This document provides an overview of experimental design and analysis of variance (ANOVA). It describes the basic principles of experimental design including randomization, replication, and error control. It defines key terms like treatments, experimental units, and experimental error. The document discusses different basic experimental designs like completely randomized design (CRD) and randomized block design (RBD). It also covers one-way and two-way ANOVA. Examples are provided to illustrate how to set up a simple CRD experiment and perform a one-way ANOVA to analyze the results. Post-hoc tests for comparing group means are also briefly mentioned.
Basic Concepts of Split-Plot Design,Analysis Of Covariance(ANCOVA)& Response ...Hasnat Israq
This gives the basic description of Analysis of Experiment . This is one of the most important topic in Statistics and also for Mathematics and for Researchers-Scientists .
Randomized complete block design - Dr. Manu Melwin Joy - School of Management...manumelwin
A completely randomized design (CRD) is one where the treatments are assigned completely at random so that each experimental unit has the same chance of receiving any one treatment.
For the CRD, any difference among experimental units receiving the same treatment is considered as experimental error.
Basic Concepts of Split-Plot Design,Analysis Of Covariance(ANCOVA)& Response ...Hasnat Israq
This gives the basic description of Analysis of Experiment . This is one of the most important topic in Statistics and also for Mathematics and for Researchers-Scientists .
Randomized complete block design - Dr. Manu Melwin Joy - School of Management...manumelwin
A completely randomized design (CRD) is one where the treatments are assigned completely at random so that each experimental unit has the same chance of receiving any one treatment.
For the CRD, any difference among experimental units receiving the same treatment is considered as experimental error.
In general, a factorial experiment involves several variables.
One variable is the response variable, which is sometimes called the outcome variable or the dependent variable.
The other variables are called factors.
DESIGN OF EXPERIMENTS (DOE)
DOE is invented by Sir Ronald Fisher in 1920’s and 1930’s.
The following designs of experiments will be usually followed:
Completely randomised design(CRD)
Randomised complete block design(RCBD)
Latin square design(LSD)
Factorial design or experiment
Confounding
Split and strip plot design
FACTORIAL DESIGN
When a several factors are investigated simultaneously in a single experiment such experiments are known as factorial experiments. Though it is not an experimental design, indeed any of the designs may be used for factorial experiments.
For example, the yield of a product depends on the particular type of synthetic substance used and also on the type of chemical used.
ADVANTAGES OF FACTORIAL DESIGN.
Factorial experiments are advantageous to study the combined effect of two or more factors simultaneously and analyze their interrelationships. Such factorial experiments are economic in nature and provide a lot of relevant information about the phenomenon under study. It also increases the efficiency of the experiment.
It is an advantageous because a wide range of factor combination are used. This will give us an idea to predict about what will happen when two or more factors are used in combination.
DISADVANTAGES
It is disadvantageous because the execution of the experiment and the statistical analysis becomes more complex when several treatments combinations or factors are involved simultaneously.
It is also disadvantageous in cases where may not be interested in certain treatment combinations but we are forced to include them in the experiment. This will lead to wastage of time and also the experimental material.
2(square) FACTORIAL EXPERIMENT
A special set of factorial experiment consist of experiments in which all factors have 2 levels such experiments are referred to generally as 2n factorials.
If there are four factors each at two levels the experiment is known as 2x2x2x2 or 24 factorial experiment. On the other hand if there are 2 factors each with 3 levels the experiment is known as 3x3 or 32 factorial experiment. In general if there are n factors each with p levels then it is known as pn factorial experiment.
The calculation of the sum of squares is as follows:
Correction factor (CF) = (𝐺𝑇)2/𝑛
GT = grand total
n = total no of observations
Total sum of squares = ∑▒〖𝑥2−𝐶𝐹〗
Replication sum of squares (RSS) = ((𝑅1)2+(𝑅2)2+…+(𝑅𝑛)2)/𝑛 - CF
Or
1/𝑛 ∑▒𝑅2−𝐶𝐹
2(Cube) FACTORIAL DESIGN
In this type of design, one independent variable has 2 levels, and the other independent variable has 3 levels.
Estimating the effect:
In a factorial design the main effect of an independent variable is its overall effect averaged across all other independent variable.
Effect of a factor A is the average of the runs where A is at the high level minus the average of the runs
Today’s overwhelming number of techniques applicable to data analysis makes it extremely difficult to define the most beneficial approach while considering all the significant variables.
The analysis of variance has been studied from several approaches, the most common of which uses a linear model that relates the response to the treatments and blocks. Note that the model is linear in parameters but may be nonlinear across factor levels. Interpretation is easy when data is balanced across factors but much deeper understanding is needed for unbalanced data.
Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician Ronald Fisher. ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the t-test beyond two means. In other words, the ANOVA is used to test the difference between two or more means.Analysis of variance (ANOVA) is an analysis tool used in statistics that splits an observed aggregate variability found inside a data set into two parts: systematic factors and random factors. The systematic factors have a statistical influence on the given data set, while the random factors do not. Analysts use the ANOVA test to determine the influence that independent variables have on the dependent variable in a regression study.
Sir Ronald Fisher pioneered the development of ANOVA for analyzing results of agricultural experiments.1 Today, ANOVA is included in almost every statistical package, which makes it accessible to investigators in all experimental sciences. It is easy to input a data set and run a simple ANOVA, but it is challenging to choose the appropriate ANOVA for different experimental designs, to examine whether data adhere to the modeling assumptions, and to interpret the results correctly. The purpose of this report, together with the next 2 articles in the Statistical Primer for Cardiovascular Research series, is to enhance understanding of ANVOA and to promote its successful use in experimental cardiovascular research. My colleagues and I attempt to accomplish those goals through examples and explanation, while keeping within reason the burden of notation, technical jargon, and mathematical equations.
A Health & Wellness Approach to Enterprise Process ManagementBrett Champlin
Don’t just fix the problems; fix the process (treat the disease, not just the symptoms). Don’t just fix the process; fix the business (treat the patient, not just the disease). Not being “sick” or in pain doesn’t mean you are healthy or will stay healthy without a wellness plan. The same thing applies to our business processes and we need to have a “wellness plan” that monitors our processes to maintain a healthy business.
In general, a factorial experiment involves several variables.
One variable is the response variable, which is sometimes called the outcome variable or the dependent variable.
The other variables are called factors.
DESIGN OF EXPERIMENTS (DOE)
DOE is invented by Sir Ronald Fisher in 1920’s and 1930’s.
The following designs of experiments will be usually followed:
Completely randomised design(CRD)
Randomised complete block design(RCBD)
Latin square design(LSD)
Factorial design or experiment
Confounding
Split and strip plot design
FACTORIAL DESIGN
When a several factors are investigated simultaneously in a single experiment such experiments are known as factorial experiments. Though it is not an experimental design, indeed any of the designs may be used for factorial experiments.
For example, the yield of a product depends on the particular type of synthetic substance used and also on the type of chemical used.
ADVANTAGES OF FACTORIAL DESIGN.
Factorial experiments are advantageous to study the combined effect of two or more factors simultaneously and analyze their interrelationships. Such factorial experiments are economic in nature and provide a lot of relevant information about the phenomenon under study. It also increases the efficiency of the experiment.
It is an advantageous because a wide range of factor combination are used. This will give us an idea to predict about what will happen when two or more factors are used in combination.
DISADVANTAGES
It is disadvantageous because the execution of the experiment and the statistical analysis becomes more complex when several treatments combinations or factors are involved simultaneously.
It is also disadvantageous in cases where may not be interested in certain treatment combinations but we are forced to include them in the experiment. This will lead to wastage of time and also the experimental material.
2(square) FACTORIAL EXPERIMENT
A special set of factorial experiment consist of experiments in which all factors have 2 levels such experiments are referred to generally as 2n factorials.
If there are four factors each at two levels the experiment is known as 2x2x2x2 or 24 factorial experiment. On the other hand if there are 2 factors each with 3 levels the experiment is known as 3x3 or 32 factorial experiment. In general if there are n factors each with p levels then it is known as pn factorial experiment.
The calculation of the sum of squares is as follows:
Correction factor (CF) = (𝐺𝑇)2/𝑛
GT = grand total
n = total no of observations
Total sum of squares = ∑▒〖𝑥2−𝐶𝐹〗
Replication sum of squares (RSS) = ((𝑅1)2+(𝑅2)2+…+(𝑅𝑛)2)/𝑛 - CF
Or
1/𝑛 ∑▒𝑅2−𝐶𝐹
2(Cube) FACTORIAL DESIGN
In this type of design, one independent variable has 2 levels, and the other independent variable has 3 levels.
Estimating the effect:
In a factorial design the main effect of an independent variable is its overall effect averaged across all other independent variable.
Effect of a factor A is the average of the runs where A is at the high level minus the average of the runs
Today’s overwhelming number of techniques applicable to data analysis makes it extremely difficult to define the most beneficial approach while considering all the significant variables.
The analysis of variance has been studied from several approaches, the most common of which uses a linear model that relates the response to the treatments and blocks. Note that the model is linear in parameters but may be nonlinear across factor levels. Interpretation is easy when data is balanced across factors but much deeper understanding is needed for unbalanced data.
Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician Ronald Fisher. ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the t-test beyond two means. In other words, the ANOVA is used to test the difference between two or more means.Analysis of variance (ANOVA) is an analysis tool used in statistics that splits an observed aggregate variability found inside a data set into two parts: systematic factors and random factors. The systematic factors have a statistical influence on the given data set, while the random factors do not. Analysts use the ANOVA test to determine the influence that independent variables have on the dependent variable in a regression study.
Sir Ronald Fisher pioneered the development of ANOVA for analyzing results of agricultural experiments.1 Today, ANOVA is included in almost every statistical package, which makes it accessible to investigators in all experimental sciences. It is easy to input a data set and run a simple ANOVA, but it is challenging to choose the appropriate ANOVA for different experimental designs, to examine whether data adhere to the modeling assumptions, and to interpret the results correctly. The purpose of this report, together with the next 2 articles in the Statistical Primer for Cardiovascular Research series, is to enhance understanding of ANVOA and to promote its successful use in experimental cardiovascular research. My colleagues and I attempt to accomplish those goals through examples and explanation, while keeping within reason the burden of notation, technical jargon, and mathematical equations.
A Health & Wellness Approach to Enterprise Process ManagementBrett Champlin
Don’t just fix the problems; fix the process (treat the disease, not just the symptoms). Don’t just fix the process; fix the business (treat the patient, not just the disease). Not being “sick” or in pain doesn’t mean you are healthy or will stay healthy without a wellness plan. The same thing applies to our business processes and we need to have a “wellness plan” that monitors our processes to maintain a healthy business.
Chapter 9 Fundamentals of Hypothesis Testing: One-Sample Tests
Chapter Topic:
Hypothesis Testing Methodology
Z Test for the Mean ( Known)
p-Value Approach to Hypothesis Testing
Connection to Confidence Interval Estimation
One-Tail Tests
t Test for the Mean ( Unknown)
Z Test for the Proportion
Potential Hypothesis-Testing Pitfalls and Ethical Issues
Chapter 8 Confidence Interval Estimation
Estimation Process
Point Estimates
Interval Estimates
Confidence Interval Estimation for the Mean ( Known )
Confidence Interval Estimation for the Mean ( Unknown )
Confidence Interval Estimation for the Proportion
Researchers use several tools and procedures for analyzing quantitative data obtained from different types of experimental designs. Different designs call for different methods of analysis. This presentation focuses on:
T-test
Analysis of variance (F-test), and
Chi-square test
OBJECTIVES:
Recognize the differences between categorical data and continuous data
Discuss assumptions of chi square distribution
Correctly interpret and use the terms:
chi-square test of independence,
contingency table
degrees of freedom,
“2x2” and “r x c” table.
Calculate expected numbers of the cells of a contingency table .
Calculate chi-square test statistic and its appropriate degrees of freedom.
Refer the chi-square table to obtain tabulated value.
Categorical variables take on values that are names or labels, such as ethnicity (e.g., Sindhi, Punjabi, Balochi etc.) and methods of teaching (e.g. lecture, discussion, activity based etc.)
Quantitative variables are numerical. They represent a measurable quantity. For example, the number of students taking Biostatistics Supplementary classes .
CHI-SQUARE TEST:
It is used to determine whether there is a significant association between the two categorical variables from a single population.
CHI-SQUARE DISTRIBUTION PROPERTIES:
As the degrees of freedom increases, the chi-square
curve approaches a normal distribution
It has many shapes which are based on its degree of freedom (df)
Distribution is skewed to the right
A chi-square distribution takes positive values only.
Commonly used approaches are:
Test for independence
Test of homogeneity
CHI-SQUARE TEST OF INDEPENDENCE:
A chi-square test of independence is used when we want to see if there is a relationship/association between two categorical variables.
EXAMPLES OF RELATIONSHIPS
BETWEEN QUALITATIVE VARIABLES:
Qualitative variables are either ordinal or nominal.
Examples:
Do the nurses feel differently about a new postoperative procedure than doctors?
Preference (Old/New) Subjects (Nurses/ Doctors)
Is there any relationship between Soya Use & Lung cancer?
Soya Intake (yes/no) Lung cancer (yes/no)
Is there any relationship between parent’s and their children Children’s Education (Illiterate/Up to Intermediate/Graduate)
education?
Parent’s Education (Illiterate/Up to Intermediate/Graduate)
CONTINGENCY TABLE:
The table which classifies categories of the qualitative
variable.
The number of individuals or items assigned to each category is called the frequency.
WHAT INFORMATION DOES CONTINGENCY TABLE REVEAL?
When we consider two categorical variables at a time, then an observation will belong to a particular category of variable one as well as a particular category of variable two. This type of table is referred as contingency table.
The simplest form of contingency table is a 2x2 contingency table with both
variables having exactly two categories.
WHAT OTHER INFORMATION DOES
CONTINGENCY TABLE REVEAL?
In this table Two independent categorical variables that
form a “r x c” contingency table, where “r” is the number of rows (number of categories in first variable e.g. helmet used at the time of accident or not?) and “c” is the number of columns (number of categories in the second variable e.g. got severe brain injury.
Basic Concepts of Experimental Design & Standard Design ( Statistics )Hasnat Israq
This gives the basic description of Design and Analysis of Experiment . This is one of the most important topic in Statistics and also for Mathematics and for Researchers-Scientists
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
5. Experimental Design
Experimental design is a set of rules used to choose
samples from populations.
The rules are defined by the researcher himself, and
should be determined in advance.
In controlled experiments, the experimental design
describes how to assign treatments to experimental units,
but within the frame of the design must be an element of
randomness of treatment assignment.
6. Experimental Design..
Treatme
nts
(populat
ion)
Size of
samples
Experime
ntal units
Sample
units
(observati
ons)
Replicatio
n
Experime
ntal error
It is necessary to define
7. Principles of Experimental Design
According to Prof R
A Fisher, the basic
principles of
Experimental
Design are
1.Randomization 2. Replication 3. Error control
Unbiased allocation of
treatments to different
experimental plot
Repetition of the
treatments to more
than one experimental
plot
Measure for
reducing the error
variance
8. The error includes all types of
extraneous variations which are due to
a) Inherent variability in the experimental
material to which the treatments are applied
b) The lack of uniformity in the methodology of
conducting experiment
c) Lack of representativeness of the sample to the
population under study
9. What is Treatment?
Different procedures
under comparison in
an experiment is
called treatment
Example
• Different varieties
of crop
• Different diets
• Different breeds of
animals
• Different dose of
drug/fertilizer
Effec
ts of
treat
ment
s are
comp
ared
in
expt
11. Types of Analysis of Variance
One way
Data are classified
into groups according
to just one
Lciaftee egoxprieccatla vnacryia ibnl e3
different races in
Malaysia
Here categorical
variable: Races
Level: L1 (Malay), L2
(Chinese), L3 (Indian)
Two way, Three way……..
Data are classified into two or more
categorical variables
CGPA of students of 4 different programmes of
FIAT in different academic years. Two-way..
1. Programmes (4)
2. Academic years (4) ; Design 4x4
Example
Example
12. Designs commonly used in Agricultural /
Biological Science
i) One-way design/single factor
design (no interaction effect)
❑ Fixed effects
❑ Random effects
ii) Factorial design/multifactor
design (interaction effect betn
treatments)
❑ Fixed effects
❑ Interaction effect
❑ Random effects
Both can be fitted into any basic design of experiment ie in CRD, RBD
or LSD
13. Some important definitions
Treatments : Whose effect is to be determined. For
example
i)You are to study difference in lactation milk yield
in different breeds of cows. ….. Treatment is
breed of cows. Breed 1, Breed 2… are levels
(1,2,..)
ii) You intend to see the effect of 3 different diets
on the performance of broilers. ….. Treatment
is diet and diet1, diet2 and diet3 are levels (1,2,3)
iii) You wish to compare the effect of different
seasons on the yield of rubber latex. Season is
treatment and season1, season2 are the levels
14. …..definitions
Experimental units: Experimental material to
which we apply the treatments and on which we
make observations. In the previous two examples
cow and broilers are the experimental materials
and each individual is an experimental unit.
Experimental error: The uncontrolled variations in
the experiment is called experimental error. In
each observation of example(i) there are some
extraneous sources of variation (SV) other than
breed of cow in milk yield. If there is no
uncontrolled SV then all cows in a breed would
give same amount of milk (!!!).
15. …..definitions
Replication (r): Repeated application of
treatment under investigation is known as
replication. In the example (i) no. of cows
under each breed (treatment) constitutes
replication.
Randomization: Independence
(unbiasedness) in drawing sample.
Precision (P): The reciprocal of the variance
of the treatment mean is termed as
P r precision. =
σ 2
16. 1. Completely Randomized Design
(CRD): Fixed Effects One-way
• CRD is the simplest type of experimental
design. Treatments are assigned
completely at random to the
experimental units, with the exception
that the number of experimental units for
each treatment may set by the
researcher.
17. are
describ
ed
with
depend
ent
variabl
e, and
the
way of
groupin
1. Completely Randomized Design
(CRD): Fixed Effects One-way ANOVA
1. Testing
hypothesis to
examine
differences
between two or
2. Each
treatm
ent
group
repres
ents a
popula
tion.
more
categorical
treatment groups.
Milk
yield
Feed
18. Designing a simple CRD
experiment
For example, an agricultural scientists wants to
study the effect of 4 different fertilizers
(A,B,C,D) on corn productivity. 4 replicates of
the 4 treatments are assigned at random to the
16 experimental units
!
➢Treatment : Types of fertilizer (A,B,C,D)
➢Experimental unit : Corn tree
➢Dependent variable : Production of corn
19. Steps
1
• Label the experimental units with number 1 to 16
2
• Find 16 three digit random number from random number table
3
• Rank the random number from smallest to largest
4
• Allocate Treatment A to the first 4 experimental units, treatment B to the next 4
experimental units and so on.
20. Random
Number
Ranking
(experimental unit)
Treatment
104 4 A
223 5 A
241 6 A
421 9 A
375 8 B
779 12 B
995 16 B
963 15 B
895 14 C
854 13 C
289 7 C
635 11 C
094 2 D
103 3 D
071 1 D
510 10 D
21. • The following table shows the plan of
experiment with the treatments have
been allocated to experimental units
according to CRD
!
!
experimental
unit number
!
!!
Treatment
A 4 5 6 9
B 8 12 16 15
C 14 13 7 11
D 2 3 1 10
22. Fixed effects one-way ANOVA..
In applying a CRD or when groups indicate a
natural way of classification, the objectives
can be
1. Estimating the mean
2. Testing the difference between
groups
23. Fixed effects one-way ANOVA..
Model
!
ij i ij Y = μ +T + e
Where
Yij = Observation of ith treatment in jth replication
= Overall mean
μ
Ti = the fixed effect of treatment i (denotes an unknown
parameter)
eij = random error with mean ‘0’ and variance σ 2
‘ ‘
!
The factor or treatment influences the value of observation
24. Designing ANOVA Table
• Suppose we have a treatment or different level of a single factor.
The observed response from each of the “a” treatments is a
random variable, as shown in the table:
Treatment
(level)
Observations Totals Mean
1 y y … y y
2 y y … y y
.
.
.
.
a y y … y y
1. y
2. y
a. y
y.. y..
25. Cont..
Source SS df MS F
Between
SSTrt a-1
treatment
Error
(within trt)
SSE N-a
Total SST N-1
❖ a= level of treatment
❖ N= number of population
❖ SS = Sum of Squares
❖ SST = Sum of Square Total
= the sample variance of the y’s
❖ SSE = Sum of Square Error
❖ SST = SSTrt + SSE
= (total variability between treatment)
+ total variability within treatment)
MS SSA
−1
MSE SSE
F MSTRT =
If the calculated value of
F with (a-1) and (N-a) df
is greater than the
tabulated value of F with
same df at 100α % level
of significance, then the
hypothesis may be
rejected
=
a
TRT
−1
=
a
MSE
26. Cont..
y y
SSTrt i
SST y y ij
2 ..
= ΣΣ −
2
N
= Σ −
N
n
2
.. 2.
1
SSE = SST – SSTrt
27. Fixed effects one-way ANOVA..
Problem 1:
An expt. was conducted to investigate the
effects of 3 different rations on post
weaning daily gains (g) on beef calf. The
diets are denoted with T1, T2, and T3. Data,
sums and means are presented in the
following table.
28. Fixed effect one-way ANOVA..: Post
weaning daily gains (g)
T T T
270 290 290
300 250 340
280 280 330
280 290 300
270 280 300
Total
1400 1390 1560 4350
n 5 5 5 15
280 278 312 290
Yi
y
29. One-way ANOVA: Hypothesis
Null hypothesis
!
Ho: There is no significant
difference between the
effect of different rations
on the daily gains in beef
calves ie Effects of all
treatments are same.
Alternative hypothesis
!
H1: There is significant
difference between the
effect of different rations
on the daily gains in beef
calves ie Effect of all
treatments are not same.
Ho: μ = μ = μHa : μ ≠ μ ≠
μ1 2 3 1 2 3
30. Level of significance or confidence interval
Commonly used level of significances (in biology/
agric)
α=0.05
• True in 95% cases
• p<0.05
α=0.01
• True in 99% cases
• p<0.01
p< 0.05, conf. interval = 95% ; p< 0.01, conf. interval =
99%
31. One-way ANOVA…
!
1. SST =
( ... ... ) (4350)
270 300 300
y
= 1268700 – 1261500 = 7200
!
2. SSTr=
( ) 2 2 2 2 2
1400 1390 1560 (4350)
y
Σ
i Σ y
N
− = + + −
5 5 5 15
..
2
i i n
!
3. SSE = SST – SSTr
15
= 7200-3640 = 3560
2
2 2 2
2
2 ..
ΣΣ − = + + − N
i j
ij y
1265140 1261500 3640
= − =
32. ANOVA for Problem 1.
Source SS df MS F
Treatment 3640 3-1=2 1820 6.13
Error
3560 15-3=12 296.67
(residual)
Total 7200 15-1=14
The critical value of F for 2 and 12 df at α = 0.05 level of
significance is
F0.05 (2,12 )= 3.89. Since the calculated F (6.13) > tabulated F or
critical value of F(3.89), Ho is rejected. It means the experiments
concludes that there is significant difference ANOVA (p<is 0.05) significant
between
the effect of different rations (at least in two) on calves’ daily
gain.
!
Now the question of difference between Difference any two betn means any two will means be solved ?????
by
MULTIPLE COMPARISON TEST(S).
33. Multiple Comparison among Group Means
(Mean separation) or Post hoc tests
There are many
post hoc tests
such as
• Least significant
difference (LSD)
34. Multiple comparison: Least Significant
Difference(LSD) test
LSD compares treatment means to see whether
the difference of the observed means of
treatment pairs exceeds the LSD numerically.
LSD is calculated by
!
t 2
MSE !
α / 2, N −
a
n
!
where is the value of Student’s t (2-tail)with
error df t α / at 2 100 α
% level of significance, n is the
no. of replication of the treatment. For unequal
replications, n1 and n2 LSD=
( 1 1 )
/ 2, t MSE r r N a × + α −
1 2
35.
36. Multiple comparison: Tukey’s test
Compares treatment means to see whether
the difference of the observed means of
treatment pairs exceeds the Tukey’s
numerically. Tukey’s is calculated by
T α =
q (a, f ) MSE
!
α
n
Where f is df error .
39. Based on problem 1
Using Tukey’s test, the mean comparison as
follows (which treatment means are differ).
40.
41. Random Effects One-way ANOVA: Difference between fixed and random effect
Fixed effect Random effect
Small number (finite)of groups or
treatment
Large number (even infinite) of
groups or treatments
Group represent distinct
populations each with its own mean
The groups investigated are a
random sample drawn from a single
population of groups
Variability between groups is not
explained by some distribution
Effect of a particular group is a
random variable with some
probability or density distribution.
Example: Records of milk production
in cows from 5 lactation order viz.
Lac 1, Lac 2, Lac 3, Lac 4, Lac 5.
Example: Records of first lactation
milk production of cows constituting
a very large population.
42. Advantages of One-Popular way analysis(CRD)
design for
its
simplicity
,
flexibility
and
validity
Can be
applied
with
moderate
Any
number
number
of
of
treatmen
ts and
treatmen
ts (<10)
any
number
of
replicatio
ns can be
Analysis
is straight
forward
even one
or more
observati
ons are
missing
43. A practical example of one-way ANOVA
Problem: Adjusted weaning weight (kg) of
lambs from 3 different breeds of sheep are
furnished below. Carry out analysis for i)
descriptive Statistics ii) breed difference.
Suffolk: 12.10, 10.50, 11.20, 12.00, 13.20,
10.90,10.00
Dorset: 11.50, 12.80, 13.00, 11.20, 12.70
Rambuillet: 14.20, 13.90, 12.60, 13.60,
15.10, 14.70, 13.90, 14.50
44. Analysis by using SPSS 14
Descriptive Statistics
N minimum maximum mean Std. dev
Suff 7 10.00 13.20 11.4143 1.09153
Dors 5 11.20 13.00 12.2400 .82644
Ramb 8 12.60 15.10 14.0625 .76520
Valid N
5
(list wise)
Mean is expressed as : X ± SD
45. ANOVA (F test)
a) One-Way ANOVA
Sum of
squares
df Means
Squares
F Sig.
Between
groups
27.473 2 13.736 16.705 .000
Within groups 13.979 17 .822
Total 41.452 19
Since the significance level of F is far below than 0.01 so breed
effect is highly significant (p<0.01)
46. Mean Separation
Post hoc tests
Homogenous subsets
Wean
Duncan
3 N Subset for alpha
=0.05
Suff 7 11.414
Dors 5 12.240
Ramb 8
14.063
Sig. .121
1.000