!EXPERIMENTAL DESIGN AND ANALYSIS
OF VARIANCE: BASIC DESIGN ☺
Prof. Dr. Md. Ruhul Amin Lecture No: 9-14
What ? Why ? How?
Concept of cause and effect
To determine/
identify
To observe/ measure
EXPERIMENTAL DESIGNThe
preplanned
statistical procedure by which samples
are drawn is called
EXPERIMENTAL DESIGN
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.
Experimental Design..
Trea
tme
nts
(pop
ulat
ion)
Size
of
sam
ples
Expe
rim
ent
al u
nits
Sam
ple
unit
s (o
bser
vati
ons)
Repl
icat
ion
Expe
rim
ent
al e
rror
It is necessary to define
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
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
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
Effe
cts
of
trea
tm
ent
s ar
e co
mp
ared
in
ex
pt
Basic Designs
1. Completely Randomized Design (CRD)
2. Randomized Block design (RBD)
3. Two Factor Factorial Design
Types of Analysis of Variance
One way
Data are classified into groups according
to just one categorical variableLife expectancy in 3 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
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
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
…..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 (!!!).
…..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 precision.
2σ
rP =
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.
1. Completely Randomized Design (CRD): Fixed Effects One-way ANOVA
1. Testing hypothesis to
examine differences
between two or more
categorical treatment groups.
2. Each treatm
ent group represents a population.
ements are
described
with depend
ent variable, and
the way of grouping by an
Milk yield
Feed
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
Steps1 • 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.
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
• 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
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
Fixed effects one-way ANOVA..
Model ! ijiij eTY ++=µ
Where
Yij = Observation of ith treatment in jth replication = Overall mean Ti = the fixed effect of treatment i (denotes an unknown parameter) eij = random error with mean ‘0’ and variance ‘ ‘ !The factor or treatment influences the value of observation
µ
σ 2
• 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:
Designing ANOVA Table
Treatment (level)
Observations Totals Mean
1 y y … y y
2 y y … y y
.
. .
.
a y y … y y
1.y2.y
.ay..y ..y
Cont..Source SS df MS FBetween treatment
SSTrt a-1
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)
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
1−=aSSMS A
TRT
1−=aSSMSE E
MSEMSF TRT=
Cont..
∑∑ −=NyySST ij
2..2
∑ −=Nyy
nSSTrt i
2..2
.1
SSE = SST – SSTrt
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.
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
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.
µµµ 321: ==Ho µµµ 321
: ≠≠Ha
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%
One-way ANOVA…!
1. SST = = 1268700 – 1261500 = 7200 !2. SSTr= !3. SSE = SST – SSTr = 7200-3640 = 3560
15)......(
)4350(300300270
2
2222..2
−++=−∑∑ Ny
i jijy
364012615001265140
15555)4350(156013901400
)( 22222..
2
=−=
−++=−
∑
∑ Nyi
i in
y
ANOVA for Problem 1.Source SS df MS F
Treatment 3640 3-1=2 1820 6.13Error (residual)
3560 15-3=12 296.67
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 (p<0.05) between the effect of different rations (at least in two) on calves’ daily gain. !Now the question of difference between any two means will be solved by MULTIPLE COMPARISON TEST(S).
ANOVA is significant
Difference betn any two means ?????
Multiple Comparison among Group Means (Mean separation) or Post hoc tests
There are many post hoc tests such as • Least significant
difference (LSD) test
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 !!!where is the value of Student’s t (2-tail)with error df at 100 % level of significance, n is the no. of replication of the treatment. For unequal replications, n1 and n2 LSD=
nMSEt aN
2,2/ −α
t 2/α α
)11(21
,2/ rrt MSEaN +×−α
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 !Where f is df error .
nMSEfaT q ),(
αα =
Multiple comparison: Duncan’s multiple range test
Based on problem 1 Using Tukey’s test, the mean comparison as follows (which treatment means are differ).
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.
Advantages of One-way analysis(CRD)Popular design for
its simplicity
, flexibility
and validity
Can be applied
with moderate number
of treatments (<10)
Any number
of treatmen
ts and any
number of
replications can be
Analysis is straight forward even one or more
observations are missing
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
Analysis by using SPSS 14Descriptive 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 (list wise)
5
Mean is expressed as : SDX ±
ANOVA (F test)
a) One-Way ANOVASum 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)
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