Download Multivariate Data Display - Basic Statistics for Behavioral Sciences - Lecture Notes and more Study notes Statistics for Psychologists in PDF only on Docsity!
Ch. 3: Multivariate Data Display
I. Mean vectors
A. For an n individual (subject) and p variable setting, we can
have n-observation vectors each of which has p-variables.
y =
ip
i
i
y
y
y
2
1
where i = 1, 2,.. n
B. The sample mean vector for all subjects for the p-variable can be expressed as:
y = (1/n) ∑
=
n
i
yi 1
y p
y
y
2
1
C. The data matrix for all n-individuals and p-variables can be expressed as:
nXp
Y =
'
' 2
' 1
y n
y
y
n n np
p
p
y y y
y y y
y y y
1 2
21 22 2
11 12 1
D. Computation of a mean vector from data matrix, Y.
y = (1/n) Y ’ j where
Y ’ = treaspose of the data matrix Y,
j = unit vector with 1s (nX1).
y = n
p p pn
n
n
y y y
y y y
y y y
1 2
21 22 2
11 12 1
=
=
=
n
i
ip
n
i
i
n
i
i
y
y
y
n
1
1
2
1
1
e.g. Y =
y =
E. Population mean vector (Expected value)
E( y ) = E
y p
y
y
2
1
2
1
E y p
E y
E y
μ p
2
1
= μ
and,
B. Matrix expression of variance
- From observation vector
S = ∑
=
n
i
yi y yi y n (^) 1
yy nyy n
∑ i i −
- From data matrix
S =
n −
[ Y’Y – Y ’(
n
J ) Y ]
where,
=
n
i
yi yi 1
' = Y’Y ,
y = (1/n) Y’j = Y ’( j /n),
n (^) y = Y’j, (^) y ’ = ( Y ’( j /n)) ’ = ( j ’/n) Y
∴n y y ’ = Y ’( jj ’/n) Y = Y ’( n
J ) Y
C. Population covariance matrix
- Σ = cov(y) =
p p pp
p
p
σ σ σ
σ σ σ
σ σ σ
1 2
21 22 2
11 12 1
= E[( y – μ )( y – μ )’] = E( yy ’) - μ μ ’
- proof: Σ = E[( y – μ )( y – μ )’]
=
- Σ = E( yy ’) - μ μ ’ and E( S ) = Σ
III. Correlation matrices
A. Sample correlation matrix, R
R = (rij) =
1 2
21 2
12 1
p p
p
p
r r
r r
r r
rij = i j
ij
ss
s
B. Computation of R and S
- Let
DS = Diag( s 11 , s (^) 22 ,.. s (^) pp )
= Daig(s 1 , s 2 ,.. sp)
s p
s
s
2
1
- Then,
R =
− 1 D (^) S S
− 1 D (^) S (c.f. rij = i j
ij
ss
s , from p. 8, DAD )
s p
s
s
2
1
p p pp
p
p
s s s
s s s
s s s
1 2
21 22 2
11 12 1
s p
s
s
2
1
And S = DSRDS. What if D = I? Then, S = R.
- Example,
- Mean vector
( pq ) X 1
m
x
y
q
p
x
x
x
y
y
y
2
1
2
1
- Covariance matrix
( pq ) X ( pq )
S
xy xx
yy yx
S S
S S
xqy xqy xqyp xqx xqx xqxq
xy xy x yp xx xx xxq
xy xy xyp xx xx xxq
ypy ypy ypyp ypx ypx ypxq
y y y y y yp y x yx y xq
yy yy yyp yx yx yxq
s s s s s s
s s s s s s
s s s s s s
s s s s s s
s s s s s s
s s s s s s
1 2 1 2
21 2 2 2 21 22 2
11 12 1 11 12 1
1 2 1 2
21 2 2 2 21 22 2
11 12 1 11 12 1
S xy =
' S yx
IV. Linear combination
A. Sample
- zi = a 1 y 1 + a 2 y 2 +.. + apyp = a’y
(c.f. y^ = b 1 x 1 + b 2 x 2 +.. + bpxp) where
ai = coefficient, yi = random variable.
a’y = [a 1 , a 2 ,.. ap]
y p
y
y
2
1
- If ai is the same vector of p-coefficients and applied to
different n subjects of y, then,
zi = a 1 yi1 + a 2 yi2 +.. + apyip = a’yi , i = 1, 2,.. n
- The mean of z ( z ) can be expressed as:
=
n
i
zi n
z 1
= a ’ y ,
where
y = sample mean vector of y 1 , y 2 ,.. y n.
And the variance of z (sz
2 ) can be expressed as:
1
2
2
−
=
n
z z
s
n
i
i
z =^ a’Sa^ ≥^0
where,
S = sample covariance matrix for y (semi-positive
definite).
- If we have a second vector of coefficient b for y , such
that,
wi = b 1 y 1 + b 2 y 2 +.. + bpyp,
then, the sample covariance between z and w is, szw = a’Sb , and
correlation between z and w is,
rzw = ( ' )( ' )
(^2 2) aSa bSb
aSb
s s
s
z w
zw
- In general, a = a 1 , and b = a 2 , if A =
' 2
' 1
a
a ,
then, z =
a y
ay
' 2
' 1 =
' 2
' 1
a
a y = Ay ,
y a
a
a y
a y
z
z z
' 2
' 1 ' 2
' 1
2
1 = A y , and
S z =
2 21 2
12
2 1
zz z
z zz
s s
s s
2
' 1 2
' 2
2
' 1 1
' 1
aSa a Sa
a Sa aSa
' 2
' 1
a
a
S [ a 1 a 2 ] = ASA’
where
VI. Distance between vectors
A. Univariate distance
Squared standard distance
d
2
2
2 ( 1 2 )
or 2
2 ( )
y
y
B. Multivariate distance (Mahalanobis distance) d
2 = ( y 1 – y 2 )’ S
- ( y 1 – y 2 ) sample distribution
_ _ D
2 = ( y – μ )’ S
- ( y – μ ) sampling distribution
_ _ Δ
2 = ( y – μ )’ Σ
- ( y – μ ) population mean distance
2 = ( μ 1 – μ 2 )’ Σ
- ( μ 1 – μ 2 ) two population mean difference
