ABSTRACT
The main objective of this study is to test and evaluate the different Methods
(or Procedures) of Multiple Comparisons by determining the conditions
under which each of them is suitable especially in terms of protection
against errors. Data used for this study were generated by means of
computer simulation. The Experiment (Simulation) was repeated 500 times
for every set of conditions, so that empirical estimates for the Error Rates
and the Correct Decision Rate can be computed for each Comparison
Procedure. The result of the study shows that the Methods of Multiple
Comparisons can be classified into two. The first group consist of LSD,
SNK and Duncan, these differ significantly from the MCPs in the second
group and are characterized by high levels of Experimentwise and
Comparisonwise type I Error Rates. The second group consists of Tukey’s
HSD, Scheffe, Bonferroni, Sidak, Gabriel and Hochberg, characterized by
relatively low type 1 error rates.
TABLE OF CONTENTS
Title page i
Dedication ii
Declaration iii
Certification iv
Acknowledgement v
Abstract vi
Table of content vii
List of tables and figures xi
Definition of Terms xiii
CHAPTER I
1.0 Introduction 1
1.1 Purpose of the study 3
1.2 Objective of the Theses 3
1.3 Significance of the study 4
CHAPTER II (Literature review)
2.1 Introduction 5
2.2 Fisher’s Least Significance Difference (LSD) 5
2.3 Duncan’s New Multiple Range Test 6
2.4 The Student-Newman-Keul’s procedure 8
2.5 Tukey’s Honestly Significant Difference (HSD) 8
2.6 Scheffe’s Method 10
2.7 Bonferroni method 11
2.8 Sidak’s Method 13
2.9 SMM or GT2 Method 14
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2.10 Gabriel’s method 14
2.11 Studies on Evaluation of Multiple Comparisons Procedure 15
2.12 Antagonists of the Multiple Comparisons Procedure 20
2.13 Summary 21
CHAPTER III (Methodology)
3.0 Introduction 23
3.1 Source of Data
25
3.2 Generation of the Experimental Data (Simulation Method )
25
3.3 Syntax used in SPSS to generate and analyze the data
27
3.4 A sample of simulation result and output for
ANOVA/Post-hoc test 28
3.5 Data Analysis
38
3.6 Coding procedure
39
3.7 Evaluation of the Error Rates
44
CHAPTER IV (Presentation and Discussion of Result)
4.0 Introduction 47
4.1 Error Rates and CDR computed from the
Simulated Experiments 48
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4.2 Summary of the Results 55
4.3 Comments on Summary of Results 58
4.4 Some Trends observed from Multiple Bar Charts of the MCPs 59
4.4.1 Multiple Bar Charts of EER by Numbers of Treatments
and Replications 59
4.4.2 Multiple Bar Charts of CER by Numbers of
Treatments and Replications for the MCPs 61
4.4.3 Multiple Bar charts of CDR by Numbers of
Treatments and Replications for the MCPs 63
CHAPTER V (Summary, Conclusion and Recommendation)
5.0 Introduction 66
5.1 Experimentwise Error Rate (EER) 66
5.2 Comparisonwise Error Rate (CER) 67
5.3 Correct Decision Rate (CDR) 68
5.4 Effect of increasing Treatment or Replication number
on EER, CER and CDR for the individual MCPs 68
5.5 Conclusion 69
5.6 Recommendations 71
5.7 Areas for further Research 71
REFERENCES 73
APPENDIX 1 76
CHAPTER ONE
1.0 Introduction
The object of an agricultural experimenter is generally to measure the
effect of varying some factor, for example the level of protein in poultry
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diets. It is logical to expect that if different levels of protein are applied to
different birds, the variation in the weight gains observed would be due
partly to the different levels of feeding and partly to the basic variation
between birds fed at the same level. The first problem for the experimenter
is to disentangle these two parts of the variation i.e. to carryout an analysis
of variance (ANOVA) so as to obtain an estimate of the true difference
caused by his treatments, i.e. the feeding levels. A significant F-value from
the ANOVA indicates that there are differences in the treatment means.
The second problem of the experimenter may be to draw some further
conclusions. He may want to decide which pairs of treatments are different,
or he may want to contrast one treatment effect with the average of some
other treatments.
To identify where the differences are, he could do a series of pairwise
t-tests. The major set back here is that the significance levels can be
misleading. If you have 6 groups for example, there will be 15 pairwise
comparisons of means; it has long been recognized, however, that if several
t-tests have been performed at 5% level of significance, say, the probability
that at least one of these is apparently significant is greater than 0.05
(Cochran & Cox 1957). If the t-tests are independent, this probability is
0.23 for 5 tests, 0.4 for 10 tests and 0.64 for 20 tests.
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Multiple Comparison Procedures (MCPs) give more detailed
information about the differences among the treatment means, while
controlling the probability of making an incorrect decision. Several multiple
comparison procedures are available to researchers. Some notable ones are:
1. Fisher’s least significant Difference;
2. Duncan’s New Multiple range test;
3. The Student-Newman Keuls’ Procedure;
4. Tukey’s Honestly Significant Difference;
5. Scheffe’s Method.
Which procedure should be used depends upon which type of error is more
serious (Schirley & Wearden 1985). Where a type I error is not serious, a
very powerful test like Fisher’s Least Significant Difference (LSD) could be
used, otherwise more conservative tests like Tukey’s or Scheffe’s are
preferable.
The Fisher’s multiple comparison procedure is based on a t-test. If
the treatment groups are all of equal size n, then two sample averages
(ӯ1 and ӯ2) can be tested for a significant difference by a t-statistic. In order
to protect the overall type I error rate for the experiment, Fisher’s procedure
requires a prior significant F-test in the analysis of variance. With this
condition, the overall error rate (comparison wise error Rate, CER) has been
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shown to be approximately the α of the F test (Shirley & Stanley 1985).
Duncan (1975) considers the error rate for each pairwise comparison
and allows a higher rate for pairs of sample averages that are further apart
when ordered by size. This method also is believed to control the C E R.
All the other procedures of multiple comparisons also aim at reducing
the error rates so that valid conclusions can be drawn at the end of the
analysis of variance.
1.1 Purpose of the Study
The purpose of this study is to come up with concise criteria for
choosing a suitable method for multiple comparisons of means in
Agricultural experiments. This will go along way towards reducing the
problem of subjective choice of method being faced by experimenters.
1.2 Objectives of the Theses
The objectives of this thesis are:
To identify the different Multiple Comparison Procedures;
To use simulation methods to generate results of Agricultural Experiments
for the purpose of testing and evaluating the different methods of multiple
comparison of means and
To determine the conditions under which each of the various methods is
suitable.
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1.3 Significance of the Study
With any of the multiple comparison procedures, the observed
difference between any two means is compared to the appropriate critical
value for that procedure. Since the magnitudes of the critical values vary
among procedures, results obtained from the application of one procedure to
a given set of data will often differ from those obtained if another procedure
is utilized. This has led to disagreement among statisticians concerning the
appropriate criteria for choosing a procedure for pairwise multiple
comparisons of means.
It is our sincere hope that at the end of this study, we will come up
with a guideline that will help Agricultural researchers and indeed all other
scientific researchers, to choose objectively from among the numerous
multiple comparison procedures, so that there will be no basis for doubt
about the appropriateness of the MCP adopted by any researcher. The
significance of this study therefore, cannot be over emphasized.
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