Transcript Document

Studentized Range Statistic q
qr 
y L  yS
MSerror
n
similar to t
2 MSerror
n
Independent Groups
yL 
yS 
Largest mean
Smallest mean
If means are selected
randomly t is approx. If
not – correct p of Type I
error why?
q t 2
Example
y1
y2
y3
8.2
8.2
11.8
n5
r # steps  1
MSerror  12
dferror  12
11.8  8.2 3.6
q3 

 2.06
15.4 1.75
5
q( r 3,df 12, .05) critical value
Fail to Reject
= 3.77
Solving for smallest significant difference
y L  yS
qr 
MSerror
n
yL  yS  qr
MSerror
n
yL  yS  q.05( r ,df )
MSerror
n
15.4
 3.77
5
 6.61
When to use
q?
When you expect:
1  2  3   4
Otherwise use F
Newman-Kewls
Uses q
1.
2.
Arrange y' s in ascending order
y1
y2
8.2
8.2
y3
11.8
Steps from yi to y j =
i  j 1
e.g. y1 & y3
3  3  1  3 steps
smallest difference required was 6.61
If 2 steps
 2  yL  yS  3.08
15.4
5
 5.41
 r  smallest significant difference
3  6.61
2  5.41
N-K
3.
Treatment Matrix
T1
T1
T2
T3
r
r
0
3.6
3
6.61
3.6
2
5.41
T2
T3
4.
Significant Difference Pattern
T1
T1
T2
T3
T2
T3
Example
y1
y2
y3
y4
y5
2
3
3
9
10
MSerror  9
n9
df  40
 2  2.86(1)
3  3.44(1)
 4  3.79(1)
5  4.04(1)
Read Right to Left
UNTIL
1.
The row is completed
2.
A nonsignificant difference is found
2.
Reaching a column which was nonsignificant on the
previous row
T1
T1
T2
T3
T4
T5
r
r
1
1
7
8
5
4.04
0
6
7
4
3.79
6
7
3
3.44
1
2
2.86
T2
T3
T4
T5
T1
T4
T5
T1
*
*
T2
*
*
T3
*
*
T4
T5
T2
T3
Unequal n’s
Tukey-Kramer
MSerror
n
Replace
with
MSerror MSerror

nL
nS
2
r  q0.05( r ,df )
L  larger y n
S  smaller y n
MSerror MSerror

nL
nS
2
Behrens-Fisher
r  yL  yS  q0.05 (r ,df  )
S L2 S S2 *

nL nS
2
2
 S L2 S S2 



n
nS 
df    L2
2
2
2
 SS 
 SL 




 n L    nS 
nL  1
nS  1
* Each particular pairing of means must be examined
with a different critical q value and their own S 2
Thus, the smallest significant difference will
vary even for a given r
r  # of steps 1
Tukey's HSD
N-K except
qHSD always  qr
(largest)
If there are 4 means, all differences are treated as 4 steps.
Tukey's WSD
qWSD
qk  qr

2
k  # of means
r = # of steps between the two
means to be compared.
Tukey's HSD
Use largest qr for all pairwise comparisons
T2
T3
T4
T5
r
r
1
1
7
8
5
4.04
T2
6
7
4
3.79
T3
6
7
3
3.44
1
2
2.86
T1
T1
T4
T5
Dunnett’s control vs. treatments
td
(even if a priori)
run standard t and use td T able(MSe )
or, solve for critical difference (CV)
CV( yc  yT j )  td
2MSe
n
yc  10 , yT1  8 , yT2  4 , MSe  30 , n  11
Go to Table for
td
(k , dfe )
td  2.32
2(30)
11
 2.32(2.34)  5.42
CV  2.32
yc  yT1  2 ns
yc  yT2  6 * p  0.05
Sheffé’s test
It sets the family-wise Type-I Error rate (   .05 in
our case) for ALL possible linear contrasts, not merely
the pair-wise comparisons.
Linear contrast
MS(contrast)
MS(error)
Evaluate at (k-1) critical value for (df treatment(k-1)),
df error
Don’t use when only doing pair-wise, because it will be
overly conservative.
Post Hoc – Sheffé test
L
a
SS con
nL2

a 2j
F
j
yj

nL2
a 2j MS error
 
F(1, df ) 
nL2
a 2j

MSerror
To evaluate
1)
consult F table and find critical value F.05 (k1, dferror) (CV)
2)
multiply CV by (k-1). (new CV)
k = # of conditions
FW will now be held at 0.05