bivariate correlation lesson 10. measuring relationships n correlation l degree relationship b/n 2...
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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
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XXs i
1
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XXXX ii
1
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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 ~
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 ~
Factors that affect size of r
Nonlinear relationships Pearson’s r does not
detect more complex relationships
r near 0 ~ Y
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 = + .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 ~
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