correlation

Correlation

Scatterplots & Correlation

Deviation & Computational Equations

Testing Significance

Intercorrelation Matrix

Partial Correlations

KEY CONCEPTS*****

Correlation

Correlation coefficientInterpretation of the concepts of magnitude and directionUse of a scatterplot to diagnose correlation Deviation score formula for the Pearson Product-Moment Correlation CoefficientKarl Pearson (1857-1936)Concepts of:

Sum of cross productsSums of squares of X & Y

Computational formula for the Pearson Product-Moment Correlation Coefficientt-test for determining the significance of r and dfNull hypothesis in determining the significance of rCoefficient of determinationCoefficient of nondeterminationAssumptions for the Pearson r

Linear relationshipX & Y are metric variablesRandomly drawn sampleX & Y are normally distributed in the population

The concept of a nonlinear relationshipIntercorrelation matrixCaveats in interpreting an intercorrelation matrixInterpretation of a partial correlationZero-order correlation1st , 2nd, etc. order correlations

Correlation: Charles M. Friel Ph.D., Criminal Justice Center,Sam Houston State University

2

Lecture Outline

The concept of correlation

Using a scatterplot to identify a correlation

Pearson Product-Moment Correlation Coefficient

Coefficients of correlation, determination & nondetermination

Intercorrelation matrix

Partial correlation coefficient


3

The Problem of Determining Relationships

Science attempts to find the "causes" of phenomena.

Q Why do some prisoners attempt to escape from prison and others do not?

Q Why do some judges have a constant backlogs of cases while others run an efficient docket?

Q What factors account for the fact that some countries have a higher rate of violent crime than others?

Q Is violence in the media related to violent behavior in society?

Q Why do some police officers become "rogue cops"?


4

The Problem in Determining Causality

How can one determined if one variable is the “cause” of another?

Example

Do liberal laws on the purchase and possession of firearms cause increases in the incidence of violent crimes involving weapons?

Principles of causality

1st Are the two variables in question related ( X & Y)? Is there a covariance between them?

2nd Is there a replicable time sequence between the two variables, the variable thought to be causal (X) always preceding the variable thought to be the effect (Y)?


5

The Problem Determining Causality (cont.)

3rd Having eliminated or controlled for all other extraneous variables, can it be demonstrated that when X occurs Y always follows?


6

Correlation and Causality

The first step in determining whether one variable (X) is correlated with another variable (Y) involves …

Determining if the two variables covary

The concept of covariance

Is it true that as X increases …

Y also increases, and to what extent?

Or, is it true that as X increases …

Y decreases, and to what extent?

Caveat

The mere fact that two variables covary (i.e. correlate) is no proof that one is the cause of the other.

Correlation does not necessarily prove causation.


7

A Correlation Coefficient

A correlation coefficient is an index number that measures …

The magnitude and

The direction of the relationship between two variables

It is designed to range in value between 0.0 and 1.0

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 +0.2 +0.4 +0.6 +0.8 +1.0

Negative PositiveRelationship RelationshipX Y X YX Y X Y

No relationship


8

Varieties of Correlational Statistics

Statisticians have developed many techniques for determining the correlation between two or more variables.

The primary difference among these techniques is a function of the types of variables being correlated (i.e. nonmetric: nominal, ordinal, or metric: interval or ratio)

Metric with metric

Metric with nonmetric

Nonmetric with nonmetric

A partial list of correlational techniques

Pearson product-moment correlation coefficient (metric with metric)

Spearman's rank-difference coefficient (rho) (ordinal with ordinal)


9

Varieties of Correlational Statistics (cont.)

Biserial coefficient (a metric variable with a metric variable that has been artificially reduced to categories)

Point biserial coefficient (a metric variable with a truly dichotomous variable)

Tetrachoric correlation coefficient (two metric variables that have been artificially reduced to dichotomous categories)

Phi Coefficient (two truly dichotomous variables)

Partial correlation (two metric variables with the intercorrelation with a third variable removed from both of them)

Kendall coefficient of concordance (three or more ordinal variables

Multiple correlation (one metric variable with two or more metric and/or nonmetric variables)


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The Scatterplot A useful tool for visually identifying the presence of a possible relationship between two metric variables.

Correlation r = +0.8257 (p 0.001)

Correlation r = -0.4174 (p 0.001)The Scatterplot (cont.)


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AGE

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AGE AT FIRST ARREST

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Correlation r = -0.0841 (P = 0.489)


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TIME TO DISPOSITION IN DAYS

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An ExampleIs There a Correlation Between

Homicide & Rape?

The incidence of homicide and rape per 100,000 population in a sample of seven

medium size cities

City Homicide (X)

Rape (Y)

A 4 16

B 6 29

C 10 43

D 5 20

E 1 3

F 2 4

G 3 6

Totals 31 121


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Scatterplot of the Homicide v Rape

Do the two variables appear to be related?

What is the magnitude of the relationship on a scale of 0.0 to 1.0?

What is the direction of the relationship, positive or negative?


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RAPE

50454035302520151050

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8

7

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Pearson Product-Moment Correlation Coefficient

Karl Pearson (1857-1936) British mathematician and statistician who also developed the Chi-Square Test

r = (X – X) (Y – Y)

(X – X)2 (Y – Y)2

Incidence of Homicide (X) and Rape (Y)

City X Y (X – X) 2 (Y - Y) 2 (X – X) (Y - Y)A 4 16 0.1849 1.6641 0.5547B 6 29 2.4649 137.124 18.4789C 10 43 31.0249 661.004 143.2047D 5 20 0.3249 5.7100 1.5447E 1 3 11.7649 204.204 49.0147F 2 4 5.9049 176.624 32.2947G 3 6 2.0449 127.464 16.1447

Totals 31 121 53.7143 1310.794 261.2371

Mean number of homicides & rapes

X = 4.43 and Y = 17.29

Pearson Product-Moment Correlation Coefficient (cont.)


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Sum of squared deviations in homicides = 53.71

Sum of squared deviations in rapes = 1310.79

Sum of cross products (SP) = 261.24

Calculation of the Pearson r

r = 261.24 = 261.24

(53.71) (1310.79) 70402.53

r = (261.24) / (265.33) = +0.985

Interpretation

The magnitude of the correlation between homicide and rape = 0.985.

The direction of the relationship is positive. As the incidence of homicide increases so does the incidence of rape.


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An Alternative Way to Calculate a Pearson Correlation

The previous equation is called a deviation score equation since the mean of each variable is subtracted from each respective case.

An alternative computational equation is given below. It will yield the same result within rounding error.

r = N(XY) – (X) (Y)

[N X2 – (X)2] [NY2 – (Y)2]


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An Alternative Way to Calculate a Pearson Correlation (cont.)

City X Y (X) 2 (Y) 2 (XY)A 4 16 16 256 64B 6 29 36 841 174C 10 43 100 1849 430D 5 20 25 400 100E 1 3 1 9 3F 2 4 4 16 8G 3 6 9 36 18

Totals 31 121 191 3407 797

r = 7 (797) – (31) (121)

[7 (191)– (31)2] [7 (3407) – (121)2]

r = 0.985

This is the same value computed with the deviation score equation.


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Determining the Significance of a Correlation Coefficient

The problem

Imagine a population in which X and Y are not related, the correlation = 0.0. ( = rho)

Is it possible to draw a random sample from that population and find that the correlation between X & Y in the sample is not 0.0?

Of course this is possible, but what is the probability of that happening?

A t-test

A t-test can be used to answer this question.


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A t-Test for the Significance of a Correlation Coefficient

t = [ r N – 2) ] / 1 – r2

df = (N – 2)

The null hypothesis H0

In the population, the correlation between X & Y is = 0.0

What is the probability, therefore that the correlation obtained in the sample came from a population where the parameter = 0.0?


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A t-Test for the Significance of a Correlation Coefficient (cont.)

For the correlation between homicide and rape

r = 0.985

t = [ 0.985 7 – 2) ] / 1 – (0.985)2

t = (2.203) / (0.1726) = 12.767

df = (N - 2) = (7 cities - 2) = 5

The critical value of t for df = 5 and = 0.05 is t = 2.571

Interpretation

Since 12.767 2.571, r is significant at p 0.05

Decision Reject the null hypothesis and affirm that the two variables are positively related in the population.


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Coefficients of Determination& Non-determination

r = the correlation between X and Y

e.g. r = 0.985

r 2 = the coefficient of determination

r2 = (0.985)2 = 0.97

This is the proportion of variance in Y that can be explained by X, in percentage terms 97%

1 - r 2 = the coefficient of nondetermination

1 - r2 = (1 - 0.9852)= 0.03

This is the proportion of variance in Y that can not be explained by X, in percentage terms 3%


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Some Examples of SPSS Correlational Output

Given a random sample of 70 felony cases. Q Is there a correlation between the age of the offender and the length of sentence?

Correlation r = 0.826, p 0.001

Correlations

1.000 .826**

. .000

70 70

.826** 1.000

.000 .

70 70

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

AGE

SENTENCE

AGE SENTENCE

Correlation is significant at the 0.01 level (2-tailed).**.


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AGE

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Some Examples of SPSS Correlational Output (cont.)

Q Is there a correlation between the age of first arrest and the length of sentence?

Correlation r = -0.417, p 0.001

Correlations

1.000 -.417**

. .000

70 70

-.417** 1.000

.000 .

70 70

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

AGE_FIRS

SENTENCE

AGE_FIRS SENTENCE



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AGE AT FIRST ARREST

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30

20

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Some Examples of SPSS Correlational Output (cont.)

Q Is there a correlation between the time to case disposition and the length of sentence?

Correlation r = -0.084, p 0.489

Correlations

1.000 -.084

. .489

70 70

-.084 1.000

.489 .

70 70

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

TM_DISP

SENTENCE

TM_DISP SENTENCE


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TIME TO DISPOSITION IN DAYS

16014012010080604020

30

20

10

0

Pearson Correlation Assumptions

That the relationship between X and Y can be represented by a straight line, i.e. it is linear.

That X and Y are metric variables, measured on an interval or ratio scale of measurement.

In using a t distribution to test the significance of the correlation coefficient …

That the sample was randomly drawn from the population, and

That X and Y are normally distributed in the population. This assumption is less important as the sample size increases


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An Intercorrelation Matrix

Multiple correlations and their significance can be computed simultaneously and reported in an intercorrelation matrix

Example

Intercorrelation of age, age at first arrest, number of prior arrests and convictions, and length of sentence

SPSS Intercorrelation ResultsCorrelations

1.000 -.312** .179 .302* .826**

. .009 .138 .011 .000

70 70 70 70 70

-.312** 1.000 -.315** -.358** -.417**

.009 . .008 .002 .000

70 70 70 70 70

.179 -.315** 1.000 .795** .246*

.138 .008 . .000 .040

70 70 70 70 70

.302* -.358** .795** 1.000 .400**

.011 .002 .000 . .001

70 70 70 70 70

.826** -.417** .246* .400** 1.000

.000 .000 .040 .001 .

70 70 70 70 70

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

AGE

AGE_FIRS

PR_ARRST

PR_CONV

SENTENCE

AGE AGE_FIRS PR_ARRST PR_CONV SENTENCE


Correlation is significant at the 0.05 level (2-tailed).*.


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Caveats in Interpreting an Intercorrelation Matrix

Are all the relationships linear?

Has each variable been checked for outliers that might lead to a Type I or II error?

Has each pair of variables (X & Y) been checked for bivariate outliers that might lead to a Type I or II error?

Can it be assumed that each variable is normally distributed in the population?

Can it be assumed that each pair of variables is a random sample from the population?


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What Is the Meaning of a Linear Relationship?

The Pearson correlation assumes that the two variables are linearly related. What does this mean?

Example

Age and length of sentence

Notice that the straight line is a "fair" representation of the relationship.

The cases are about evenly distributed above and below the line.

This is called homogeneity of the variance of Y over levels of X.


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AGE

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What Is the Meaning of a Linear Relationship? (cont.)

Example

Age at first arrest and length of sentence

Notice that the straight line is not a "fair" representation of the relationship.

The cases are not evenly distributed above and below the line.

This is called heterogeneity of the variance of Y over levels of X.


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AGE AT FIRST ARREST

24222018161412

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20

10

0

The Problem of Multiple Intercorrelation

Imagine three variables that are interrelated with each other.

How can the correlation between two of them be computed …

Eliminating the intercorrelation that both have with the third variable?

Example

Age

Age at first arrest

Length of sentence

Correlations

1.000 -.312** .826**

. .009 .000

70 70 70

-.312** 1.000 -.417**

.009 . .000

70 70 70

.826** -.417** 1.000

.000 .000 .

70 70 70

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

AGE

AGE_FIRS

SENTENCE

AGE AGE_FIRS SENTENCE



31

The Problem of Multiple Intercorrelation (cont.)

Q What is the correlation between age and sentence …

Eliminating the intercorrelation of both variables with age at first arrest?

A This problem can be solved by computing the partial correlation between age and sentence, controlling for age at first arrest.

Q How is a partial correlation computed?


32

Partial Correlation Coefficient

rXY.Z = [ rXY – (rXZ) (rYZ) ] / [ 1 - r2XZ 1 - r2

YZ ]

What is the correlation of X and Y taking out the intercorrelation of both variables with Z?

X Y

Z

rXY.Z = the partial correlation between X and Y, partialling out the inter-relationship between X and Z, and Y and Z


33

Partial Correlation Coefficient (cont.)

Example

What is the correlation between age (X) and length of sentence (Y), partialling out or controlling for age at first arrest (Z)?

Correlations

1.000 -.312** .826**

. .009 .000

70 70 70

-.312** 1.000 -.417**

.009 . .000

70 70 70

.826** -.417** 1.000

.000 .000 .

70 70 70

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

AGE

AGE_FIRS

SENTENCE



rXY.Z = [ .826 - (-.312) (-.417) ]

[ 1 - (-.312 )2 1 - ( -.417) 2 ]

rXY.Z 0.806

Notice the difference between the correlation (0.826) and the partial correlation (0.806) when controlled for age at first arrest.


34

Partial Correlation Coefficient (cont.)

Example

What is the correlation between age at first arrest (X) and length of sentence (Y), partialling out or controlling for age (Z)?

Correlations

1.000 -.312** .826**

. .009 .000

70 70 70

-.312** 1.000 -.417**

.009 . .000

70 70 70

.826** -.417** 1.000

.000 .000 .

70 70 70

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

AGE

AGE_FIRS

SENTENCE



rXY.Z = [ -.417 - (-.312) (.826) ]

[ 1 - (-.312 )2 1 - ( .826) 2 ]

rXY.Z -0.298

Notice the difference between the correlation (-0.417) and the partial correlation (-0.298) when controlled for age.


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SPSS Partial Correlation Results

Age and sentence controlling for age at first arrest

- - - P A R T I A L C O R R E L A T I O N C O E F F I C I E N T S - - -

Controlling for.. AGE_FIRS

AGE SENTENCE

AGE 1.0000 .8055 ( 0) ( 67) P= . P= .000

SENTENCE .8055 1.0000 ( 67) ( 0) P= .000 P= .

(Coefficient / (D.F.) / 2-tailed Significance)

" . " is printed if a coefficient cannot be computed

Age at first arrest and sentence controlling for age - - - P A R T I A L C O R R E L A T I O N C O E F F I C I E N T S - - -

Controlling for.. AGE

SENTENCE AGE_FIRS

SENTENCE 1.0000 -.2980 ( 0) ( 67) P= . P= .013

AGE_FIRS -.2980 1.0000 ( 67) ( 0) P= .013 P= .




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Multiple Partial Correlation

rxy.zz

More than one variable can be partialled out of a bivariate correlation.

X Y

Z Z

Example

What is the correlation between age (X) and sentence (Y) …

Partialling out prior arrests, time to disposition, prior convictions, drug use and the seriousness of the offense?

Multiple Partial Correlation (cont.)


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The bivariate correlations among the seven variables.

Correlations

Correlations

1.000 .826** .179 .048 .302* .252* .609**

. .000 .138 .692 .011 .036 .000

70 70 70 70 70 70 70

.826** 1.000 .246* -.084 .400** .346** .744**

.000 . .040 .489 .001 .003 .000

70 70 70 70 70 70 70

.179 .246* 1.000 -.072 .795** -.003 .502**

.138 .040 . .556 .000 .979 .000

70 70 70 70 70 70 70

.048 -.084 -.072 1.000 -.066 -.024 .032

.692 .489 .556 . .589 .841 .794

70 70 70 70 70 70 70

.302* .400** .795** -.066 1.000 .056 .578**

.011 .001 .000 .589 . .645 .000

70 70 70 70 70 70 70

.252* .346** -.003 -.024 .056 1.000 .279*

.036 .003 .979 .841 .645 . .019

70 70 70 70 70 70 70

.609** .744** .502** .032 .578** .279* 1.000

.000 .000 .000 .794 .000 .019 .

70 70 70 70 70 70 70

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

AGE

SENTENCE

PR_ARRST

TM_DISP

PR_CONV

DR_SCORE

SER_INDX

AGE SENTENCE PR_ARRST TM_DISP PR_CONV DR_SCORE SER_INDX


Correlation is significant at the 0.05 level (2-tailed).*.


39

Multiple Partial Correlation (cont.)

The partial correlation of age and sentence, controlling for five other variables.

- - - P A R T I A L C O R R E L A T I O N C O E F F I C I E N T S - - -

Controlling for.. PR_ARRST TM_DISP PR_CONV DR_SCORE SER_INDX

AGE SENTENCE

AGE 1.0000 .7044 ( 0) ( 63) P= . P= .000

SENTENCE .7044 1.0000 ( 63) ( 0) P= .000 P= .



Notice the difference between the correlation and the partial correlation between age and sentence.

Correlation = +0.826

Partial correlation = +0.7044

The correlation is lower when the intercorrelation with the other five variables is removed.


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correlation

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