CORRELATION - Elder Laboratoryelderlab.yorku.ca/~elder/teaching/psyc3031/lectures/Lecture 3... ·...

PSYC 3031 INTERMEDIATE STATISTICS LABORATORY J. Elder

LAST UPDATED: October 4, 2012

CORRELATION

Correlation

J. Elder PSYC 3031 INTERMEDIATE STATISTICS LABORATORY

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Acknowledgements

¨  Some of these slides have been sourced or modified from slides created by A. Field for Discovering Statistics using R.

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Outline

¨  Measuring relationships ¤ Scatterplots ¤ Covariance ¤ Pearson’s correlation coefficient

¨  Nonparametric measures ¤ Spearman’s rho ¤ Kendall’s tau

¨  Interpreting correlations ¤ Causality

¨  Partial correlations

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Textbook Errata

¨  6.5.1. As of Oct 4, 2012, the ggm package appears to be broken. ¤  As a substitute, we will use the ppcor package.

¨  6.5.3 ¤  cor(examData, use = “complete.obs”, method = “pearson”)

should be ¤  cor(examData[,c(“Revise”, “Exam”, “Anxiety”)], use =

“complete.obs”, method = “pearson”) ¤  I’m not sure what “2 d.p. only” means in Table 6.2.

¨  6.5.5 ¤  rcorr(liarMatrix)

should be ¤  rcorr(liarData$Position, liarData$Creativity, type = "spearman”)

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Textbook Errata

¨  6.6.2 ¤ We will use package ppcor instead of ggm. ¤ The implementation of ppcor has the format

n pcor(x, method = c("pearson", "kendall", "spearman"))

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What is a Correlation?

¨  It is a way of measuring the extent to which two variables are related.

¨  It measures the pattern of responses across variables.

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Weak Relationship

Slide 7 Age

10 20 30 40 50 60 70 80 90

App

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n of

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gir

-20

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Strong Positive Relationship

Slide 8

Age

10 20 30 40 50 60 70 80 90

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mu

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Strong Negative Relationship

Slide 9

Age

10 20 30 40 50 60 70 80 90

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Measuring Relationships

¨  We need to see whether as one variable increases, the other increases, decreases or stays the same.

¨  This can be done by calculating the covariance. ¤ We look at how much each score deviates from the

mean. ¤  If both variables deviate from the mean by the same

amount, they are likely to be related.

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Revision of Variance

¨  The variance tells us by how much scores deviate from the mean for a single variable.

¨  It is closely linked to the sum of squares. ¨  Covariance is similar – it tells is by how much scores

on two variables differ from their respective means.

sample variance sx =xi−x( )2∑N−1 = xi−x( )∑ xi−x( )

N−1

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Covariance

¨  Calculate the deviation of each subject’s score from the mean for the first variable (x).

¨  Calculate the deviation of each subject’s score from the mean for the second variable (y).

¨  Multiply these deviations for each subject. ¨  Now simply take the average of the resulting

values. This is the sample covariance.

( )( )1cov( , ) i ix x y y

Nx y − −∑−=

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Problems with Covariance

¨  It depends upon the units of measurement. ¤  e.g. the covariance of two variables measured in miles might be 4.25,

but if the same scores are converted to kilometres, the covariance is 11.

¨  Solution: standardize it! ¤  Divide by the standard deviations of both variables.

¨  The standardized version of covariance is known as the correlation coefficient. ¤  It is unaffected by the units of measurement.

r = covxy

sxsy= xi−x( )∑ yi− y( )

N−1( )sxsy

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Correlation: Example

¨  Anxiety and exam performance ¨  Participants:

¤ 103 students

¨  Measures ¤ Time spent revising (hours) ¤ Exam performance (%) ¤ Exam Anxiety (the EAQ, score out of 100) ¤ Gender

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Doing a Correlation with R Commander

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General Procedure for Correlations Using R

¨  To compute basic correlation coefficients there are three main functions that can be used: cor(), cor.test() and rcorr().

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Correlations using R

¨  Pearson correlations: ¤  cor(examData[,c("Revise","Exam","Anxiety")], use =

"complete.obs", method = "pearson") ¤  rcorr(examData$Revise, examData$Exam, type = "pearson") ¤  cor.test(examData$Exam, examData$Anxiety, method =

"pearson")

¨  If we predicted a negative correlation: ¤  cor.test(examData$Exam, examData$Anxiety, alternative =

"less"), method = "pearson")

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Pearson Correlation Output

Exam Anxiety Revise Exam 1.0000000 -0.4409934 0.3967207 Anxiety -0.4409934 1.0000000 -0.7092493 Revise 0.3967207 -0.7092493 1.0000000

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Reporting the Results

¨  Exam performance was significantly correlated with exam anxiety, r = -.44, and time spent revising, r = .40; the time spent revising was also correlated with exam anxiety, r = -.71 (all ps < .001).

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Things to Know about the Correlation

¨  It varies between -1 and +1 ¤ 0 = no relationship

¨  It measures the size of the effect ¨  Coefficient of determination, r2

¤ By squaring the value of r you get the proportion of variance in one variable shared by (e.g., “explained by”) the other.

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Correlation and Causality

¨  The third-variable problem: ¤  In any correlation, causality between two variables

cannot be assumed because there may be other measured or unmeasured variables affecting the results.

¨  Direction of causality: ¤ Correlation coefficients say nothing about which

variable causes the other to change.

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Non-parametric Correlation

¨  Spearman’s rho ¤ Pearson’s correlation on the ranked data

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Example

¨  World’s Biggest Liar competition ¤ 68 contestants ¤ Measures

n Where they were placed in the competition (first, second, third, etc.)

n Creativity questionnaire (maximum score 60)

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Spearman’s Rho

¨  You can use cor() to compute Spearman’s Rho: cor(liarData$Position, liarData$Creativity, method = "spearman")

¨  The output of this command will be: [1] -0.3732184

¨  To get the significance value use rcorr(): rcorr(liarData$Position, liarData$Creativity, type = "spearman")

¨  Or cor.test(): cor.test(liarData$Position, liarData$Creativity, alternative = "less", method = "spearman")

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cor.test() Output

Spearman's rank correlation rho data: liarData$Position and liarData$Creativity S = 71948.4, p-value = 0.0008602 alternative hypothesis: true rho is less than 0 sample estimates: rho -0.3732184

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Kendall’s Tau

¨  Kendall’s tau ¤ Better than Spearman’s for small samples with many tied

scores. cor(liarData$Position, liarData$Creativity, method = "kendall") cor.test(liarData$Position, liarData$Creativity, alternative = "less", method = "kendall")

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cor.test() Output

¨  The output is similar to that for Spearman’s correlation. Kendall's rank correlation tau

data: liarData$Position and liarData$Creativity z = -3.2252, p-value = 0.0006294 alternative hypothesis: true tau is less than 0 sample estimates: tau -0.3002413

Oct 8, 2012

End of Lecture

Partial Correlation

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Pearson Correlation Output


Exam Performance

Exam AnxietyVariance Accounted for by Exam Anxiety (19.4%)

Exam Performance

Revision Time

Variance Accounted for by Revision Time (15.7%)

Exam Performance

Exam AnxietyUnique variance accounted for by Exam Anxiety

Revision Time

Unique variance accounted for by Revision Time

Variance accounted for by both Exam Anxiety and

Revision Time

1

2

3

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Slide 32

Partial and Semi-partial Correlations

¨ Partial correlation: ¤ Measures the relationship between two variables,

controlling for the effect that a third variable has on them both.

¨ Semi-partial correlation: ¤ Measures the relationship between two variables

controlling for the effect that a third variable has on only one of the others.

8-Nov-12

Partial Correlation Semi-Partial Correlation

Revision

Exam Anxiety

Revision

Exam Anxiety

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Doing Partial Correlation using R

¨  NB: we are using the package ppcor instead of ggm (ggm is broken).

¨  The general form of pcor() is: pcor(x, method = c("pearson", "kendall", "spearman"))

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Example

¨  pcor(examData[,c("Revise","Exam","Anxiety")], method = "pearson”)

$estimate Revise Exam Anxiety Revise 1.0000000 0.1326783 -0.6485301 Exam 0.1326783 1.0000000 -0.2466658 Anxiety -0.6485301 -0.2466658 1.0000000 $p.value Revise Exam Anxiety Revise 0.000000e+00 0.18069538 1.596071e-17 Exam 1.806954e-01 0.00000000 1.091818e-02 Anxiety 1.596071e-17 0.01091818 0.000000e+00 $statistic Revise Exam Anxiety Revise 0.000000 1.338617 -8.519961 Exam 1.338617 0.000000 -2.545307 Anxiety -8.519961 -2.545307 0.000000 …

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Comparison

¨  Bivariate Correlations

¨  Partial Correlations

Exam Anxiety Revise Exam 1.0000000 -0.4409934 0.3967207

Anxiety -0.4409934 1.0000000 -0.7092493 Revise 0.3967207 -0.7092493 1.0000000


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