research design notes weeks 7-12
TRANSCRIPT
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Topic 1: Basic Conceptsof Experimental Design
Dr Amirul Islam
Acknowledged to: Dr Jahar Bhowmik
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Contents
1.1 Topic introduction 3
1.2 Topic learning objectives 4
1.3 Important Terms and Definitions of Experimental Design 4
1.4 Principles of an Experimental Design 8
1.5 Design of Experiments in Marketing 11
1.6 Sample Surveys versus Experimental Design 12
1.7 The Parallels between Experimental Designs & Sample Surveys 12
1.8 Study Design in Medical Research 13
1.9 Guidelines for Designing Experiments 14
1.10 Research questions and hypotheses 15
Revision Exercises 17
Solution to Revision Exercises 20
References 22
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Note: Some of the materials are adapted from standard texts and guides (see references).
1.1 Topic introduction
“The formulation of a problem is often more essential than its solution which may be
merely a matter of mathematical or experimental skill”. --------------Albert Einstein
Design of Experiment is a structured, organized method that is used to determine the
relationship between the different factors affecting a process and the output of that
process. This method was first developed in the 1920s and 1930, by Sir Ronald A. Fisher,
the renowned mathematician and geneticist.
This chapter examines the basic concepts of experimental design. Experimentation and
making inferences are twin features of general scientific methodology. The subject-matter
of experimental design includes:
(i) Planning the experiment,
(ii) Obtaining relevant information from it regarding the statisticalhypothesis under study, and(iii) Making a statistical analysis of the data.
Experimental design is a term which includes efficient methods for planning for the
collection of data, in order to obtain the maximum amount of information for the least
amount of work. Data are everywhere. Anyone can collect and analyse data, be it in the
lab, the field, or the production plant, can benefit from knowledge about experimental
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design. Directed experimentation generates critical events. An experiment is “an
invitation for an informative event to occur” (Box et al., 2005).
Experience has shown that proper consideration of statistical analysis before the
experiment is conducted, forces the experimenter to plan more carefully the design of the
experiment. The observations obtained from a carefully planned and well-designed
experiment give entirely valid inferences.
Experiments are usually more structured than sample surveys and include the additional
step of treating the elements. In Sample Survey we select elements from frames and then
take measurements (such as responses to a questionnaire) but in Experimental Designs
we select experimental units, allocate treatments and then take measurements (either a
few or all elements).
1.2 Topic learning objectives
Learning objectives
When you have worked through this topic you should:
• Understand the idea of experimental design.
• Know the basic definition of experimental design.
• Understand the basic concepts that underlie scientific investigations.
1.3 Important Terms and Definitions ofExperimental Design
Observation (Correlational) and experimental studies
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A study in which the researcher observes and records what has already happened is called
an observational approach. On the other hand, an experimental study or trial is initiated
by a researcher. In an "ideal" experiment, the researcher manipulates the independent
variable (s), holds all other variables constant, and observes the changes in the dependent
variable. In experimental studies or trials we determine which experimental units receive
which treatment, whereas in observational studies we have to take what is observed.
Observational studies often show an association between two variables, but they cannot in
themselves prove cause and effect.
For example, consider the hypothesis:
"Driving ability varies with blood alcohol level".
The researcher would manipulate the amount of alcohol given to the drivers and then
observe changes in their driving skills. If all other variables are held constant, then any
changes in driving skill must be caused by the effects of the alcohol.
Consider an alternative means of collecting data. The researcher stands outside the pub
on Friday night and asks for volunteers leaving the pub. Each volunteer undergoes a
driving test and also has his/her blood alcohol level measured. The researcher then
compares the driving skills of volunteers with zero blood alcohol level to the driving
skills of those drivers whose alcohol level is over .05. This is an observational design.The researcher is merely observing the blood alcohol level of each subject, rather than
controlling or manipulating it.
Experiment
An experiment is the device or the means of getting the answer to the problem under
investigation, e.g. comparison of different manures or fertilizers, different varieties of a
crop, different cultivation processes, or different diets or medicines in a dietary or medical
experiment.
An experiment is a planned inquiry to discover new facts, or to confirm or deny the
results of previous investigations (Petersen, 1985).
Nuisance variables
Nuisance variables are associated with variation in an outcome (dependent variable) that
is extraneous to the effects of independent variables that are of primary interest to the
researcher. In our description of an "ideal" experiment, we stated that "all other variables"
should be held constant. If, for example, we are interested in the effects of alcohol on
driving ability, then any other variable which may influence driving ability is known as a
nuisance variable. Such things as the type of car, the driving course, temperature,
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humidity, time of day, and the driver's, age, reflexes and level of experience would all
have an influence on a driving test score. These are all referred to as nuisance variables.
Confounding variables
Variables that are not controlled for that systematically change experimental results, they
are called confounding variables. A confounding variable has two properties. First, a
confounding variable is related to the explanatory (independent) variable in the sense thatindividuals who differ due to the explanatory variable are also likely to differ for the
confounding variable. Second, a confounding variable affects the response (dependent)
variable.
Suppose we are interested in the effects of alcohol on driving ability. If all of the zero
alcohol level driving tests were performed in the morning, and all of the .05 alcohol level
driving tests were completed in the evening, we could not tell if the resulting differences
in driving abilities were due to differences in the alcohol level, or if they were due to
differences in the time of day of the test. In this case, "time of day" is known as a
confounding factor , because it literally confounds our interpretation of the experiment.
Treatments
Various objectives of comparison in a comparative experiment, are known as treatments,
e.g., in field experimentation different fertilizers or different varieties of crop or different
methods of cultivation are treatments.
A treatment is one or a combination of categories of the explanatory variable(s) assigned
by the experimenter. The plural term treatments incorporates a collection of conditions,
each of which is one treatment.
Factor and Level
A factor of an experiment is a controlled independent variable; the levels of the variable
are set by the experimenter.
A factor is a general type or category of treatment. Different treatments constitute
different levels of a factor. For example, three different groups of runners are subjected to
three different training methods. The runners are the experimental units, the training
methods are the treatments. Where the three types of training methods constitute three
levels of the factor e.g. 'type of training'. The states of a factor, i.e., the treatments within
the class, are called the levels of the factor.
Experimental Units
The individuals in an experiment are referred to as experimental units. The smallest
division of the experimental material, to which we apply the treatments and on which we
make observations on the variable under study, is termed an experimental unit.
Experimental units can be people, animals, batteries, etc. In field experiment the plot of
‘land’ is the experimental unit. In other examples may be a patient in a hospital, pigs in a
pen, or a batch of seeds. With animal trials, an experimental unit can be a paddock of
animals, a single animal, or even part of an animal.
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Blocks
In agricultural experiments, most of the time we divide the whole experimental unit
(field) into relatively homogenous sub-groups (shown in the following diagram) or strata.
These strata, which are uniform amongst themselves, are known as blocks. That means, a
block is a group of experimental units that are similar in a way that is expected to affect
the response to the treatments. A group of homogenous experimental units is called ablock.
The term blocking was first used by R. A. Fisher in agronomic experiments (1920). In the
statistical theory of the design of experiments, blocking is the arranging of experimental
units in groups (blocks) that are similar to one another. Typically, a blocking factor is a
source of variability that is not of primary interest to the experimenter. Blocking is
sometimes used for nuisance factors that can be controlled. Nuisance factors are those
that may affect the measured result, but are not of primary interest. For example, in
applying a treatment, nuisance factors might be the specific operator who prepared the
treatment, the time of day the experiment was run, or the room temperature. All
experiments have nuisance factors. The experimenter will typically need to spend some
time deciding which nuisance factors are important enough to keep track of or control if
possible, during the experiment.
Figure 1: Non-homogenous experimental units
Figure 2: Blocking into homogenous groups
Replication
Replication means the repetition of a test or an experiment more than once. In other
words, the repetition of treatments under investigation is known as replication.
Precision
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The reciprocal of the variance of the mean is termed as the precision, or the amount of
information of a design. Thus for an experiment replicated r times, the precision is given
by
2
1
var( )
r
x σ = .
Experimental ErrorLet us suppose that a large homogenous field is divided into different plots (of equal
shape and size) and different treatments are applied to those plots. If the yields from some
of the treatments are more than those of the others, the experimenter is faced with the
problem of deciding if the observed differences are really due to treatment effects or they
are due to chance (uncontrolled) factors. In field experiments, it is a common experience
that the fertility gradient of the soil does not follow any systematic pattern but behaves in
an erratic fashion. Experience tells us that even if the same treatment is used on all the
plots, the yields would still vary due to the differences in soil fertility. Such variation
from plot to plot, which is due to random (or chance or non-assignable) factors beyond
human control, is spoken of as experimental error . It may be pointed out that the term
‘error’ used here in not synonymous with ‘mistake’ but is a technical term which includes
all types of extraneous variations due to:
(i) the inherent variability in the experimental material towhich treatments are applied,
(ii) the lack of uniformity in the methodology of conducting theexperiments, or in other words failure to standardise the
experimental technique, and
(iii) lack of representativeness of the sample to the populationunder study.
Blind Experiment
The blind method is a part of some scientific methods, used to prevent research outcomes
from being influenced by either the placebo effect or the observer bias. In a blind
experiment, the subjects do not know whether they are in the treatment group or the
control group. The idea is that the groups studied, including the control, should be
unaware of the group they are placed in. In medicine, when researchers are testing a new
medicine, they ensure that the placebo looks, and tastes, the same as the actual medicine.
There is strong evidence of a placebo effect with medicine, where, if people believe that
they are receiving a medicine, they show some signs of improvement in health. A blind
experiment reduces the risk of bias from this effect, giving an honest baseline for theresearch, and allowing a realistic statistical comparison.
Ideally, the subjects should not be told that a placebo was being used at all, but this is
regarded as unethical.
Natural sources of error in field experiments
Plant variability
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– type of plant, larger variation among larger plants – competition, variation among closely spaced plants is smaller – plot to plot variation because of plot location (border effects)
Seasonal variability – climatic differences from year to year – rodent, insect, and disease damage varies – conduct tests for several years before drawing firm conclusions
Soil variability
– differences in texture, depth, moisture-holding capacity, drainage, availablenutrients
– since these differences persist from year to year, the pattern of variability can bemapped with a uniformity trial
1.4 The Three basic Principles of Experimental
Design
Professor Ronald A. Fisher pioneered the study of experimental designs with his classicalbook, The Design of Experiments. According to him, the basic principles of the design of
experiments are:
(i) Randomisation(ii) Replication, and(iii) Local Control or Error Control or Blocking.
The roles they play in data collection and interpretation are discussed below.
Randomisation
By randomisation we mean that both the allocation of the experimental material and the
order in which the individual runs or trials of the experiment to be performed, are
randomly determined. After the treatments and the experimental units are decided the
treatments are allotted to the experimental units at random to avoid any type of personal
or subjective bias which may be conscious or unconscious. This brings to the
experimenter the question of allocation of treatments to experimental units so that each
treatment gets an equal chance of showing its worth. In the absence of prior knowledge of
the variability of the experimental material, this objective is achieved through‘randomisation’, a process of assigning the treatments to various experimental units in a
purely chance manner. The following are the main objectives of randomisation:
(i) To eliminate bias,(ii) To ensure independence among the observations.
Criteria for randomisation in clinical trial studies
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1. Unpredictability
– Each participant has the same chance of receiving any of the interventions.– Allocation is carried out using a chance mechanism so that neither the
participant nor the investigator will know in advance which will be
assigned.
2. Balance
– Treatment groups are of a similar size & constitution; groups are alike in all
important aspects and only differ in the intervention each group receives.3. Simplicity
– Easy for investigator/staff to implement.
Replication
As pointed out earlier, replication means the execution of an experiment more than once.
In other words, the repetition of treatments under investigation is known as replication.
An experimenter resorts to replication in order to average out the influence of the chance
factors on different experimental units. Thus, the repetition of treatments results in a more
reliable estimate than is possible with a single observation. Replication is necessary to
increase the accuracy of estimates of the treatment effects. Although, the more the
number of replications the better the estimate is, it cannot be increased indefinitely as it
increases costs of experimentation.
Replication serves a number of purposes in an experimental design:
(i) It allows the experimenter to obtain an estimate of the experimentalerror.
(ii) It permits the experimenter to increase precision by reducingstandard errors.
(iii) It can expand the base for making inference.
Local Control or Blocking
Blocking means to arrange the experimental materials into groups, or blocks, of more or
less uniform experimental units. If the experimental material, say field for agriculture
experimentation, is heterogenous and different treatments are allocated to various units
(plots) at random over the entire field, the soil heterogeneity will also enter the
uncontrolled factors and thus increase the experimental error. It is desirable to reduce the
experimental error as far as practicable without unduly increasing the number of
replications or without interfering with the statistical requirement of randomness, so that
even smaller differences between treatments can be detected as significant.
In addition to the principles of replication and randomisation discussed earlier, the
experimental error can further be reduced by making use of the fact that neighbouring
areas in a field are relatively more homogenous than those widely spread. In order to
separate the soil fertility effects from the experimental error, the whole experimental area
(field) is divided into homogenous groups (blocks) row-wise or column-wise or both,
according to the fertility gradient of the soil such that the variation within each block is
minimum and between the blocks is maximum. The treatments are then allocated at
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random within each block. The process of reducing the experimental error by dividing the
relatively heterogenous experimental area (field) into homogenous blocks is known as
local control.
Example 1.1
Consider the very simple agricultural problem of comparing two varieties of tomatoes.The purpose of the comparison is to find the variety which produces the greater quantityof marketable quality fruit from a given area for large scale commercial planting. Whatshould we do? A simple approach would be to plant a block of land of each variety andmeasure the total weight of marketable fruit produced. However, there are some obviousdifficulties. The variety that cropped most heavily may have done so simply because itwas growing in better soil. There are a number of factors which affect growth: soilfertility, soil acidity, irrigation and drainage, wind exposure, exposure to sunlight(e.g. shading, north-facing or south-facing hillside). Unfortunately no one knows exactlyto what extent changes in these factors affect growth. So unless the two blocks of land arecomparable with respect to all of these features, we won't be able to conclude that themore heavily producing variety is better as it may just be planted in a block that is bettersuited to growth.
If it was possible (and it never will be) to find two tracts of land that were identical inthese respects, using just those two blocks for comparison would result in a faircomparison but the differences found might be so special to that particular combination ofgrowing conditions that the results obtained were not a good guide to full scaleagricultural production anyway.
Why randomise? Let us think about it another way. Suppose we took a large block ofland and subdivided it into smaller plots by laying down a rectangular grid. By usingsome sort of systematic design to decide what variety to plant in each plot, we may comeunstuck if there is a feature of the land like an unknown fertility gradient. We may stillend up giving one variety better plots on average. Instead, let's do it randomly bynumbering the plots and randomly choosing half of them to receive the first variety. Therest receive the second variety. We might expect the random assignment to ensure that
both varieties were planted in roughly the same numbers of high fertility and low fertilityplots, high pH and low pH plots, well drained and poorly drained plots etc.
In that sense we might expect the comparison of yields to be fair. Moreover, although wehave thought of some factors affecting growth, there will be many more that we, and eventhe specialist, will not have thought of. And we can expect the random assignment oftreatments to ensure some rough balancing of those as well!
Why replicate? Random sampling gives representative samples, on average. However, insmall samples, it may occur, just by chance, that your sample may be a 'bit weird'.Unfortunately, we can only expect the random allocation of treatments to lead to balancedsamples (e.g. a fair division of the more and less fertile plots) if we have a large numberof experimental units to randomise. In many experiments this is not true (e.g. using plots
to compare varieties) so that in any particular experiment there may well be a lack ofbalance on some important factor. Random assignment still leads to fair or unbiasedcomparisons, but only in the sense of being fair or unbiased when averaged over a wholesequence of experiments. This is one of the reasons why there is such an emphasis inscience on results being repeatable.
Why block? Partly because random assignment of treatments does not necessarily ensurea fair comparison when the number of experimental units is small. In this case morecomplicated experimental designs are available to ensure fairness with respect to thosefactors which we believe to be very important. Suppose with our tomato example that,
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because of the small variation in the fertility of the land we were using, the only thing thatwe thought mattered greatly was drainage. We could then try and divide the land into twoblocks, one well drained and one badly drained. These would then be subdivided intosmaller plots, say 6 plots per block. Then in each block, 3 plots are assigned at random tothe first variety and the remaining 3 plots to the second variety. We would then onlycompare the two varieties within each block so that well drained plots are only comparedwith well drained plots, and similarly for badly drained plots. This idea is called blocking.By allocating varieties to plots within a block at random we would provide some
protection against other extraneous factors.
1.5 Design of Experiments in MarketingDesign of experiments, or conjoint analysis as it is known in a marketing context, is
known to be the most powerful statistical method for establishing the linkage between a
customer's decision-making process and the service or product being offered. After
effective application of design of experiments, companies find it easier to gain an insight
into the significant variables affecting a customer's decision-making ability.
Marketing Problems
Eventually, the primary aim of marketing is to calculate the upcoming market share netsales, or profitability of an offering, thus, allowing a company to:
• Foretell customer buying tendency
• Boost customer retention
• Ascertain trade-off strategies during contract negotiation
• Ascertain competitive pricing
• Predict sales
• Control brand equity
• Devise product elements
• Establish price sensitivity
• Forecast and reduce customer switch rates
• Ascertain best market position for new product introductions.
1.6 Sample Surveys versus Experimental DesignExperiments are usually more structured than sample surveys and include the additionalstep of treating the elements in some way.
Sample Survey Experimental Design
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Select elements from frame. Select experimental units
Take measurements. Allocate treatments.
Take measurements.
1.7 The Parallels between Experimental Designs &Sample Surveys
Sample Survey Experimental Design
Random selection is the method used to
choose units from the population for the
sample.
Randomisation is used to assign treatments
to experimental units.
The sampling error can be minimized
by stratification.
Method of blocking/local control is
common to reduce error.
Partial grouping is useful in cluster
sampling.
Partial grouping is useful in split-plot
designs.
For analysis regression techniques are
useful.
For analysis ANOVA (analysis of
variance) and ANCOVA (analysis of
covariance) are useful.
1.8 Study Design in Medical Research
(Taken from Dawson, B. & Trapp, R.G. (2004): Basic & Clinical Biostatistics, p.7)
Study designs in medicine fall into two categories: studies in which subjects are observed
(observational), and studies in which the effect on an intervention in observed
(experimental).
Classification of Study Designs
With a little practice, the classification of study designs outlined below would help us to
read medical articles and classify studies with little difficulty.
1. Observational Studies
a. Descriptive or case-series
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b. Case-control studies (retrospective studies)
i. Case and incidence of disease
ii. Identification of risk factors
c. Cross-sectional studies, surveys (prevalence)
i. Disease description
ii. Diagnosis and stagingiii. Disease processes, mechanisms
d. Cohort studies (prospective studies)
i. Causes and incidence of disease
ii. Natural history, prognosis
iii. Identification of risk factors
e. Historical cohort studies
2. Experimental studies
a. Controlled trials
i. Parallel or concurrent controls
1. Randomised
2. Not randomised
ii. Sequential controls
1. Self-controlled
2. Crossover
iii. External controls (including historical)
b. Studies with no controls
3. Meta-analysis.
1.9 Guidelines for Designing Experiments
(Taken from Montgomery, D.C. (2005): Design and Analysis of Experiments).
To use the statistical approach in designing and analysing an experiment, it is necessary
for everyone involved in the experiment to have a clear idea in advance of exactly what is
to be studied, how the data are to be collected, and at least a qualitative understanding of
how these data are to be analysed. An outline of the recommended procedure byMontgomery (2005) is as below:
Step 1: Recognition of and statement of the problem
Step 2: Selection of the response variable*
Step 3: Choice of factors, levels and ranges*
Step 4: Choice of experimental design
Step 5: Performing the experiment
Pre-experimental planning.
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Step 6: Statistical analysis of the data
Step 7: Conclusions and recommendations.
*In practice, steps 2 & 3 are often done simultaneously or in reverse order.
Step 1: The first step for designing an experiment is to develop all ideas about the
objectives of the experiment. It is usually helpful to prepare a list of specific problems or
questions that are to be addressed by the experiment. A clear statement of the problem
often contributes substantially to better understanding of the phenomenon being studied
and the final solution of the problem. It is also important to keep an overall objective in
mind; for example, is this a new process or system-in which case the initial objective is
likely to be characterization or factor screening-or is it a mature or reasonably well-
understood system that has been previously characterized-in which case the objective may
be optimization.
Step 2: In selecting the response variable, the experimenter should be certain that this
variable really provides useful information about the process under study. Most often, the
average or standard deviation (or both) of the measured characteristic will be the response
variable. Multiple responses are not unusual. It is usually critically important to identify
issues related to defining the responses of interest and how they are to be measured before
conducting the experiment.
Step 3: When considering the factors that may influence the performance of a process or
system, the experimenter usually discovers that these factors can be classified as either
potential design factors or nuisance factors. The potential design factors are those factors
that the experimenter may wish to vary in the experiment. Nuisance factors are often
classified as controllable, uncontrollable, or noise factors. Once the experimenter has
selected the design factors, he or she must choose the ranges over which these factors will
be varied and the specific levels at which runs will be made. We reiterate how crucial it is
to bring out all points of view and process information in steps 1 through 3. We refer to
this as pre-experimental planning.
Step 4: If the above pre-experimental planning activities are done correctly, this step is
relatively easy. Choice of design involves consideration of sample size (number of
replicates), selection of a suitable run order for the experimental trials, and determination
of whether or not blocking or other randomisation restrictions are involved. In Topic 2 we
discusses some of the important types of experimental designs for a wide variety of
problems. In selecting design, it is important to keep the experimental objectives in mind.
Step 5: When running the experiment, it is vital to monitor the process carefully to ensure
that everything is being done according to plan. Errors in experimental procedure at this
stage will usually destroy experimental validity. Up-front planning is crucial to success. It
is easy to underestimate the logistical and planning aspects of running a designed
experiment in a complex manufacturing or research and development environment. Thisstep suggests re-visiting the decisions made in steps 1-4, if necessary.
Step 6: Statistical methods should be used to analyse the data so that results and
conclusions are objective rather than judgemental in nature. If the experiment has been
designed correctly and performed according to the design, the statistical methods required
are not elaborate. Remember that statistical methods cannot prove that a factor (or
factors) has a particular effect. They only provide guidelines as to the reliability and
validity of results.
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Step 7: Once the data have been analysed, the experimenter must draw practical
conclusions about the results and recommend a course of action. Graphical methods are
often used in this stage, particularly in presenting the results to others. Follow-up runs and
confirmation testing should also be performed to validate the conclusions from the
experiment.
1.10 Research questions and hypothesesThe development of a research question from a research idea is largely a matter of
organising one’s thoughts into a concise statement of what one intends to do and why.
Research questions and hypotheses are closely related but are not quite the same. A
hypothesis is a statement, at a higher level, in which an attempt is made to generalise
about the nature of the universe in which we live.
Research begins with a question. Such questions may come about talking with friends,
reading the scientific literature, or through an untold number of ways. When reading the
current literature as a means to inform your research, you will need to ask three questions:
1. ‘Is my idea based solidly in theory?’; 2. ‘Is this idea the next most obvious step for the
discipline to take?’; and 3. ‘Is my idea novel in some way?’ Having satisfied yourself
that your idea is worth pursuing it is necessary to turn it into a specific research question.
In doing so, you will have to tease out various parts of your idea, making each a more
focused question. Through this process there is the genesis of experimental/research
hypotheses.
There is an art to devising good experimental/research hypotheses. As a general rule there
should be one hypothesis per experiment. Put another way, each experiment should have
only one question to answer. As to how we state an experimental/research hypothesis, it is
more or less convention to treat it as a proposition of only one sentence. Begin with the
word ‘That….’. Within the hypothesis include the general sort of manipulation you will
be performing, known as the independent variable, and what it is you will be measuring,
now referred to as the dependent variable. However, an excellent hypothesis goes one
step further by suggestion how specific treatments known as the levels of the independent
variable, will affect the dependent variable.
Research design and analysis is a method of thought. It begins with a good idea that is
then refined into an experimental/research hypothesis but does not conclude until the
experiment is completed and the results published. At its heart is an experimental design
that limits error and thus promotes a simple and honest analysis of the data.
(Taken from Edwards, T. 2008).
Example: A suitable research question might be:
“Does drug treatment of hypertension reduce the morbidity associated with cardiovasculardisease?”
A suitable hypothesis for the above research question might be:
“Participants with hypertension who are treated with a specific drug will experience less
morbidity associated with cardiovascular disease than participants who were not.”
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Revision Exercises
1. Explain the differences between sample survey and experimental design.
2. A pharmaceutical company wishes to test a new medication it thinks will reducecholesterol. A group of 20 volunteers is formed and each person has their cholesterol
level measured. Half the group is randomly assigned to take the new drug and the other
half is given a placebo. After 6 months the volunteers’ cholesterol is measured again and
any change from the beginning of the study is recorded. In this experiment, identify the
experimental unit, factors, treatments, and response variable.
3. An agricultural researcher is interested in determining how much water andfertilizer are optimal for growing a certain plant. Twenty four plots of land are available
to grow the plant. The researcher will apply three different amounts of fertilizer (low,
medium, and high) and two different amounts of water (light and heavy). These will be
applied at random in equal combination to each of four plots. After 6 weeks, the plants’heights in each plot will be recorded.
Identify the experimental units, factors and their levels, treatments (treatment
combinations), and response variable in this study.
4. In 1930, it was decided to carry out an experiment in Lanarkshire schools to assess the
possible beneficial effects of giving the children free milk during the school day. Twenty
thousand children took part and over the course of five months, February to June, half of
them had three-quarters of a pint of either raw or pasteurised milk while the remainder
did not have milk. All the children were weighed and had their heights measured before
and after the experiment, but contrary to expectation the average increase for the children
who had not had milk exceeded that for the children who had milk. This unexpected
result was later attributed to unconscious bias in the formation of the groups being
compared. In each school the division of the children into a "milk" or a "no-milk" group
was made either by ballot or by using an alphabetic system, but if the outcome appeared
to give groups with an undue preponderance of well-nourished or ill-nourished children,
some arbitrary interchange was carried out in an effort to balance them. In this
interchange the teachers must have unconsciously tended to put a preponderance of ill-
nourished children into the group receiving milk. The results of the experiment were
further complicated by the fact that the children were weighed in their clothes and this
probably introduced a differential effect as between winter and summer and children from
poorer and wealthier homes. Because of the deficiencies in design the results of theexperiment were ambiguous despite the very large sample of children concerned.
(a) Suggest an appropriate research hypothesis.
(b) What is the independent/predictor variable?
(c) What is the dependent/outcome variable?
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(d) Is it an observational study or a designed experiment?
5. From the shelf of a fresh juice shop, all the bottles of a certain brand of orange
juice on the shelf on a particular day were taken and analysed to observe the
vitamin C in orange juice. There were 21 bottles, and their vitamin C readings
(mg/100gm) were as follows:15, 21, 20, 21, 18, 17, 15, 17, 13, 22, 23, 16,
13, 19, 23, 20, 25, 14, 26, 22, 23.
(a) Is it an observational study or a designed experiment?
(b) Is the random variable discrete or continuous?
(c) What parameters are we likely to be interested in estimating?
(d) What null hypothesis might be taken?
6. A researcher conducted an experiment to examine the efficiency of three types of
fungal sprays (T1, T2, & T3) in controlling fungal rots on blueberries. Threeadjacent rows of blueberries are available, each with 24 plants. Sprays can be
applied to individual blueberry plants. The outcome/response variable is the
proportion of blueberries with rot. For the following two designs, specify the
experimental unit, blocking factor, and number of replications of the treatments.
(a) The sprays are randomly allocated to rows and 8 blueberry plantsrandomly selected from each row for assessment.
(b) Each row is divided into 3 plots of 8 plants each. The sprays arerandomly allocated to plots within each row.
7. A new drug was given to a group of 20 patients who suffered hay fever. Of these,
15 reported that the remedy was very helpful in treating their hay fever. From theinformation we can conclude
(A) The new drug is effective for the treatment of hay fever;
(B) Sample size is too small to make a decision;
(C) This result is not valid because there was no control group forcomparison.
8. Why is randomisation important?
9. Suppose a toy company wants to know if certain colors are more appealing and
attractive to toddlers than others. They decide to measure this by choosing fivecolors of blocks and making sets of blocks in each of the five colors. Then they
found 30 toddlers to participate in the study, and they randomly assigned each
toddler a block color. They observed each toddler separately at the same time of
the day, and gave them no other toys to play with. They recorded the length of
time each toddler played with the blocks, to see if some colors of blocks were
played with longer than other colors. All toddlers in the experiment were the same
age (2 years old) and an equal number of girls and boys played with each color of
blocks.
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(A) What is the explanatory variable (IV) and what is the response variable(DV)?
(B) Is this study an observational study or an experiment?
(C) Name one confounding variable that was controlled for in this study.
(D) Give two reasons why we must sometimes use an observational studyinstead of an experiment.
10. A common mistake made by the media, the general public, and some researchers,is to think that a link between two variables in any study implies that one variable
causes the other. Explain what is wrong with this automatic conclusion.
11. How can a researcher try to address the problem of confounding variables whendesigning an observational study?
12. Explain why each of the following is used in experiments:
a) Placebo treatmentsb) Blindingc) Control groups.
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Solution to revision exercises
1. The sample survey focuses on the selection of individuals from the population. We
discover the effect of applying a stimulus to subjects from experiments. The experimental
design focuses on the formation of comparison groups that allow conclusions about the
effect of the stimulus to be drawn.
2. The experimental units (subjects) in this study are the 20 volunteers. There is onefactor, the medication and it has two levels, the active pill and the placebo. There are two
treatments; the active pill and the placebo. The response variable is the change in
cholesterol over the period of the study.
3. The experimental units in this study are plots of land. There are two factors, fertilizer
and water. Fertilizer has three levels: low, medium, and high. Water has two levels: light
and heavy. There are a total of six treatments of fertilizer-water combinations: low-light,
low-heavy, medium-light, medium-heavy, high-light, high-heavy. The response variable
is the height of the plants at the end of the study.
4. (a) Average height will increase for the children who had milk as compared to thechildren who had not had milk.
(b) Milk
(c) Height
(d) Designed experiment.
5. (a) Observational study.
(b) Continuous random variable.
(c) Example: mean vitamin C concentration in all bottles of that brand of orange juice
stocked at the fresh juice shop over a period of time.(d) Example: mean vitamin C equal to 20 mg/100gm.
6. (a) Experimental unit = row,
No Blocking factor,
Number of replications is one.
Lay out of the design:
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Single replication
(b) Experimental unit = plot of 8 plants,
Blocking factor is row,
Number of replication is 3.
-
Row-2
-
Plant
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Lay out of the design:
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 replications
7. (C) This result is not valid because there was no control group for comparison.
8. Why randomisation?
The basic benefits of randomisation include
i. Elimination of selection bias.
ii. Formation of basis for statistical tests, a basis for an assumption-free statistical test of
the equality of treatments.
In general, a randomised trial is an essential tool for testing the efficacy of the treatment.
9. (A) Explanatory variable is Block Colour and response variable is Playing Time.
(B) An experiment.
(C) Any of the following: age, time of the day, other toys, interaction with other childrenetc.
(D) 1) It is unethical or impossible in certain situations to assign people to receive a
specific treatment (such as smoking); 2) certain explanatory variables, such as left vs.
right handedness, are inherent traits and cannot be randomly assigned.
10. If the link is based on an observational study, there is simply no way to rule out all
potential confounding factors, so cause and effect cannot be established.
11. Measure all the potential confounding variables he/she can think of and include them
in the ANALYSIS to see whether they are related to the response variable; or use a case-
control study and choose the controls to be as similar as possible to the cases.
12. a) The power of suggestion may lead to changes in the participants, and those changes
would be mistakenly attributed to the treatment or drug.
b) Participants are kept blind so they don't alter their behavior or outcome to please the
experimenter. Those collecting the measurements are kept blind so they don't
inadvertently bias the measurements in the desired direction.
c) Control groups are used to compare the effect of the treatment with what would have
happened under similar circumstances without the treatment.
Row-1
Row-2
Row-3
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References
Box G.E.P., Hunter W.G. & Hunter J.S. (2005). Statistics for Experimenters: Design,
Innovation and Discovery. 2nd
edition. New York: Wiley.
Cox D.R. (1958). Panning of Experiments. New York: Wiley.
Das M.N. & Giri N.C. (1986). Design and Analysis of Experiments. 2nd
edition. New
Delhi: Wiley Eastern Ltd.Dawson B. & Trapp R.G. (2004). Basic and Clinical Biostatistics. New York: McGraw-
Hill.
Edwards T. (2008). Research Designs and Statistics. New York: McGraw-Hill.
Gupta S.C and Kapoor V.K. (1984). Applied Statistics. Sultan Chand & Sons, New Delhi.
Hinkelmann, K. and Kempthorne, O. (2008). Design and Analysis of Experiments, John
Wiley & Sons, Inc.
Jones B. & Kenward M.G. (2003). Design and Analysis of Crossover Trials. 2nd
edition.
London: Chapman & Hall.
Montgomery D.C. (2005). Design and Analysis of Experiments. 6th
edition. New York:
Wiley.
Petersen, R.G. (1985). Design and Analysis of Experiments. New York: Marcel Dekker,INC.
Utts J.M. (2005). Seeing Through Statistics. Third Edition. Brooks/Cole Cengage
Learning, CA, USA.
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Topic 2: Common Designs
Dr Amirul Islam
Acknowledged to: Dr Jahar Bhowmik
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Contents
2.1 Topic introduction 3
2.2 Topic learning objectives 3
2.3 Completely Randomised Designs 3
2.4 Randomised Block Designs 8
2.5 Latin Square Designs 14
2.6 Factorial Experiments 17
2.7 Nested Designs 19
2.8 Repeated Measures Design 20
Revision Exercises 21
Solutions to Revision Exercises 22
References 23
Note: Some of the materials are adapted from standard texts and guides (see references).
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2.1 Topic introduction
In the previous chapter we have explored the fundamental principles of good
experimental design. In this chapter we apply these principles to some of the basic
designs that are commonly used in practice. These are: (i) Completely Randomised
Designs (CRD), (ii) Randomised Block Designs (RBD) and (iii) Latin Square Designs(LSD). These designs are described below one by one. We also consider the analysis of
data from these basic designs. In practice most experimental data are continuous, so we
will try to restrict our attention to continuous response (outcome) variable.
2.2 Topic learning objectives
Learning objectives
When you have worked through this topic you should:
• Recognise the designs commonly used in practice.
• Understand the principles of basic designs.
• Understand which design would be useful for a particular researchproject.
2.3 Completely Randomised Designs (CRD)
The completely randomised design is the simplest of all the designs, based on principles
of randomisation and replication. In this design treatments are allocated at random to the
experimental units over the entire experimental material and each treatment is repeated an
equal number of times. This design is very flexible in that any number of treatments and
any number of replications may be used. A completely randomised design is one in which
all experimental units are assigned treatments solely by chance. No grouping of
experimental units is done prior to assignment of treatments. In general, an equal number
of replications for each treatment should be made except in particular cases when some
treatments are of greater interest than others or when practical limitations dictate
otherwise.
In this design treatments are assigned to the experimental units completely at random.
There are a variety of ways that this is done in practice, usually using computer programs
are easy but all have the feature that each observation has an equal chance of being
allocated to each group. Suppose we want to conduct an experiment with four treatments,
each replicated five times. This will require 20 experimental units, which we number
from 1 to 20 as in Figure 2.1 below. We can now assign different experimental units to
various treatments many ways. For example, we are explaining two methods. Method 1 is
not used much practically but the Method 2 is always used.
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Method 1:
1. Obtain 20 identical pieces of paper (this is the experimental unit). Label five ofthem “Treatment A”, five of them “Treatment B”, five of them “Tret C” and five
of them “Tret D”.
2. Place the pieces of paper in a box and mix thoroughly.
3. Pick a piece of paper at random. The treatment named on this piece is assigned toexperimental unit 1.
4. Without returning the first piece of paper to the box, select another piece. Thetreatment named on this piece is assigned to experimental unit 2.
5. Continue this way until all 20 pieces of paper have been drawn.
6. This is just an example, the allocation will vary according to what you getrandomly.
Treatment Experimental Unit Total 5 units in eachtreatment
Treatment A 1 2 3 4 5 5
Treatment B 6 7 8 9 10 5
Treatment C 11 12 13 14 15 5
Treatment D 16 17 18 19 20 5
Figure 2.1: Assignment of numbers to experimental units.
Method 2: Using EXCEL
1. Put the numbers 1 to 20 in column A.
2. Enter the formula =RAND ( ) in cell B1 and fill down to B20. It will generate 20random number with 4-5 decimal places. You can make it one decimal places or it
does not matter if you keep it. You have to do exactly
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3. Copy column B onto itself using Paste Special Values
4. Select a cell in either column A or B, and sort the worksheet by Data SortColumn B. (from points 3 and 4, you will find the following)
5. We now have the numbers 1 to 20 in column A in random order.
6. Give the first five to treatment A, and so on, which gives
Treatment Experimental Unit Total 5 units
in each
treatment
Treatment A 2 20 16 13 6 5
Treatment B 10 12 19 4 17 5
Treatment C 3 9 15 5 8 5
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Treatment D 18 14 7 11 1 5
Figure 2.2: Assignment of numbers to experimental units.
2.3.1 Analysis
A completely randomised design provides a one-way classified data according to levels of
a single factor, “treatment”. The data from this design can be analysed by a one-wayanalysis of variance (ANOVA). The ANOVA results help us to answer the following:
How much variation is due to differences between treatments?
How much variation is due to differences within each set of observations for thesame treatment?
It provides solution of the hypotheses to test if there is any difference across thetreatments, i.e., Treatment A vs. Treatment B and so on.
An appropriate linear statistical model for a one-way classified data is
Response = general mean effect (overall mean)+ effect of treatment i + error
ij i ij y e µ α = + + ; i=1,2,…,p & j=1,2,….,r.
Where yij is the yield or response from the jth unit receiving the ith treatment, µ is the
general mean effect, αi is the effect due to the ith treatment, and eij is the error component
due to chance. The error components are assumed to be independently and normally
distributed with 0 mean and constant variance σ2.
The general form of the ANOVA table for a completely randomised design with p
treatments each replicated r times with N (rp) experimental units is given below.
Table 2.1: ANOVA for CRD
Source of
variation (SV)
Degrees of
freedom (df)
Sum of
squares (SS)
Mean square
(MS)
F Statistic
Treatment
Error
Total
p-1
N- p [N-1-p+1,
i.e., total df –
treatment df]]
N-1
SST
SSE
SSTot
MST=SST/( p-1)
MSE=SSE/(N- p)
FT=MST/MSE
SST= Between treatments sum of squares (or between groups sum of squares) which is
the sum of squares of the differences between the treatment means and the overall mean.
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SSE= Residual sum of squares or error sum of squares (or within groups sum of squares)
which is the sum of the squares of the differences between the observations and their
respective treatment means.
SSTot= Total sum of squares which is the sum of the squares of the differences between
the observations and the overall mean. Note that SSTot=SST+SSE.
In this design the total variation is partitioned into two components:
(a) Variation among treatment means (treatments).
(b) Variation among units within treatments (error).
Example 2.1
The following table shows some of the results of an experiment on the effect of
applications of sulphur [S3, S6, S12] in reducing scale disease of potatoes. The object in
applying sulphur is to increase the acidity of the soil since scale does not thrive in very
acid soil. In addition to untreated plots which serve as controls [O]- 3 [F3, F6, F12]
amounts of dressing are compared-300, 600 and 1200 lb. per acre. Both a spring and fall
application of each treatment was tested so that in all there were seven distinct treatments.
Field plan and scale indices for a completely randomized experiment on potatoes
F3
9
O
12
S6
18
F12
10
S6
24
S12
17
S3
30
F6
16
O
10
S3
7
F12
4
F6
10
S3
21
O
24
O
29
S6
12
F3
9
S12
7
S6
18
O
30
F6
18
S12
16
F3
16
F12
4
S3 9
O18
S12 17
S6 19
O32
F12 5
O26
F3 4
Results grouped by treatments for data analysis
Totals
Means
O F3 S3 F6 S6 F12 S12
12
10
24
29
30
18
32
26
9
9
16
4
30
7
21
9
16
10
18
18
18
24
12
19
10
4
4
5
17
7
16
17
181 38 67 62 73 23 57
22.6 9.5 16.8 15.5 18.2 5.8 14.2
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Figure 2.1: Mean (acidity level) plots for Sulphur (S3, S6 and S12:treatment) and
controls (O, F3, F6 and F12)
The figure shows the highest mean acidity level for control “O” but in general application
of sulphur increased the acidity level, especially the mean deferences were higher in case
of S3 vs F3 and S12 vs F12.
Example 2.2 (taken from Petersen R.G. 1985, p.14)
An anthropologist was interested in studying physical differences, if any, among the
various races of people inhabiting Hawaii. As a part of her study she obtained a random
sample of eight 5-year-old girls from each of three races: Caucasian, Japanese, andChinese. She made a number of anthropometric measurements on each girl. She wanted
to determine whether the Oriental races differ from the Caucasian, and whether the
Oriental races differ from each other. The results of the head width measurements (cm)
are given in the following table. The anthropologist is interested in knowing whether or
not head width means differ among the races.
Head width (cm)
Caucasian Japanese Chinese
14.20
14.30
15.00
14.60
14.55
15.15
14.60
14.55
12.85
13.65
13.40
14.20
12.75
13.35
12.50
12.80
14.15
13.90
13.65
13.60
13.20
13.20
14.05
13.80
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Total: 116.95 105.50 109.55
Mean: 14.619 13.188 13.694
Grant mean 13.83
AOVA table of head width
SV d.f. SS MS F
Race (Treatment)
Error
Total
3-1=2
24-3=21
24-1=23
8.43
3.84
12.27
4.21
0.18
23.39
Calculations:
Error sum square calculation
(14.2-14.619)2+……………..+(14.55-14.619)
2+(12.85-
13.188)2+………..+(13.80-13.694)
2= 3.84
Sum square total = (14.2-13.83)2+………………(13.80-13.83)
2=12.27
In this case, (every treatment units –grant total)2
Sum square treatment = SS total – SS error = 12.27-3.84 = 8.43.
In case of CRD, the total variation is due to treatment and error.
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If we do the analysis in SPSS, then the data entry should be like this.
Here, 0 = Caucasian; 1= Japanese; and 2= Chinese
If the data are in SPSS, the analysis will produce the following output.
ANOVA
headwidth
Sum of Squares df Mean Square F Sig.
Between Groups 8.428 2 4.214 23.041 .000
Within Groups 3.841 21 .183
Total 12.268 23
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Explanation of the ANOVA Table:
Between groups degrees of freedom is 2. This is because there are 3 ethnic groups, i.e.,
the number of group minus 1. Eight girls in each ethnic groups, i.e., the total degrees of
freedom equal 3×8 – 1 = 23. The error degree of freedom = 23-2 = 21. Mean sum square
equals sum squares divided by the number of degrees of freedom. F = (4.214/0.183)= 23.04.was supposed to be significant with F(2, 21) degrees of freedom if F was greater than
2.57. Please get this information from the F table (available online from the link).
http://www.socr.ucla.edu/applets.dir/f_table.html.
Descriptives
Headwidth
N Mean Std. Deviation
Caucasian 8 14.6188 .31953
Japanse 8 13.1875 .56553
Chinese 8 13.6938 .35601
Total 24 13.8333 .73035
2.3.2 Advantages of CRD
There are a number of advantages of a completely randomised design:
(i) The design is very flexible. Any number of treatments can be used anddifferent treatments can be used unequal number of times without unduly
complicating the statistical analysis in most of the cases. The number of
replications need not be the same from one treatment to another, although
comparisons are most precise when the treatments are equally replicated.
(ii) The statistical analysis remains simple if some or all the observations forany treatment are rejected or lost or missing for some purely random
accidental reasons. Moreover the loss of information due to missing datais smaller in comparison with any other design.
(iii) The design provides maximum degrees of freedom for the estimation ofthe error variance, which increases the sensitivity or the precision of the
experiment for small experiments, i.e., for experiments with small
number of treatments.
(iv) This design results in the maximum use of the experimental units since allthe experimental material can be used.
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2.3.3 Disadvantages of CRD
There is one principal disadvantage of this design:
(i) If the experimental material is not uniform its precision is low. Since
randomisation is not restricted in any direction to ensure that the units receivingone treatment are similar to those receiving the other treatment, the whole
variation among the experimental units is included in the residual variance. This
makes the design less efficient and results in less sensitivity in detecting
significant effects.
2.3.4 Applications of CRD
Although other designs have more precision, the CRD has a number of uses:
(i) It is most useful in laboratory techniques and methodological studies, e.g.,in physics, chemistry or cookery, in chemical and biological experiments,
in some green house studies, etc., where the experimental material isuniform.
(ii) This design is also recommended in situations where a large fraction ofunits is likely to be destroyed or to fail to respond.
(iii) This design may be useful for experiments in which the total number ofunits is limited.
2.4 Randomised Block Designs (RBD)
The second commonly used design is the randomised block design. If a researcher has to
believe that subgroups of the experimental units will respond differently to treatments
because of some characteristic, the units are sorted into those subgroups before treatments
are assigned. In an experiment these subgroups are called blocks. Once units are assigned
to blocks, treatments are randomly assigned to the units in each block. Blocking is a form
of control to reduce unwanted variability in the response variable due to some variable
other than the treatment (s). In field experimentation, if the whole of the experimental
area is not homogenous (uniform) and the fertility gradient is only in one direction, then a
simple method of controlling the variability of the experimental material consists in
stratifying or grouping the whole area into relatively homogenous strata or sub-groups (or
blocks), perpendicular to the direction of the fertility gradient. Now if the treatments are
applied at random to relatively homogenous units within each strata or block and
replicated over all the blocks, the design is a randomised block design (RBD). In CRD no
such local control measure is adopted except that the experimental units should be
homogenous and treatments allocated at random to the experimental units. But in
randomised block designs treatments are allocated at random within the units of each
stratum or block, i.e. randomisation is restricted. Therefore, homogenous grouping of
experimental units and the random allocation of the treatments separately in each block
are the two main characteristic features of randomised block designs. RBD is the
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improvement of CRD obtained by providing error control measures. The error control
measures in RBD consist of making the units in each of these blocks homogenous.
Layout of RBD: In the RBD the experimental units are first grouped into blocks or
strata. Treatments are then randomly assigned to the units within the blocks. A separate
randomisation is used in each block. To illustrate the procedure, suppose we want to run
an experiment with five treatments (A, B, C, D and E) replicated four times in an
agricultural field with a fertility gradient (see Petersen R.G, 1985, p. 36). We construct a
RBD using the following steps:
BLOCK
I II III IV
Treatment A 1 1 1 1
Treatment B 2 2 2 2
Treatment C 3 3 3 3
Treatment D 4 4 4 4
Treatment E 5 5 5 5GRADIENT
Figure 2.3: Assignment of numbers to units blocked to remove effects of a gradient
Step 1: Form four blocks of five plots each perpendicular to the gradient. Number the
plots from 1 to 5 within each bock as shown in Figure 2.3.
Step 2: Use a table of random numbers or some other procedure (e.g. using EXCEL), to
assign treatments to the units in the first block. To illustrate for
Block I:
Sequence Treatment Random number
(generated in Excel)
Random No
sorted and
ranked the plot
Treatment
according to the
rank, e.g., 1=A,
2=B, 3=C, 4=D,
5=E
1 A 293 (second smallest)=2 2 B
2 B 078 (smallest) =1 1 A
3 C 721 (largest)=5 5 E
4 D 569 (3rd
smallest)=3 3 C
5 E 612 (4th
smallest)=4 4 D
Step 3: Repeat step 2 for the reaming three blocks:
Block II
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Sequence Random number Rank (plot) Treatment
1 962 4 D
2 036 1 A
3 844 3 C
4 963 5 E
5 097 2 B
Block III
Sequence Random number Rank (plot) Treatment
1 675 3 C
2 936 5 E
3 709 4 D
4 591 1 A5 665 2 B
Block IV
Sequence Random number Rank (plot) Treatment
1 230 1 A
2 981 5 E
3 687 4 D
4 604 3 C
5 454 2 B
The final plan of the RBD is given in the following figure.
Block I II III IV
Treatment 1 B 1 D 1 C 1 A
2 A 2 A 2 E 2 E
3 E 3 C 3 D 3 D
4 C 4 E 4 A 4 C
5 D 5 B 5 B 5 B
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Figure 2.4: Final experimental plan with treatments randomly assigned to units within
blocks in a RBD
Example 2.3
Suppose we are interested in how weight gain (Y) in rats is affected by source of protein
(Beef, Cereal, and Pork) and by level of protein (High or Low). There are a total of 6
(3x2) treatment combinations of the two factors (Beef -High Protein, Cereal-High
Protein, Pork-High Protein, Beef -Low Protein, Cereal-Low Protein, and Pork-LowProtein) . Suppose we have available to us a total of N = 66 experimental rats to which
we are going to apply the different diets based on the t = 6 treatment combinations. Prior
to the experimentation the rats were divided into n = 11 homogeneous groups of size 6.
The grouping was based on factors that had previously been ignored (Example - Initial
weight size, appetite size etc.). Within each of the 11 blocks a rat is randomly assigned a
treatment combination (diet). The weight gain (in grams) after six month is measured for
each of the test animals and is tabulated in the following table.
6101 70 98 82 77 79
(1) (2) (3) (4) (5) (6)
Example 2.4
A group of researchers are interested in comparing the effects of four different chemicals
(A, B, C and D) in producing water resistance (y) in textiles. A strip of material,
randomly selected from each bolt, is cut into four pieces (samples) the pieces are
randomly assigned to receive one of the four chemical treatments. This process is
replicated three times producing a Randomised Block (RB) design. Moisture resistance
(y) was measured for each of the samples. (Low readings indicate low moisture
penetration). The data is given below.
Blocks (Bolt Samples)
Completed Design
Block Block
1 107 96 112 83 87 90 7 128 89 104 85 84 89
(1) (2) (3) (4) (5) (6) (1) (2) (3) (4) (5) (6)
2 98 72 101 82 70 94 8 56 70 71 64 62 67
(1) (2) (3) (4) (5) (6) (1) (2) (3) (4) (5) (6)
3 102 76 101 85 95 89 9 99 91 92 80 71 85
(1) (2) (3) (4) (5) (6) (1) (2) (3) (4) (5) (6)
4 97 70 93 65 71 61 10 82 63 87 87 81 61
(1) (2) (3) (4) (5) (6) (1) (2) (3) (4) (5) (6)
5 109 79 101 75 75 81 11 101 102 110 83 93 83
(1) (2) (3) (4) (5) (6) (1) (2) (3) (4) (5) (6)
9.9 C 13.4 D 12.7 B
10.1 A 12.9 B 12.9 D
11.4 B 12.2 A 11.4 C
12.1 D 12.3 C 11.9 A
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Example 2.5
An experiment was carried out on wheat. Three varieties of wheat A, B, C were tested for
their yield in four randomised blocks. Each of four blocks were divided into three plots
and plots of each block were assigned at random to the three varieties. The plan and yield
per plot in kg are given below:
Block 1 Block 2 Block 3 Block 4
A
8
C
10
A
6
B
10
C12
B8
B9
A8
B
10
A
8
C
10
C
9
Wheat yield Block 1 Block 2 Block 3 Block 4
A 8 8 6 8
B 10 8 9 10
C 12 10 10 9
Example 2.6
A researcher is carrying out a study of the effectiveness of four different skin creams forthe treatment of a certain skin disease. He has eighty subjects and plans to assign them
into 4 treatment groups of twenty subjects each. Using a randomised block design, the
subjects are assessed and put in blocks of four according to how severe their skin
condition is; the four most severe cases are the first block, the next four most severe cases
are the second block, and so on to the twentieth block. The four members of each block
are then randomly assigned, one to each of the four treatment groups.
( Example taken from Valerie J. Easton and John H. McColl's Statistics Glossary).
2.4.1 Analysis
If in an RBD a single observation is made on each of the experimental units, then the data
from an RBD can be analysed by a two-way ANOVA. In this design the ANOVA enables
us to partition the total variation into blocks, treatments and error. A randomised block
experiment is assumed to be a two-factor experiment. The factors are blocks and
treatments.
An appropriate linear statistical model for RBD is
Response = general mean effect (overall mean)+ treatment effect + block effect + error
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ij i j ij y b e µ α = + + + ; i=1,2,…,p & j=1,2,….,r.
Where yij is the yield or response of experimental unit from ith treatment and jth block, µ
is the general mean effect, αi is the effect due to the ith treatment, b j is the effect due to jth
block or replicate and eij is the error component due to chance. The error components are
assumed to be independently and normally distributed with 0 mean and constant variance
σ2
.
The general form of the ANOVA table with p treatments each replicated r times in a
randomised block design with r blocks of p units each, is given below.
Table 2.2: ANOVA for RBD
Source of
variation (SV)
Degrees of
freedom (df)
Sum of
squares (SS)
Mean square (MS) F Statistic
Treatment
Block
Error
Total
p-1
r-1
( p-1)(r -1)
rp—1
SST
SSB
SSE
SSTot
MST=SST/( p-1)
MSB=SSB/(r-1)
MSE=SSE/( p-1)(r -1)
FT=MST/MSE
Analysis output using SPSS from example 2.5 (four blocks and 3 varieties of wheat)
Analysis summary
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Between-Subjects Factors
Value Label N
BLOCK 1 3
2 3
3 3
4 3TREATNUM 1 A 4
2 B 4
3 C 4
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Tests of Between-Subjects Effects
Dependent Variable: YEILD
Source
Type III Sum of
Squares df Mean Square F Sig.
Corrected Model 26.000a 11 2.364 . .
Intercept 972.000 1 972.000 . .
BLOCK 4.667 3 1.556 . .
TREATNUM 15.500 2 7.750 . .
BLOCK *
TREATNUM5.833 6 .972 . .
Error .000 0 .
Total 998.000 12
Corrected Total 26.000 11
a. R Squared = 1.000 (Adjusted R Squared = .)
Interpretation of the Table
There are four blocks, so theoretically df is expected to be 4-1 = 3; similarly for
variety/treatment, 2 and for interaction (4-1)× (3-1) = 6 or simply 3×2=6. Since there is
no error df, no F value was able to compute.
2.4.2 Advantages of RBD
There are a number of advantages of a randomised block design. Chief advantages of
RBD can be outlined as follows:
(i) This design is more efficient or accurate than CRD for most types of experimentalworks. Blocking can increase precision by removing one source of variation from
experimental error.
(ii) In this design no restrictions are placed on the number of treatments or the numberof replicates. Any number of blocks and any number of treatments can be used so long
as each treatment is replicated the same number of times in each block. However, for
better management of the experiment, it is suitable not to use a large number of
treatments.
(iii) Statistical analysis is relatively simple and rapid.
2.4.3 Disadvantages of RBD
There are a few disadvantages of RBD:
(i) RBD is not suitable for large number of treatments. The efficiency of the designdecreases as the number of treatments and, hence, block size increases.
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(ii) In the analysis, missing data can cause some difficulty.
2.4.4 Applications of RBD
This design has a number of applications:
(i) RBD provides unbiased estimates of the means for blocking categories, providingadditional information from the experiment.
(ii) This design can remove one source of variation from the experimental error andthus increase precision.
2.5 Latin Square Designs (LSD)
In RBD the whole of the experimental area is divided into relatively homogenous groups
(blocks) to control one source of variation, and treatments are allocated at random to units
within each block. But in field experimentation, it may happen that an experimental area
(field) exhibits fertility in strips, e.g., cultivation might result in alternative strips of high
and low fertility. RBD will be effective if the blocks happen to be parallel to these stripsand would be extremely inefficient if the blocks are across the strips. Initially the fertility
gradient is seldom known. A useful method of eliminating fertility variations consists of
an experimental layout which will control variation in two perpendicular directions. One
design which controls two sources of variations is called a Latin square design. A Latin
square design incorporates two blocking factors, which are usually represented as rows
and columns.
Layout of LSD: In this design the number of treatments is equal to the number of
replications. To construct a Latin square design for p treatments we require p×p = p2
experimental units. The whole of the experimental area in divided into p2
experimental
units (plots) arranged in a square so that each row as well as each column contains p units
(plots). The p treatments are then allocated at random to these rows and columns in such away that every treatment occurs once and only once in each row and in each column.
Such a layout is known as p×p Latin Square Design (LSD) and is extensively used in
agricultural experiments, e.g. if we are interested in studying the effects of p types of
fertilizers (treatments) on the yield of a certain variety of wheat, it is customary to
conduct the experiment on a square field with p2-plots of equal area and to associate
treatments with different fertilizers and row and column effects with variations in fertility
of soil. A Latin Square Design incorporates two blocking factors, which are usually
represented as rows and columns.
The basic pattern of a Latin square design with p = 5 treatments, A, B, C, D, and E, in a
5x5 square is given below which enables both blocking factors (rows: say treatment and
column: say soil fertility):
Column
Row 1 2 3 4 5
1
2
3
A B C D E
B C D E A
C D E A B
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4
5
D E A B C
E A B C D
Figure 2.4: Basic design for a 5x5 Latin square.
Example 2.7
An experiment was conducted to compare the effectiveness of four types of foodsupplements for increasing the milk yield of dairy cows in a farm. The supplements (A,
B, C and D) were given to four cows, and repeated in four successive time periods while
rotating the cows. Milk yields, in grams/day, are recorded. The cow (1, 2, 3 or 4) was one
blocking factor and the time period (I, II, III or IV) was the other. The plan and yields are
given in the following table:
Cow
Time Period
I II III IV
1 A
882
B
605
C
947
D
772
2 B1078
C705
D712
A756
3 C
702
D
659
A
824
B
644
4 D
690
A
789
B
930
C
762
Example 2.8
An experiment was conducted to compare the effectiveness of five manorial treatments
A, B, C, D and E on the yield of sugarcane (in kg/plot). The following are the results of
the Latin Square experiment.
B
405
A
525
E
463
D
441
C
481
C
325
D
445
B
429
A
513
E
493
E
471
B
492
A
472
C
381
D
410
A
552
C
431
D
425
E
572
B
410
D
430
E
469
C
432
B
467
A
460
Example 2.9 (taken from Petersen R.G, page 57)
A ceramics engineer wanted to test the strength of high-tension insulators made from four
clay mixtures A, B, C, D and a control, E. He made five insulators from each mixture. He
suspected that there was a temperature gradient from front to back and from top to bottom
in his oven. He decided to use a Latin square design with shelves (top to bottom) as rows
and positions on the shelves (front to back) as columns. The insulators were placed in the
oven in the Latin square arrangements. After firing, the strength of each insulator was
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measured. The experimental layout and strength measurements were as shown in the
following table:
Front Back
A
33.8
B
33.7
D
30.4
C
32.7
E
24.4
D35.0
E28.8
B33.5
A26.7
C33.4
C
35.8
D
35.6
A
36.9
E
26.7
B
35.1
E
33.2
A
37.1
C
37.4
B
38.1
D
34.1
B
34.8
C
39.1
E
32.7
D
37.4
A
36.4
2.5.1 Analysis
In Latin square design there are three factors: row, column and treatment. The datacollected from this design can be analysed by a three-way ANOVA.
An appropriate linear statistical model for the ith row, jth column and the pth treatment
is:
Response = general mean effect (overall mean) + row effect + column effect + treatment
effect + error
ijk i j k ijk y r c e µ α = + + + + ; i=j=k=1,2,…,p.
Where yijk is the yield or response of experimental unit from ith row, jth column and kthtreatment, µ is the general mean effect, r i is the effect due to the ith row, c j is the effect
due to jth column and αk is the effect due to k th treatment and eijk is the error component
due to chance. As usual the error components are assumed to be independently and
normally distributed with 0 mean and constant variance σ2.
The general form of the ANOVA table for a Latin square design with p treatments is
presented in the following table.
Table 2.2: ANOVA for RBD
Source of
variation (SV)
Degrees of
freedom (df)
Sum of
squares (SS)
Mean square (MS) F Statistic
Rows
Columns
Treatment
Error
p-1
p-1
p-1
( p-1)( p-2)
SSR
SSC
SST
SSE
MSR=SSR/( p-1)
MSC=SSC/( p-1)
MST=SST/( p-1)
MSE=SSE/( p-1)( p-1)
FR=MSR/MSE
FC=MSC/MSE
FT=MST/MSE
Top
Bottom
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Total P2--1 SSTot
De