Lecture 18 LSD and HSD

8/11/2019 Lecture 18 LSD and HSD

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MULTIPLE COMPARISONOF TREATMENT MEANS

Comparing means ofdifferent treatments i.e.which two treatments are

different significantly?



Comparison of means is

necessary only when there issignificant difference amongtreatment means as indicatedby ANOVA

Ho: u1 = u2 = u3 = ……= ut

When Ho

is rejected, we conclude that

the means differ significantly (at least 2

means differ significantly), but does not

specify which pairs of treatment means

differ. LSD, DNMRT and HSD can beapplied.



LSD (Least significantdifference) Test

Designed by Fisher

It’s a t-test.

More suitable forcomparing paired means.



d s

Y Y t

21

Therefore:

If the observed difference between the two means exceeds LSD the

difference is declared as significant, thus:

d

st LSD .

r

MSE sd

2

Where:

d st Y Y .

21



Arrange data according to treatments and replications:

Treatment RI RII RIII RIV Total Mean

V1 47 52 62 51 212 53

V2 50 54 67 57 228 57

V3 57 53 69 57 236 59

V4 54 65 74 59 252 63

Total 208 224 272 224 928

Mean 52 56 68 56



ANOVA Table

Source Variation

df SS MS F Ftable

(α=0.05)

Total 15 854

Rep. 3 576 192 24.7* 3.86

Treatment 3 208 69.3 8.9* 3.86

Error 9 70 7.78

Calculation :



Obtain value of t from table at df = dfe = 9 and

α=0.05, t = 2.262

therefore LSD = 2.262 x 1.972

= 4.46 t/ha

d st LSD .

r MSE sd

/2

972.14/78.72 d s



Compare difference between treatment means with LSD:

Larger significant

Smaller not significant

Variety Var1 Var2 Var3 Var4

Mean 53 57 59 63

V1 vs V4 = 53 – 63 = 10 > 4.46 significant difference


V1 vs V2 = 53 – 57 = 4 < 4.46 no significant difference






A CRD experiment on nitrogen content of red clover plantsinnoculated with combination cultures of Rhizobium trifolii

strains and Rhizobium meliloti strains with 6 treatments

and 5 replications results the following:

a b c d e f

19.4 17.7 17.0 20.7 14.3 17.3

32.6 24.8 19.4 21.0 14.4 19.4

27.0 27.9 9.1 20.5 11.8 19.1

32.1 25.2 11.9 18.8 11.6 16.9

33.0 24.3 15.8 18.6 14.2 20.8

Exercise:

Test the different combinations using LSD.



a b c d e f Y .j

R1 19.4 17.7 17.0 20.7 14.3 17.3 106.4

R2 32.6 24.8 19.4 21.0 14.4 19.4 131.6

R3 27.0 27.9 9.1 20.5 11.8 19.1 115.4

R4 32.1 25.2 11.9 18.8 11.6 16.9 116.5

R5 33.0 24.3 15.8 18.6 14.2 20.8 126.7

Y i. 144.1 119.9 73.2 99.6 66.3 93.5 596.6

Y 28.8 24.0 14.6 19.9 13.3 18.7

Calculation:



Source Variation

df SS MS F Ftable(α=0.05/0.01)

Total 29 1129.98

Treatments 5 847.05 169.41 14.37** 2.62/

3.90

Error 24 282.93 11.79

Calculation :

CF = 11864.39





Compare difference between treatment

means with LSD= 4.483 mg N :

a vs e = 28.8 – 13.3 = 15.5 > 4.483 significant difference

a vs c = 28.8 – 14.6 = 14.2 > 4.483 significant difference

a vs f = 28.8 – 18.7 = 10.1 > 4.483 significant difference

a vs d = 28.8 – 19.9 = 8.9 > 4.483 significant difference

a vs b = 28.8 – 24.0 = 4.8 > 4.483 significant difference

b vs e = 24.0 – 13.3 = 10.7 > 4.483 significant difference

b vs c = 24.0 – 14.6 = 9.4 > 4.483 significant difference

b vs f = 24.0 – 18.7 = 5.3 > 4.483 significant difference

b vs d = 24.0 – 19.9 = 4.1 < 4.483 insignificant difference

Contd .......

cultures e c f d b amean 13.3 14.6 18.7 19.9 24.0 28.8



d vs e = 19.9 – 13.3 = 6.6 > 4.483 significant difference

d vs c = 19.9 – 14.6 = 5.3 > 4.483 significant difference

d vs f = 19.9 – 18.7 = 1.2 < 4.483 insignificant difference

f vs e = 18.7 – 13.3 = 5.3 > 4.483 significant difference

f vs c = 18.7 – 14.6 = 2.1 < 4.483 insignificant difference

c vs e = 14.6 – 13.3 = 1.3 < 4.483 significant difference



HSD (Honestly Significant

Difference) - Tukey Tukey (1949) proposed a multiplecomparison procedure where all pairs

of treatment means can be compared Like LSD, a single value is used for

judging significant difference, called

HSD HSD is higher than LSD, thus moreconservative.



Obtain Q from table refering to df = dfe = 24,

k = 6, α=0.05, where k = number of treatments i.e

Q = 4.37.

Thus

r MSE Q HSD /

mgN HSD 73.654.137.45/79.1137.4



Compare difference between treatment

means with HSD = 6.75 mg N :

a vs e = 28.8 – 13.3 = 15.5 > 6.73 significant difference

a vs c = 28.8 – 14.6 = 14.2 > 6.73 significant difference

a vs f = 28.8 – 18.7 = 10.1 > 6.73 significant difference

a vs d = 28.8 – 19.9 = 8.9 > 6.73 significant difference

a vs b = 28.8 – 24.0 = 4.8 < 6.73 insignificant difference

b vs e = 24.0 – 13.3 = 10.7 > 6.73 significant difference

b vs c = 24.0 – 14.6 = 9.4 > 6.73 significant difference

b vs f = 24.0 – 18.7 = 5.3 < 6.73 insignificant difference

b vs d = 24.0 – 19.9 = 4.1 < 6.73 insignificant difference

Contd .......

cultures e c f d b amean 13.3 14.6 18.7 19.9 24.0 28.8



d vs e = 19.9 – 13.3 = 6.6 < 6.73 insignificant difference

d vs c = 19.9 – 14.6 = 5.3 < 6.73 insignificant difference

d vs f = 19.9 – 18.7 = 1.2 < 6.73 insignificant difference

f vs e = 18.7 – 13.3 = 5.3 < 6.73 insignificant difference

f vs c = 18.7 – 14.6 = 2.1 < 6.73 insignificant difference

c vs e = 14.6 – 13.3 = 1.3 < 6.73 insignificant difference



Treatments Mean

a 28.8ab 24.0ab

C 14.6c

d 19.9bc

e 13.3c

f 18.7bc

HSD = 6.73 mg N

Treatments Mean

a 28.8ab 24.0b

c 14.6c

d 19.9bd

e 13.3c

f 18.7cd

LSD = 4.48 mg N

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Lecture 18 LSD and HSD