Correlation

CJ 526 Statistical Analysis in Criminal Justice

Correlation and Prediction

1. If a relationship exists between two variables

2. Usually used with ex post facto designed

3. No manipulation of an IV by the researcher

Requirements for Correlation

1. Requires two scores for each unit of analysis:

1. X

2. Y

Represented by a scatterplot

Graphical representation of relationship between the two variables

GPA

ACT

Characteristics of a Relationship

1. Direction (sign)1. +: Positive

2. -: Negative

Direction

1. Positive

As one variable increases, the other increasesScatterplot goes to the right

Negative

As one variable increases, the other decreases

Scatterplot goes to the left

Magnitude

1. Strength of a relationship

Closer to 1 or to -1, stronger the relationship

Less predictive error

Closer to 0, the weaker the relationship

More error in prediction

Magnitude -- continued

Zero correlation1. Result of no systematic relationship between X

and Y

2. Knowing X would be of no value in predicting Y

Magnitude -- continued

Perfect correlations can be positive or negative

Strong relationships can be either positive or negative

The negative sign only indicates the direction of the relationship, not the strength or ability to predict

Interpretation Heuristic for Magnitude: Positive Correlation

Correlation Coefficient Range Description

0 to 0.4

0 to -.4

No to weak relationship

0.4 to 0.8

-.4 to -.8

Moderate relationship

0.8 to 1.0

-.8 to -1.0

Strong relationship

Form

1. Form:

Linear and non-linear relationships

Linear: every change in X is accompanied by a corresponding change in Y

Nonlinear Relationship

1. No linear relationship1. A change in X does not correspond to any

predictable change in Y

Example: 0 correlation

Parabola

Nonlinear Relationships

1. Exponential1. Time and retention

Retention

Time

Performance

Arousal

Use of Correlation

1. Reliability

Test-retest and split-half

2. Personality

Correlating test scores on personality tests: scales with similar traits should have high correlations, and scales with differing or opposite traits should have lower correlations

Pearson Product-Moment Correlation

1. Measures the direction and strength of the linear relationship between two variables

Pearson Product-Moment Correlation -- continued

degree to which X and Y vary together (covariance)

1. divided by the variations in X and the variation in Y

2. See p. 462 for the computational formula

Correlation and Causality

Correlation does not imply causality

Cause requires 3 criteria: (1) temporal; (2) correlation; and (3) nonspuriousness—relationship cannot be explained by a third variable

Cause: relationship between x (presumed cause) and Y (effect)

Poverty and Crime

1. Poverty and crime are related, as arrest statistics indicate

Does poverty “cause” crime? There are poor people who do not commit crime and non-poor people who do

Factors Affecting Pearson Correlation

Restricted range1. Could overestimate or underestimate

Example

The correlation between ACT and GPA will be much lower if you look at the range between 24 and 30

Interpreting Correlation in Terms of Variance

Coefficient of Determination1. Proportion of variance of Y that is explained or

accounted for by the variance of X

r squared

Coefficient of Nondetermination

Proportion of variance of Y that is not explained or accounted for by the variance of X

r r2%

Explained 1 - r2%

Unexplained0.0 0.0 0 1.0 100.2 .04 4 .96 96.4 .16 16 .84 84.6 .36 36 .64 64.8 .64 64 .36 36.9 .81 81 .19 19

SPSS Procedure Graphs

• Use to generate scatterplot– Determine whether the relationship is linear

• Graphs, Scatter– Simple

• Define

SPSS Procedure Correlate

• Analyze, Correlate, Bivariate– Move variables over– Options

• Statistics– Means and standard deviations

SPSS Procedure Correlate Output

• Descriptive Statistics– Variables

– Mean

– Standard Deviation

– N

• Correlations– Pearson Correlation

– Sig (2-tailed)

– N

Hypothesis Tests With Pearson Correlations

• H0: The population correlation is zero

• H1: The population correlation is non-zero

(rho)

• df = N - 2

Report Writing

• A correlation for the data revealed that population and crime rate were significantly related, (r = .97, p < .01).