H o : σ 12 = σ 22
Fmax Test: (a.k.a. Hartley’s test)
2
2
Fmax, s = smax
smin
vs. H a : σ 1 ≠ σ 2
2
2
vs. H a : σ 1 ≠ σ 2
2
2
2
(for 2 samples)
•
Uses recoded data: z1 j =| x1 j − x1 | , z2 j =| x2 j − x2 | where
•
ÆHo: “deviations from the median are the same in the 2 populations”
∑ n (z − z )
=
∑ ∑ n (z − z
2
Ls
2
i =1
L*s = Ls =
j =1 i
ii
ii
ij
⎛ z1 − z2
=
⎜⎜
2
)
/(
N
2)
−
⎝ SEz1 − z2
ii
2
/(2 − 1)
xi
is the median of sample i.
2
⎞
⎟⎟ is defined on p. 314 (with t=2)
⎠
z1 − z2
2
with df = n1 + n2 − 2 (Note: This is just an equal variance T-statistic with s p in the SE)
SEz1 − z2
Rejection Region at the α level of significance:
•
•
(for 2 samples)
df = n * −1 where n* = max(n1 , n2 ) (Table 10)
Brown & Forythe’s HOV Test H o : σ 1 = σ 2
i =1 i
ni
2
| L*s |≥ tα / 2,
n1 + n2 − 2
This test is more robust to non-normal populations than the Fmax test and F-test (from section 7.3, which we skipped).
The text refers to this as Levene’s test, but Levene originally used z1 j =| x1 j − x1 | with x rather than x
JMP:
•
•
There is no Fmax test in JMP.
Levene’s and the B&F test is under: (Fit Y by X: <red▼> UnEqual Variances)
o Results from the F-test (section 7.3, which we skipped) are also given, but these are not robust to non-normality.
Results for: ch6-1-MouseDiet-v1-1.jmp (Cholesterol reduction after 21 days on a special diet: 14 mice/group)
Oneway Analysis of Z(median) By Diet
t Test (Oat-Bean)
Assuming equal variances
Difference
Std Err Dif
Upper CL Dif
Lower CL Dif
Confidence
1.5262
1.9151
5.4627
-2.4103
0.95
t Ratio
DF
Prob > |t|
Prob > t
Prob < t
0.79695
26
0.4327
0.2163
0.7837
---------------------------------------------------------------------------------Oneway Analysis of Reduction By Diet
Tests that the Variances are Equal
Level
Bean
Oat
Count Std Dev
14
6.122964
14
9.163964
Test
Brown-Forsythe**
Levene
MeanAbsDif to Mean
5.045698
6.483653
F Ratio DFNum DFDen p-Value
26
0.4327
0.6351 1
0.5934 1
26
0.4480
**the text calls this test based on
z1 j =| x1 j − x1 |
Example:
x1
3
5
6
x2
2
6
14
MeanAbsDif to Median
4.948514
6.474729
z1
z2
Levene’s Test