What ? Why ? How?

What ? Why ? How?

EXPERIMENTAL DESIGN

The preplanned 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. It is necessary to define

Experimental Design..

Treatme

nts (popula

tion)

Size of samples

Experimental units

Sample units

(observations)

Replication Experimental error

Basic Designs

1. Completely Randomized Design (CRD)2. Randomized Block design (RBD)3. Latin Square Design

CRD is known as “One-way design”

Designs commonly used in Animal Science

i) One-way design (no interaction effect)

a. Fixed effectsb. Random effectsii) Factorial design (interaction effect)

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

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)

…..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: 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.

Randomization, replication and error control are three principles of experimental design.

Fixed Effects One-way ANOVA1. Testing

hypothesis to examine

differences between two or

more categorical

treatment groups.

2. Each treatment

group represents a population.

3. Measurements are described with

dependent variable, and the way of grouping

by an independent variable (factor).

Fixed effects one-way ANOVA

• Consider an experiment with 15 steers and 3 treatments (T1, T2, T3)

• Following scheme describes a CRDSteer No 1 2 3 4 5 6 7 8Treatment T2 T1 T3 T2 T3 T1 T3 T2Steer No 9 10 11 12 13 14 15Treatment T1 T2 T3 T1 T3 T2 T1

NB: One treatment appeared 5 times. Equal no. of replication/treatment – not necessary in one-way ANOVA

Fixed effects one-way ANOVA..

Data sorted by treatment for RANDOMIZATION

Steer Measurement

Steer Measurement

Steer Measurement

2 y11 1 y21 3 y31

6 y12 4 y22 5 y32

9 y13 8 y23 7 y33

12 y14 10 y24 11 y34

15 y15 14 y25 13 y35

T1 T2 T3


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


Model

ijiij etY

WhereYij = Observation of ith treatment in jth replication = Overall meanti = 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

2


Treatment 1 Treatment 2

Look the difference


Problem 1: An expt. was conducted to investigate the effects of 3 different rations on post weaning daily gains (g) in 3 different groups of beef calf. The diets are denoted with T1, T2, and T3. Data, sums and means are presented in the following table.

Fixed effects one-way ANOVA.. T1 T2 T3

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

y

One-way ANOVA: Hypothesis

Null hypothesis

Ho: There is no significant difference between the effect of rations on the daily gains in beef calves ie Effects of all treatments are same.

Alternative hypothesis

Ha: There is significant difference between the effect of rations on the daily gains in beef calves ie Effect of all treatments are not same.

321

: Ho 321: Ha

Level o

f sig

ni

fica

nce

or c

onf id

en

ce i

nterval

Commonly used level of significances

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%

Calculation of different Sum of Squares(SS)

Total SS =

Treatment SS =

Error SS = Total SS – Treatment SS = T0-T= E say

CF stands for correction Factor

N

CFWheresayCFy

Ty ij

i jij

2

0

2,

sayTCFi i

i

ny

,

2

.

One-way ANOVA TableSource of variation

Degrees of freedom (df)

Sum of squares (SS)

Means square (MS)

F

Treatment k-1T’/E’

Error N-k T0 –T = E E’ = E/(N-k)

Total N-1 T0 =

CFTi

i

i

ny

2

)1/(' kTT

CFyij 2

If the calculated value of F with (k-1) and (N-k) df is greater than the tabulated value of F with same df at 100α % level of significance, then the hypothesis may be rejected ie the effects of all the treatments are not same. Otherwise the hypothesis may be accepted. (N=Total no of observation, k=no of treatments)

One-way ANOVA…1. Grand Total (GT) = 2. CF =

3. Total Corrected SS = = 1268700 – 1261500 = 7200

4. Treatment SS =

5. Error SS = Total SS – Treatment SS = 7200-3640 = 3560

4350)300......300270( i j ijy

CFCFi j

ijy )......( 300300270 2222

364012615001265140555

156013901400)( 222

2

CFCFj ij

i in

y

126150015

2)( )4350( 2

Ni j ijy

ANOVA for Problem 1.Source SS df MS F

Treatment 3640 3-1=2 1820 6.13

Error (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 F 0.05 (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 different rations (at least in two) of calves causing daily gain.

Now the question of difference between any two means will be solved by MULTIPLE COMPARISON TEST(S).

Multiple Comparison among Group Means (Mean separation)

There are many tests such as

•Least significant difference (LSD) test•Tukey’s W-test•Newman-Keul’s sequential range test•Duncan’s New Multiple Range Test (DMRT)•Scheffe 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 with error df at 100 % level of significance, s2 is the MS of error and r is the no. of replication of the treatment. For unequal replications, r1 and r2 LSD=

rst 2

t

)11(21 rrt s

Duncan’s Multiple Range Test(DMRT)

Duncan (1995) made , the level of significance a variable from test to test. The Least Significant Range (LSR) is defined by

The value of significant studentized range (SSR) is given in Duncan (1955).In case, a pair of means differs by more than its LSR, they are declared to be significantly different.

k

rsSSRLSR

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.

One-way ANOVA, random effectSource SS df MS=SS/df Expected Means

Square(EMS)Between groups or treatments

SSTRT a-1 MSTRT

Residual (within groups or treatments)

SSRES N-a MSRES

2

22

Tn

For unbalanced cases n is replaced with

N

Na

iin 2

11

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 treatments and any number of replications can be carried out

Analysis is straight forward

even one or more

observations are missing

Two-way ANOVA

Suppose you intend to study the effectiveness of 3 different types of feed in 4 different strains of hybrid broilers. You need to distribute your treatments (3, feed) in a way so that birds of each of the strains (4, blocks) receive each type of feed. Randomization of the samples are to be ensured in an efficient way. Total no. of records = No. of treatments x No. of Blocks x No. of replication (2 in this case) per treatment (3x4x2=24)

Why doing this kind of expt. ? 1.Effect of type of feed on

the final live weight in broilers (treatment effect)

2.Effect of strain on the final live weight in broilers (block

effect)

3.Joint effect of feed x strain on the final live weight of

broilers ( interaction effect)

You want to know

Two-way ANOVAB L O C K S

I II III IV

No. 1 (T3) No. 7 (T3) No. 13 (T3) No. 19 (T1)

No. 2 (T1) No. 8 (T2) No. 14 (T1) No. 20 (T2)

Broiler No. No. 3 (T3) No. 9 (T1) No. 15 (T2) No. 21 (T3)

(Treatment) No. 4 (T1) No. 10 (T1) No. 16 (T1) No. 22 (T3)

No. 5 (T2) No. 11 (T2) No. 17 (T3) No. 23 (T2)

No. 6 (T2) No. 12 (T3) No. 18 (T2) No. 24 (T1)

Two-way ANOVAObservations can be shown sorted by treatments and blocks

Blocks

Treatments I II III IV

T1 y111

y112

y121 y122

y131

y132

Y141

y142

T2 y211 y212

y221 y222

y231y232

y241 y242

T3 y311 y312

y 321 y322

y331 y332

y341 y342

y ijk in

dica

tes e

xper

imen

tal u

nit ‘

k’ in

trea

tmen

t’ i ‘

and

bloc

k’ j ‘

Statistical model in two-way ANOVA

etty ijkijjiijk

i = 1,…,a; j = 1,…,b; k = 1,….,n

Whereyijk= observation k in treatment i and block jμ= overall meanti = effect of treatment iβj = effect of block jtβij = the interaction effect of treatment I and block jeijk = random error with mean 0 and variance Ϭ2 a = no. of treatments; b= no. of blocks; n= no. of obs in each treatment x block combination.

Sum of Squares, Degrees of Freedom and Mean Squares in ANOVA

Source SS df MS= SS/df

Block SSBlk b-1 MSBLK

Treatment SSTRT a-1 MSTRT

TreatmentxBlock SSTRTXBLK (a-1)(b-1) MSTRTxBLK

Residual SSRES ab(n-1) MSRES

Total SSTOT abn-1

Example: Two-way design

Recall that the objective of the experiment previously described was to determine the effect of 3 treatments (T1, T2, T3) on average daily gain of steers, and 4 blocks were defined. However, in this example 6 animals (3x2) are assigned to each block. Therefore, a total of 4x3x2 = 24 steers were used. Treatments were assigned randomly to steers within block.

Example: Two-way design

The data are as follows Blocks

Treatments I II III IV

T1 826 864 795 850 806 834 810 845T2 827 871 729 860 800 881 709 840T3 753 801 736 820 773 821 740 835

Two-way: Computations

1. Grand Total = 2. Correction term for the mean =

3. Total SS= 4. Treatment SS=

19426)835......806826( i j k

ijky

17.1572372824

2)(194262

abn

C i j kijky

83.52039

17.1572372815775768........2 835806826 222

Ci j k

ijkTOT ySS

58.802517.15723728888

2)(627965176630 222

Cnbi

j kijk

TRT

ySS

Two-way: Computations…

5. Block SS =

6. Interaction SS

7. Residual SS =

83.3381617.157237286666

5050451950724785 2222

2

j

Cj k ijk

na

y

42.8087

17.1572372883.3381658.80252

........22

)835820()871864()806826(

)(222

2

Ck ijk SSSSySS BLKTRT

i jTRTxBLK

00.2110 SSSSSSSSSS TRTxBLKBLKTRTTOTRES

ANOVA TABLESource SS df MS

Block 33816.83 4-1 = 3 11272.28

Treatment 8025.58 3-1 = 2 4012.79

TreatmentxBlock 8087.42 2x3=6 1347.90

Residual 2110.00 3x4x(2-1)=12 175.83

Total 52039.83 23

F value for treatment : F = 4012.79/175.83 = 22.82F value for interaction: F = 1347.90/175.83 = 7.67

Conclusion

The critical value for testing the interaction is F0.05,6,12 = 3.00, and for testing treatments is F0.05,2,12 = 3.89. So at p = 0.05 level of significance, H0 is rejected for both treatments and interaction.

Inference: There is an effect of treatments and the treatment effects are different in different blocks.

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, 12.70Rambuillet: 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

ANOVA (F test)

a) 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

Mean Separation

Post hoc testsHomogenous subsetsWeanDuncan

3 N Subset for alpha =0.05 1 2

suff 7 11.414

dors 5 12.240

ramb 8 14.063

Sig. .121 1.000

Interpretation of results

i) Null hypothesis (μ1=μ2=μ3) is rejected ie

there is significant (p<0.001) difference in weaning wt. between

breeds.

ii) Rambuillet has significantly (p<0.05)

highest weaning wt. among the 3 breeds and there is no

significant difference (p>0.05) between weaning wt.s of Suffolk and Dorset.

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