two types: planned and unplanned pairwise comparison.

Planned pairwise comparison - in which the specific pair of treatments to be compared

is identified before the start of the experiment. A common example is comparison of the

control treatment with each of the other treatments.

Example 1:

In an experiment where the objective is to determine the effectiveness of a new herbicide

in controlling weeds relative to three (3) existing herbicides where the treatments are as

follows:

Treatments: H1 = non-weeded treatment

H2 = Butachlor 1.0

H3 = 2,4-D

H4 = Thiobencarb 1.0

H5 = New Herbicide

Then planned comparisons are :

H1 vs H5

H2 vs H5

H3 vs H5

H4 vs H5


Example 2:

As another example, if the objective is to determine if any of the following three neem

seed kernel water extract (NSKWE) treatments are effective in controlling the incidence

of major insect pests:

Treatments:

T1 = 5000 ppm NSKWE (systemic) & 250 ppm Citowett (foliar)

T2 = 5000 ppm NSKWE (systemic) & 10000 ppm NSKWE-Citowett (forthnightly

application)

T3 = 5000 ppm NSKWE (systemic) & 10000 ppm NSKWE-Citowett (weekly)

T4 = control (250 ppm Citowett)

Then the planned comparisons are:

T1 vs T4

T2 vs T4

T3 vs T4

Unplanned Pairwise Comparison 184

Unplanned pairwise comparison - in which no specific pairs of treatments are

identified for comparison before the start of the experiment. Instead, every possible pair

of treatment means is compared to identify those that are significantly different

Example 3:

In an experiment where the objective is to evaluate extracts of 4 promising plants for

insecticidal properties against primary pests of rice after harvest, all possible pairs of

treatments are to be compared:

Neem Oil vs Margosan Oil

Neem Oil vs Acorus Oil

Neem Oil vs Turmeric Oil

Margosan Oil vs Acorus Oil

Margosan Oil vs Turmeric Oil

Acorus Oil vs Turmeric Oil

This situation of unstructured qualitative treatments is, however, extremely rare and

unplanned pairwise comparisons are hardly ever appropriate.

The two most commonly used test procedures for pair comparisons are the least

significant difference (LSD) test which is suited for planned pair comparisons, and the

Duncan’s multiple range test (DMRT) which is suited for unplanned pair comparisons.

LSDα is the value against which the difference between any two means is compared. If

the difference exceeds this value, then it is significant at a probability α. The LSD is

computed as LSDα = tα sd. In the LSD test, a single LSD value, at a prescribed level of

significance α serves as the critical value between significant and non-significant

difference between any pair of treatment means. This test is not suitable for comparing

all possible pairs of means when number of treatments is large (say, greater than 5)

because in experiments where no real difference exists among the treatments, the

probability that at least one pair will have a difference that exceeds the LSD value,

increases with the number of treatments being tested.


For example, it can be shown that the numerical difference between the largest and the

smallest treatment means is expected to exceed the LSD.05 even when there is no

difference between treatments with the following frequency:

Number of treatments

Percent of the time

Range exceeds LSD(.05)

5 29%

10 63%

15 83%

Hence, the LSD test must be used only when the F-test for treatment effect is significant

and the number of treatments tested is small, say less than six.

Duncan’s Multiple Range Test (DMRT) is useful in experiments that require the

comparison of all possible pairs of treatment means. The procedure for applying the

DMRT is similar to that for the LSD test but DMRT involves the computation of critical

values that allow for the rank differences between the two treatment means being

compared at each stage.

Multiple comparison procedures like LSD and DMRT are best suited for experiments

with unstructured qualitative treatments. Usually, the objective for this type of

experiment is to select the best treatment among a set of qualitative treatments. Example

of this is variety trials. However, multiple comparison is never appropriate when 1) the

treatments are graded levels of a quantitative variable, 2) for factorial combination of two

or more factors at two or more levels, and 3) for quantitative treatments where previously

formulated linear combinations of treatments are of particular interest. Indiscriminant use

of multiple comparison tests may lead to loss of information and reduced efficiency when

more appropriate procedures are available.

186 Partitioning of Sum of Squares

Partitioning of Sum of Squares

Main effects or interaction sums of squares in the Analysis of Variance (ANOVA) can

often be partitioned into two or more components based on the structure of the

treatments. Each component is designed to test a specific hypothesis about the

treatments. Such procedure is called the partitioning of sum of squares or PSS.

If the decision was made before the experiment is performed, i.e., without studying the

data, then tests based on PSS are perfectly valid no matter what the status of the overall

F-test for the partitioned effect. These tests are independent provided that the contrasts

are orthogonal. When factor levels are amounts of quantitative input, the form of the

response to that factor can be tested through PSS using orthogonal polynomials.

For example, in a variety trial with three varieties A, B, and C, variety A may be the

standard variety and B and C new varieties. Immediately, the researcher would like to

know if the new varieties are different from the standard one, and whether the new ones

differ between themselves. These are special questions that have been explicitly or

implicitly asked by the researcher prior to doing the experiment.

Such questions are expressed as contrasts between treatment means as follows: Is the

value of (2 x mean of A - mean of B - mean of C) significantly different from zero?

These comparisons are termed contrasts because the coefficients of the means: 2, -1, -1,

always sum to zero. In general given a factor with k levels and it is the intention to test a

contrast among these levels with coefficients C1, C2, ..., Ck, then ΣCi = 0 and proceed as

follows:

The SS for the contrast is computed as: CSS = (Σ CiTi)2 / (r Σ Ci

2) where Ti are the

treatment totals and r is the number of replications of each treatment. A contrast SS has 1

dF, and test is made about the null hypothesis that the contrast is zero using F =

CSS/EMS with 1 dF and error dF.


Example 4:

In the RCB analysis, it is of interest to compare the standard variety A with the new

varieties B and C.

Variety A B C

Totals 16.0 18.8 18.0

Contrast Coefficients for standard against new

varieties

2 -1 -1

Contrast Coefficients for variety B against C 0 1 -1

CSS = (2*16.0 - 1*18.8 - 1*18.0)2/(4*6) = 0.960

F = 0.960/0.089 = 10.7 with 1 and 6 dF which is significant at 0.01

The other obvious contrast among these varieties is to compare the two new varieties, B

and C. The coefficients for this contrast would be: 0, +1 , -1 so that CSS = (0*16.0 +

1*18.8 - 1*18.0)2/(4*2) = 0.080 which leads to an F computed less than one and so

cannot be significant.

Notice, in this case the two contrast SSs add up to the variety SS, 0.960 + 0.080 = 1.040.

In effect the variety SS has been partitioned into two separate SSs for testing specific

hypotheses.

In general any SS with k dF can be partitioned into k contrast SSs provided the contrasts

are orthogonal. If the contrasts are orthogonal then the sum of products of the

coefficients of any two of the contrasts will always be zero, as in this case: (2*0) + (-1*1)

+ (-1*-1) = 0. When the contrasts are orthogonal the resulting tests are statistically

independent and so the test level (α) is preserved over the set of tests.


Frequently the contrast tests are included in the ANOVA table and for the example above,

the table would look as follows:

SV dF SS MS F

Blocks 3 0.587 0.196

Varieties 2 1.040 0.520 5.84*

A vs B&C 1 0.960 0.960 10.70**

B vs C 1 0.080 0.080 0.89

Error 6 0.533 0.089

Total 11 2.160

A special case of these comparisons is always suggested when the factor has a set of

quantitative levels such as the amount of fertilizer, or the temperature of a mixture. It is

almost always desired to consider whether the response to the levels of the factor follows

a straight line or some other specific curve, perhaps quadratic or cubic.

The contrast coefficients for these tests are called orthogonal polynomials and provided

the levels of the factor are equally spaced, the coefficients can be found in common

statistical tables. They are different for each number of different levels. The most

common ones are given below. They are used exactly the same way as the examples

above.

No. of

Levels

Trend SS

3 Linear -1 0 +1 2

3 Quadratic +1 -2 +1 6

4 Linear -3 -1 +1 +3 20

4 Quadratic +1 -1 -1 +1 4

4 Cubic -1 +3 -3 +1 20

5 Linear -2 -1 0 +1 +2 10

5 Quadratic +2 -1 -2 -1 +2 14

5 Cubic -1 +2 0 -2 +1 10

After ANOVA ... What Then? 189

There are three (3) ways of Partitioning Sums of Squares

1. Group Comparison

• Between-Group Comparison, in which treatments are classified into s meaningful

groups, each group consisting of one or more treatments, and the aggregated mean

of each group is compared to that of the others.

• Within-Group Comparison, which is primarily designed to compare treatments

belonging to a subset of all the treatments tested.

2. Factorial Comparison is applicable only to factorial treatments in which specific sets

of treatment means are compared to investigate the main effects of the factors tested

and, in particular, the nature of their interaction.

3. Orthogonal Polynomials are partitioning designed to examine the functional

relationship between factor levels and treatment means. As a result, it is suited for

treatments that are quantitative, such as rate of herbicide application, rate of fertilizer

application, and planting distance or spacing.

Example 5:

For group comparison, consider a varietal trial involving 5 varieties and 4 replications

layed out in RCB.

Varieties : V1

Japonica group

V2

V3

V4 Indica group

V5

The following group comparisons are of interest :

• Japonica group vs Indica group (V1, V2) vs (V3, V4, V5)

• Within Japonica group V1 vs V2

• Within Indica group V3 vs V4 vs V5


With these, the ANOVA table will be of the following form:

SV dF

Block 3

Variety 4

(V1, V2) vs (V3, V4, V5) (1)

V1 vs V2 (1)

V3 vs V4 vs V5 (2)

Error 12

Total 19

Example 6:

For factorial comparison, consider an experiment where in the objective is to determine

the effect of prilled urea (PU) sulfur coated urea (SCU) and urea super granules (USG) at

three rates of application on rice yield.

Treatment No. N-rate (kg/ha) Form of Urea

1 (control) 0 -

2 29 PU

3 29 SCU

4 29 USG

5 58 PU

6 58 SCU

7 58 USG

8 87 PU

9 87 SCU

10 87 USG


And further assume that this experiment is layed-out in RCB with 4 replications, then the

detailed analysis of variance table should have the following form:

SV dF MS F

Rep 3 0.7064 5.00**

Treatment 9 1.1638 8.23*

Ctrl vs Trtd (1) 4.0541 28.68**

Among Trtd (8) 0.8025 5.68**

Rate (R) 2 1.0653 7.54**

Form (F) 2 0.2028 1.43ns

RxF 4 0.9710 6.87**

Error 27 0.1413

Example 7:

For orthogonal polynomials, consider the following yield trial involving 4 N-rates and

3 replications layed out in RCB. The objective is to determine the optimum N-rate that

will maximize rice yield.

Treatments: N-rates

1. 0 kg N/ha

2. 60 kg N/ha

3. 90 kg N/ha

4. 120 kg N/ha

The following is an outline of the ANOVA table for this trial:

SV dF MS F

Block 2 0.104381 < 1

N-rate 3 3.055244 21.07**

Linear (1) 8.974925 61.09**

Quadratic (1) 0.018254 < 1

Cubic (1) 0.172552 1.19ns

Error 6 0.145011

c.v. = 5.7%

192 Exercise

Results show that N-rate is highly significant which indicate that much of the variation in

yield is affected by the differential rates in N application. The partitioning of the N-rate

SS verifies a linear effect of N-rate on yield within the range of 0 kg N/ha to 120 N kg/ha.

References :

Cochran, W.G., and Cox, G.M. (1957). Experimental Designs, Second Edition. John

Wiley, New York, New York.

Gomez, K.A., and Gomez, A.A. (1984). Statistical Procedures for Agricultural Research.

John Wiley, New York, New York.

Petersen, R.G. (1977), Use and Misuse of Multiple Comparison Procedures, Agronomy

Journal, Vol. 69, March-April, 205-208.

Searle, S.R. (1987). Linear Models for Unbalanced Data. John Wiley, New York, New

York.

Snedecor, G.W., and Cochran, W.G. (1989). Statistical Methods, Eighth Edition. Iowa

State University, Ames, Iowa.

After ANOVA ... What Then? 193

After ANOVA

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