correlational methods and statistics...strength of correlation strength of correlation = r2 the...

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