BivariateCorrelation

Lesson 10

Measuring Relationships Correlation

degree relationship b/n 2 variables linear predictive relationship

Covariance If X changes, does Y change also? e.g., height (X) and weight (Y) ~

Covariance Variance

How much do scores (Xi) vary from mean? (standard deviation)2

Covariance How much do scores (Xi, Yi) from their

means

1)( 2

2

N

XXs i

1))((

N

XXXX ii

1))((

),cov(

NYYXX

yx ii

Covariance: Problem How to interpret size

Different scales of measurement Standardization

like in z scores Divide by standard deviation Gets rid of units

Correlation coefficient (r)

YX

ii

YX ssNYYXX

ssYXr

)1())((),cov(

Pearson Correlation Coefficient Both variables quantitative (interval/ratio) Values of r

between -1 and +1 0 = no relationship Parameter = ρ (rho)

Types of correlations Positive: change in same direction

X then Y; or X then Y Negative: change in opposite direction

X then Y; or X then Y ~

Correlation & Graphs Scatter Diagrams Also called scatter plots

1 variable: Y axis; other X axis plot point at intersection of values look for trends

e.g., height vs shoe size ~

Scatter Diagrams

Height

Shoe size6 7 8 9 10 11 12

60

66

72

78

84

Height

Shoe size6 7 8 9 10 11 12

60

66

72

78

84

Slope & value of r

Determines sign positive or

negative From lower left to

upper right positive ~

Weight

Chin ups3 6 9 12 15 18 21

100

150

200

250

300

Slope & value of r

From upper left to lower right negative ~

Width & value of r Magnitude of r

draw imaginary ellipse around most points Narrow: r near -1 or +1

strong relationship between variables straight line: perfect relationship (1 or -1)

Wide: r near 0 weak relationship between variables ~

Width & value of r

Weight

Chin ups3 6 9 12 15 18 21

100

150

200

250

300

Strong negative relationship

r near -1

Weight

Chin ups3 6 9 12 15 18 21

100

150

200

250

300

Weak relationship

r near 0

Strength of Correlation R2

Coefficient of Determination Proportion of variance in X

explained by relationship with Y Example: IQ and gray matter volume

r = .25 (statisically significant) R2 = .0625 Approximately 6% of differences in

IQ explained by relationship to gray matter volume ~

Factors that affect size of r Nonlinear relationships

Pearson’s r does not detect more complex relationships

r near 0 ~ Y

X

Height

Shoe size6 7 8 9 10 11 12

60

66

72

78

84

Factors that affect size of r Range restriction

eliminate values from 1 or both variable

r is reduced e.g. eliminate

people under 72 inches ~

Hypothesis Test for r H0: ρ = 0 rho = parameter

H1: ρ ≠ 0 ρCV

df = n – 2 Table: Critical values of ρ PASW output gives sig.

Example: n = 30; df=28; nondirectional ρCV = + .335 decision: r = .285 ? r = -.38 ? ~

Using Pearson r Reliability

Inter-rater reliability Validity of a measure

ACT scores and college success? Also GPA, dean’s list, graduation rate,

dropout rate Effect size

Alternative to Cohen’s d ~

Evaluating Effect Size

Cohen’s d

Small: d = 0.2

Medium: d = 0.5

High: d = 0.8

Note: Why no zero before decimal for r ?

Pearson’s r

r = ± .1

r = ± .2

r = ±.5 ~

Correlation and Causation Causation requires correlation, but...

Correlation does not imply causation! The 3d variable problem

Some unkown variable affects both e.g. # of household appliances

negatively correlated with family size Direction of causality

Like psychology get good grades Or vice versa ~

Point-biserial Correlation

One variable dichotomous Only two values e.g., Sex: male & female

PASW/SPSS Same as for Pearson’s r ~

Correlation: NonParametric

Spearman’s rs

Ordinal Non-normal interval/ratio

Kendall’s Tau Large # tied ranks Or small data sets Maybe better choice than Spearman’s ~

Correlation: PASW Data entry

1 column per variable Menus

Analyze Correlate Bivariate Dialog box

Select variables Choose correlation type 1- or 2-tailed test of significance ~

Correlation: PASW Output

Figure 6.1 – Pearson’s Correlation Output

Guidelines1. No zero before decimal point2. Round to 2 decimal places3. significance: 1- or 2-tailed test4. Use correct symbol for correlation type5. Report significance level

There was a significant relationship between the number of commercials watch and the amount of candy purchased, r = +.87, p (one-tailed) < .05.

Creativity was negatively correlated with how well people did in the World’s Biggest Liar Contest, rS = -.37, p (two-tailed) = .001.

Reporting Correlation Coefficients