Expermental design by gemechu fufa arfasa pptx 2021 wolaita sodo university

PART II
Experimental design
Chapter 1 (one)
Research and Sampling Design
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8/23/2021 Gemechu Fufa M.Sc in NRM

What is Research Design?
Refers to the plan, structure, and strategy of research
The blueprint that will guide the research process.
Is arrangement of conditions for data collection and analysis in a
manner that aims to combine relevance to the research purpose
Quality of research design is tested by researches conclusion
validity.
Conclusion validity refers to extent of researcher's ability to draw
accurate conclusions from the research.
degree of internal ad external validity
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Gemechu Fufa M.Sc in NRM

Cont…
Research design decisions can be in respect of:
 What is the study about?
 Why is the study being made?
 Where will the study be carried out?
 What type of data is required?
 Where can the required data be found?
 What periods of time will the study include?
 What will be the sample design?
 What techniques of data collection will be used?
 How will the data be analyzed?
 In what style will the report be prepared?
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Cont…
Generally, research design has the following parts:
The sampling design deals with the method of selecting items to be observed for
the given study
The observational design which relates to the conditions under which the
observations are to be made;(controlled or uncontrolled)
The statistical design concerns with the question of how many items are to be
observed and how the information and data gathered are to be analyzed; and
The operational design deals with the techniques by which the procedures
specified in the sampling, statistical and observational designs can be carried out.
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Cont…
Important features of a research design:
1. It is a plan that specifies the sources and types of information relevant to the
research problem.
2. It is a strategy specifying which approach will be used for gathering and
analyzing the data.
3. It also includes the time and cost budgets since most studies are done under these
two constraints.
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1.2. Sampling
Sampling:- is process of obtaining information about an entire population by
examining only a part of it.
A sample should be true representative of the population so that the inferences
derived from sample can be generalized back to the population of interest
In social science/survey based research is generally about inferring patterns of
behaviors/phenomena within specific populations
Improper and biased sampling is the primary reason for divergent and erroneous
inferences
The purpose of sampling is to draw inferences about a population of interest.
E.g. the average height of trees in a forest.
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Cont…
 Sampling Process
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Cont…
 Sampling process comprises of several stage;
1. Defining the target population- define unit of analysis
(person/groups of peoples, scope of watershed, forest types, land use
types..) with the characteristics that one wishes to study.
2. Choose a sampling frame- list accessible the target population
from where a sample can be drawn. “sample frame that is not
representative of target population leads to wrong sample &
conclusion”
3. Choosing a sample from the sampling frame using a well defined
sampling technique
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Cont…
 The quality of research designs can be defined in terms of four key
design attributes:
1. Internal validity
2. External validity
3. Construct validity and
4. Statistical conclusion validity
Validity;- refers to the credibility of experimental results and the
degree to which the results can be applied to the general population of
interest.
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Cont…
1. Internal validity
Known as causality
Examines observed change in dependent variable is really caused by a
corresponding change in hypothesized independent variable, and not by variables
extraneous to the research context.
Internal validity Requires three conditions:
1. Co variation of cause and effect - if cause happens, then effect also happens; if
cause does not happen, effect does not happen),
2. Temporal precedence: cause must precede effect in time,
3. No reasonable alternative explanation (or false correlation).
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Cont…
 Laboratory experiments- strong in internal validity by virtue of
their ability to manipulate the independent variable via a treatment
and observe the effect (dependent variable) of that treatment after a
certain point in time, while controlling for the effects of extraneous
variables.
Field surveys - poor in internal validity because of their inability to
manipulate the independent variable (cause), and cause and effect are
measured at the same point in time.
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Cont….
2. External validity
Known as generalizability
Examines whether the observed associations can be generalized
from sample to population (population validity), or to other people,
organizations, contexts, or time (ecological validity).
• Field experiments have higher degrees of internal and external
validities
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Cont….
3. Construct validity
• examines how well a given measurement scale is measuring the
theoretical concept/ hypothesis that it is expected to measure
4. Statistical conclusion validity
• examines the extent to which conclusions derived using a statistical
procedure is valid
• examines whether the right statistical method was used for
hypotheses testing, whether the variables used meet the assumptions
of that statistical test
(e.g., sample size or distributional requirements),
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1.2.1. Sample Design
Is a definite plan for obtaining a sample from a given population,
The procedure the researcher would adopt in selecting items for the
sample,
determined before data are collected
Researcher must pay attention to the following point when
developing a sampling design;
• Type of universe: finite or in finite?
• Sampling unit : e.g., geographical area- regions, zones,
districts, village, households
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Cont…
A sampling frame(source list):- from which sample is to be drawn
-Contains the names of all items of a universe
- if source list is not available, researcher has to prepare it.
•Size of sample: is the number of items to be selected from the universe to
constitute a sample
• should neither be excessively large, nor too small,
• should be optimum- fulfills the requirements of efficiency, representativeness,
reliability and flexibility
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Cont…
 While deciding the size of sample, researcher must determine the
desired precision and acceptable confidence level for the estimate
• Parameters of interest: specific population characteristic
• Budgetary constraint: determiners sample size & type
• Sampling procedure: must decide the type of sample to be used.

Criteria of Selecting Sampling procedure
 In this context one must remember that two costs are involved in a
sampling analysis:
• the cost of collecting data
• the cost of an incorrect inference resulting from the data
 Researcher must keep in view the two causes of incorrect
inferences
 systematic bias and
 sampling error
A systematic bias:
•Results from errors in the sampling procedures and it cannot be
reduced or eliminated by increasing the sample size.
•At best the causes responsible for these errors can be detected and
corrected.
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Sampling Error
Population
sampling frame
sample
Sampling error = Frame error + chance error + response error
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Res
pon
se

Characteristics of A Good Sample Design
 Truly representative sample
 Small sampling error
 The context of funds available for the research study
 systematic bias can be controlled in a better way
 The results of the sample study can be applied in general, for the
universe with a reasonable level of confidence.
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1.2.3. Sampling Techniques
Sampling techniques can be grouped as:
1) probability sampling ( random sampling or representative
sampling)
 In which every unit of the population has a equal chance of being
sampled, and the chance can be accurately determined
 Involves random selection at some point
Advantages:- reduce the chance of systematic errors
:- minimize chance of sampling biases
:- produce better representative sample
:- inferences drawn from sample are generalizable to the population.
Draw backs:- the techniques need a lot of efforts
:- A lot of time is consumed
:- they are expensive
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Type of Probability Sampling Techniques
1. Simple random sampling
 each and every element of the population has an equal chance of
being selected in the sample.
 The population must contain a finite number of elements that can
be listed or mapped.
 Every element must be mutually exclusive i.e. able to distinguish
from one another and does not have any overlapping
characteristics.
 The population must be homogenous i.e. every element contains
same kind of characteristics that meets the described criteria of
target population.
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Simple random sampling…
Benefits:- there is no possibility of sampling biases
:- the sample is a good representative of the population
Draw backs:- very costly and time consuming ( participants are
widely spread geographically)
:- lots of efforts especially for large population
:- not possible to get or prepare an exhaustive list of elements
Application of simple random sampling method involves two
stages:
 list of all members of population is prepared ( marked 1- N)
 n items are chosen among a population size of N.
 n item selection can be made;
– lottery method
– random numbers method
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cont…
2. Stratified random sampling
This type of sampling method is used when population is
heterogeneous i.e. every element of population does not matches all
the characteristics of the predefined criteria.
Intends to guarantee sample representativeness of specific
subgroups or strata.
involves dividing population into different subgroups (strata) and
selecting subjects from each strata in a proportionate manner.
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cont…
Benefits :
For a heterogeneous population it produces a representative sample as it captures
the diversity which otherwise is likely to be undermined through simple random or
systematic random sampling.
Draw backs
:- Lot of efforts
:- Costly and time consuming
:- If the criterion characteristic/ variable used for classification is not selected
correctly, the whole research may go in vain.
The basis of your strata is the purpose of studying the population
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cont…
Application of stratified random sampling contains two stages;
– Identification of relevant stratums and ensuring their actual
representation in the population
– Application of random sampling method to choose sufficient
numbers of subjects from each stratum.
 As compared with simple random sampling stratified random
sampling reduces sampling error through stratification processes
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cont…
Stratified sampling has two divisions;
 Proportionate stratified random sampling- technique involves
determining sample size in each stratum in a proportional to entire
population.
 Disproportionate stratified random sampling- numbers of subjects
recruited from each stratum doesn’t proportionate to the total size of
population.
NB: application of proportionate stratified random sampling
generates more accurate data than disproportionate sampling.
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cont…
3. Cluster sampling
 used when the elements of population are spread over a wide
geographical area
 the population is divided in to sub-groups called as clusters on the
basis of their geographical allocation.
involves cluster of participants that represent the population are
identified and included in the sample.
 main aim of cluster sampling is to reduce cost and increasing
efficiency of sampling.
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cont…
Advantages
•Reduce costs, less time and efforts
•The list of elements of the population is not required
•Instead of going place to place over a widely spread area for
randomly selecting elements, you get a group of elements in one
geographical region.
Draw backs
•Some times lead to sampling biases
and systematic errors.
-if clusters are not homogeneous among
them, the final sample may not be
representative of the population.
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cont…
it can applied in integration with multi-stage sampling
Difference between cluster and stratified sampling
– in cluster sampling a cluster is perceived as a sampling unit,
whereas
– in stratified only specific elements of strata are accepted as
sampling unit.
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cont…
4. Systematic Sampling
 Used for homogenous population
Unlike simple random sampling, there is not an equal of being selected.
The elements are selected at a regular interval; may be in terms of time, space or order
Thus, this regularity and uniformity in selection makes the sampling systematic.
Every Nth member of population is selected to be included in the study.
 Requires an approximated frame for a priori but not the full list.
 To apply systematic sampling in your thesis:
– the first sample should be chosen randomly.
– other members of sample group are chosen by employing each Nth subject
among the population. 31
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cont…
 Suppose your topic is: Impact Leadership Style on Employee
Motivation in ABC Company and you have chosen semi-structured
in-depth interview as primary data collection method.
 ABC Company has 200 employees who could be potentially
interviewed but you identified your sample size as 12 .
– You may start from 10
– Nth is 16
– Accordingly, your sample group will comprise : 10,26, 42,
58, 74, 90, 106, 122, 138, 154, 170, 186.
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cont…
 There is an alternative way of applying systematic sampling
method;
– first you identify the sample size and divide the total number of
population to sample size to get sampling fraction (N).
– Then, the value of N is used as a constant (like Nth) difference
between the numbers of sample group members.
Benefits :-ensures the extension of sample to the whole population
:-provides the way to get a random and representative sample in the
situation where prior listing up the element is not possible.
Draw backs
•Very costly and time consuming,
•Lots of efforts especially for a large population
•If the order of the list is biased in some way, systematic error may
occur. 33
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cont…
5. Multi-stage sampling
it is a sampling technique where two or more probability techniques
are combined.
used when the elements of population are spread over a wide
geographical region.
large clusters of population are divided into smaller clusters in
several stages in order to make data collection more manageable.
Implementation of multi-stage sampling method in household
survey in one regional states;
1. random number to districts within the regional states need
to be selected as primary clusters.
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cont…
2. Random number of villages within district need to be selected as
secondary clusters.
3. Finally a number of houses need to be selected as
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cont…
Advantages
•Increases cost and time efficiency
•Useful in overcoming the heterogeneity problem within the clusters
Draw backs
•If the selected clusters do not capture the characteristic diversity of
population, the sample would not be representative of the population
•If the characteristic variable used for making strata (in case of
heterogeneity) at any stage is not appropriately selected depending on
the nature of investigation, the whole research may go in vain.
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1.3.2. Non-probability sampling techniques
Non-probability sampling (judgment or non-random sampling)
Every unit of population does not get an equal chance of
participation in the investigation.
Advantages:
•Less effort
•Less time to finish up
•Not much costly
Disadvantages:
•Prone to encounter with systematic errors and sampling biases.
•Cannot be claimed to be a good representative of the population
•Inferences drawn from sample are not generalizable to the
population
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Type of Non-probability sampling
1. Quota sampling
Used when population is heterogeneous i.e., every element of population does not
matches all the characteristics of the predefined criteria.
Subgroups are formed that are homogenous i.e., all the elements within a group
contains same kind of characteristics (keep in mind, those characteristics
are to be taken into account that defines the target population)
Common criterions used for quota are gender, age, ethnicity, socioeconomic, etc.
The participants are selected non-randomly from each sub group on the basis of
some fixed quota.
The approach useful for small-scale studies
Advantages:
•The presence of every subgroup of the population in the sample
•It is less time consuming and low in cost than stratified random sampling
Drawback:
•The sample is not representative and thus encounters the problem of
generalizability.
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Quota sampling...
There are two type of quota: proportionate and non-proportionate.
Proportionate quota sampling
The percentage of every subgroup is set on the basis of their actual proportion
present in the population.
Non-Proportionate quota sampling
 the percentage of quota sampling does not go with the proportion of the sub-
group present in the population rather a minimum percentage is set that is to be
included.
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Quota sampling …
What is difference between quota and stratified sampling ?
• In quota sampling, researchers use non-random sampling methods to
gather data from one stratum until the required quota fixed by the
researcher is fulfilled
• The quota is based on the proportion of subclasses in the population.
Application of quota sampling method contain the following
stages:
– Identification of stratums and their proportions as they are
represented in the population.
– Selection of convenience or judgment sampling method to recruit
required number of subjects from each stratum.
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2. Convenience sampling
2. Convenience sampling ( accidental or opportunity sampling)
as the name implies sampling it relies on data collection from
population members who are conveniently available to participate in
study.
Any member of the target population who is available at the
moment is approached.
Advantages:
•It consumes fewer efforts
•It is inexpensive
•It is less time consuming as the sample is quick and easy to apptoach
Draw backs:
•Subjected to sampling biases and systematic errors.
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2. Convenience sampling …
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3. Judgment sampling
…
researcher relies on his or her own judgment when choosing
members of population to participate in the study.
Researchers often believe that they can obtain a representative
sample by using a sound judgment
E.g., TV reporters stopping certain individuals on the street and ask
their opinions about certain conditions (political, economic ...)
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4. Snowball sampling (chain sampling)
The investigator selects a person who matches the criteria of the research
The first participant is now asked to refer the investigator to another person who
meets the same criteria.
Now the second participant approached is asked to refer the researcher to another
one. In this way a chain is made.
sample group are identified a chain referral.
Advantages:
•Useful in approaching the type of population which is not readily available or
present in a very small quantity.
Draw backs:
•Subjected to sampling biases and systematic errors due to network connection.
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Cont…
Patterns of snowball sampling:
– Linear snowball sampling
• formation of sample group starts with one subject and the subject
provides only one referral-
-> provides one new referral- -> continued until the sample group is
fully formed.
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Cont…
_Exponential non-discriminative snowball sampling.
• The first subject recruited to the sample group provides multiple
referrals.
• Each new referral is explored until primary data from sufficient
amount of samples are collected.
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Cont…
_Exponential discriminative snowball sampling.
• Subjects give multiple referrals, however, only one new
subject is recruited among them.
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CHAPTER TWO (2)
EXPERIMENTAL DESIGNS
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Introduction
 There are two keywords in the topic: experiment and design.
 An experiment is the act of conducting a controlled test or investigation.
 In the field of agriculture, an experimental research is conducted to
answer a particular question or solve a particular problem.
 In experimental research, different kinds or levels of a particular factor
or several factors are evaluated.
 The second keyword is design which may mean arrangement.
 In agricultural research, proper design is important because we want to
establish or find the true results without any doubt in mind.
 With improper or wrong design, results may not be convincing or
reliable.

 Research means systematic investigation to establish facts.
 It is systematic because all the activities are planned and executed
based on rules so everything can be repeated.
 The term established facts indicates that research is done to prove
something that has been done before.
 The aim in conducting research is find out if significant differences exist
among the levels or kinds of treatments (or factor combinations).
 This is accomplished by using the technique of ANALYSIS OF
VARIANCE.
 ANOVA uses two basic estimates: mean and deviation from the mean
cont…

 Before going into the computation of mean and deviation, some
notations need to be understood and remembered.
 Given a set of 10 values assigned to a variable X and to variable Y:
X1 = 4 Y1 = 7
X2 = 3 Y2 = 8
X3 = 5 Y3 = 6
X4 = 2 Y4 = 5
X5 = 6 Y5 = 7
X6 = 7 Y6 = 9
X7 = 3 Y7 = 5
X8 = 6 Y8 = 9
X9 = 4 Y9 = 8
X10 =5 Y10 = 6
Cont…

Cont…
1. Population is a group of individuals, objects, or things possessing at least one
common character or trait not possessed by any other individuals, objects or
things.
• Examples are human population, plant population, population of one
country, bacterial population, students enrolled in a particular university,
books in a library, etc.
• In studying statistics, it is not always possible to deal with populations because
of their enormous size.
• For example, if we want to know the average height of the human population,
• it is impossible to measure each and every human being.
• If we want to know the average yield of wheat in a particular country, we
cannot weigh all the wheat harvested in that country.
• In this case, we have to deal with a sample where proper statistical analysis
can be done.

Cont…
2. Sample is a representative group taken at random from a population. When a
sample is properly taken, the statistics from that sample can be applied to the
population. The 10 values given above is an example of a sample.
3. X (or Y) is a symbol used to designate a variable.
 A variable is something that changes in value.
 In contrast, a constant is something that remains the same.
• For example, if there are 10 children in the group, they can be designated X1,
X2, X3, X4, X5, X6, X7, X8, X9, and X10
• whose ages are 4, 3, 5, 2, 6, 7, 3, 6, 4, and 5 years, respectively.
4. Σ is the symbol to designate summation or sum. In the example, the sum of Xi
where i goes from 1 to 10 is:
ΣXi

Cont…
5. N refers to the number of observations in a population.
6. refers to the population mean. It is the average value of all observations in a
population.
7. n refers to the number of observations in a sample. In the example, n = 10.
8. X refers to sample mean. It is the average value of all observations in a sample.
By formula:

Cont…
9. ΣX2 refers to the sum of the squares of individual observation. In the formula,
each value is squared first and then added.
10. (ΣX)2 refers to the square of the sum of individual observations. In the formula,
all the values are added first and the total is squared.
11. ΣXY refers to the sum of the product of X & Y, that is, the combinations of X &
Y are multiplied first before adding.
(ΣX)2

12. ΣXΣY refers to the product of the sum of X and sum of Y, that is, all X’s are
added first as well as all Y’s and the sums are multiplied
13. The sum of individual numbers each multiplied by a constant is equal to the
over-all sum of the numbers multiplied by a constant.
= Σ[(X1 x c) + (X2 x c) + (X3 x c) + (X4 x c) + (X5 x c) + (X6 x c) + (X7 x c) +
(X8 x c) + (X9 x c) + (X10 x c)]
= Σ(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) x c
14. The sum of individual numbers each divided by a constant is equal to the over-
all sum of the numbers divided by a constant.
Cont…

15. Deviation is the difference between an observation and the mean.
Deviation is positive if the observation value is larger than the mean and negative if
it is smaller than the mean.
The sum of the deviation of all observations from the mean is zero.
• Sample Mean Deviation
Cont…

 Since the total deviation is zero, the use of deviation alone is not advisable in
determining how much variation is there between the observations and the mean.
 There is a need to make all the deviations positive in value.
 One of the techniques commonly used is the Method of Least Squares.
 In this method, the deviation value is first squared (hence all values will be
positive) and then added to become Sum of Squares.
Cont…

 When the Sum of Squares is divided by the number of observations, the value
obtained is known as Variance (designated by the symbol σ2 when dealing with
a population and by s2 when mdealing with a sample).
 By definition, Variance is the average of the squared deviation from the mean.
 By formula:
Cont…

 Standard Deviation (designated by the symbol σ when dealing with a
population and by s when dealing with a sample) is the square root of Variance.
It is used as a measure of average deviation from the mean.
In the example, the standard deviation is √(2.40) = 1.55
Cont…

Terms useful in experimental design
1. Hypothesis:-a proposition or an assumption that one attempts to verify through
experimentation or observation.
Two types of hypothesis:- Null Hypothesis and Alternative Hypothesis:
 The null hypothesis refers to the statement that changes in the independent
variable have no effect on the dependent variable and that therefore whatever
difference was found between the experimental and control groups simply
occurred by chance through the influence of random error variables.
Rejecting a true null hypothesis = type one error
Accepting a false null hypothesis = type two error

Cont…
 The alternative hypothesis refers to the statement that changes in the
independent variable really have an effect on the dependent variable and that
the difference in performance between the experimental and control groups
was greater than what would be expected by chance through only the
influence of random error variables.
 Example:
 The null hypothesis is that changes in land cover have no effect on climate
change
 The alternative hypothesis is that changes in land cover have an effect climate
change

Cont…
2. Experiment:
• In an experiment, the experimenter deliberately manipulates one or more
variables (factors) in order to determine the effect of this manipulation on
another variable (or variables).
3. Independent Variable:
• Is the "treatment" variable that the experimenter hypothesizes "has an
effect" on some other variable.
• In an experiment, the independent variable is directly manipulated by the
experimenter.

Cont…
4. Dependent Variable:
 Is the variable that the experimenter hypothesizes is "affected by," or "related to,"
the independent variable.
 It is the "outcome" or "effect" variable, usually a measure of the subjects‘
performance resulting from changes in the independent variable.
5. Error Variable:
 In an experiment, the independent variable is likely not the only variable
potentially responsible for observed changes in the dependent variable.
 Error variables account for all the individual differences in responding not
accounted for specifically by changes in the independent variable.
 Error variables must be controlled, for example by randomization.

Cont…
6. Confounding Variable ( Confound):
 A particular error variable whose possible effects on the dependent variable are
completely consistent with the effects of the independent variable.
 The presence of a confounding variable precludes being able to describe the
changes in the dependent variable exclusively to the independent variable.
 The changes could also be due to the confounding variable.
 Confounding variables must be controlled for, for example, through
randomization or by holding them constant.
7.Control Variable:
 One can prevent the effects of a specific, identifiable error variable from
clouding the results of an experiment by holding this error variable constant.
 For example, if all subjects are the same age, then variations in age cannot act as
an error variable.
 A variable that is thus held constant is called a control variable.

6.3. Basic principles of experimental design
 The basic principles of experimental designs are:
1. Randomization
2. Replication and
3. Local control.
 These principles make a valid test of significance possible. Each of them is
described briefly in the following subsections.
(1) Randomization.
 The first principle of an experimental design is randomization, which is a
random process of assigning treatments to the experimental units.
 The random process implies that every possible allotment of treatments has the
same probability.
 An experimental unit is the smallest division of the experimental material,
and a treatment means an experimental condition whose effect is to be
measured and compared.

Cont…
 The purpose of randomization is to remove bias and other sources of
extraneous variation which are not controllable.
 Another advantage of randomization (accompanied by replication) is that it
forms the basis of any valid statistical test.
 Hence, the treatments must be assigned at random to the experimental units.
 Randomization is usually done by drawing numbered cards from a well-
shuffled pack of cards, by drawing numbered balls from a well-shaken
container or by using tables of random numbers.

(2) Replication.
 The second principle of an experimental design is replication, which is a
repetition of the basic experiment.
 In other words, it is a complete run for all the treatments to be tested in the
experiment.
 In all experiments, some kind of variation is introduced because of the fact that
the experimental units such as individuals or plots of land in agricultural
experiments cannot be physically identical.
 This type of variation can be removed by using a number of experimental units.
 We therefore perform the experiment more than once, i.e., we repeat the basic
experiment.
 An individual repetition is called a replicate.
 The number, the shape and the size of replicates depend upon the nature of the
experimental material.

Cont…
 A replication is used to:
 Secure a more accurate estimate of the experimental error, a term which
represents the differences that would be observed if the same treatments were
applied several times to the same experimental units;
 Decrease the experimental error and thereby increase precision, which is a
measure of the variability of the experimental error; and
 Obtain a more precise estimate of the mean effect of a treatment, since ,
where denotes the number of replications.

(3) Local Control.
 It has been observed that all extraneous sources of variation are not removed by
randomization and replication.
 We need to choose a design in such a manner that all extraneous sources of
variation are brought under control.
 For this purpose, we make use of local control, a term referring to the amount of
balancing, blocking and grouping of the experimental units.
 Balancing means that the treatments should he assigned to the experimental
units in such a way that the result is a balanced arrangement of the treatments.

Cont…
 Blocking means that like experimental units should be collected together to
form a relatively homogeneous group.
 A block is also a replicate.
 The main purpose of the principle of local control is to increase the
efficiency of an experimental design by decreasing the experimental error.
 The point to remember here is that the term local control should not be
confused with the word control.
 The word control in experimental design is used for a treatment which does not
receive any treatment when we need to find out the effectiveness of other
treatments through comparison.

6.4. Overview of some experimental designs
 There are four types of experimental design
1. CRD (Completely random design )
2. RCBD (Randomized Complete Block Design )
3. LSD/Split plots (Latin square design )
4. Factorial experiment

COMPLETELY RANDOM DESIGN (CRD)
 Simplest design to use.
 Design can be used when experimental units are essentially homogeneous.
 Because of the homogeneity requirement, it may be difficult to use this
design for field experiments.
 It is common in laboratory experiment , green house
 The CRD is best suited for experiments with a small number of treatments.
Randomization Procedure
 Treatments are assigned to experimental units completely at random.
 Every experimental unit has the same probability of receiving any treatment.
 Randomization is performed using a random number table, computer program,
etc.

Cont…
Example of Randomization
• -Given you have 4 treatments (A, B, C, and D) and 5 replicates, how many
experimental units would you have?
• Note that there is no “blocking” of experimental units into replicates.
• -Every experimental unit has the same probability of receiving any treatment.

Cont…
Advantages of a CRD
1. Very flexible design (i.e. number of treatments and replicates is only limited by
the available number of experimental units).
2. Statistical analysis is simple compared to other designs. (simple and easy)
3. Loss of information due to missing data is small, compared to other designs due
to the larger number of degrees of freedom for the error source of variation.
Disadvantages
1. If experimental units are not homogeneous and you fail to minimize this variation
using blocking, there may be a loss of precision.
2. Usually the least efficient design unless experimental units are homogeneous.
3. Not suited for a large number of treatments.

ANOVA for Any Number of Treatments with Equal Replication
 Given the following data:

ANOVA for Any Number of Treatments with Unequal
Replication
Given the following data:

RANDOMIZED COMPLETE BLOCK DESIGN (RCBD)
 The RCBD is perhaps the most commonly encountered design that can be
analyzed as a two-way ANOVAs.
 In this design, a set of experimental units is grouped (blocked) in a way that
minimizes the variability among the units within groups (blocks).
 The objective is to keep the experimental error within each block as well as
possible.
 Each block contains a complete set of treatments, therefore differences among
blocks are not due to treatments, and this variability can be estimated as a
separate source of variation.
 The greater the variability among blocks the more efficient the design becomes.
In the absence of appreciable block differences the design is not as efficient as a
completely randomized design (CRD).
 The CRD has more degrees of freedom for error and a smaller F value is
required for significant difference among treatments.

Use of Randomized Complete Block Design
 It is the most frequently used experimental design in field experiments.
 Completely Randomized Design is appropriate when no sources of variation,
other than treatment effects are known or anticipated, i.e. the experimental units
should be similar.
 However, in many experiments certain experimental units if treated alike will
behave differently,
 e.g in field experiments adjacent plots are more alike in response than those some
distant part.
 Heaviest animals in a group of the same age may show a different rate of weight
gain than lighter animals.

Cont…
 In such situations, designs and layouts can be constructed so that the portion of
variability attributable to the known source can be measured and excluded from
experimental error.
 Thus, difference among treatment means will contain no contribution to the known
source.
 Randomized Complete Block Design (RCBD) can be used when the experimental
units can be meaningfully grouped, the number of units in a group being equal to the
number of treatments.
 Such a group is called a block and equals to the number of replications.
 Each treatment appears an equal number of times usually once, in each block and
each block contains all the treatments.
 Blocking (grouping) can be done based on soil heterogeneity in a fertilizer or variety
trials; initial body weight, age, sex, and breed of animals; slope of the field, etc.

 The primary purpose of blocking is to reduce experimental error by eliminating the
contribution of known sources of variation among experimental units. By blocking
variability within each block is minimized and variability among blocks is
maximized.
 During the course of the experiment, all units in a block must be treated as uniformly
as possible.
 For example, if planting, weeding, fertilizer application, harvesting, data recording,
etc, operations cannot be done in one day due to some problems, then all plots in any
one block should be done at the same time. if different individuals have to make
observations of the experimental plots, then one individual should make all the
observations in a block.
 This practice helps to control variation within blocks, and thus variation among
blocks is mathematically removed from experimental error.
Cont…

Advantages & Disadvantages of RCBD
Advantages
1. Precision: More precision is obtained than with CRD because grouping
experimental units into blocks reduces the magnitude of experimental error.
2. Flexibility: Theoretically there is no restriction on the number of treatments or
replications. If extra replication is desired for certain treatments, it can be
applied to two or more units per block.
3. Ease of analysis: The statistical analysis of the data is simple.
If as result of change or misshape, the data from a complete block or for certain
treatments are unusable, the data may be omitted without complicating the
analysis.
If data from individual units (plots) are missing, they can be estimated easily so that
simplicity of calculation is not lost.
Disadvantages:
 The main disadvantage of RCBD is that when the number of treatments is large
(>15), variation among experimental units within a block becomes large,
resulting in a large error term.
 In such situations, other designs such as incomplete block designs should be
used.

Randomization and layout of RCBD
Step 1: Divide the experimental area (unit) into r-equal blocks, where r is the number
of replications, following the blocking technique. Blocking should be done
against the gradient such as slope, soil fertility, etc.
Step 2: Sub-divide the first block into t-equal experimental plots, where t is the
number of treatments and assign t treatments at random to t-plots using any of the
randomization scheme (random numbers or lottery).
Step 3: Repeat step 2 for each of the remaining blocks.
Three treatments each replicated four times

Cont…
 The major difference between CRD and RCBD is that in CRD, randomization
is done without any restriction to all experimental units but in RCBD, all
treatments must appear in each block and different randomization is done for
each block (randomization is done within blocks).
Analysis of variance of RCBD
Step 1. State the model
The linear model for RCBD:
Yij =  + i + j + ij
where, Yij = the observation on the jth block and the ith treatment;
 = common mean effect;
i = effect of treatment i;
j = effect of block j; and
ij = experiment error for treatments i in block j.

Cont…
Step 2. Arrange the data by treatments and blocks and calculate treatment totals (Ti),
Block (rep) totals (Bj) and Grand total (G).
Example: Oil content of linseed treated at different stages of growth with N-
fertilizes. Oil content (g) from sample of 20 g seed
Treatment
(Stages of
application)
Block 1 Block 2 Block 3 Block 4 Treat.
Total (Ti)
Seeding 4.4 5.9 6.0 4.1 20.4
Early blooming 3.3 1.9 4.9 7.1 17.2
Half blooming 4.4 4.0 4.5 3.1 16.0
Full blooming 6.8 6.6 7.0 6.4 26.8
Ripening 6.3 4.9 5.9 7.1 24.2
Unfertilized
(control)
6.4 7.3 7.7 6.7 28.1
Block total (Bj) 31.6 30.6 36.0 34.5
Grand total 132.7

Cont…
From this table we can test:
1. Does fertilizer application have effect on oil content? Comparing treatment 6
versus 1-4.
2. Which stage of application is best in increasing oil content in linseed?
Comparing the difference between 1, 2, 3, 4
Step 3: Compute the correction factor (C.F.) and sum of squares using r as number
of blocks, t as number of treatments, Ti as total of treatment i, and Bj as total of
block j.
a. C.F. =
b. Total Sum of Square (TSS) = Yij
2- C.F. = (4.4)2 + (3.3)2 + …. + (6.7)2 – C.F.
= 788.23 – 733.72 = 54.51
  72
.
733
6
4
7
.
132
2
2



rt
G

Cont…
c. Block Sum of Squares (SSB) =
--733.72 = 736.86 – 733.72 = 3.14
d. Treatment Sum of Squares (SST) =
--733.72 = 765.37 – 733.72 = 31.652
e. Error Sum of Squares (SSE) = Total SS – SSB – SST
= 54.51 – 3.14 – 31.65 = 19.72
Step 4: Compute the mean squares for block, treatment and error by dividing each
sum of squares by its corresponding d.f.
a. Block Mean Square (MSB) =

Cont…
b. Treatment Mean Square (MST) =
c. Error Mean Square (MSE) =
Step 5: Compute the F-value for testing block and treatment differences.
F-block = for block =
F-treatment = for treatment =
MSE
MSB
MSE
MST
80
.
0
31
.
1
05
.
1

83
.
4
31
.
1
33
.
6


Cont….
Step 6: Read table F- and compare the computed F-value with tabulated F-value
and make decision.
 F-table for comparing block effects, use block d. f. as numerator (n1) and error
d.f. as denominator (n2); F (3, 15) at 5% = 3.29 and at 1% = 5.42.
 F-table for comparing treatment effects, use treatment d.f. as numerator and error
d.f. as denominator F (5, 15) at 5% = 2.90 and at 1% = 4.56. If the calculated F-
value is greater than the tabulated F-value for treatments at 1%, it means that
there is a highly significant (real) difference among treatment means.
 In the above example, calculated F-value for treatments (4.83) is greater than the
tabulated F-value at 1% level of significance (4.56). Thus, there is a highly
significant difference among the stages of application on nitrogen content of the
linseed.

Cont…
Step 7: Compute the standard error of the mean and coefficient of variability.
Standard error (SE)
Step 8: Summarize the results of computations in analysis of variance table for quick
assessment of the result.
%
7
.
20
100
53
.
5
31
.
1
100
57
.
0
4
31
.
1









mean
Grand
MSE
CV
g
r
MSE
SE

Source of
variation
Degree of
Freedom
Sum of
Squares
Mean
Squares
Computed
F
Tabulated F
5% 1%
Block (r-1) = 3 3.14 1.05 0.80 3.29 5.42
Treatment (t-1) = 5 31.65 6.33 4.83** 2.90 4.56
Error (r-1) (t-1) =
15
19.72 1.31
Total (rt-1) = 23 54.51
Analysis of variance for oil content of the linseed.

Missing Data in RCBD
 Sometimes data for certain units may be missing or become unusable. For
example,
 When an animal becomes sick or dies but not due to treatment
 When rodents destroy a plot in field
 When a flask breaks in laboratory
 When there is an obvious recording error
 A method is available for estimating such data.
 Note that an estimate of a missing value does not supply additional information to
the experimenter; it only facilitates the analysis of the remaining data.

Cont…
 When a single value is missing in RCBD, an estimate of the missing value can
be calculated as:
 Where:
Y = estimate of the missing value
t = No. of treatments
r = No. of replications or blocks
Bo = total of observed values in block (replication) containing the missing
value
To = total of observed values in treatment containing the missing value
Go = Grand total of all observed values.
)
1
)(
1
( 




t
r
Go
tTo
rBo
Y

Cont…
 The estimated value is entered in the table with the observed values and the
analysis of variance is performed as usual with one d. f. being subtracted from
both total and error d. f. because the estimated value makes no contribution to
the error sum of squares.
 When all of the missing values are on the same block or treatment the simplest
solution is to consider as if the block or treatment had not been included in the
experiment.

Cont…
Varieties
I II III IV
Treatment
total (Ti)
Bukuri 18.5 15.7 16.2 14.1 64.5
Katumani 11.7 ___ 12.9 14.4 39(To)
Rarree-1 15.4 16.6 15.5 20.3 67.8
Al-composite 16.5 18.6 12.7 15.7 63.5
Block total 62.1 50.9 (Bo) 57.3 64.5
Example: In the table given below are yields (kg) of 4-varieties of maize (Al-
composite, Rarree-1, Bukuri, Katumani) in 4-replications planted in RCBD on a
plot size of 10m x 10 m of which one plot yield is missing. Estimate the missing
value and analyze the data
Go (Grand total) = 234.8

Cont…
Solution
a. Estimate the missing value
b. Enter the estimated value and carry out the analysis following the usual
procedure:
 Corrected treatment total = 39 + 13.9 = 52.9
 Corrected block total = 50.9 + 13.9 = 64.8
 Corrected grand total = 234.8 + 13.9 = 248.7
)
1
)(
1
( 




t
r
Go
tTo
rBo
Y
  
9
.
13
1
4
1
4
]
8
.
234
)
39
4
(
)
9
.
50
4
[(








Y

Cont…
c. Analysis of variance
C.F. =
TSS=
Trss =
BSS =
Error SS= Total SS- Treatment SS- Block SS = 79.18 – 31.21 – 9.02 = 38.95
 
 
73
.
3865
4
4
7
.
248
2


rt
        18
.
79
.
.
7
.
15
........
9
.
13
.....
7
.
11
5
.
18
.
.
2
2
4
1
4
1
2
2
2










 
F
C
F
C
Yij
i j
        21
.
31
73
.
3865
4
5
.
63
8
.
67
9
.
52
5
.
64
.
.
2
2
2
2
4
1
2









F
C
r
Ti
i
      02
.
9
73
.
3865
4
)
5
.
64
(
3
.
57
8
.
64
1
.
62
.
.
2
2
2
2
4
1
2









F
C
t
Bj
i

Cont…
d. Compute the correction factor for bias (B) for treatment sum of squares as the
treatment SS is biased upwards.
B=
Bo = Total of observed values in blocks (replication) containing the missing value
Y = estimated value
B= =
e. Subtract the computed B value from Total SS & Treatment SS
Adjusted Treatment SS = Treatment SS – B
= 31.21 – 7.05= 24.16
Adjusted Total SS = Total SS-B
= 79.18 – 7.05 = 72.13
f. Subtract 1 from error d. f. and total d. f. and complete the analysis of variance table.
 
 
 
1
1
2



t
t
y
t
Bo
 
 
 
1
4
4
9
.
13
1
4
9
.
50
2




   05
.
7
12
7
.
41
9
.
50
2



Cont…
Source of
variation
Degree of
freedom
Sum of
squares
Mean
squares
Computed
F cal.
Table F
5% 1%
Block 3 9.02 3.01 0.61ns 4.07 7.59
Treatment 3 24.16 8.05 1.64ns 4.07 7.59
Error (t-1) (r-1)-1 = 8 38.95 4.9
Total rt-1-1= 14 72.13
%
1
.
14
100
)
15
/
8
.
234
(
65
.
15
9
.
4
100 



mean
Grand
MSE
CV=

Latin Square Design (LSD)
6.1. Uses Latin Square Design
 The major feature of the LSD is its capacity to simultaneously handle two known
sources of variation among experimental units unlike Randomized Complete
Block Design (RCBD), which treats only one known source of variation.
 The two directional blocking in a LSD is commonly referred as row blocking
and column blocking.
 In Latin Square Design the number of treatments is equal to the number of
replications that is why it is called Latin Square.

Advantages & Disadvantages of LSD
Advantages:
 you can control variation in two direction
 hopefully you increase effeciency as compared to the RCBD
 Greater precision is obtained than Completely Randomized Design & Randomized Complete
Block Design (RCBD) because it is possible to estimate variation among row blocks as well as
among column blocks and remove them from the experimental error.
Disadvantages:
 As the number of treatments is equal to the number of replications, when the number of
treatments is large the design becomes impractical to handle.
 On the other hand, when the number of treatments is small, the degree of freedom associated
with the experimental error becomes too small for the error to be reliably estimated.
 Thus, in practice the Latin Square Design is applicable for experiments in which the number of
treatments is not less than four and not more than eight.
• Randomization is relatively difficult.
• Expensive

Randomization and layout of LSD
Step 1: To randomize a five treatment Latin Square Design, select a sample of 5 x 5
Latin square plan from appendix of statistical books. We can also create our own
basic plan and the only requirement is that each treatment must appear only once
in each row and column. For our example, the basic plan can be:
Step 2: Randomize the row arrangement of the plan selected in step 1, following one
of the randomization schemes (either using lottery method or table of random
numbers).
A B C D E
B A E C D
C D A E B
D E B A C
E C D B A

Cont…
 Select from table of random numbers, five three digit random numbers avoiding
ties if any
Random numbers: 628 846 475 902 452
Rank: (3) (4) (2) (5) (1)
 Rank the selected random numbers from the lowest (1) to the highest (5)
 Use the ranks to represent the existing row number of the selected plan and the
sequence to represent the row number of the new plan.
 For our example, the third row of the selected plan (rank 3) becomes the first row
(sequence) of the new plan, the fourth becomes the second row, etc.
1 2 3 4 5
3 C D A E B
4 D E B A C
2 B A E C D
5 E C D B A
1 A B C D E

Cont…
Step 3: Randomize the column arrangement using the same procedure. Select five
three digit random numbers.
Random numbers: 792 032 947 293 196
Rank: (4) (1) (5) (3) (2)
 The rank will be used to represent the column number of the above plan (row
arranged) in step 2.
 For our example, the fourth column of the plan obtained in step 2 above
becomes the first column of the final plan, the first column of the plan becomes
2, etc.
E C B A D
A D C B E
C B D E A
B E A D C
D A E C B

Cont…
 Note that each treatment occurs only once in each row and column
 Sample layout of three treatments each replicated three times in Latin Square
Design

Analysis of variance for LSD
 There are four sources of variation in Latin Square Design, two more than that of
CRD and one more than that for the RCBD.
 The sources of variation are row, column, treatment and experimental error.
The linear model for Latin Square Design:
Yijk =  + i + j + k + ijk
where,
Yijk = the observation on the ith treatment, jth row & kth
column
 = Common mean effect
i = Effect of treatment i
j = Effect of row j
k = Effect of column k
ijk = Experiment error (residual) effect

Cont…
Example: Grain yield of three maize hybrids (A, B, and D) and a check variety, C,
from an experiment with Latin Square Design
Grain yield (t/ha)
Row number C1 C2 C3 C4 Row total (R)
R1 1.640(B) 1.210(D) 1.425(C) 1.345(A) 5.620
R2 1.475(C) 1.15(A) 1.400(D) 1.290(B) 5.350
R3 1.670(A) 0.710(C) 1.665(B) 1.180(D) 5.225
R4 1.565(D) 1.290(B) 1.655(A) 0.660(C) 5.170
Column total(C) 6.350 4.395 6.145 4.475
Grand total (G) 21.365
Treatment Total (T) Mean
A 5.855 1.464
B 5.885 1.471
C 4.270 1.068
D 5.355 1.339

STEPS OF ANALYSIS
Step 1: Arrange the raw data according to their row and column designation, with the
corresponding treatments clearly specified for each observation and compute row
total (R), column total (C), the grand total (G) and the treatment totals (T).
Step 2: Compute the C.F. and the various Sum of Squares
a. C.F. =
b. Total SS = y2 – C.F. = --28.53 = 1.41
c. Row SS = 28.53 = 0.03
d. Column SS= = 0.83
e. Treatment SS= = 0.43
f. Error SS =Total SS–Row SS–Column SS–Treatment SS =1.41–0.03–0.83–0.43
= 0.12
  53
.
28
16
365
.
21
2
2
2


t
G
        53
.
28
4
475
.
4
145
.
6
395
.
4
35
.
6
2
2
2
2




        53
.
28
4
355
.
5
270
.
4
885
.
5
855
.
5
2
2
2
2





Cont…
Step 3: Compute the mean squares for each source of variation by dividing the sum
of squares by its corresponding degrees of freedom.
a) Row MS = = = 0.01
b) Column MS = = = 0.276
c) Treat MS =
d) Error MS =
Step 4: Compute the F-value for testing the treatment effect and read table F-value
as:
As the computed F-value (7.15) is higher than the tabulated F-value at 5% level of
significance (4.76), but lower than the tabulated F-value at the 1% level (9.78),
the treatment difference is significant at the 5% level of significance.
143
.
0
3
43
.
0
1



t
SS
Treatment
  
02
.
0
2
3
12
.
0
2
1




 t
t
SS
Error

Cont…
Step 4: Compute the F-value for testing the treatment effect and read table F-value as:
 As the computed F-value (7.15) is higher than the tabulated F-value at 5% level of
significance (4.76), but lower than the tabulated F-value at the 1% level (9.78), the
treatment difference is significant at the 5% level of significance.
Compute the CV as: = =
 Note that although the F-test on the analysis of variance indicates significant
differences among the mean yields of the 4-maize varieties tested, it does not identify
the specific pairs or groups of varieties that differed significantly.
 For example, the F-test is not able to answer the question whether every one of the
three hybrids gave significantly higher yield than that of the check variety.
 To answer these questions, the procedure for mean comparison should be used.
15
.
7
02
.
0
143
.
0




MS
Error
MS
Treatment
calculated
F
100

mean
Grand
MS
Error
%
6
.
10
100
335
.
1
02
.
0



Cont…
Step 5: Summarize the results of the analysis in ANOVA table

Cont…
Source D.F. SS MS Computed F Table F
5% 1%
Row (t-1)=3 0.03 0.01 0.50ns 4.76 9.78
Column (t-1)=3 0.83 0.275 13.75** 4.76 9.78
Treatment (t-1)=3 0.43 0.142 7.15* 4.76 9.78
Error (t-1)(t-2)=6 0.13 0.02
Total (t2-1) =15 1.41
Step 5: Summarize the results of the analysis in ANOVA table

7. Split-Plot Design
8.3.1 Uses of Split-Plot Design
 Split-plot design is frequently used for factorial experiments where the nature of
experimental material makes it difficult to handle all factor combination.
 The principle underlying is that the levels of one factor are assigned at random to
large experimental units.
 The large units are then divided into smaller units and then the levels of the
second factor are assigned at random to small units within large units.
 The large units are called the whole units or main-plots whereas the small units
are called the split-plots or sub-plots (units).
 Thus, each main plot becomes a block for the sub-plot treatments.
 In split-plot design, the main plot factor effects are estimated from larger units,
while the sub-plot factor effects and the interactions of the main-plot and sub-plot
factors are estimated from small units.

Cont…
 As there are two sizes of experimental units, there are two types of experimental
error, one for the main plot factor and the other for the sub-plot factor.
 Generally, the error associated with the sub-plots is smaller than that for the
whole plots due to the fact that error degrees of freedom for the main plot are
usually less than those for the sub-plots.
 In split-plot design, the precision for the measurement of the effect of main plot
factor is sacrificed to improve the precision of the measurement of the sub-plot
factors.

Situations when to use split-plot design?
A. When the level of one or more of the factors require larger amounts of
experimental units than another.
For instance, in field experiments, one of the factors could be method of land
preparation (tractor, oxen, hand) and method of fertilizer application (broad
cast, drill). These factors usually require larger experimental plots (units).
B. When an additional factor is to be incorporated in an experiment to increase its
scope.
For example, if the major purpose of an experiment is to compare the effect of
several vaccines as a protectant against infection from certain disease of
animals, to increase the scope of the experiment, several breeds of animals can
be included which are known to differ in their resistance to disease.

Cont…
 Here, the breeds of animals could be arranged in main units and the vaccines to
the subunits.
C. When greater precision is desired for comparison of certain factors than others.
 Since in a split-plot design, plot size and precision of measurement of the effects
are not the same for both factors, the assignment of a particular factor to either
the main-plot or to the sub-plot is extremely important.

Guidelines to apply factors either to main-plots or sub-plots:
A. Degree of precision required: Factors which require greater degree of
precision should be assigned to the sub-plot.
 For example, animal breeder testing three breeds of dairy cows under different
types of feed stuff, will assign the breeds of animals to sub-units and the feed
stuffs to the main unit.
 On the other hand, animal nutritionist may assign the feeds to the sub-units and
the breeds of animals to the main-units as he is more interested on feed stuffs
than breeds.
B. Relative size of the main effect: If the main effect of one factor (factor A) is
expected to be much larger and easier to detect than factor B, then factor A can
be assigned to the main unit and factor B to the sub-unit.
 For instance, in fertilizer and variety experiments, the researcher may assign
variety to the sub-unit and fertilizer rate to the main-unit, because he expects
fertilizer effect to be much large and easier to detect than the varietal effect.
C. Management practice: The factors, which require smaller amounts of
experimental material, should be assigned to sub-plots.
 For example, in an experiment to evaluate the frequency of irrigation (5, 10, 15
days), on performance of different tree seedlings on nursery, the irrigation
frequency factor could be assigned to the main plot and the different tree
species to the sub-plots to minimize water movement to adjacent plots.

Advantages &Disadvantages Split-Plot Design
Advantages
a) It permits the efficient use of some factors, which require large experimental units in
combination with other factors, which require small experimental units.
b) It provides increased precision in comparison of some of the factors (sub-plot
factors).
c) It promotes the introduction of new treatments into an experiment, which is already
in progress.
Disadvantages:
a) Statistical analysis is complicated because different factors have different error mean
squares.
b) Low precision for the main plot factor can result in large differences being non-
significant, while small differences on the sub-plot factor may be statically
significant even though they are of no practical significance.

8. Factorial Experiments
 Factorial Experiment is an experiment whose design consists of two or more
factors, each with discrete possible values or levels and whose experimental
units take on all possible combinations of these levels across all such factors.
 A factorial treatment arrangement is one in which the effects of a number of
different factors are investigated simultaneously.
 The treatments include all of the possible combinations of levels that can be
formed from the factors being investigated.
 Factor refers to a kind of treatment
 Level refers to several treatments with in any factor.
 Note that factorial is not a design but an arrangement

Cont…
 The simplest factorial experiment contains two levels for each of two factors
 Total number of treatments = lever of factor A * level of factor B
 Suppose if you want to know the amount of (nitrogen and phosphorus) and if
you apply each fertilizer at the rate of 10kg/ha and 20kg/ha then you have 2
levels. There fore the total treatment are 2*2 = 4.
 Another example:- if you want to test the effect of variety and harvesting
method on the quality of tomato you will have two factors (variety and
harvesting methods ). If you have four varities then four variety you will have 4
levels and if you have 3 methods of harvesting then you will have three levels for
harvesting method. Then you will have 4*3 = 12 number of treatments.

Cont…
8.1 Simple Effects, Main Effects and Interaction
 Factorial experiments are experiments in which two or more factors are studied
together.
 Factor is a kind of treatment and in a factorial experiment any factor will supply
several treatments.
 In factorial experiment, the treatments consist of combinations of two or more
factors each at two or more levels.
 Factorial experiment can be done in CRD, RCBD and Latin Square Design as
long as the treatments allow.
 Thus, the term factorial describes specific way in which the treatments are
formed and it does not refer to the experimental design used, e.g. nitrogen &
phosphorus rates:
N = 0, 50, 100, 150 kg/ha
P = 0, 50, 100, 150 kg/ha

Cont…
 E.G. if 5-varieties of sorghum are tested using 3-different row spacing the
experiment is called 5 x 3 factorial experiment with 5 levels of variety factor (A)
and three levels of spacing factor (B).
 An experiment involving 3 factors (variety, N-rate, weeding method) each at 2
levels is referred as 2  2  2 or 23 factors; 3 refers to the number of factors and
2 refers to levels.
 Here we have 8 treatment combinations variety (x, y), N-rate (0, 50 kg/ha),
weeding (with or without weeding).
 If the above 23 factorial experiment is done in RCBD, the correct description of
the experiment will be 23 factorial experiment in RCBD.

Cont…
 There are two types of factorial experiments: these are:
1. Complete factorial experiment and
2. Incomplete factorial experiment.
 Complete factorial experiment:- is when all combination of the treatments is
included in the experiment.
 Incomplete factorial experiment:-is when only a fraction of all combinations
is used in the experiment.

Advantages &Disadvantages of Factorial Experiment
 Advantages of Factorial Experiment
 All expermental units are utilized in evaluating effects, resulting in the most
efficient use of resources
 The range of conditions tested can be increased with a minimum outlay of
resources
 The interaction between factors can be estimated
 A factorial arrangement of treatments is optimum for estimating main effects
and interactions.
 Disadvantages of Factorial Experiment
 The disdvantages of factorial experiments as the number of factors increase
they become costly, complex and they require large size.
 The large size of the replicates decreases the efficiency of the experiment
 Reduction of the size of the replicates by fractional replication or other means
make statistical analysis more complex and can create confounding

Interaction between two factors
 Interaction:- is defined as the failure of one factor to give the same response at
all levels of the second factor.
 Factorial treatment arrangements allow assessment of the interaction between
two or more factors.
 If the effect of one factor change and the effect of another not change there
is no interaction.
 If the effect of one factor change with change in the effect of another factor,
there is interaction.
 When factors interact, the factors are not independent and a single factor
experiment will lead to disconnected or misleading information.
 However, if there is no interaction it is concluded that the factors under
consideration act independently of each other.
 Thus, results from separate single factor experiments are equivalent to those
from a factorial experiment.

Cont…
For example:- we have two factors A &B. A has two levels (a1 and a2) similarly B has
two levels (b1 and b2). Then we have four treatment combination (a1b1, a2b1, a1b2,
a2 b2). Therefore we can study four different simple effects.
The simplest effect of A at b1and b2., the simple effect of B at a1 and a2 ( in other word
we have to see either the effect of A changes with in B or the effect of B changes
with change in A. this is what interaction is).
 Effect of A at b1 = a1b1-a2b1
 Effect of A at b2 = a1b2- a2b2
 Effect of B at a1 = a1b1 – a1b2
 Effect of B at a2 = a2b1 – a2 b2
If there is no interaction between treatments means when the level of one factor changes
it, the effect of another factor will not change. Then effect of A at b1 -effect of A at
b2 = 0 similarly effect of B at a1- effect of B at a2 = 0.

Cont…
 Let’s see the following example: we have two factors variety with two levels and
nitrogen fertilizer with two levels then we have 2x2= 4 treatments. The effect of
each treatment on yield of rice is given below:
 Let’s see the effect of variety at nitrogen fertilizer ( i.e does the response of
nitrogen varies with different varieties ?).
 Simple effect of variety at N1: variety 1N1-variety 2N1= 1.0 - 2.0 = -1
 Simple effect of variety at N2: variety 2N2-variety 2N2 = 3.0 - 4.0 = -1
10kg/ha N(N1) 60kg/ ha (N2) Average
Variety 1 1.0 3.0 2.0
Variety 2 2.0 4.0 3.0
Average 1.5 2.5

Cont…
 Similarly Let’s see if change in nitrogen has an effect on varities or not ( i.e does
the response of verities change with change in nitrogen?).
 Simplest effect of nitrogen at variety1: V 1 N1-V1N 2 = 1-3 = -2
 Simplest effect of nitrogen at variety 2: V 2 N 1-V2 N 2 = 2-4 = -2
 Again the difference is equal, therefore from this we can conclude that changes
in nitrogen levels do not result change in yield of variety. In other word there is
no interaction between treatments.

Cont…
Let’s see another example:
 Let’s see the effect of variety at nitrogen fertilizer (i.e does the response of
nitrogen with verities?).
 Simple effect of variety at N1: variety 1N1-variety 2N1= 2.0 - 1.0 = 1
 Simple effect of variety at N2: variety 2N2-variety 2N2 = 4.0 - 1.0 = 3
 Similarly Let’s see if change in nitrogen has an effect on varities or not ( i.e does
the response of verities change with change in nitrogen?).
 Simplest effect of nitrogen at variety1: V 1 N 1-V1N2 = 2-4 = 2
 Simplest effect of nitrogen at variety 2: V 2N1- V1N 2 = 1-1 = 0
 Since the difference is not equal in both cases, we can conclude that change
in the level of nitrogen affects the response of varities and vice versa 1.
10kg/ha N(N1) 60kg/ ha (N2) Average
Variety 1 2.0 4.0 1.0
Variety 2 1.0 1.0 3.0
Average 1.5 2.5

Cont…
 In factorial experiments, the following points should be noted:
 An interaction effect between two factors can be measured only if the two factors
are tested together in the same experiment.
 When interaction is absent, the simple effect of a factor is the same for all levels
of the other factors and equals to the main effect.
 When interaction is present, the simple effect of a factor changes as the level of
the other factor changes.
 Uses of factorial experiments
 In exploratory experiments, where the aim is to examine a large number of
factors to determine as which ones are important and which are not.
 To study relationships among several factors, to determine the presence and
magnitude of interaction
 In experiments designed to lead to recommend over a wide range of conditions.

Cont…
 Disadvantages of factorial experiments
 As the number of factors increase, the size of experiment becomes very large,
 e.g: with 8 factors each at 2-levels, there are 28, 256 treatment combinations.
Thus, experiments with this many treatments are costly to run.
 Large factorial experiments are difficult to interpret especially when there are
interactions.

10. REGRESSION AND CORRELATION ANALYSIS
 Regression analysis describes the effect of one or more variables (designated as
independent variables) on a single variable (designated as the dependent variable).
 It expresses the dependent variable as a function of independent variable(s).
 For regression analysis, it is important to clearly distinguish between the dependent
and independent variables.
Examples:
 Weight gain in animals depends on feed
 Number of growth rings in a tree depends on age of the tree
 Grain yield of maize depends on a fertilizer rate
In the above cases, weight gain, number of growth rings and grain yield are dependent
variables, while feed, age and fertilizer rates are independent variables.
The independent variable is designated by X and the dependent variable by y.

Cont…
Correlation analysis, It provides a measure of the degree of association between the
variables, e.g. the association between height and weight of students; body weight of
cows and milk production; grain yield of maize and thousand kernel weight.
Regression and correlation analysis can be classified:
a. Based on the number of independent variables as:
 Simple: one independent variable and one dependent variable.
 Multiple: if more than one independent variables and a dependent variable is involved
b. Based on the form of functional relationship as:
 Linear: if the form of underlying relationship is linear
 Non-linear: if the form of the relationship is non-linear

Cont…
 Thus, regression and correlation analysis can be classified into 4:
Simple linear regression and correlation analysis
Multiple linear regression and correlation analysis
Simple non-linear regression and correlation analysis
Multiple non-linear regression and correlation analysis
Reading Assignmnent
Read each type of regression
and correlation analysis

Expermental design by gemechu fufa arfasa pptx 2021 wolaita sodo university

Expermental design by gemechu fufa arfasa pptx 2021 wolaita sodo university

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