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
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
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
Fixed effects one-way ANOVA..
Treatment 1 Treatment 2
Look the difference
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) 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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are
goin
g to
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an A
nim
al
Scie
ntist
!!!!
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ou k
now
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stics
????
?
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