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Chapter 6
Correlational Methods
and Statistics
Scatterplots and correlations
Relationship between time spent studying and performance
on exam
Horne and Ostberg (1976)
Scale of “morningness” Considering only your own “feeling best” rhythm, at what time would
you get up if you were entirely free to plan your day?
5-6:30am; 6:30-7:45am; 7:45-9:45am; 9:45-11am; 11-12
At what time would you go to bed if you were entirely free to plan?
8-9pm; 9-10:15pm, 10:15-12:30am; 12:30-1:45am; 1:45-3
Assuming normal circumstances, how easy do you find getting up in the morning?
Not at all easy; slightly easy; fairly easy; very easy
How alert do you feel during the first half-hour after waking?
Not at all; slightly alert, fairly alert, very alert
If you had to be at peak performance for a test that is going to be mentally exhausting and lasting 2 hours. You are free all day what time do you choose?
8-10am; 11am-1pm; 3-5pm; 7-9pm
Correlational studies
Why do people act the way they do?
Examine patterns and relationships to predict behavior
e.g. Guthrie, Ash, & Bendapudi (1995)
Examine relationship between college students’ GPA and “tendency of morningness”
Scale by Smith, Reilly & Midkiff (1989) Evening type: 22 or lower; Intermediate: 23-43; Morning type; 44 or higher
Method 454 undergrads; records of gpa and time of day first class scheduled
Is the measure of “morningness” predictive of patterns of sleep, studying and class schedule?
Guthrie, Ash, & Bendapudi (1995)
What are the conclusions you can make from the results?
Scatterplots Graphical tool for exploring the relationship between 2
quantitative variables
http://www.stat.berkeley.edu/~stark/Java/Html/Correlation.htm
Correlations
Direction of relationship:
Positive: As value of 1 variable increases, so
does the other
Direct correlation
Negative: As value of 1 variable increases,
the other decreases
Indirect correlation
No relationship
Magnitude, size or strength of relationship:
-1.00 to 0 to +1.00 (“correlation
coefficient”)
0 = no relationship
1 = perfect predicted relationship
What is size of correlation?
Time spent studying and exam performance
r = +.58 r = -.58
http://www.stat.berkeley.edu/~stark/Java/Html/Correlation.htm
Lang & Heckhausen (2001) Examine relationship between perceived control over development
(PCD) and subjective well-being (SWB)
Study 1: 480 adults 20-90 yrs
4 PCD items – 5 strongly agree to 1 strongly disagree: “I am able to make my goals come true.” “My abilities and efforts are significant to my success.”
4 Life satisfaction - 5 strongly agree to 1 strongly disagree: “I am satisfied with my life these days.” “As I get older, life is better than I thought it would be.”
20 Positive and negative affect – 5 very often to 1 not at all: How often they felt each of 10 pos (interested, inspired, excited,
attentive) or neg states (nervous, guilty, distressed, irritated)
Also examined: SES, “negative social support”, cognitive functioning, health functioning
Lang & Heckhausen (2001)
Correlations Examples from Lang & Heckhausen (2001)
Direction of relationship:
Positive: As value of 1 variable increases, so does the other
Direct correlation
e.g.: Perceived control over life with life satisfaction (r = .35)
Negative: As value of 1 variable increases, the other decreases
Indirect correlation
e.g.: # negative events in life and perception of control (r = -.13)
No relationship
e.g.: Life satisfaction and gender (r = -.02)
Types of relationships Linear
Amount of change on X, same amount of change on Y
Nonlinear Curvilinear
Amount of change on X, smaller change on Y (or vice versa)
U or V function Multilinear Other
150100500
20
15
10
5
Speed(km/h)
Fu
el u
sed
Guthrie, Ash, & Bendapudi (1995) Examine relationship between college students’ GPA and “tendency of
morningness” Morningness scale by Smith, Reilly & Midkiff (1989) Method: 454 undergrads; records of gpa and time of day first class
Interpretation of correlations
Causality
Correlation does not imply causation
Directionality
Unsure of whether A causes B or reverse
Third variable problem
Another factor causing relationship
Bushman & Anderson (2001)
Relationship between media violence and aggressive behavior
Many assume causal relationship b/c high correlation!
What are other interpretations?
Interpretation of correlations
Causality
Correlation does not imply causation
Directionality
Unsure of whether A causes B or reverse
Third variable problem
Another factor causing relationship
Other considerations:
Restricted range
Heterogeneous subgroups
Outliers
Effect of restricted range
If no correlation, is it b/c have restricted range?
e.g. GPA and SAT
Possible to get invalid high correlation b/c of restricted range?
Restricted range
Heterogeneous subgroups
Invalid correlation due to presence of subgroups
Example: Correlation of height & weight = .78, but…
Correlation for men only = .60
Correlation for women only = .39
Draw the scatterplot!
Effect of outliers
6.56.05.55.04.54.0
4.5
3.5
2.5
Alcohol
Tob
acco
r=0.22
Remove outlier and
r jumps to 0.79
Outlier!
Correlational analyses
Pearson’s product-moment (r)
Degree and direction of linear relationship between two variables
Interval or ratio scales
Spearman’s rank-order correlational coefficient
Ordinal scale
Point-biserial correlation coefficient
One variable is dichotomous, other is interval
Phi coefficient
Both variables are dichotomous and nominal
Correlational analyses
Theoretical calculation:
Convert raw score to z-score
Computational formula
N
ZZr
YX
S
MXz
separately vary Y and X which todegree
ther vary togeY and X which todegreer
N
MXS
2)(
N
YY
N
XX
N
YXXY
r2
2
2
2)(
())(
(
))((
Example calculation
X X2 Y Y2 XY
10 100 3 9 30
9 81 1 1 9
8 64 3 9 24
7 49 4 16 28
6 36 7 49 42
5 35 7 49 35
0 0 7 49 0
N=
N
YY
N
XX
N
YXXY
r2
2
2
2)(
())(
(
))((
= = = = =
Example calculation
X X2 Y Y2 XY
10 100 3 9 30
9 81 1 1 9
8 64 3 9 24
7 49 4 16 28
6 36 7 49 42
5 35 7 49 35
0 0 7 49 0
=45 355 32 182 168
N=7
N
YY
N
XX
N
YXXY
r2
2
2
2)(
())(
(
))((
)7
32182)(
7
45355(
7
)32)(45(168
22
r
)714.35)(714.65(
714.37r
778.044.48
714.37
r = = = = =
Correlation table
df = degrees of freedom
df for correlations = n-2
Is correlation higher than
value given for df and
significance level (p = .05)?
Ex.: df 7-2 = 5
For 2-tailed p .05, critical
value = .754
Our calculation r = -.778
Conclusion: significant r
Hospital example
Examine hospital
acquired infection and
efficient treatment of
patients (N = 90)
Infection rate and length
of stay: r = .55
Conclusion?
Letter from your HMO
“To improve service to our valued customers, we have determined it is beneficial to manage inpatient recovery by accelerating standard patient discharge.”
“Results suggest this will not have any adverse impact on patient care, and may in fact reduce the chance of hospital acquired infection, r(90) = + .55, p < .05.”
Is it an accurate conclusion?
Reporting results
What test is used
Report variables investigated
Sample size
Value of statistic
Probability level
If it is significant or not
Example of correlation write-up:
The correlation between IQ and SAT scores was found to be
statistically significant, r(30) = 0.65, p < .01.
Write-up
“Pearson correlations were used to examine the relationship
between the ages of younger and older participants’ first
memories and their scores on three psychometric measures.”
“Results indicated an inverse relationship between the age of
first memories and the scores on the WAIS-R digit span for
younger adults, r(46) = -0.31, p < .02, and older adults,
r(46) = -0.29, p < .02.
This suggests that smarter individuals have earlier first
memories!
Strength of correlation
Strength of correlation = r2
The proportion of variability explained
Used to evaluate the strength or effect size
Example: r = +.80 = 64% of variability in Y can be predicted by relationship with X
What is the strength of the relationships for the following examples?
“The correlation between TAT (personality inventory) and behaviors are in the neighborhood of +.30.”
“The correlation between 1st and 2nd administration of a personality inventory ranged from +.59 to +.87.”
“The results showed average correlations of +.50 between identical twins on scores both of extroversion/introversion and neuroticism/emotional stability. The correlations corresponding for fraternal twins were +.21 and +.23.”
Partial correlation
Measure 3+ variables
Statistically remove 1 to see effect on relationship