biometry MTH 201

Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
0.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
0.1.1 Course Objective . . . . . . . . . . . . . . . . . . . . . . . . . v
0.1.2 Course Description . . . . . . . . . . . . . . . . . . . . . . . . v
0.1.3 Pre-requisite . . . . . . . . . . . . . . . . . . . . . . . . . . . v
0.1.4 Course requirement . . . . . . . . . . . . . . . . . . . . . . . v
0.1.5 Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
0.1.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
1 Terminologies in Experimental Designs 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Principles of experimental designs 5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Randomization principle . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Replication principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Local control principle . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Analysis of Variance 9
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Assumptions in the analysis of variance . . . . . . . . . . . . . . . . 10
3.3 Analysis of variance for one-way classification . . . . . . . . . . . . . 11
3.3.1 Analysis of variance for one-way classification with unequal
replication (unbalanced data) . . . . . . . . . . . . . . . . . . 11
3.3.2 Linear additive model for one-way classification . . . . . . . . 12
3.3.3 Fixed vs. random effects . . . . . . . . . . . . . . . . . . . . . 12
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ii TABLE OF CONTENTS
3.3.4 Calculation of sums of squares . . . . . . . . . . . . . . . . . 13
3.3.5 ANOVA for one-way classification with equal replication (bal-
anced data) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 ANOVA for two-way classification (Without Replication) . . . . . . 21
3.4.1 Linear additive model for two-way classification . . . . . . . . 21
3.5 The least significance difference (LSD) . . . . . . . . . . . . . . . . . 25
4 Introduction to SPSS 29
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Starting SPSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 Data entry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.4 Keying data into SPSS . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.4.1 Osteopathic manipulation data set . . . . . . . . . . . . . . . 32
4.5 Opening an existing dataset . . . . . . . . . . . . . . . . . . . . . . . 34
4.6 Importing data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.7 Exporting data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.8 ANOVA for one-way classification in SPSS . . . . . . . . . . . . . . . 35
5 Completely Randomized Design 39
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.3 Statistical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.3.1 Statistical hypotheses . . . . . . . . . . . . . . . . . . . . . . 40
5.3.2 Test procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.4 Advantages and disadvantages of CRD . . . . . . . . . . . . . . . . . 41
5.4.1 Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.4.2 Disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6 Randomised Block Design 45
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.2 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.3.1 Statistical hypotheses . . . . . . . . . . . . . . . . . . . . . . 47
6.3.2 Test procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 47

TABLE OF CONTENTS iii
6.4 Advantages and disadvantages of RBD . . . . . . . . . . . . . . . . . 48
6.4.1 Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.4.2 Disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.6 Reasons for blocking in RBD . . . . . . . . . . . . . . . . . . . . . . 52
7 Latin Square Design 55
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7.2 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7.2.1 Linear additive model . . . . . . . . . . . . . . . . . . . . . . 57
7.3.1 Calculation of sums of squares . . . . . . . . . . . . . . . . . 57
7.4 Advantages and disadvantages of LSD . . . . . . . . . . . . . . . . . 58
7.4.1 Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
7.4.2 Disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
7.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
8 Factorial Experiments 65
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
8.2 Main effects and interaction effects . . . . . . . . . . . . . . . . . . . 66
8.3 The 22 factorial experiments . . . . . . . . . . . . . . . . . . . . . . . 66
8.4 The 23 factorial experiments . . . . . . . . . . . . . . . . . . . . . . . 68
8.5 Sum of squares due to factorial effects . . . . . . . . . . . . . . . . . 69
8.6 Tests of significance of factorial effects . . . . . . . . . . . . . . . . . 71
8.7 Yates’ method of computing factorial effect totals . . . . . . . . . . . 74
9 Multiple Comparisons 77
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
9.2 Multiple comparisons procedures . . . . . . . . . . . . . . . . . . . . 78
9.2.1 Duncan’s new multiple range-test . . . . . . . . . . . . . . . . 78
10 Multiple Linear Regression 85
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
10.2 Multiple Regression Models . . . . . . . . . . . . . . . . . . . . . . . 85
10.2.1 Estimation of the Model Parameters . . . . . . . . . . . . . . 86

iv TABLE OF CONTENTS
10.2.2 Interpretation of Regression Parameters in Multiple Regres-
sion Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
10.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
10.3.1 Prediction of the Dependent Variable Y for Given Values of
the Explanatory Variables . . . . . . . . . . . . . . . . . . . . 89
10.3.2 Assessing the Fitted Multiple Regression Model . . . . . . . . 91
10.3.3 Testing for the Signiﬁcance of the Relationship Between the
Dependent Variable and the Explanatory Variables . . . . . . 92
10.4 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
11 Data Transformation 97
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
11.2 Parameters of normal distribution . . . . . . . . . . . . . . . . . . . 97
11.2.1 Shape of the normal distribution . . . . . . . . . . . . . . . . 98
11.3 Reasons for data transformation . . . . . . . . . . . . . . . . . . . . 98
11.4 Testing for normality . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
11.5 Common data transformations . . . . . . . . . . . . . . . . . . . . . 99
12 Analysis of Frequency Data 103
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
12.2 Objective of two-way classiﬁcation . . . . . . . . . . . . . . . . . . . 103
12.3 The Chi-square test of independence . . . . . . . . . . . . . . . . . . 105
13 Review Exercises 113
13.1 Exercise I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
13.2 Exercise II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
13.3 Exercise III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
13.4 Exercise IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
13.5 Exercise V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
13.6 Ecercise VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
13.7 Exercise VII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
13.8 Exercise VIII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
13.9 Exercise IX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

0.1. PREFACE v
0.1 Preface
0.1.1 Course Objective
Focuses on the use of statistical/mathematical techniques to problems in agricul-
tural, environmental, and biological sciences. It is concerned with the design of
experiments, analysis and interpretation of results.
0.1.2 Course Description
Principles of experimental designs, analysis of variance (ANOVA): one way classiﬁca-
tion, e.g., completely randomised design (balanced and unbalanced data), multiway
classiﬁcation, e.g., randomised complete block design, Latin square design; factorial
experiments. Multiple comparisons, data transformation; multiple linear regression,
analysis of frequency data, e.g., contingency tables.
0.1.3 Pre-requisite
MTH 106-Introductory Statistics.
0.1.4 Course requirement
I: Coursework- 2 quizzes and 2 tests: contribute 40% of the total credits allotted
to this course.
II: Final (End of Semester) Exam: contribute 60%
0.1.5 Computing
Where necessary, for illustration purposes, the Statistical Package for the Social
Sciences (SPSS) and the SAS software packages will be frequently used. However,
use of SPSS or SAS in the course is considered optional.
0.1.6 References
Cody, R.P. and Smith, J.K. (1997). Applied Statistics and the SAS Programming
Language. Fourth Ed. Prentice Hall.
Der, G. and Everitt, B.S.(2002). A Handbook of Statistical Analyses using SAS.
Second Ed. Chapman & Hall/CRC.
Levine, D. M., Berenson, M. L. and Stephan, D. (1998). Statistics for Managers
Using Microsoft Excel. Prentice-Hall, Inc.
Miller, I. and Miller, M. (1999). John E. Freund’s Mathematical Statistics. Prentice-
Hall, Inc.
Montgomery, D. (2001). Introduction to Linear Regression Analysis. Wiley and

vi TABLE OF CONTENTS
Sons, Inc.
Montgomery, D. (2001). Design and Analysis of Experiments. Wiley and Sons, Inc.
Neter, J., Kutner, M., Nachetsheim, C. J. and Wasserman, W. (1996). Applied Lin-
ear Statistical Models. Irwin, Chicago.

Chapter 1
Terminologies in Experimental
Designs
1.1 Introduction
Real-life scientific investigations often involve comparisons between several sets of
data collected from basically similar conditions, e.g., groups of plants of the same
type which have been grown under conditions alike except that different fertilizers
were used for each group, different doses of a drug administered to the same or
different groups of patients, different varieties of rations given to a group of homo-
geneous animals, same instructor teaching students with different background about
the subject being taught, etc.
Many types of biological data are collected through planned (well designed) experi-
ments. Designing an experiment requires adherence to some rules or principles and
procedures if valid conclusions are to be drawn. For example, the data from an ex-
periment set up according to a particular design should be analysed according to the
appropriate procedure for that design. No type of statistical method, no matter how
sophisticated, can compensate for a poorly designed study or improve the quality
of results obtained from an improperly designed experiment. Thus, an important
aspect in this respect is that design of experiment determines the quality of the
results! Before we embark on the contents of the course, let us first, briefly distin-
guish between biometry and some related specializations/fields within the statistics
domain.
1.1.1 Definitions
Biometry. As alluded to above, biometry: is a subject that is concerned with the
application of statistics and mathematics to problems in the agricultural, environ-
mental, and biological sciences. Hence, biometrics: is the application of statistics
and mathematics to problems with a biological component, including the problems
in agricultural, environmental, and biological sciences as well as medical science.
1

2 CHAPTER 1. TERMINOLOGIES IN EXPERIMENTAL DESIGNS
These include statistical methods, computational biology, applied mathematics, and
mathematical modeling.
Biostatistics: is a field of study that is concerned with the application of statis-
tics to the biological sciences, especially those relating to medical sciences. Med-
ical colleges/universities (for example, Muhimbili University of Health and Allied
Sciences-MUHAS, International Medical and Technological University-IMTU, Kili-
manjaro Christian Medical College-KCMC, Catholic University of Health and Allied
Sciences - CUHAS, and so on) often have biostatistics as one of the core courses to
students enrolled in various degree programmes with a major in medical sciences.
Described below are key terminologies in the notion of experimental designs.
Experiment: is an investigation set up to provide answers to a question or ques-
tions of interest. For example, we may wish to conduct an experiment to test the
efficacy of a certain newly developed drug for curing a certain skin condition in hu-
mans or aminals. We may also conduct an experiemnt to invest whether or not three
varieties of feeds give same are different in terms of amount of milk produced per day.
In this context, an experiment is more likely to involve comparison of treatments
(defined below) e.g., drugs, rations, methods, varieties, fertilizers, etc. However, in
some cases experiments do not involve comparison of one treatment with another
treatment. Hence, experiments can be absolute or comparative. If we conduct
an experiment to examine the usefulness of a newly developed drug for curing a
certain skin condition in animals without comparing its effect with other drugs, the
experiment will be an absolute experiment. On the other hand, if we conduct an
experiment to assess the effectiveness of one drug as compared to the effects of other
drugs, the experiment is said to be a comparative experiment.
Experimental design or designing of an experiment: a design is a plan for
obtaining relevant information to answer the research question of interest. In other
words, we define designing of an experiment as the complete sequence of steps laid
down in advance to ensure that maximum amount of information relevant to the
problem under investigation will be collected.
Treatment or treatment combination: procedure whose effect is to be mea-
sured and compared with other procedures. For example, in a dietary or medical
experiment, the different diets or medicines are the treatments, in an agricultural
experiment, the different varieties of a crop or the different fertilizers will be the
treatments.
Experimental unit: the unit of experimental material to which one application of
the treatment is applied and on which the variable under study is measured or an
experimental unit is that unit to which a single treatment (which may be a combi-

1.1. INTRODUCTION 3
nation of many factors as in factorial experiments) is applied in one replication of
the basic experiment.
Examples
In an agricultural experiment, the plot of land will be the experimental unit; in a
dietary experiment the whole animal is the experimental unit, in medical experi-
ments for which treatments (or medications) are assigned to individuals and effects
measured, the individual is the experimental units.
Response (yield/outcome): is a result observed for a particular experimental
unit.
Examples
One may be interested to know the amount of a crop (in kg) produced when different
types of fertilizers are applied to a piece of land, or number of students who pass
MTH 201 when different instructors are used for each degree programme taking the
course, or the amount of milk (in litres) that will be produced when different types of
feeds are used to a group of supposedly homogeneous cows, or number of customers
who will visit a particular supermarket in Dar es Salaam when different marketing
strategies are used by the company operating the supermarket.
Exercise
In the agricultural field experiment of assessing the effects of different varieties of
fertilizers on crop production described above, to illustrate the notion of response
identify:
i. the experimental unit;
ii. the treatments; and
iii. the response or yield or outcome.
Factor: Is a variable, which is believed to affect the outcome of an experiment, e.g.
humidity, pressure, time, concentration, etc.
Level: the various values or classifications of the factors are known as the levels of
the factor (s). For example, suppose we wish to compare the efficacy of three med-
ications (M1, M2, and M3) for lowering blood pressure among middle aged women,
thus, there are three levels of the factor medication. Assume also that a researcher
is interested in comparing four different doses (D1, D2, D3 and D4) of a drug admin-
istered to rats of the same type; here there are four levels of the factor drug.
Experimental error: is a measure of the variation among experimental units that
measures mainly inherent variation among them. Thus, experimental error is a
technical term and does not mean a mistake, but includes all types of extraneous
variation due to:

4 CHAPTER 1. TERMINOLOGIES IN EXPERIMENTAL DESIGNS
i. inherent variability in the experimental units;
ii. error associated with the measurements made; and
iii. lack of representativeness of the sample to the population under study.
Therefore, based on the above reasons particularly the first one, we cannot completely
control experimental error, but we can always think of how to reduce it. Variations
among experimental units sometimes cannot be avoided in practice, some variations
are controllable, and some are beyond the control of the experimenter. If we can
control the magnitude of experimental error we would be in a better position to
detect differences among treatments if really exists.
Exercises
1 Suppose the following experiment is conducted, with the aim of comparing three
feeds (I, II, II) in cows. Three cows are obtained. One cow is given feed I, another
feed II and the last cow feed III. 300 observations are taken on each cow.
i. What is the experimental unit?
ii. What are the treatments?
iii. How many replicates per treatment? (to be answered later)
2 An experiment is to be undertaken to compare growth patterns obtained in mice
given three different types of drug. The drug may be administered orally or by
injection. 72 identical mice are available for study. Two different experimental plans
are proposed:
(i) The 72 mice are to be allocated to 12 cages, 6 mice per cage. Each cage is assigned
at random to one of the three drugs, 4 cages per drug. For each cage, the drug is
administered to the animals within the cage by mixing it into the daily shared food
supply for the 6 mice.
(ii) The 72 mice are to be allocated to 12 cages, 6 mice per cage. Within each cage,
each mouse is assigned to receive one of the drugs by daily injection, 2 mice per drug
in each cage.
i. What are the treatments under investigation?
ii. In each of plans or designs (i) and (ii), identify the experimental units.

Chapter 2
Principles of experimental
designs
2.1 Introduction
Designing an experiment to obtain relevant data in a way that permits subjective
analysis leading to valid inferences/conclusions with respect to the problem(s) un-
der investigation is often a challenging step in practice. Correctly identifying the
relevant experimental units, their size or number, and the way the treatments are
assigned to the experimental units are some of the most important aspects of design
of experiments. In this section we describe the principles that depending on the
design chosen must be adhered to when planning an experiment to answer a speciﬁc
problem. There are three main principles of experimental designs, namely:
i. Randomisation;
ii. Replication; and
iii. Error/local control
2.2 Randomization principle
Randomisation is an essential component/principle in experimental design. Ran-
domisation involves the assignment of treatments to the experimental units, based
on the chosen design, by some chance mechanism or probabilistic procedures, e.g.,
random numbers, so that each experimental unit has the same chance of receiving
any one of the treatments, which are under study. Conscious allocation of the treat-
ments to the experimental units has been criticised by many researchers, in fact
results from studies which had not allocated treatments to the experimental units
at random have left useless and thus contributed nothing to the literature avail-
able to date. Brieﬂy speaking, randomization is the use of a known, understood
probabilistic procedure for the assignment of treatments to experimental units.
5

6 CHAPTER 2. PRINCIPLES OF EXPERIMENTAL DESIGNS
As we will discuss later in the course, randomisation been an important principle of
experimental designs is utilised in all designs that we will discuss in this course.
Question
Why do we really need to adhere to this principle?
Recap: as we explained in chapter 3, treatment is a procedure whose effect is to
be measured and compared with other procedures. Goal: Based on the fact that
our intention is to measure and compare effects of one treatment in comparison to
another treatment(s), thus, one obvious goal of randomisation is to ensure that no
treatment is somehow favoured or handicapped.
Randomisation ensures that observations represent random samples (independence of
observations) from population of interest. This insures validity of statistical methods
leading to valid conclusions/inferences.
Illustration
The following example illustrates the importance of randomisation. A study is to
be conducted to compare the efficacies of two drugs (I and II) for treating a certain
skin condition. It is decided that patients will be assigned to drug I if they have
had previous outbreaks of the condition and to drug II if the current outbreak of the
condition is the first for the patient. Comment on this experimental design. If you
feel that the design has drawbacks, state how you would improve it.
From our discussion above, clearly this design lacks randomisation. The design had
allocated the drugs depending on whether the patient has had a previous outbreak
of the condition. This is not a proper way of assigning treatments to experimental
units. It may be, for example, that patient with first-time outbreaks are more or
less difficult to treat than repeat outbreaks. This may put one drug or the other at
a disadvantage in evaluation of efficacy.
How to improve?
A better design would be one that assigns patients at random to the drugs regardless
of outbreak status. An even better design would be one which assigns patients with
first-time outbreaks to each drug randomly, and similarly for patients with repeat
outbreaks, so that each drug is seen on patients of each type.
2.3 Replication principle
The term replication refers to the number of experimental units on each treatment.
A treatment is said to be replicated if it is applied to more than one experimental
unit. Literally speaking, replication means the number of times a treatment appears
on experimental units.
Question
What do we replicate and why? The first part of this question is answered above.

2.4. LOCAL CONTROL PRINCIPLE 7
We replicate treatments to experimental units. Perhaps of most interest here at least
in my views is the question of why do we need to replicate the treatments.
We repeat a treatment a number of times in order to obtain more reliable estimate
than is possible from a single observation. If you can recall our discussion of statis-
tical inference in MTH 106, we mentioned that the sample size n is a key factor
in determining precision and power. This is the case because if we increase the
sample size, we decrease the magnitude of s ¯D which is a measure of how precisely
we can estimate the difference and determine the size of our test statistic (and thus
power of the test). In the context of experimental design, the number of replicates
per treatment is also a key factor in determining precision and power.
Example
Suppose an experiment is conducted with the goal of comparing two diets in weight
gain in sheep. Two sheep are available for experimentation. One sheep is given diet
A, the other; diet B. 400 observations are taken on each sheep.
In this example very little can be learned about how the treatments compare in the
population of such sheep. We have only one sheep on each treatment (diet); thus,
we do not know if observed differences we might see are due to a real difference
in the treatments we are comparing or just the fact that these two sheep are quite
different. This is perhaps an artificial example, but it illustrates a general point of
why replication is an important principle in experimental designs.
A practical advice
If we have a fixed number of experimental units available for experimentation, an
obvious challenge is how to make the best use of them for detecting treatment dif-
ferences. In this situation of limited resources we would be better off with a few
treatments with lots of replicates on each treatment rather than many treatments
with fewer replicates on each.
Thus, if limited resources are available, it is better to reduce the number of treat-
ments to be considered or postpone the experiment rather than to proceed with
too few replicates. So randomisation plus replication will be necessary for the
validity of the experiment.
Exercise
In your own words explain why you think replication is an important concept to
keep in mind when designing an experiment in your own field of study.
2.4 Local control principle
Experimental design is founded on the principle of reducing what we regard as ex-
perimental error by meaningful grouping of experimental units into small non-
overlapping units. As we discussed above, we cannot eliminate inherent variability

8 CHAPTER 2. PRINCIPLES OF EXPERIMENTAL DESIGNS
completely but if we try to be careful enough about what we consider to be inherent
variability we should be in a position to separate systematic variation among exper-
imental units from inherent variation and hence arrive at the stated goal (s) of the
experiment.
Local control are techniques for reducing the error variance. One such measure is to
make experimental units homogeneous, i.e. to form units into several homogeneous
groups called blocks. This is done particularly in situations where the experimen-
tal units are assumed to be non-homogeneous. Thus, to reduce the magnitude of
experimental error one needs to group the experimental units.
It should be understood that in order to detect treatment differences if they really,
we must strive to control the effects of experimental error, so that any variation we
observe can be attributed mainly to the effects of the treatments we are comparing
rather than to differences among the experimental units to which the treatment are
applied.
Summary: From what we have discussed so far, it is clear that a good experimental
design attempts to:
i. ensure sufficient replication of treatments to experimental units; and
ii. reduce the effects of experimental error by meaningful grouping of experimental
units –application of local/error control.

Chapter 3
Analysis of Variance
3.1 Introduction
Among the most extremely useful statistical procedures in the fields of agriculture,
economics, psychology, education, sociology, business/industry and in researches of
several other disciplines is the analysis of variance. This technique is particularly
used when multiple sample cases are involved.
Recap: Tests of significance discussed in MTH 106 between the means of two sam-
ples can easily be judged through either the standard normal distribution, z-test or
the student’s t- test. Just to remind you one of the popular t-test is the two sample
pooled t- test used when the two unknown population variances are assumed to be
equal.
Problem: When there are more than two samples, performing all possible pairwise
comparisons especially if n is large becomes a wearying exercise. The analysis of
variance technique enables us to perform this simultaneous test. Using this tech-
nique one can draw inferences about whether the samples have been drawn from
populations having the same mean.
Example
Comparison of yields of a certain crop from several varieties of seeds, the smoking
habits of six groups of SUA students, and so on.
If we are to use either the z or t-tests, one need to consider all possible combinations
of two varieties of seeds at a time and also two groups of students. This would take
some time before one arrives at a decision. In such circumstances, one quite often
utilizes the analysis of variance technique and through it investigates the differ-
ences among the means of all the populations simultaneously.
Acronym: The popular acronym for ANalysis Of VAriance is ANOVA.
9

10 CHAPTER 3. ANALYSIS OF VARIANCE
Definition: Montgomery (2001) defines ANOVA, as a procedure for testing the
difference among different groups of data for homogeneity.
Target: To partition the total amount of variation in a set of data into two compo-
nents:
i. The amount which can be attributed to chance; and
ii. The amount, which can be attributed to specified causes
If we take only one factor (recall the definition in Chapter two) and investigate dif-
ferences amongst its various categories (or levels) having numerous possible values,
we are said to use one-way ANOVA (more to come later) and in case we investi-
gate two factors at the same time, then we use two-way ANOVA (more to come
later).
3.2 Assumptions in the analysis of variance
When one employs the ANOVA technique has to be satisfied that the basic as-
sumptions underlying the technique are fulfilled if he/she is to give valid inferences.
There are three basic assumptions underlying the analysis of variance technique:
i. The observations, and hence the errors, are normally distributed
ii. All observations both across and within samples, are unrelated (independent)
iii. The observations have the same variance σ2
Important: The assumptions stated above are not necessarily true for any given
situation. In fact, they are probably never exactly true. For many data sets in
practice, they may be a reasonable approximation in which case the results will be
fairly reliable. In other cases, they may be badly violated; in this case, the resulting
conclusions may not be valid or may be misleading.
If the data really are not normal, hypothesis test may be imperfect, leading to
invalid inferences.
Strategy: In some data it is possible to get around these issues somewhat. One
of the most commonly used approaches to deal with the problem of non-normality
of data is the so called data transformation, an aspect which with be dealt with
later in the course.

3.3. ANALYSIS OF VARIANCE FOR ONE-WAY CLASSIFICATION 11
Important: For the reminder of our discussion of analysis of variance in this and
subsequent chapters, we will assume that the above assumptions are reasonable either
on the original scale of measurement of the data or transformed scale. Keep this in
mind at all times that these are just assumptions, and must be confirmed before the
methods may be considered suitable.
3.3 Analysis of variance for one-way classification
Under the one-way (or single factor) ANOVA, we randomly obtain the experimen-
tal units for the experiment and randomly assign them to the treatments so that
each experimental unit is observed under one of the treatments. In this situation,
the only way in which experimental units may be classified is with respect to which
treatment they received. Basically, the experimental units are viewed alike in this
experiment. Thus, when experimental units are thought to be alike and are thus
expected to exhibit a small amount of variation from unit to unit, grouping then
would be pointless in the sense that doing so would not add much precision to an
experiment.
It can be shown that the total variation in the observed responses can be subdivided
into two components:
i. Due to the differences in the level of factor (say A)
ii. The residual variation (error term)
3.3.1 Analysis of variance for one-way classification with unequal
replication (unbalanced data)
We will first consider the case where unequal number of replication of the treatments
to the experimental units is observed.
Notation: To facilitate the development of methods that we will require in our
discussion, we will change slightly our notation of sample mean we discussed in
MTH 106. As we will see shortly, we will be dealing with several different types of
means for the data.
Let t denote treatment and k the number (levels) of treatments. Let also Yij be the
response of the jth experimental unit receiving the ith treatment level
We will also denote the sample mean for treatment i (mean of all plots receiving
treatment i) by
Y i. = 1
ni
ni
j=1
Yij

Also we define Y .. =
k
i=1
ni
j=1
yij
k
i=1
ni
or Y.. = G
N as the grand mean yield (sample mean
of all the data) in the whole experiment. Note that because we have unequal
replications, the total number of observations is
k
i=1
ni = N
3.3.2 Linear additive model for one-way classification
For a one way classification with unequal replication we may classify an individual
observation as being on the jth experimental unit in the ith treatment level as:
Yij = µ + ti + eij, i=1, 2,. . . k, j=1, 2,. . . ni
Where:
µ = the general mean effect
ti = the effect of level i or the ith treatment effect
eij = the error term
Yij as defined above
Remark: In our discussion we will consider only cases where a single observation
of response is made on each experimental unit; however, it is common practice to
take more than one observation on an experimental unit.
3.3.3 Fixed vs. random effects
In the above model for one way classification with unequal replication, ti represents
the ith treatment effect. However, interpretation of timay differ depending on the
situation.
To better understand the notions of fixed and random effects, consider the following
examples.
Example 1
Suppose there are three varieties of wheat for which mean yields are to be compared.
Here, we are interesting in comparing 3 specific treatments. If we repeated the ex-
periment again, these 3 varieties of wheat would always constitute the treatments of
interest.
Example 2
Suppose a factory operates a large number of machines to produce a product and
wishes to determine whether the mean yield of these machines differs. It is unfea-
sible for the company to keep track for all of the many machines it operates, so a
random sample of 5 such machines is selected, and observations on yield are made
on these 5 machines. The hope is that the results for the 5 machines involved in the

experiment may be generalized to gain insight into the behaviour of all the machines.
In the first example, there is a particular set of treatments of interest. If we started
the experiment next week instead of this week, we would still be interested in this
same particular set of treatments. It would not vary across other possible experi-
ments we might do.
In the second example, the treatments are the 5 machines from all machines at the
company, chosen at random. If we started the experiment next week instead of this
week, we might end up with a different set of 5 machines with which to do the
experiment. Here interest focuses on the population of all machines operated by
the company. The question of interest is not about the particular treatments in the
experiment, but the population of all such treatments.
We thus make the following distinction in our model:
In the case like example 1, the ti are best regarded as fixed quantities, as they de-
scribe a particular set of conditions. Thus, ti are referred to as fixed effects
In a case like example 2, the ti are best regarded as random variables. Here the par-
ticular treatments in the experiment may be thought of as drawn from a population
of all such treatments, so there is a chance involved. In this situation, the ti are
referred to as random effects.
3.3.4 Calculation of sums of squares
As we described above, the fundamental nature of the ANOVA is that the total
amount of variation in a set of data is broken down into two components, that
amount which can be attributed to chance and that amount which can be attributed
to specified causes.
Thus, based on the above linear additive model we partition the total variation in
the data as:
Total variation = Variation due to factor A (treatment) + Residual/Error term or
Total sum of squares = Sum of squares due to factor A + Sum of squares due to error
In short we have,
SST = SSA + SSE

Algebraic facts show that the total sum of squares (SST) can be partitioned as:
k
i=1
ni
j=1
Yij − Y..
2
=
k
i=1
ni Y i. − Y..
2
+
k
i=1
ni
j=
Yij − Y i.
2
Y i.and Y .. as defined in Section 3.3.1
Thus, SST=
k
i=1
ni
j=1
Yij − Y..
2
, SSA=
k
i=1
ni
¯Yi. − ¯Y..
2
and SSE=
k
i=1
ni
j=1
Yij − ¯Yi.
2
For calculation we express the SSs as follows:
Define C.F=Correction factor =
k
i=1
ni
j=1
Yij
2
k
i=1
ni
= G2
N Here, G is the grand total=
k
i=1
ni
j=1
Yij and N as defined in Section 2.3.1
It can be shown that:
SST=
k
i=1
ni
j=
Y 2
ij −
k
i=1
ni
j=1
Yij
2
k
i=1
ni
or
k
i=1
ni
j=
Y 2
ij − G2
N or
k
i=1
ni
j=
Y 2
ij − C.F
Treatment SS or Factor A SS=
k
i=1
Y 2
i
ni
− C.F
Where: Yi =
ni
j=1
Yijis the total yield of all the njplots which carried treatment i
Error SS (SSE) =SST-SSA
Since we have k levels of factor (A) or treatment then SSA will have k-1 independent
comparisons possible (degrees of freedom). Similarly SST will have N-1 independent
comparisons (degrees of freedom), and SSE will have (N-1)-(k-1) =N−k independent
comparisons (degrees of freedom).
We summarize the computations in a table known as the ANOVA table.
The calculated F-value MSA
MSE is compared with the F-tabulated value
(Fα, [(k − 1) , (N − k)]) at α level of significance for k-1 and N −k degrees of freedom.

Table 3.1: One way ANOVA table with unequal replication
Source of Degrees of Sum of Mean square
variation (S.V) freedom (D.F) squares (S.S) (M.S) F- ratio
Between treatments k-1 SSA SSA
K−1 = MSA MSA
MSE
Error(within treat.) N − k SSE SSE
N−k = MSE
Total N-1 SST
Statistical Hypotheses
The question of interest in this setting is to determine if the means of the different
treatment populations are different.
Mathematically we write:
Ho : µ1 = µ2 = ... = µk That is, the µi are all equal
H1 : µi = µj for at least one i = j That is, the µi are not all equal
Or simply
Ho :There is no variation among the treatments
H1 :Variation exists
Test procedure
At level of significance α, if F> Fα, [(k − 1) , (N − k)] then there is evidence for no
significance variation (i.e. we reject the null hypothesis).
Note that the alternative hypothesis stated above does not specify the way in
which the treatment means (or deviation) differ. The best we can say based on our
statistic is that they differ somehow.
Example
Four Machines are used for filling plastic bottles with a net volume of 16.0 cm3.
The quality-engineering department suspects that both machines fill to the same net
volume whether or not this volume is 16.0 cm3. A random sample is taken from the
output of each machine.
Table 3.2: Machine data set
Machines
A B C D
16.03 16.01 16.02 16.03
16.04 15.99 15.97 16.04
15.96 16.03 15.96 16.00
16.05 15.05 16.02
16.04
Total 64.08 48.03 79.04 64.09
Assume that the measurements are approximately normally distributed, with ap-
proximately constant variance σ2. Do you think the quality-engineering department

is correct? Use α = 0.05
Statistical hypotheses:
Ho :There is no signiﬁcant variation among the levels of machines
or
Ho : µ1 = µ2 = µ3 = µ4 (all means are equal)
H1 : µi = µj for at least one i = j (the means are not all equal)
Calculation
Here we have 4 treatment levels (A, B, C, D).
N =
k
i=1
ni = n1 + n2 + n3 + n4 = 4 + 3 + 5 + 4=16
Grand Total (G) =
4
i=1
ni
j=1
Yij=16.03+16.04+. . . +16.00+16.02=255.24
Thus, C.F = G2
N = (255.24)2
16 =4071.7161
Uncorrected total SS
=
4
i=1
ni
j=1
Y 2
ij = (16.03)2
+ (16.04)2
+ ... + (16.00)2
+ (16.02)2
=4072.5976
Total Sum of Squares (SST) =
4
i=1
ni
j=1
Y 2
ij − C.F=4072.5976-4071.7161=0.8815
Totals: A=64.08, B=48.03, C=79.04, D=64.09
Machine (treatment) sum of squares (SSM):
SSM= 1
ni
k
i=1
Y 2
i − C.F= 1
4(64.08)2 + 1
3(48.03)2 + 1
5(79.04)2 + 1
4(64.09)2 − 4071.7161
=4071.868245-4071.7161=0.152145
Error sum of squares (SSE) =Total SS-Treatment (Machine) SS or
SST-SSM=0.8815-0.152145=0.729355
We also have k-1=4-1, N − k =16-4=12, so that:
Treatment (machine) MS=0.152145
5 =0.051, Error MS=0.729355
12 =0.061,
F = 0.051
0.061 =0.83

We summarize the computation in an analysis of variance table:
Table 3.3: ANOVA Table
Source of variation DF SS MS F- ratio
Between treatments (machines) 3 0.152145 0.051 0.83
Error(within treatments) 12 0.729355 0.061
Total 15 0.881500
To perform the hypothesis test for differences among the means of machines, we
compare the F-calculated value (0.83) from the appropriate value from the F table.
For level of significance α = 0.05, we have F0.05; 3, 12=3.49. Since 0.83 does not
exceed F0.05; 3, 12=3.49. We thus do not reject Ho and hence conclude that the
quality-engineering department is correct, that is, no significant variations among
the machines. In other words, all machines fill to the same net volume.
In SAS
SAS Program
data notes;
input Machines $ Volume @@;
cards;
A 16.03 A 16.04 A 15.96 A 16.05
B 16.01 B 15.99 B 16.03 C 16.02
C 15.97 C 15.96 C 15.05 C 16.04
D 16.03 D 16.04 D 16.00 D 16.02
;
run; proc print; run; quit;
proc anova;
class Machines;
model Volume=Machines;
run; quit;
%newpage
Selected SAS outputs
%begin{verbatim}
Obs Machines Volume
1 A 16.03
2 A 16.04
3 A 15.96
4 A 16.05
5 B 16.01
6 B 15.99
7 B 16.03
8 C 16.02
9 C 15.97
10 C 15.96
11 C 15.05
12 C 16.04
13 D 16.03
14 D 16.04
15 D 16.00

16 D 16.02
The ANOVA Procedure
Dependent Variable: Volume
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 3 0.15214500 0.05071500 0.83 0.5004
Error 12 0.72935500 0.06077958
Corrected Total 15 0.88150000
R-Square Coeff Var Root MSE Volume Mean
0.172598 1.545433 0.246535 15.95250
Source DF Anova SS Mean Square F Value Pr > F
Machines 3 0.15214500 0.05071500 0.83 0.5004
Exercise
The following data comes from an experiment conducted to investigate the effect
of 4 diets on weight gain in pigs. 19 pigs were randomly selected and assigned at
random to one of the 4 diet regimes. The data are the body weights of the pigs, in
pounds, after having been raised on the diets.
Diet 1 Diet 2 Diet 3 Diet 4
133.8 151.2 225.8 193.4
125.3 149.0 224.6 185.3
143.1 162.7 220.4 182.8
128.9 145.8 212.3 188.5
135.7 153.5 198.6
Assume that the measurements are approximately normally distributed, with con-
stant variance: Is there any evidence in these data to suggest that the mean weights
are different under the different diets? Use α = 0.05. Compare your ANOVA table
with the one below from SAS.
The ANOVA Procedure
Dependent Variable: BWeight
Sum of
Model 3 20461.40576 6820.46859 164.38 <.0001
Error 15 622.39950 41.49330
R-Square Coeff Var Root MSE BWeight Mean
0.970480 3.753460 6.441529 171.6158
Diets 3 20461.40576 6820.46859 164.38 <.0001

3.3.5 ANOVA for one-way classification with equal replication (bal-
anced data)
In the above exercise the diets 1, 2, and 4 are each replicated 5 times while diet 3 is
replicated 4 times. In this case as we have discussed above, the sample mean for
treatment i is Y i. = 1
ni
ni
j=1
Yij. We now discuss the case where ni = n for all i,
i=1, 2,. . . , k
Since each treatment is replicated the same number of time (say n), then the total
number of observations, N=nk.
Thus, with this new notation, we define the quantities Y i.,Y .., Total SS, Treatment
SS, and Error SS and their degrees of freedom as follows:
Y i. = 1
n
n
j=1
Yij, Y .. =
k
i=1
n
j=1
yij
nk orY.. = G
N , Total SS (SST) =
k
i=1
n
j=1
Y 2
ij − G2
N ,
Treatment SS or Factor A SS=
k
i=1
Y 2
i
n − C.F where C.F =
k
i=1
n
j=1
Yij
2
nk = G2
N
G = the grand total=
k
i=1
n
j=1
Yij
The degrees of freedom for Treatment SS, Error SS and Total SS, are respectively
(k-1), (N − k) or (nk-k) or k(n-1) and (N-1) or (nk-1).
Table 3.4: One way ANOVA table with equal replication
Source of DF SS MS F- ratio
variation
Between treat. k-1 SSA SSA
k−1 = MSA MSA
MSE
Error (within treat.) k(n-1) SSE SSE
k(n−1) = MSE
Total N-1 SST
The calculated F-value MSA
MSE is compared with the F-tabulated value
(Fα, [(k − 1) , (k(n − 1))]) at α level of significance for k-1 and k(n-1) degrees of free-
dom.
Statistical Hypotheses as given above
Test procedure
At level of significanceα, if F> Fα, [(k − 1) , (k(n − 1))] then there is evidence for no
significance variation (i.e. we reject the null hypothesis).
Example
The following data record the length of pea sections, in ocular units (×0.114 mm),

grown in tissue culture with auxin present. The purpose of the experiment was to
test the eﬀects of the addition of various sugars on growth as measured by length.
Pea plants were randomly assigned to one of 5 treatment groups: control (no sugar
added), 2% glucose added, 2% fructose added, 1% glucose + 1% fructose added, and
2% sucrose added. 10 observations were obtained for each group of plants.
Control 2% glucose 2% fructose 1% fructose 2% sucrose
1 75 57 58 58 62
2 67 58 61 59 66
3 70 60 56 58 65
4 75 59 58 61 63
5 65 62 57 57 64
6 71 60 56 56 62
7 67 60 61 58 65
8 67 57 60 57 65
9 76 59 57 57 62
10 68 61 58 59 67
Total 701 593 582 580 641
We assume that the measurements are approximately normally distributed, with the
same variance σ2. Use α = 0.05 and perform the relevant hypothesis test to these
data.
Calculations show that (check):
C.F=
k
i=1
n
j=1
Yij
2
nk = (701+...+641)2
10×5 = (3097)2
50 = 191828.18
k
i=1
n
j=1
Y 2
ij = 752
+ 672
+ ... + 672
= 193151.00
Thus, Total SS=
k
i=1
n
j=1
Y 2
ij − G2
N =193151.00-191828.18=1322.82
Treatment SS=
k
i=1
Y 2
i
n − C.F =
(7012+...+6412
)
10 − 191828.18=192905.50-191828.18
=1077.32
Error SS= Total SS-Treatment SS=1322.82-1077.32=245.50
We also have (k-1) =5-1=4, k(n-1) =5(10-1)=45 so that
Treatment MS=1077.32
4 = 269.33, Error MS=245.50
45 = 5.46, F = 269.33
5.46 = 49.31
We summarize the computations in an analysis of variance table:

3.4. ANOVA FOR TWO-WAY CLASSIFICATION (WITHOUT REPLICATION)21
Table 3.5: ANOVA table-Pea section data
Between treatments 4 1077.32 269.33 49.33
Error (within treatments) 45 245.50 5.46
Total 49 1322.82
F0.05; 4, 45 = 2.61
Comparing the calculated F value (49.33) with the F value from F table (2.61) at
0.05 level of significance we see that 49.33 >F0.05; 4, 45=2.61. We thus reject H0.
There is evidence in these data to suggest that the mean lengths of pea sections are
different depending upon which sugar was added.
3.4 ANOVA for two-way classification (Without Repli-
cation)
As the name suggests, two-way classification means the data are classified on the
basis of two factors. Thus, two-way ANOVA technique is used when the data are
classified on the basis of two factors.
Suppose the two factors are A and B which have h and g levels respectively in an
experiment without replication. Using the ANOVA technique we can partition the
variation of the response about their mean into three different components.
3.4.1 Linear additive model for two-way classification
For two-way classification without replication, let Yij be the response for the ith
level of factor A and jth level of factor B. Thus, the model can be written as:
Yij = µ + ti + bj + eij, i=1, 2, . . . , h; j=1, 2, . . . g
Where:
µ is the overall mean
ti is the effect of level i for factor A
bj is the effect of level j for factor B and eij is the residual (error term).
The ANOVA technique allows us to partition the total SS as:
Total SS = Factor A SS + Factor B SS + Residual SS
or simply,
SST= SSA+SSB + SSE
As in one-way classification, the short methods of computing sum of squares are
given as follows:

Let N (=hg) be the total number of experimental observations
Let G = the sum of yields over all the N (=hg) plots. So that G =
h
i=1
g
j=
Yij,
Correction factor (C.F) =G2
N =
h
i=1
g
j=1
Yij
2
hg
Total SS (SST) =
h
i=1
g
j=1
Y 2
ij − C.F
Factor A SS (SSA) =1
g
h
i=1
Y 2
i − C.F where Yi =
g
j=1
Yij is the total yield of all the
g plots which carried treatment i.
Factor B SS (SSB) =1
h
g
j=1
Y 2
j − C.F where Yj =
h
i=1
Yij is the total yield of all
the h plots which carried treatment j.
Error SS (SSE) = SST – (SSA +SSB)= SST –SSA- SSB
Table 3.6: ANOVA table for two-way classification
Source of variation DF SS MS F-ratio
Factor A h-1 SSA SSA
h−1 = MSA MSA
MSE
Factor B g-1 SSB SSB
g−1 = MSB MSB
MSE
Residual (h-1)(g-1) SSE SSE
(h−1)(g−1) = MSE
Total N-1 SST
Factor A:
Ho: t1 = t2=. . . =th
H1: ti =tj for at least one i = j
Factor B:
Ho: b1=b2=. . . =bg
H1: bi =bj for at least one i = j
Test procedure
Reject Ho for factor A, if the calculated F-value MSA
MSE > the tabulated F-value
Fα, [(h − 1) , (h − 1)(g − 1)] at α-level of significance. Otherwise, we do not reject
Ho.
Similarly, reject Ho for factor B, if the calculated F-value MSB
MSE > the tabulated
F-value Fα, [(g − 1) , (h − 1)(g − 1)] at α-level of significance. Otherwise, we do not
reject Ho

3.4. ANOVA FOR TWO-WAY CLASSIFICATION (WITHOUT REPLICATION)23
Example
Three different methods of analysis M1, M2, and M3 are used to determine in parts
per million the amount of a certain constituent in a sample. Each method is used
by five analysts and the results are given below.
Analyst
1 2 3 4 5 Total
7.0 6.9 6.8 7.1 6.9 34.7
Method 6.5 6.7 6.5 6.7 6.6 33.0
6.6 6.2 6.4 6.3 6.4 31.9
Total 20.1 19.8 19.7 20.1 19.9 99.6
Do these results indicate a significant variation either between the methods or be-
tween analysts? Use α = 0.01
For analyst
Ho: analysts do not differ
H1:Analysts differ
For method
Ho: methods do not differ
H1:methods differ
Calculation
C.F =G2
N =
h
i=1
g
j=1
Yij
2
hg = (99.6)2
15 =661.344
Total SS (SST) =
h
i=1
g
j=1
Y 2
ij − C.F=662.32-661.344=0.976
Analyst SS (SSA) =1
g
h
i=1
Y 2
i − C.F
=1
3 (20.1)2 + (19.8)2 + (19.7)2 + (20.1)2 + (19.9)2 − 661.344
=661.3866667-661.344=0.0426667
Method SS (SSM)
=1
h
g
j=1
Y 2
j − C.F =1
5 (34.7)2 + (33.0)2 + (31.9)2 − 661.344=662.14-661.344=0.796

Error SS (SSE):
= SST –SSA- SSM=0.976-0.0426667-0.796=0.1373333
Table 3.7: ANOVA table
Analyst 4 0.04267 0.01067 0.620
Method 2 0.79600 0.39800 23.18
Error 8 0.13733 0.01717
Total 14 0.9760
Comparing the F calculated values, (0.62) and (23.18) for analyst and method with
the critical F values, (7.01) and (8.65) respectively, we do not reject the null hy-
pothesis for analyst while for method the null hypothesis is rejected. Hence, we
conclude that there is not enough evidence in these data to suspect that the analysts
differ. On the other hand, the data indicates significant differences in methods at
the 1% level of significance.
In SAS
SAS Program
data twoway;
input Analyst $ Method $ ppm @@;
cards;
A1 M1 7.0
A1 M2 6.5
A1 M3 6.6
A2 M1 6.9
A2 M2 6.7
A2 M3 6.2
A3 M1 6.8
A3 M2 6.5
A3 M3 6.4
A4 M1 7.1
A4 M2 6.7
A4 M3 6.3
A5 M1 6.9
A5 M2 6.6
A5 M3 6.4
;
run; proc print;run;quit;
proc anova;
class Analyst Method;
model ppm=Analyst Method;
run;quit;
The SAS System
Obs Analyst Method ppm
1 A1 M1 7.0
2 A1 M2 6.5
3 A1 M3 6.6
4 A2 M1 6.9

3.5. THE LEAST SIGNIFICANCE DIFFERENCE (LSD) 25
5 A2 M2 6.7
6 A2 M3 6.2
7 A3 M1 6.8
. . . .
. . . .
The ANOVA Procedure
Dependent Variable: ppm
Sum of
Model 6 0.83866667 0.13977778 8.14 0.0046
Error 8 0.13733333 0.01716667
R-Square Coeff Var Root MSE ppm Mean
0.859290 1.973217 0.131022 6.640000
Analyst 4 0.04266667 0.01066667 0.62 0.6601
Method 2 0.79600000 0.39800000 23.18 0.0005
3.5 The least signiﬁcance diﬀerence (LSD)
If we reject the null hypothesis by the use of the F-test, we can carry out further
analyses, i.e., carry out pairwise comparisons of the levels of the factor (s) by the
use of t-test. We consider the situation where we have planned in advance of the
experiment to make certain comparisons among treatment means. In this case, each
comparison is important in its own right, and thus is to be viewed as separate, i.e.,
cannot be combined.
Suppose we have t treatments in the experiment, and we are interested in comparing
two treatments 1 and 2, with means µ1 and µ2 respectively. That is, we wish to test
the hypotheses:
H0 : µ1 = µ2 vs. H1 : µ1 = µ2
Test statistic: As our test statistic for H0 vs. H1, we use:
| ¯Y1. − ¯Y2.|
s¯Y1.− ¯Y2.
, s¯Y1.− ¯Y2.
= s
1
r1
+
1
r2
, s =
√
MSE
That is, instead of basing the estimate of σ2 on only the two treatments in question,
we use the estimate from all t treatments in the experiment. Here, r1 and r2 are
respectively the replicates of samples 1 and 2.

Test procedure: Reject H0 in favour of H1 if
| ¯Y1. − ¯Y2.|
s¯Y1.− ¯Y2.
> tN−t,α/2
Here, N − t are the degrees of freedom for estimating σ2 (experimental error)
Note that the test procedure above for testing H0 against H1 may be rewritten as
follows:
Reject H0 if:
| ¯Y1. − ¯Y2.| > s¯Y1.− ¯Y2.
× tN−t,α/2, s =
√
MSE
Terminology: In comparing two treatment means from large experiments involving
t treatments, the value
s¯Y1.− ¯Y2.
× tN−t, α/2 = s
1
r1
+
1
r2
× tN−t,α/2, s =
√
MSE
is called the least significance difference (LSD) for the test of H0 vs. H1 based
on the entire experiment. Thus, from the above expression, we reject H0 in favour
of H1 at level α if
| ¯Y1. − ¯Y2.| > s¯Y1.− ¯Y2.
× tN−t,α/2
The case of equal replication: If all treatments are replicated equally, that is,
ri = r the value of the LSD is the same and is given by:
s¯Y1.− ¯Y2.
= s
2
r
, s =
√
MSE, LSD = s
2
r
× tN−t,α/2
Thus, in case of equal replications, all pairwise comparisons of interest require only
a single calculation.
Example
Consider the pea section data we discussed in Section 5.3.5. In this data we had
equal replications (r=10) and 5 treatments (t=5). Suppose it was decided in advance
that one investigator was interested in the particular question of whether 2% glucose
(treatment 2) differs from control.
Let µ1 denote the mean for the control andµ2, µ3, µ4,µ5 denote the means for the
sugar treatments, 2% glucose, 2% fructose, 1% fructose and 2% sucrose
respectively.

3.5. THE LEAST SIGNIFICANCE DIFFERENCE (LSD) 27
In this situation, we want to test the hypotheses:
H02 : µ1 = µ2 vs. H12 : µ1 = µ2
From the information given, we have
¯Y1. = 70.1, ¯Y2. = 59.3, s =
√
MSE=2.3357, N − t = 45, tN−t,α/2 = t45,0.025 = 2.01.
Thus
LSD = s
2
r
× tN−t,α/2 = (2.3357) 2/10(2.01) = 2.10
| ¯Y1. − ¯Y2.|=10.8 > 2.10
Conclusion
Since | ¯Y2. − ¯Y1.| = 10.8>LSD (= 2.10), we reject H02 at level of significance α
=0.05; there is sufficient evidence to suggest that the glucose treatment yields mean
pea section lengths different from the control.
Exercise
1. Suppose that another investigator was interested in the specific question of
whether the 2% fructose (treatment 3) differs from the control. That is, test for:
H03 : µ1 = µ3vs. H13 : µ1 = µ3 Use α=0.05
2. Test whether the means of the 2% glucose and 2% fructose differ significantly at
5% level of significance.
3. Calculate 99% confidence limits for the mean of treatment 4 (1% fructose)

Chapter 4
Introduction to SPSS
4.1 Introduction
SPSS is an extremely useful statistical software package. It provides full statistical
analysis capabilities including data management, an analysis tool which embraces
both plain and sophisticated but interesting and easy to learn statistical techniques
one cannot afford to ignore in the analysis of real-life data! SPSS has historically
been applied extensively in the areas of social science, however, these days it is also
widely being used in other fields of study. The current version of SPSS is 21.
As mentioned, this text of SPSS is not part of MTH 201 course coverage (require-
ment) as you have seen in Section 1.1 but is meant to make you understand that all
computations of the different theoretical aspects that we have discussed and those
still to be discussed, though some have been illustrated using SAS, can also be done
in other software packages, SPSS being one of them. Other software packages from
which statistical analyses may be carried out include STATA, S-Plus/R. However,
S-Plus/R requires good programming knowledge to be able to use it!.
Unlike many software packages, SPSS is a user-friendly (easy to use), widely avail-
able and well documented such that one can quickly make reference to available and
easily accessible citation. These are among the reasons why I have chosen to give
you this text! Don’t forget that there is always no free lunch! Like SAS, S-Plus, and
STATA, SPSS is not free! You have to pay something to get it!
It is important to note that the SPSS statistical analyses presented in this text are
specific. That is, does not cover all features available in SPSS but focuses on only
few of the many analysis tools that SPSS can offer to the analyst. Consequently, to
sharpen your competency in using SPSS especially in carrying out more advanced
statistical analyses you are urged to refer to any SPSS Manual.
In SPSS, unlike with the other software packages, getting output is relatively easy;
however, one needs to be cautious-remember that “there is always no free lunch”.
29

30 CHAPTER 4. INTRODUCTION TO SPSS
James Steven, 1996 points out that because it is easy to get output, it is also easy to
get “garbage.” Hence, knowing exactly what to focus on in the printout so as to be
able to give a practical interpretation of the problem at hand is an important aspect
one needs to bear in mind when selecting the output to concentrate on.
Throughout all our illustrations in this text it is assumed that the reader will be ac-
cessing the data from disk or CD ROM already saved as an SPSS file. Meaning that
the data has already undergone through important treatments like editing, coding
etc. This is not always the case in practice. Often in practice analysts receive raw
data and do the required treatments themselves. Data management (e.g., merging,
interleaving or matching files) in SPSS is out of the scope of this text. For those
who are interested however, SAS is nicely set up for ease of file manipulation. In
this text I will however, briefly describe how data entry is done.
It is worth mentioning that coming up with a valid conclusion or answer to a spe-
cific scientific question of interest requires not only one’s competency in the software
package of analysis but also an understanding of several other facets such as knowing
what assumptions are important to check for a given analysis, adequate sample size,
and careful selection of variables.
SPSS do a wide range of analyses from simple descriptive statistics to various analysis
of variance designs and to all kinds of complex multivariate analyses (multivariate
analysis of variance –MANOVA-, factor analysis, multiple regression, discriminant
analysis, etc.). Multivariate analyses as listed above are complete arenas I do not
wish to enter into in this text. I limit myself into only those aspects expected to be
covered by the target group(s), i.e., some important SPSS environments or analysis
tools. However, I refer any reader interested with both the theoretical and practical
treatment of the complex multivariate analyses to the books by Johnson, R.A. and
Wichern, D.W. (1998) and Steven, J. (1996).
4.2 Starting SPSS
You can start SPSS in two different ways depending on how it is set up on your
computer. You can either double-click on the SPSS icon on the desktop of your
computer or click on the start button “normally” located at the lower left corner of
your computer then on programs, etc, as indicated in the root below:
Start>Programs>SPSS for Windows>SPSS 11.0 for Windows
When you click on the last option (SPSS 11.0 for Windows) of the above root you
will see the “Data Editor Window”
In general SPSS has four different types of windows namely:

4.2. STARTING SPSS 31
i. Data Editor;
ii. An output Window;
iii. A syntax Window; and
iv. A Chart Editor
We brieﬂy describe each of these windows in turn.
The Data Editor Window
The Data Editor Window is where data can be entered and edited. The Data Editor
is further divided into a data view and a variable view.
At the top of the Data Editor you can see a menu line consisting of the following
options: File, Edit, View, Data, Transform, Analyze, Graphs, Utilities, Window and
Help.
Figure 1. Data Editor menu
For more details on how you can use each one of these options I refer you to any
SPSS manual. I focus my attention on the “Analyze Menu” Here is where all the
statistical analyses are carried out.
The Output Window
Through this window you can read the results of your analysis. Depending on the
analysis you are carrying out, you can also see graphs and you can copy your results
into another document (e.g., word) for further description.
The Syntax Window
The syntax window is used for coding your analysis manually. Through this window
the user can code more advanced analyses, which may not be available in the stan-
dard menu. To open the syntax window select File>New>Syntax. In the window
you can enter the program code you want SPSS to perform. This requires a little
more programming. However, when the code is ready to be run you mark it (with
your mouse) and select Run> Selection.
The Chart Editor
The chart editor is used when editing a graph in SPSS. To be able to edit your graph
you need ﬁrst to double-click your graph.

4.3 Data entry
There are basically two ways to enter data into SPSS. One is to manually enter
directly into the Data Editor the other is to import data from a different program
or a text file. For example from Excel, SAS, etc. I will illustrate both options here.
For importing data, I will restrict myself to importing data from excel.
4.4 Keying data into SPSS
As we have seen above, when SPSS is opened, by default the Data Editor is opened
and this is where you can enter your data. Alternatively, to enter data go to
File>New>Data.
Before you start entering your data it is always a good idea to first give names to your
variables. This is done by selecting the variable view in the Data Editor window.
4.4.1 Osteopathic manipulation data set
The following is part of the data collected from a clinical trial1 whose prime objec-
tive was to compare the effect of an osteopathic manipulation with a control group
in measuring influence on blood flow at two different time points. This effect was
assessed in 80 volunteer healthy subjects aged between 17 and 69 years. Blood flow
(in ml/min) was measured from the right superficial femoral artery using Duplex-
Doppler while subjects lying down on a research table at baseline (minute-0: M1),
one minute after manipulation (M2) and four minutes after manipulation (M3).
The variable Patid in the table below represents patient’s identity number. The
variables initials, age, weight, height and gender carry the usual meaning. M1, M2
and M3 are as described above. Use this simple data set to practice data entry in
SPSS.
1
A clinical trial is study that investigates the efficacy of drug (s).

4.4. KEYING DATA INTO SPSS 33
Table 4.1: Osteopathic Manipulation data set
Patid Initials Age Weight Height Gender M1 M2 M3
1 SJ 31 75 178 M 109.5 262.1 136.4
2 TG 30 69 178 M 103.2 145.7 121.3
3 SF 38 73 176 M 221.2 231.7 111.9
4 WD 24 78 179 M 230.0 281.3 196.2
5 VF 54 73 162 F 112.4 120.4 139.7
6 SM 64 75 168 M 226.8 369.4 247.1
7 DWM 61 65 160 F 103.7 84.9 109.5
8 GM 34 60 166 F 178.5 139.6 154.6
9 VM 38 64 165 F 103.7 132.4 107.1
10 DWF 47 63 167 M 150.6 158.8 110.4
11 BE 49 85 172 M 149.1 72.0 96.2
12 CV 55 91 177 M 193.5 286.1 245.5
13 CG 25 69 170 F 183.4 270.6 183.7
Exercise
Enter these data in your SPSS Data Editor window without labelling the variables.
If you enter the data in SPSS without first giving names to the variables, SPSS labels
the variables as var00001, var00002, etc. Do you see this?
Next try to give names to the variables. As described above, you can give names to
your variables via the variable view in the Data Editor window. Alternatively you
can double click the variable.
Now click on the first cell of the first column “Name” and type the name of the first
variable as indicated in Table 3. That is, Patid, and then move on to the second cell
and type the name of the second variable and so on.
Under Type you define which type your variable is (numeric, string etc.). If you
place the marker in the Type cell, a button like the one in Figure 1 below appears.
Figure 1: Defining variables
This button indicates that you can click it and a window like the one below in Figure
2 will show:

Figure 2: Variable Type
Numeric is selected if your variable exists of numbers. String is selected if your
variable is a text (Male/Female). The same way you can specify Values and Missing.
By selecting Label you get the possibility to further explain the respective variable in
a sentence or so. This is often a very good idea since the variable name is restricted
to only 8 characters. Missing is selected when deﬁning if missing values occur among
the observations of a variable.
In Values you can enter a label for each possible response value of a discrete variable
(e.g. 1 = Male and 2 = Female).
When entering a variable name the following rules must be obeyed in SPPS for it to
work:
i. The name has to start with a letter and not end with a full stop (.).
ii. No more than 8 characters can be entered.
iii. Do not enter space or other characters like e.g.! ? ‘, and *.
iv. No two variable names must be the same.
When all data are entered and variable names are given you can save your data via
select File>Save As. . . in the menu.
4.5 Opening an existing dataset
If the dataset already exists in SPSS ﬁle you can easily open it. Select File>Open. . . and
the dataset will automatically open in the Data Editor.

4.6. IMPORTING DATA 35
4.6 Importing data
Sometimes the data are available in a different format than an SPSS data file. E.g.
the data might be available as an Excel, SAS, or text file. As already mentioned we
describe how to import data from excel.
Importing data from Excel
If you want to use data from an Excel file in SPSS there are two ways to import the
data.
i. One is to simply mark all the data in the Excel window (excluding the variable
names) you want to enter into SPSS. Then copy and past them into the SPSS
data window. The disadvantage by using this method is that the variable
names cannot be included meaning that you will have to enter these manually
after pasting the data.
ii. The other option (where the variable names are automatically entered) is to
do the following:
• Open SPSS, select File> Open>Data. Choose the drive where the data are
stored and then double click on the file you want to open or mark the file and
click on the open icon on the open file menu.
Under Files of type you select Excel, press ‘Open’, and the data now appear
in the Data Editor in SPSS.
4.7 Exporting data
Exporting data from SPSS to a different program is done by selecting File Save
As. . . Under Save as type you select the format you want the data to be available in
e.g. Excel.
4.8 ANOVA for one-way classification in SPSS
Let us now see how we can use SPSS to perform analysis of variance for one way
classification. I will use the machine data set discussed in Section 5.3.4 to illustrate
the construction of the analysis of variance table. As described above, getting out-
puts in SPSS is simple. Assuming that you have already entered the data, what you
need to do next is to analyze the data by following the root below:
Analyze>Compare Means>One-Way ANOVA
If you click on the last option (One-Way ANOVA) you will see a window like the
one below:

The only dependent variable in this example is “volume” and the factor is “machine”.
You can also include descriptive statistics in your outputs by clicking on the ”Op-
tions” and then select descriptive.
Below is the SPSS ANOVA table for the machine data set.
SPSS ANOVA table-Machine data
Source of Sum of df Mean Square F Sig.
Variation Squares
Between Groups 0.152 3 5.072E-02 0.834 0.500
Within Groups 0.729 12 6.078E-02
Total 0.882 15
From the above ANOVA table we see that, the results presented in SPSS are approx-
imated to three decimal places. By default, SPSS, like any other software package
gives the p-value (s) of the test (s)-indicated as Sig. in the last column of the table.
The p-value indicates “how much evidence against the null hypothesis” has been
observed in the outcomes of the experiment. Based on the given p-value (> 0.5) we
do not reject the null hypothesis (remember we are testing the hypotheses at 0.05
level of signiﬁcance), the conclusion we reached by comparing the F calculated value
(0.83) and the critical F-value (3.49) from the table. For comparison purposes of the
two ANOVA tables-the one we obtained before through mathematical calculations
and the one from SPSS above, I reproduce below the ANOVA table obtained by
mathematical computations. Are they similar?
Between treatment 3 0.152145 0.051 0.83
Within treatment 12 0.729355 0.061
Total 15 0.881500
Exercise (optional)
Use the Pea section data set to perform analysis of variance for one way classiﬁcation

4.8. ANOVA FOR ONE-WAY CLASSIFICATION IN SPSS 37
with equal replication. Compare your ANOVA table with the one obtained through
mathematical computations.
Note: The above two examples-the machine and pea section data sets illustrates
respectively what is termed as unbalanced and balanced data. Unbalanced in the
ﬁrst case in the sense that there are unequal numbers of replications of machines
and balanced in the second case in the sense that the various sugar types are all
replicated equal number of times (10 times).

Chapter 5
Completely Randomized Design
5.1 Introduction
When the experimental units are assumed to be fairly uniform or homogeneous,
that is, no sources of variations other than the treatments are expected, grouping
them (applying error/local control principle) will be pointless in the sense that very
little (in terms of precision) may be gained. Thus, the simplest experimental design,
which incorporates only the ﬁrst two principles (randomisation and replication) of
experimental designs, is the completely randomized design or CRD.
CRD is a design in which the treatments are assigned completely at random to the
experimental units, or vice-versa. Since we assume that there are no other sources of
variations in the experiment except the treatments under investigation, then CRD
imposes no restrictions, such as blocking on the allocation of the treatments to the
experimental units.
5.2 Layout
Suppose that we have t treatments under investigation and that the ithtreatment is
to be replicated ri times, i =1, 2, . . . , t. For an experiment with t treatments each
one replicated ri times, the total number of experimental units N =
t
i=1
ri. When
ri = r, that is, the case of equal replication, N=rt.
Deﬁnition: layout refers to the placement of treatment to the experimental units
subject to conditions of the design. Randomisation in CRD can be carried out by
using a random number table or any other probabilistic procedures.
Example
Suppose there are three treatments to be compared in a CRD. Suppose further
that the treatments are replicated 4, 3 and 5 times respectively. Thus, a total of
39

40 CHAPTER 5. COMPLETELY RANDOMIZED DESIGN
N =
t
i=1
ri=4 + 3 +5=12 experimental units. One possible layout of this experiment
is as follows:
T2
1
T1
2
T2
3
T3
4
T3
5
T1
6
T3
7
T1
8
T1
9
T3
10
T2
11
T3
12
5.3 Statistical analysis
The analysis of CRD is the same as that of one way classification. Let Yij be the
yield on the jthplot receiving treatment i. Thus, the model is:
Yij = µ + ti + eij, i=1, 2, . . . , t; j=1, 2, . . . r
Where:
µ is the grand mean (average) yield over all the N plots,
ti is the ith treatment effect
eij is the experimental error
Sums of squares are computed in the same way we discussed in one way classification.
5.3.1 Statistical hypotheses
The statistical hypotheses of interest as we stated before are:
Ho : µ1 = µ2 = ... = µt That is, the µi are all equal
H1 : µi = µj for at least one i = j. That is, the µi are not all equal.
Or simply
Ho :There is no variation among the treatments

5.4. ADVANTAGES AND DISADVANTAGES OF CRD 41
Source of Variation DF SS MS F-ratio
Treatments t-1 SSA SSA
t−1 = MSA MSA
MSE
Error N − t SSE SSE
N−t = MSE
Total N-1 SST
5.3.2 Test procedure
At level of significanceα, if F = MSA
MSE > Fα, [(t − 1) , (N − t)] then there is evidence
for no significance variation, i.e. we reject the null hypothesis. Otherwise, e do not
reject.
5.4 Advantages and disadvantages of CRD
5.4.1 Advantages
• Useful in small preliminary experiments and also in certain types of animal or
laboratory experiments where the experimental units are homogeneous.
• Flexibility in the number of treatments and the number of their replications.
• Provides maximum number of d.f. for the estimation of experimental error-
The precision of small experiment increases with error d.f.
5.4.2 Disadvantages
• Its use is restricted to those cases in which homogeneous experimental units
are available- local control not utilised. Thus, presence of entire variation may
inflate the experimental error.
• Rarely used in field experiments because the plots are not homogeneous.
5.5 Example
A sample of plant material is thoroughly mixed and 15 aliquots taken from it for
determination of potassium contents. 3 laboratory methods (I, II, and III) are em-
ployed. “I” being the one generally used. 5 aliquots are analysed by each method,
giving the following results (µg/ml).
I 1.83 1.81 1.84 1.83 1.79
Method II 1.85 1.82 1.88 1.86 1.84
III 1.80 1.84 1.80 1.82 1.79
Examine whether methods II and III give results comparable to those of method I.
Use α = 0.05

Calculations
The statistical model for this problem is Yij = µ + ti + eij. Here, i=1, 2, 3, j=1,2,
3, 4, 5.
Grand total, G =
3
i=1
5
j=1
Yij=1.83+1.81+. . . +1.79=27.4
Total number of observations, N =
k
i=1
ni =
3
i=1
ni = n1 + n2 + n3=5+5+5=15. In
the particular situation at hand (equal replication), N =rt =5×3=15
Correction factor, C.F=
3
i=1
5
j=1
Yij
2
rt = (27.4)2
15 = 750.76
15 =50.0507
Total sum of squared observations or uncorrected total sum of squares
3
i=1
5
j=1
Y 2
ij = (1.83)2
+ (1.81)2
+ ... + (1.79)2
= 50.0602
Total SS (SST) =
3
i=1
5
j=1
Y 2
ij − G2
rt =50.0602-50.0507=0.0095
=0.0095
Treatment (Method) totals: I=9.10, II=9.25, III=9.05
Treatment SS (SSTr) =1
r
k
i=1
Y 2
i − G2
rt , Yi =
r
j=1
Yij
=
1
5
(9.10)2
+ (9.25)2
+ (9.05)2
− 50.0507
=50.055-50.0507
=0.0043
Error SS (SSE) =SST-SSTr
=0.0095-0.0043
=0.0052
Source of Variation DF SS MS F-ratio
Between treatments 2 0.0043 0.00215 4.9654
Error (within treatments) 12 0.0052 0.00043
Total 14 0.0095

5.5. EXAMPLE 43
F0.05, 2, 12 = 3.89
Statistical hypothesis
Ho : µ1 = µ2 = µ3That is, the µi are all equal
H1 : µi = µjfor at least one i = j. That is, the µi are not all equal
Or
Ho: Methods do not differ
H1: Methods differ
Decision
Since the F calculated value (4.9654)> the critical F-value (3.89) at 0.05 level of
significance, we reject the null hypothesis and thus conclude that the laboratory
results depends on the method of analysis. That is, there exist significance variations
among the three laboratory methods.
To examine whether methods II and III give results comparable to those of method
I, we need to carry out further analysis using the t-test (LSD) as follows:
Let the mean of method I be denoted by ¯Y1., of method II by ¯Y2.and that of method
III by ¯Y3.
Thus, ¯Y1. = 9.10
5 = 1.82, ¯Y2. = 9.25
5 = 1.85, ¯Y3. = 9.05
5 = 1.81
Statistical Hypotheses
Here we need to test the hypotheses:
H02 : µ1 = µ3 vs. H12 : µ1 = µ3
H03 : µ1 = µ3 vs. H13 : µ1 = µ3
Test procedure
Reject: H02 if
| ¯Y2. − ¯Y1.| > LSD = s 2
r × tN−t,α/2
and H03 if | ¯Y3. − ¯Y1.| > LSD = s 2
r × tN−t,α/2
Exercises
1. Complete the test.
2. Eight varieties, A − H, of black currant cuttings are planted in square plots in a
nursery, each plot containing the same number of cuttings. Four plots of each variety
are planted, A and the shoot length made in the first growing season measured.

The plot totals are:
A: 46 29 39 35 E: 16 37 24 30
B: 37 31 28 44 F: 41 28 38 29
C: 38 50 32 36 G: 56 48 44 44
D: 34 19 29 41 H: 23 31 29 37
B and C are standard varieties; assess the remaining six for vigour in comparison
with B and C. Use α = 0.05

Chapter 6
Randomised Block Design
6.1 Introduction
CRD discussed in Chapter 3 will seldom be used if the experimental units are not
alike. Hence, when experimental units may be meaningfully grouped, e.g., by
area of ﬁeld, device, hospital, salesmen, etc, clearly a completely randomised design
(CRD) will be insuﬃcient. In this situation an alternative strategy for assigning
treatments to the experimental units, which takes advantage of the grouping, may
be used. The alternative strategy that we are going to discuss is what we call the
randomised block design or Randomised Complete Block Design (RCBD).
In the randomised block design:
• The groups are called blocks
• Each treatment appears the same number of times in each block; hence the
term complete block design
• The simplest case is that where each treatment appears exactly once in each
block. Here, because the number of replicates=number of experimental
units for each treatment,
we therefore have: number of replicates=number of blocks=r
• Blocks are often called replicates for this reason
• To set up such randomised block design the following steps are involved:
(i) Divide the units into r more homogeneous groups commonly known as blocks.
(ii) Assign the treatments at random to the experimental units within each block.
This randomisation has to be done afresh for each block.
Hence, the term randomised block design
45

46 CHAPTER 6. RANDOMISED BLOCK DESIGN
Motivation: experimental units within blocks are alike as possible, so observed
differences among them should be mainly attributed to the treatments. To ensure
this interpretation holds, in the conduct of the experiment, all experimental units
within a block should be treated as uniform as possible.
Intuitively speaking, randomised block design is an improvement over the CRD.
In the RBD the principle of local control can be applied along with the other two
principles of experimental design (randomisation and replication).
Number of experimental units (N )
Suppose we want to compare the effects of t treatments, each treatment being repli-
cated an equal number of times, say r times. Then we need N =rt experimental
units.
6.2 Layout
To illustrate the layout of an RBD, consider 4 treatments, each replicated 3 times.
So we need N =rt= 3×x4=12 experimental units which are grouped into 3 blocks
of 4 units.
Suppose the blocks formed after grouping the experimental units are labelled as 1,
2, and 3. To ensure randomness in every process involved in the experiment we
select the block to start with in allocating the treatments to the experimental units
at random. Assume the blocks are selected in the order 3, 1, 2. Thus, we start with
the third block and assign the 4 treatments at random to it. As we have discussed,
to assign the treatments, we may use any probabilistic procedure.
Permutations is one of the probabilistic procedures that may be used to allocate
treatments to experimental units. Suppose one of the permutations of the digits 1
to 4 for the treatment is 4, 1, 3, 2. Therefore we allocate treatment 4 in the first
unit of block 3, treatment 1 in the second unit of block 3, up to treatment 2 in the
fourth unit of block 3. That is, we have the following layout for block 3 (first selected
block).
T4 T1 T3 T2
Repeating the same procedure, suppose we select the permutations 3, 4, 2, 1 for
block 1 and 2, 3, 4, 1 for block 2, finally get the following complete layout.

6.3. STATISTICAL ANALYSIS 47
Block 1 T3 T4 T2 T1
Block 2 T2 T3 T4 T1
Block 3 T4 T1 T3 T2
The analysis of the design is the same as that of two-way classified data with one ob-
servation per cell-experimental unit- (without replication) we discussed in Section
5.5.
We use the same model we have discussed,
Yij = µ + ti + bj + eij, i=1, 2, . . ., t, j=1, 2, . . ., r
In words:
Observation of the ith treatment from the jth block =general mean +ith treatment
effect + jth block effect + experimental error component
RECAP: we partition the total sum of squares into different components:
Total SS=Treatment SS + Block SS + Error SS
6.3.1 Statistical hypotheses
The hypotheses of interest are:
HO1 : t1 = t2 = ... = tk
Ho2 : b1 = b2 = ... = bk
Against their alternative that tis, bjs are not all equal.
ANOVA table
Blocks r-1 SSB SSB
r−1 = MSB MSB
MSE = FB
Treatment t-1 SSTr
SSTr
t−1 = MSTr
MSTr
MSE = FT r
Error (r-1)(t-1) SSE SSE
(r−1)(t−1) = MSE
Total N-1 SST
6.3.2 Test procedure
The calculated F-values for treatments and blocks (FBand FTr) are compared with
the tabulated (critical) F-values at (t-1) and (r-1)(t-1) and (r-1) and (r-1)(t-1) de-
grees of freedom respectively.

In symbols
Fα, [(t − 1), (r − 1)(t − 1)] and Fα, [(r − 1), (r − 1)(t − 1)]
Thus, if FB>Fα, [(r − 1), (r − 1)(t − 1)] we reject the null hypothesis, otherwise we
do not reject. Also if FTr>Fα, [(t − 1), (r − 1)(t − 1)] we reject the null hypothesis,
otherwise we do not reject.
6.4 Advantages and disadvantages of RBD
6.4.1 Advantages
• Greater precision
• Increased scope of inference is possible because more experimental conditions
may be included
6.4.2 Disadvantages
• Large number of treatments increases the block size; as a result the block may
loose homogeneity leading to large experimental error.
• Any missing observation in a unit in a block will lead to either:
(i) discard the whole block
(ii) estimate the missing value from the unit by special missing plot technique.
6.5 Example
The following data are yields in bushels/acre from an agricultural experiment set out
in a randomised complete clock design. The experiment was designed to investigate
the differences in yield for seven hybrid varieties of wheat, labelled A-G here. A field
was divided into 5 blocks, each containing 7 plots. In each plot, the seven plots were
assigned at random to be planted with the seven varieties, one plot for each variety.
A yield was recorded for each plot. Examine whether varieties affect the yield. Use
α=0.05.
Variety
Block A B C D E F G Total
I 10 9 11 15 10 12 11 78
II 11 10 12 12 10 11 12 78
III 12 13 10 14 15 13 13 90
IV 14 15 13 17 14 16 15 104
V 13 14 16 19 17 15 18 112
Total 60 61 62 77 66 67 69 G=462

6.5. EXAMPLE 49
We assume that the measurements are approximately normally distributed, with the
same variance σ2.
Calculations
Total number of experimental observation (N) = r × t= 7×5=35
Grand total (G) =
t
i=1
r
j=1
yij=10 + 9 +. . . + 15 + 18=462
Correction factor (C.F) =G2
N = (462)2
35 =6098.4
Uncorrected total sum of squares
t
i=1
r
j=1
y2
ij=102 +92 +. . . + 152 + 182 =6314.0
Total sum of squares (SST) =
t
i=1
r
j=1
y2
ij-C.F= 6314.0 - 6098.4=215.6
Treatment (variety) sum of squares (SSTr) = 1
r
t
i=1
Y 2
i -C.F
=1
5(602 + ... + 692)-C.F
=6140.0-6098.4
=41.6
Block sum of squares (SSB) = 1
t
r
j=1
Y 2
j -C.F
=1
7(782 + ... + 1122)-C.F
=6232.6-6098.4
=134.2
Error sum of squares (SSE) =Total SS-Treatment SS-Block SS
= 215.6-134.2-41.6=39.8
Treatment (variety) mean squares (MSTr) = SSTr
t−1 = 41.6
6 =6.93
Block mean squares (MSB) = SSB
r−1 = 134.2
4 =33.54
Error mean squares (MSE) = SSE
(t−1)(r−1) = 39.8
24 =1.66.
Finally, we estimate the F-values. For block the F-calculated value is

=MSB
MSE = 33.54
1.66 =20.21 and
for treatments the F-calculated value is MSTr
MSE = 6.93
1.66=4.18
We summarize the calculations in an ANOVA table as follows:
ANOVA Table
Source of variation D.F SS MS F-ratio
Blocks 4 134.2 33.54 20.21
Treatments 6 41.6 6.93 4.18
Error 24 39.8 1.66
Total 34 215.6
To perform the hypothesis test for differences among the treatment means, we com-
pare the F calculated values to the appropriate value from the F table. For level of
significance α=0.05, we have
F0.05;6,24 = 2.51. F-calculated (= 4.18)> F0.05;6,24 = 2.51.
Therefore, we reject H0. There is evidence in these data to suggest that there are
differences in mean yields among the varieties.
To test the hypothesis on block differences, we find F0.05,4,24=2.78. We have 20.21>2.87,
thus, we also reject H0. There is strong evidence in these data to suggest differ-
ences in mean yield across blocks at the 5% level of significance.
In SAS
data hybrid;
input Block $ Variety $ Yield @@;
cards;
I A 10 I B 9 I C 11 I D 15 I E 10 I F 12 I G 11
II A 11 II B 10 II C 12 II D 12 II E 10 II F 11 II G 12
III A 12 III B 13 III C 10 III D 14 III E 15 III F 13 III G 13
IV A 14 IV B 15 IV C 13 IV D 17 IV E 14 IV F 16 IV G 15
V A 13 V B 14 V C 16 V D 19 V E 17 V F 15 V G 18
;
run;
proc print;run;
proc anova;
class Block Variety ;
model Yield=Block Variety;
run;quit;
The SAS System
Obs Block Variety Yield
1 I A 10

6.5. EXAMPLE 51
2 I B 9
3 I C 11
4 I D 15
5 I E 10
6 I F 12
7 I G 11
8 II A 11
9 II B 10
10 II C 12
11 II D 12
12 II E 10
13 II F 11
14 II G 12
15 III A 12
16 III B 13
17 III C 10
18 III D 14
19 III E 15
20 III F 13
21 III G 13
22 IV A 14
23 IV B 15
24 IV C 13
25 IV D 17
26 IV E 14
27 IV F 16
28 IV G 15
29 V A 13
30 V B 14
31 V C 16
32 V D 19
33 V E 17
34 V F 15
35 V G 18
Class Level Information
Class Levels Values
Block 5 I II III IV V
Variety 7 A B C D E F G
Number of observations 35
Dependent Variable: Yield
Sum of
Model 10 175.7714286 17.5771429 10.59 <.0001
Error 24 39.8285714 1.6595238
R-Square Coeff Var Root MSE Yield Mean
0.815266 9.759281 1.288225 13.20000

Block 4 134.1714286 33.5428571 20.21 <.0001
Variety 6 41.6000000 6.9333333 4.18 0.0052
6.6 Reasons for blocking in RBD
Note from these results that the blocking served to explain much of the overall vari-
ation. To appreciate this further, suppose that we had not blocked the experiment,
but instead had just conducted the experiment according to a completely random-
ized design. Suppose that we ended up with the same data as in the experiment
above.
Under these conditions variety is the only classification factor for the plots, and we
would construct the following analysis of variance table for one way-classification as
discussed in the previous sections.
ANOVA Table
Between treatments (varieties) 6 41.6 6.93 1.12
Within treatments (error) 28 174.0 6.21
Total 34 215.6
The test for differences in mean yield for the varieties (treatments) would be to
compare F=1.12 to F0.05,6,28=2.45. Note that we would thus not reject H0 of no
treatment differences at the 5% level of significance.
Concluding Remark
From the above example, it is immediately clear that if the different sources of
variation are not properly identified (e.g., due to erroneously accounting for the
experimental design), then invalid conclusions will be drawn.
The example discussed above clearly demonstrates the aspect of wrongly identify-
ing the experimental design. In the one-way classification experiment and analysis
presented above, there is no accounting for the variation in the data that is actually
attributable to a systematic source, position in the field (the factor used to block
the experiment). The one-way analysis has no choice but to attribute this variation
to experimental error; that is, it regards this variation as just part of the inherent
variation among experimental units that we cannot explain. The result is that the
Error SS in the one-way analysis contains both variation due to position in the field
(which is actually systematic variation) and inherent variation.
Here, note that 143.2+39.8=174.0 and 4+24=28.
This is the Error SS for the one-way classification analysis. Which actually may be
regarded as ignoring the blocks (because what we really did was to pretend that the

6.6. REASONS FOR BLOCKING IN RBD 53
blocks didn’t exist) and thus resulting into big MSE in which we could not reject the
H0. By blocking the experiment, and explicitly acknowledging position in the field
as a potential source of variation, MSE was sufficiently reduced so that we could
identify variety differences.
It can be learned from this example that:
• Blocking may be an effective means of explaining variation (increasing preci-
sion) so that differences among treatments that may really exist are more likely
to be detected.
• The data from an experiment set up according to a particular design should
be analysed according to the appropriate procedure for that design. The
above shows that if we set up the experiment according to a randomised com-
plete block design, but then analyse it as if it had been set up according to
a completely randomised design, erroneous inferences results, in this case,
failure to identify real treatment differences. Remember, the design of an
experiment dictates the analysis!!
Exercise
Four different plant densities A-D are included in an experiment on the growth of
lettuce. The experiment is laid out as a randomised block, and the same number of
plants is harvested from each plot, giving the weights (recorded) below. Examine
whether density appears to affect the yield. Use α = 0.01

Block
Density I II III IV V VI
A 2.7 2.6 3.1 3.0 2.5 3.0
B 3.0 2.8 3.1 3.2 2.8 3.1
C 3.3 3.3 3.5 3.4 3.0 3.2
D 3.2 3.0 3.3 3.2 3.0 3.1

Chapter 7
Latin Square Design
7.1 Introduction
There are often situations where it may be necessary to account for two sources
of variation by blocking. If the number of treatments and levels of each blocking
factor is large, the size of the experiment may become unwieldy or resources may be
limited. Thus, in agricultural field experiments (and other situations), a particular
setup is often used that allows differences among treatments to be assessed with less
recourses.
The principle of local control was used in the RBD by grouping the units in one
way; i.e. according to blocks. The grouping can be carried one step forward and
we can group the units in two ways, each way corresponding to a known source of
variation among the units, and get the Latin Square Design (LSD). This design is
used with advantage in agricultural experiments where the fertility contours are not
always known. It has also been used successfully in industry and in the laboratory.
Latin square design is a design, which uses the principle of local control twice.
RBD removes one systematic source of variation in addition to treatments, but LSD
removes two such sources. Hence, LSD is a three-way classification.
In a field experiment if two probable fertility trends can be thought of, in directions
at right angles, both need to be made the basis of blocking. Thus when there is a
slope of the land being used, and also a climatic trend (e.g. effects of wind, rain) at
right angles to this, a randomised block cannot take out all the known variation.
7.2 Layout
In field experiments, the physical layout is that of a square with rows of plots. In
this set up the layout is such that every letter (A, B, C,. . . ) the set of treatments
occurs exactly once in each row and in each column. For four letters A, B, C, and
D the layout would be as shown below.
55

56 CHAPTER 7. LATIN SQUARE DESIGN
Column 1 2 3 4 5
1 E B A D C
2 C A D E B
Row 3 B E C A D
4 A D B C E
5 D C E B A
This type of setup would be useful when for example variability due to soil differences,
etc, arises in two directions. Each plot would constitute a single experimental unit.
This particular kind of setup with two blocking variables (rows and columns), in
which the numbers of rows, columns and treatments are the same, is known as a
Latin square.
Notation
Because the number of treatments, rows and columns are the same, the number
of replicates on each treatment is equal to the number of treatments, rows, and
columns. We will denote this as t. For given value of t, there may be several ways
to construct a Latin square. This is actually as mathematical exercise. Extensive
listings of ways to construct Latin squares for different values of t are often given in
texts on experimental design (see for example, Montgomery, 2001).
Selection of LSD
The totality of LSD’s obtained from a single LSD by permuting rows, columns and
treatments (letters) is called a transformation set.
e.g. A B C D fixed
B C D A
C D A B
D A B C
A B C D
C D A B
D A B C
B C D A
A k×k Latin Square with k letters A, B, C, . . . in the natural order occurring in the
first column is called a standard square (square in canonical form).
e.g. A B C D
B C D A
C D A B
D A B C
From a standard k × k Latin Square, we may obtain k! (k-1)! Different LSD’s by
permuting all the k columns and the (k-1) rows except the first row. Hence there

7.3. STATISTICAL ANALYSIS 57
are in all k! (k-1)! Different LSDs with the same standard square. Thus the total
number of different LSDs in a transformation set is k! (k-1)! times the number of
standard LSDs in the set. In order to give all k × k LSDs equal probability of being
selected, we select one LSD from all k × k LSDs and then randomise the columns
and rows, excluding the first row (if it is the fixed one).
Randomisation consists of choosing one of the possible designs for given t at random.
Then randomly assign the letters A, B, C, etc to the treatments of interest
7.2.1 Linear additive model
To write down a model, we need to be a bit careful with the notation. The key is
that, although we have three classifications (row, column and treatment) we do not
have t × t × t = t3 observations; rather we only have t × t = t2
The mathematical model will now be,
yijk = µ + ti + rj + ck + eijk
where:
yijk observation for the ith treatment appearing in row j, column k
µ is an overall mean
rj, ck represents the effects of the jth row and kth column
ti represents the effect of the treatment appearing at position, j,k
eijk error associated with the experimental unit appearing at position j,k
The analysis proceeds along the same lines as for RBD, but instead of the one sum
of squares for blocks; systematic variation is now taken out by two sums of squares,
which are always called Rows (SSR) and columns (SSC).
7.3.1 Calculation of sums of squares
To setup the analysis of variance we define Rjand Ck as the totals of all plots in the
jth row and kth column respectively in the layout, the sums of squares required are:
Total SS SST =
t
i=1
t
j=1
t
k=1
y2
ijk − G2
t2
Row SS (SSR)=1
t
t
j=1
R2
j − G2
t2
Column SS (SSC)=1
t
t
k=1
C2
k − G2
t2

58 CHAPTER 7. LATIN SQUARE DESIGN
Treatment SS (SSTr) = 1
t
t
i=1
T2
i − G2
t2
Error SS (SSE)= SST –(SSR+SSC+SSTr) or
SSE =SST – SSR –SSC -SSTr
The degrees of freedom for SSR, SSC, SSTr are each (t-1); for SST, (t2-1), and so
for SSE, (t2-1)-3 (t-1), reducing to (t-1)(t-2).
Table 7.1: Three way ANOVA table
Source of DF SS MS F-ratio
variation
Rows t-1 SSR SSR
t−1 = MSR MSR
MSE = FR
Columns t-1 SSC SSC
t−1 = MSC MSC
MSE = FC
Treatments t-1 SSTr SSTr
t−1 = MSTr
MSTr
MSE = FTr
Error (t-1)(t-2) SSE SSE
(t−1)(t−2) = MSE
Total t2-1 SST
7.4 Advantages and disadvantages of LSD
7.4.1 Advantages
• Eliminates from the error two major sources of variation. Hence LSD is an
improvement over RBD in controlling error by planned grouping just as the
RBD is an improvement over CRD.
• LSD is a 3-way incomplete layout since LSD considers treatments, rows, and
columns at the same number of levels t, we would need a complete three way
layout of t3 number of experimental units. However, since we are using t2
number of experimental units, then it is said to be a 3-way incomplete layout.
7.4.2 Disadvantages
• A serious limitation of the LSD is that the number of replicates must be the
same as the number of treatments, the larger the square the more is the repli-
cates, hence the bigger the blocks (columns and rows). Hence larger squares
(over 12×12 ) are seldom used in the sense that the squares does not remain
homogeneous. On the other hand, small squares provide only a few degrees of
freedom for the error. Preferable LSDs are form 5×5 to 8×8.
• The analysis depends heavily on the assumption that there are no interactions
present.
• Analysis becomes very diﬃcult where there are missing observations.

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