2
2
df = n * −1 where n* = max(n1 , n2 )
(Table 10)
Yes
o Fmax, s = smax smin
No
• Fmax Test: (a.k.a. Hartley’s test)
o Not robust to non-normal populations
• Each individual in sample 1 is matched to one individual
ts =
d −0
sd n
in sample2 in order to control for other factors and reduce variability
o di = x1i − x2i
Nonparametric Tests
• Do not rely on the assumption of normally distributed populations Æ Nonparametric
• Use ranks of the observations as opposed to the original values
Æ not influenced by outliers
• 2 Independent Samples: Wilcoxon Rank Sum Test (a.k.a. Mann-Whitney Test)
o TWMW [= Sum of the ranks in sample 1] with a sampling dist. that is approx. normal with
ƒ µT =
n1 (n1 + n2 + 1)
,
2
σ T2 =
n1n2
(n1 + n2 + 1) , Æ
12
zs =
TWMW − µT
σT
• Paired Samples: Wilcoxon Signed Rank Test
o TWSR [= minimum (T+ , T-)] with a sampling distribution that is approx. normal with
* *
T −µ
n* (n* + 1)(2n* + 1)
σ T2 =
ƒ µT = n (n + 1) ,
, Æ zs = WSR T
σT
24
4
Yes
Æ
No
H o : µd = 0
T-Test for 2 Paired Samples
Assumptions
met for T-test ?
2
o The test involves a T-test on z1 and z2 (Equal variance with s p in the SE)
o Robust to non-normal populations
No
o Uses recoded data: z1 j =| x1 j − x1 | , z2 j =| x2 j − x2 | (Absolute deviations from the median)
Paired
Samples ?
• Brown & Forsythe’s Test (a.k.a. Levene’s median): (for 2 samples)
Equal Variance T-test
(sp2 in the SE)
df = n1+n2-2
H a : σ 12 ≠ σ 22
Yes
H o : σ 12 = σ 22
Tests of Homogeneity of Variance (HOV)
Unequal Variance T-test
(s12 & s22 in the SE)
df by Satterthwaite approx
df by Satterthwaite’s approximation
No
2
σ1 2 = σ 2 2 ?
from Levene’s or
Fmax test
1
s12 s22
+
n1 n2
equivalent to a
T-test on the ranks
o SEx − x =
Paired T-test
(sd2 in the SE)
df = nd -1
Assumes σ 1 ≠ σ 2
• Unequal (or Unpooled) Variance T-test:
Non-Parametric test:
Wilcoxon Signed Rank
df = n1 + n2 − 2
Parametric T-test
1⎞
2
Assumptions
met for T-test ?
⎛1
1
( x1 − x2 ) − D0
SEx1 − x2
Yes
Assumes σ 1 = σ 2
• Equal (or Pooled) Variance T-test:
o SEx − x = s 2p ⎜ + ⎟
⎝ n1 n2 ⎠
ts =
Æ
Non-Parametric test:
Mann-Whitney
(aka: Wilcoxon Rank Sum)
H o : µ1 − µ2 = D0
T-Tests for 2 Independent Samples