Comparing Two Populations
Result 6.2:
If X1,1, X1,2, . . . , X1,n1 is a random sample from Np(µ1, Σ) and
X2,1, X2,2, . . . , X2,n2 is an independent random sample from Np(µ2, Σ)
(note common Σ) then
T2 =
h
X̄1 − X̄2 − (µ1 − µ2)
i0
"
!
1
1
+
Sp
n1
n2
#−1
h
X̄1 − X̄2 − (µ1 − µ2)
is distributed as
(n1 + n2 − 2) p
× Fp,n1+n2−p−1,
n1 + n2 − p − 1
where Sp is the pooled variance matrix:
Sp =
n1 − 1
n2 − 1
S1 +
S2 .
n1 + n2 − 2
n1 + n2 − 2
1
i
Example: turtle carapace dimensions
• Responses: length, width, height (mm).
• Factor: gender
• n1 = n2 = 24.
• > pairs(turtle[,1:3],
col = ifelse(turtle$Gender == "male",
"red", "blue"))
• SAS proc glm program and output.
2
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130
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3
• Unequal variance matrices: if n1 and n2 are large, T 2 is
approximately χ2
p.
• Note: so is
(n1 + n2 − 2) p
× Fp,n1+n2−p−1,
n1 + n2 − p − 1
so the analysis for equal variances is approximately valid.
4
Comparing Several Populations
• Xl,1, Xl,2, . . . , Xl,nl is a random sample from Np(µl , Σ), l =
1, 2, . . . , g.
• Samples are independent.
5
MANOVA table:
Source of
variation
Matrix sum of squares and
cross products (SSCP)
Degrees of
freedom
Treatment
Pg
= l=1 nl (x̄l − x̄) (x̄l − x̄)0
0
Pg
Pnl = l=1 j=1 xl,j − x̄l xl,j − x̄l
g−1
0
Pnl Pg
= l=1 j=1 xl,j − x̄ xl,j − x̄
Pg
l=1 nl − 1
Residuals
B
W
Pg
l=1 nl − g
Corrected
Total
B+W
Note: in proc glm, B and W are called H and E, respectively.
6
• Likelihood ratio test of H0 : µ1 = µ2 = · · · = µg is based on
Wilks’ Λ∗:
|W |
1
Λ∗ =
=
|B + W |
|W−1B + I|
• Other proposed statistics, all functions of W−1B:
– Pillai’s trace, trace B (B + W)−1 = trace
h
i
I + B−1W
h
i
−1
– Hotelling-Lawley trace, trace W B ;
– Roy’s greatest root, maximum eigenvalue of W−1B.
7
−1
;
Example: Egyptian skulls
• Responses: max breadth, base height, base length, nasal
height (mm);
• Factor: period (1 = 4000 BC, 2 = 3300 BC, 3 = 1850 BC);
• g = 3, n1 = n2 = n3 = 30.
• > pairs(skulls[, 1:4],
col = c("blue", "red", "green")[skulls$Period])
• SAS proc glm program and output.
8
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BHeight
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9
SAS 9.1
The GLM procedure now compute[s] exact p-values for three
of the four multivariate tests (Wilks’ Lambda, the HotellingLawley Trace, and Roy’s Greatest Root) and an improved Fapproximation for the fourth (Pillai’s Trace).
Specifying
MSTAT=EXACT computes exact p-values for three of the four
tests (Wilks’ Lambda, the Hotelling-Lawley Trace, and Roy’s
Greatest Root) and an improved F-approximation for the fourth
(Pillai’s Trace).
10