Fisher’s Least Significant Difference  ( Fisher’s LSD )

Why not using  t test to compare each two population means?  Overall error rate 1 – ( 1- a )c .

1)   Perform an analysis of variance to test Ho: m1 = m2 =…= mt

2)   If there is insufficient evidence to reject Ho – proceed no further

3)   If Ho is rejected, calculate LSD

LSD  = ta/2 S2w ( 1/ni + 1/nj )

If ni = nj = n

LSD  = ta/2 √ 2S2w/n

Note: df used in t is for S2w

4)   If absolute value  |Yi – Yj| ³  LSD ,  the population means are different

e. g.

 Treatment ( n= 5) Sample mean 1 505 2 528 3 564 4 498 5 600 6 470

The ANOVA table is then

 Source Df SS MS F Between 5 56360 11272 4.60 Within 24 58824 2451 Total 29

From tables F 5,24 (0.05) = 2.62

LSD  = t a/2 √ 2S2w / n

1) Rank the sample means from lowest to highest

 Pop 6 4 1 2 3 5 Mean 470 498 505 528 564 600

2) Compute sample differences

Y5  -  Y6  ,  Y3  -  Y6  ,   Y2  -  Y6  ….

6    4    1    2    3    5

Y5  - Y4  ,  Y3  -  Y4 ,   Y2  -  Y4

6    4    1    2    3    5

Tukey’s W procedure

1)    rank the t sample means

2)   two populations are different if

| Yi  -  Yj | ³ W

W  =  q a (t,v) √ S2w / n

Note:   v is df for mean square within samples

Tukey’s W is more conservative compared to LSD

If different number of samples per treatment

W ij  =  q a (t,v) √ S2w/2 ( 1/ni + 1/nj )

Student-Newman-Keuls Procedure ( SNK )

The SNK procedure make use of a critical value

Wr  =  qa (r,v) √ S2w/n

For means that are r steps apart when the sample means are ranked from lowest to highest

 Pop 6 4 1 2 3 5 Mean 470 498 505 528 564 600

e. g.  W6 = qa (6,v) √ S2w /n

 r 2 3 4 5 6 qa(r,v) 2.92 3.53 3.9 4.17 4.37 Wr 64.65 78.16 86.35 92.33 96.75

1)   rank the t sample means from lowest to highest

2) for two means Yi and Yj that are  r  steps apart, mI and mj are different if

| Yi  - Yj  |  ³ Wr

Wr as defined above, n is the number of observations per sample , v is the df for mean square for within samples

Note: replace t by r when finding the q values from the table

SNK is less conservative than Tukey’s W procedure – will declare more significant differences than W