Multivariate Data Analysis

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Multivariate Data Analysis

• Sociology 315, Winter 2002
• Week 4 Jan. 28 - Feb. 1

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When considering one variable, the standard
deviation provides us with an indicator of
average error in terms of differences in scores
from the mean
s2 ? (Xi X ) 2 N
66 10 6.6 s ? 6.6 2.57
(From Mayors years of service example.)
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Standard Error
s 5.5
s 9.5
40 45 50 55 60 65 70
x 55
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Spread of scores around the regression line
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Y
Y
Y
? (Y - Y)2 regSS max
X
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Y
Y
Y
? (Y - Y)2 RegSS 0
X
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Error (res_1)
X
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Error (res_1)
X
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Error (res_1)
Y (pre_1)
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Error (res_1)
Y (pre_1)
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Error (res_1)
Y (pre_1)
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3. Errors are distributed normally.
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Unstandardized and Standardized Slopes
BETAyx Change in Zy Change in
Zx Betayx (Byx) The magnitude of BETA ranges
from -1 to 1.
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Only in Bivariate Regression Byx ryx Bxy
rxy Standardized Bivariate Equation Zy Byx
(zx)
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b r sy b B sy sx sx B
b sx sy
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1. The introduction of a third variable may
intervene between the original two variables.
Hourly Wage(X) Job Satisfaction(Y)
r .56 Control variable Seniority Low
Seniority High Seniority r .81 r
.35 Wage (X) Seniority(Z) Job. Satis. (Y)
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Hourly Wage(X) Job Satisfaction(Y) r
.56 Control variable Seniority Low
Seniority High Seniority r .07 r
.09 Job Satisfaction Job
Seniority Hourly Wage
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Hourly Wage(X) Job Satisfaction(Y) r
.07 Control variable Seniority Low
Seniority High Seniority r .4 r
.65
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(Mothers Educ.)
by1.2
X1
Y
(Resps Educ.)
X2
by2.1
(Fathers Educ.)
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The Logic of Partial Correlation
Original Relationship X Y Control
Variable W W Y Y a b
W W X X a b W
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Example Suppose we want to predict current test
scores (Y) from previous test scores (X).
X Y W We believe that current
test scores are not simply a function of previous
scores, and that aptitude (W) may have something
to do with the relationship.
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Current Previous Test Test Score Score Aptitude
(Y) (X) (W) 3 4 3 1
2 6 2 1 4 4 6 5 6
5 1
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Current Test Score Aptitude (Y) Y2 (W)
W2 (YW) Y Y-Y 3 9 3 9 9
3.8 -.8 1 1 6 36 6 1.4
-.4 2 4 4 16 8 3.0 -1
4 16 5 25 20 2.2 1.8 6 36
1 1 6 5.4 .6 16 66 19 87
49
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Prev Test Score Aptitude (X) X2 (W)
W2 (XW) X X-X 4 16 3 9 12
3.9 .1 2 4 6 36 12 2.8 -.8
1 1 4 16 4 3.5 -2.5 6 36
5 25 30 3.2 2.8 5 25 1 1 5
4.6 .4 18 82 19 87 63
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(With (With W indep.) W indep.)
(Y - Y) Y - Y (Y - Y)2 X-X (X -
X)2 (X - X) 1 -.8 .7 .1
.01 -.08 2 -.4 .2 -.8 .64 .32 3 -1
1.08 -2.5 6.39 2.5 4 1.8 3.09
2.8 8.05 5.04 5 .6 .32 .4
.14 .24 .2 5.39 0 15.23 8.02
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r xy.w N ?XY - (?X)(?Y) ?N ?X2 -
(?X)2 N ? Y2 - (?Y)2
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r xy.w N ?XY - (?X)(?Y) ?N ?X2 -
(?X)2 N ? Y2 - (?Y)2 5 (8.02) - (.2)
(0) ?5 (15.23) - (.2) 25 (5.39) - (0)
2 40.1/45.29 .885
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A partial correlation coefficient of
.885, symbolized as rxy.w means that the
correlation of X and Y, controlling for the
impact of W is .885.
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(Y)
(X)
(W)
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r xy.w rxy - rxw ryw ?(1- rxw2) (1
- ryw2) .777 - (-.338)(-.797) ?1 -
(-.338)21-(.797)2 .777 - .269
?(1-.114)(1-.635) 0.508/0.569 .893
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A partial correlation coefficient of .893, when
squared provides the percent of variation that
two variables share after a third variable is
controlled for. For this example, X and Y
share nearly 80 of the variation once W is
controlled for.