Transcript 00t0holtgrav.iweb.bsu.edu

Chapter 9
Correlation
Fundamental Statistics for the
Behavioral Sciences, 5th edition
David C. Howell
©2003 Brooks/Cole Publishing Company/ITP
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Chapter 9 Correlation
Major Points
• The problem
• Scatterplots
• An example
• The correlation coefficient
 Correlations on ranks
• Factors affecting correlations
Cont.
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Chapter 9 Correlation
Major Points--cont.
• Testing for significance
• Intercorrelation matrices
• Other kinds of correlations
• Review questions
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Chapter 9 Correlation
The Problem
• Are two variables related?
 Does one increase as the other increases?
• e. g. skills and income
 Does one decrease as the other increases?
• e. g. health problems and nutrition
• How can we get a numerical measure of
the degree of relationship?
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Chapter 9 Correlation
Scatterplots
• Examples from text
 See next three slides
• Infant mortality and number of physicians
• Life expectancy and health care expenditures
• Cancer rate and solar radiation
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Chapter 9 Correlation
Figure 9.1
Infant Mortaility and Number of Physicians
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Infant Mortality
6
4
2
0
-2
-4
-6
10
12
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16
Physicians per 100,000 Population
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Chapter 9 Correlation
Figure 9.2
Life Expectancy and Health Care Costs
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200
400
600
800
1000
1200
Health Care Expenditures
1400
1600
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Chapter 9 Correlation
Figure 9.3
Cancer Rate and Solar Radiation
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Breast Cancer Rate
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200
300
400
Solar Radiation
500
600
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Chapter 9 Correlation
An Example
• An actual course with both a lab and an
exam component of final grades
• Plotting exam component against lab
component
 Fairly weak relationship
 Relationship is positive
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Chapter 9 Correlation
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Rs q = 0.1368
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To tal Points in L ab
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200
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Chapter 9 Correlation
Exams and Labs
• Note relationship is weak, but real.
• Note most data cluster on right.
• Why do we care about relationship?
 What would students conclude if there were
no relationship?
 What if the relationship were near perfect?
 What if the relationship were negative?
Chapter 9 Correlation
Heart Disease and Cigarettes
• Landwehr & Watkins report data on heart
disease and cigarette smoking in 21
developed countries
• Data have been rounded for
computational convenience.
 The results were not affected.
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Chapter 9 Correlation
The Data
Cigarette Consumption and Coronary Heart Disease Mortality for 21 Countries
Cig. 11 9 9 9 8 8 8 6 6 5 5
CHD 26 21 24 21 19 13 19 11 23 15 13
Cig. 5 5 5
CHD 4 18 12
5 4 4
3 11 15
4 3
6 13
3 3
4 14
Cig. = Cigarettes per adult per day
CHD = Cornary Heart Disease Mortality per 10,000 population
Surprisingly, the U.S. is the first country on the list--the country
with the highest consumption and highest mortality.
Chapter 9 Correlation
Scatterplot of Heart Disease
• CHD Mortality goes on ordinate
 Why?
• Cigarette consumption on abscissa
 Why?
• What does each dot represent?
• Best fitting line included for clarity
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Chapter 9 Correlation
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{X = 6, Y = 11}
0
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4
6
8
10
Cigarette Consumption per Adult per Day
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Chapter 9 Correlation
What Does the Scatterplot
Show?
• As smoking increases, so does coronary
heart disease mortality.
• Relationship looks strong
• Not all data points on line.
 This gives us “residuals” or “errors of
prediction”
• To be discussed later
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Chapter 9 Correlation
Correlation Coefficient
• A measure of degree of relationship.
• Sign refers to direction.
• Based on covariance
 Measure of degree to which large scores go
with large scores, and small scores with
small scores
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Chapter 9 Correlation
Covariance
• The formula
Cov XY
( X  X )(Y  Y )

N 1
• How this works, and why
• When would covXY be large and positive?
• When would covXY be large and negative?
Chapter 9 Correlation
Correlation Coefficient
• Symbolized by r
• Covariance ÷ (product of st. dev.)
Cov XY
r
s X sY
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Chapter 9 Correlation
Calculation
• CovXY = 11.13
• sX = 2.33
• sY = 6.69
cov XY
11.13
11.13
r


 .71
s X sY (2.33)(6.69) 15.59
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Chapter 9 Correlation
Correlation--cont.
• Correlation = .71
• Sign is positive
 Why?
• If sign were negative
 What would it mean?
 Would not alter the degree of relationship.
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Chapter 9 Correlation
Factors Affecting r
• Range restrictions
 See next slide
• Data only for countries with low consumption
• Nonlinearity
 e.g. age and size of vocabulary
• Heterogeneous subsamples
 Everyday examples
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Chapter 9 Correlation
Countries With Low
Consumptions
Data With Restricted Range
Truncated at 5 Cigarettes Per Day
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CHD Mortality per 10,000
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4
2
2.5
3.0
3.5
4.0
4.5
Cigarette Consumption per Adult per Day
5.0
5.5
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Chapter 9 Correlation
Testing r
• Population parameter = 
• Null hypothesis H0:  = 0
 Test of linear independence
 What would a true null mean here?
 What would a false null mean here?
• Alternative hypothesis (H1)   0
 Two-tailed
Chapter 9 Correlation
Tables of Significance
• Table in Appendix E.2
• For N - 2 = 19 df, rcrit = .433
• Our correlation > .433
• Reject H0
 Correlation is significant.
 Greater cigarette consumption associated with higher
CHD mortality.
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Chapter 9 Correlation
Computer Printout
• Printout gives test of significance.
• See next slide.
 Double asterisks with footnote indicate
p < .01.
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Chapter 9 Correlation
SPSS Printout
Correlations
Cigarette
Consumption
per Adult per
Day
CHD
Mortali
ty per
10,000
Cigarette
Pearson
Consumption per Correlation
Adult per Day
Sig.
(2-tailed)
N
CHD Mortality
Pearson
.713**
per 10,000
Correlation
Sig.
.000
(2-tailed)
N
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**. Correlation is significant at the 0.01 level
(2-tailed).
Chapter 9 Correlation
Intercorrelation Matrix
• Matrix of correlations of several variables
at once.
• Example from Kliewer et al (1998) JCCP
 99 young children
 Measured level of
• Witness violence, Intrusive thoughts, Social
support, and Internalizing symptoms
 Define these variables
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Chapter 9 Correlation
Wit
ness
Witness 1.00
Intrus
.37
SocSup .08
Internal .20
Intrus
ions
.37
1.00
-.08
.39
Social
Support
.08
-.08
1.00
-.17
Internal
izing
.20
.39
-.17
1.00
Cont.
Chapter 9 Correlation
Intercorrelation Matrix--cont.
• Describe the table.
• What does this tell us about the effects
of witnessing violence?
• What role does social support play?
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Chapter 9 Correlation
Review Questions
• What determines what goes on which
axis of a scatterplot?
• What would a correlation of 0 tell us
about the relationship between lab
grades and exam grades?
• What factors might affect the relationship
between smoking and CHD Mortality?
Chapter 9 Correlation
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Review Questions--cont.
• Indicate level (high, med., or low) and
sign of the correlation for:
 number of guns in community and number
firearm deaths
 robberies and incidence of drug abuse
 protected sex and incidence of AIDS
 community education level and crime rate
 solar flares and suicide
Cont.
Chapter 9 Correlation
Review Questions--cont.
• Why would the size of the correlation
required for significance decrease with
N?
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