!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