Analysis of Treatment Means
KNNL – Chapter 17
Cell Means Model – Sampling Distributions and Graphs
Model: Yij  i   ij
 ij ~ NID  0,  2  i  1,..., r

Fixed Effects  Yij ~ N i ,  2

j  1,..., ni
independent
1
2
2
 i  Y i    Yij
E Y i   i
 Y i 
ni
j 1  ni 

Y i   i
2 
Y i  ~ N  i ,
independent
~ t  nT  r 

ni 
s Y i

 
ni
^
 
 
s 2 Y i 
MSE
ni
 
Bar Graph - Package Design
Main Effects Plot - Package
Design
30
25
30
20
Cases Sold
25
20
15
15
TrtMean
10
10
AllMean
5
5
0
1
2
3
Design
4
0
1
2
3
4
Inference for Individual Treatment Means
Y i   i
 
s Y i
 
~ t  nT  r  where s Y i 
MSE
ni
1   100% Confidence Interval for i :
 
Y i  t 1   2  ; nT  r  s Y i
Test of H 0 : i  c vs H A : i  c
Test Statistic: t * 
Y i  c
Reject H 0 if t *  t 1   2  ; nT  r 
 
s Y i
Note: The t-distribtion arises from:
1)
Y i   i
 
 Y i
~ N  0,1
2)
SSE

2
~  n2T  r
3) Y i , SSE are independent
Comparing Two Treatment Means
Parameter: D  i  i ' Difference between 2 Treatment Population Means
^
Estimator: D  Y i   Y i ' Difference between 2 Treatment Sample Means


^
E D  i  i '
s
2

2

1
1 
D   2 Y i    2 Y i '   2   
 ni ni ' 
^
1
1 
D  MSE   
 ni ni ' 
^
 

^
s D 
 
1
1 
MSE   
 ni ni ' 
^
D D

^
~ t  nT  r  
1   100% CI for D :
s D
Test of H 0 : D  0   i  i '  vs H A : D  0   i  i ' 
^
Test Statistic: t * 
D

^
s D

D  t 1   2  ; nT  r  s D
^
Reject H 0 if t *  t 1   2  ; nT  r 
^
Contrasts among Treatment Means
Contrast: A Linear Function of Treatment with their coefficients summing to 0:
r
L   ci i
r
such that
i 1

r
^
c
i 1
 0 (Note: A difference between 2 means is contrast)
r
^
L   ci Y i
i
E L  L   ci i
i 1

2
i 1

^
r
L  c
i 1
2
i
^
L L

^
~ t  nT  r 
1   100% CI for L :

s L
r
r
i 1
i 1
2
ni

r
ci2
MSE
s L  c
 MSE 
n
i 1
i 1 ni
i
2
^
r
2
i

L  t 1   2  ; nT  r  s L
^
^
Test of H 0 : L   ci i  0 vs H A : L   ci i  0
r
^
Test Statistic: t * 
L

^
s L

c Y
i 1
i
i
ci2
MSE 
i 1 ni
r
Reject H 0 if t *  t 1   2  ; nT  r 
r
Note: this method applies to any linear combination of means, that is we do not need
c
i 1
i
0
Simultaneous Comparisons
• Confidence Coefficient (1-) applies to only one estimate
or comparison, not several comparisons simultaneously.
Confidence Coefficient for a “family” of tests/intervals
will be smaller than confidence coefficient for
“individual” tests/intervals
 If we construct five independent confidence intervals, each
with confidence level = 0.95, Pr{All Correct} = (0.95)5 = 0.774
• Confidence Coefficient (1-) applies to only pre-planned
comparisons, not those suggested by observed samples
(referred to as “data snooping”).
 If we wait until after observing the data, then decide to test
whether most extreme means are different, actual  too high
Tukey’s Honest Significant Difference (HSD) - I
Background:
1) Suppose Y1 ,..., Yr ~ NID   ,  2  and the range is w  max Y1 ,..., Yr   min Y1 ,..., Yr 
2) s 2 is an estimate of  2 , based on  degrees of freedom
3) s 2 is independent of Y1 ,..., Yr
w
 q  r ,  is the studentized range, with selected critical values in Table B.9
s
 Y Y

w

5) P   q  r ,   q 1   ; r ,    1    P  i i '  q 1   ; r ,    1   for all i, i '
s

 s

4) Then:
Application to All Pairwise Comparisons (Under Assumption of equal means and equal sample sizes):
 2 
1a) Y 1 ,..., Y r ~ NID   , 
n 

MSE
2
2a)
is an estimate of
, based on   nT  r degrees of freedom
n
n
3a) MSE independent of Y 1 ,..., Y r
 Y i  Y i'



4a) P 
 q 1   ; r ,    1   for all i, i '
 MSE n

Y i  Y i'
Conclude any two population means are different if:
MSE n
 q 1   ; r , 
Tukey’s Honest Significant Difference (HSD) - II
Simultaneous Confidence Intervals (all pairs of treatments):

1 1
D  i  i '
D  Y i   Y i '
s D  MSE   
 ni ni ' 
Tukey's multiple confidence intervals with family level of 1    :
^
^

^
D  Ts D
^
1
where: T 
q 1   ; r , nT  r 
2
Simultaneous tests of H 0 : i  i '  0 vs H A : i  i '  0
^
Test Statistic: q 
*
2D

^
s D
Reject H 0 if q*  q 1   ; r , nT  r 
Scheffe’s Method for Multiple Comparisons
Very Conservative Method, but can be applied to all possible contrasts among treatment means
r
L   ci i such that
i 1
^
r
L   ci Y i
i 1

r
c
i 1
0
i
ci2
s L  MSE 
i 1 ni
r
^
Simultaneous 1   100% Confidence Intervals:
^

^
L  Ss L
S
 r  1 F 1   ; r  1, nT  r 
r
r
i 1
i 1
Testing H 0 : L   ci i  0 vs H A : L   ci i  0
Test Statistic: F * 
^
 L
 
 r  1 s
2
2

^
L
Reject H 0 if F *  F 1   ; r  1, nT  r 
Bonferroni’s Method for Multiple Comparisons
Can be used for any number ( g ) of pre-planned comparisons, contrasts, linear combinations
r
L   ci i
i 1
^
r
L   ci Y i
i 1

ci2
s L  MSE 
i 1 ni
r
^
Simultaneous 1   100% Confidence Intervals for g linear combinations of means:
^

^
L  Bs L

B  t 1    2 g   ; nT  r
r
r
i 1
i 1

Testing H 0 : L   ci i  0 vs H A : L   ci i  0
^
Test Statistic: t * 
L

^
s L

Reject H 0 if t *  t 1    2 g   ; nT  r
