BivariateCorrelation

Lesson 11

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

)( 22

N

XXs i

1

))((

N

XXXX ii

1

))((),cov(

N

YYXXyx 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 ssN

YYXX

ss

YXr

)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 size

6 7 8 9 10 11 12

60

66

72

78

84

Height

Shoe size

6 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 ups

3 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 ups

3 6 9 12 15 18 21

100

150

200

250

300

Strong negative relationship

r near -1

Weight

Chin ups

3 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 ~

Guidelines for interpreting strength of correlation

Table 5.2 Interpreting a correlation coefficient

Size of Correlation (r) General coefficient interpretation

.8 to 1.0 Very strong relationship

.6 to .8 Strong relationship

.4 to .6 Moderate relationship

.2 to .4 Weak relationship

.0 to .2 Weak to no relationship

*The same guidelines apply for negative values of r

*from Statistics for People Who (Think They) Hate Statistics: Excel 2007 Edition By Neil J. Salkind

Factors that affect size of r

Nonlinear relationships Pearson’s r does not

detect more complex relationships

r near 0 ~

Peeps (Y)

Stress (X)

Height

Shoe size

6 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 = + .361 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

Large: d = 0.8

Note: Why no zero before decimal for r ?

Pearson’s r

r = ± .1

r = ± .3

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: SPSS Data entry

1 column per variable Menus

Analyze Correlate Bivariate Dialog box

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

Guidelines1. No zero before decimal point

2. Round to 2 decimal places

3. significance: 1- or 2-tailed test

4. Use correct symbol for correlation type

5. 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

Correlation: ExampleCorrelations

  WorkHours ExCurrHours

WorkHours Pearson Correlation 1 -.313

Sig. (2-tailed)   .081

N 32 32

ExCurrHours Pearson Correlation -.313 1

Sig. (2-tailed) .081  

N 32 32

Correlation: Example

Analysis using the Pearson’s r correlation indicated that the there was moderately strong negative relationship between the number of work hours and the number of hours spent on extracurricular activities, but the relationship was not statistically significant, r = -.31, p (two-tailed) = .08. The R2 = .097, indicating that the relationship accounts for approximately 9.7% of the variance in the number of hours spent in each activity.