Correlation

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Correlation

• CJ 526 Statistical Analysis in Criminal Justice

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Introduction

1. Correlation

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Correlation and Prediction

1. If a relationship exists between two variables

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Correlation and Ex Post Facto Designs

1. Usually used with ex post facto designs
2. No manipulation of independent variable by the
   researcher

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Requirements for Correlation

1. Requires two scores for each unit of analysis
2. X
3. Y

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Scatterplot

1. Graphical representation of relationship between
   the two variables

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GPA
ACT
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Characteristics of a Relationship

1. Direction (sign)
2. Positive
3. - Negative

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Direction

• Positive
• As one variable increases, the other increases
• Scatterplot goes to the right

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Direction -- continued

• Negative
• As one variable increases, the other decreases
• Scatterplot goes to the left

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Magnitude

1. Strength

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Magnitude -- continued

• Closer to 1, stronger the relationship
• Less predictive error

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Magnitude -- continued

• Zero correlation
• Result of no systematic relationship between X
  and Y
• Knowing X would be of no value in predicting Y

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Magnitude -- continued

1. Perfect correlations can be positive or negative

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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
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Form

• Form
• Linear and non-linear relationships

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Linear Relationship

1. Linear relationship
2. Every change in X is accompanied by a
   corresponding change in Y

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Nonlinear Relationship

• No linear relationship
• A change in X does not correspond to any
  predictable change in Y
• Example 0 correlation
• Parabola

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Nonlinear Relationships

1. Exponential
2. Time and retention

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Retention
Time
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Performance
Arousal
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Use of Correlation

• Reliability
• Test-retest and split-half

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Pearson Product-Moment Correlation

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

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Pearson Product-Moment Correlation -- continued

• degree to which X and Y vary together
  (covariance)
• divided by

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Correlation and Causality

1. Correlation does not imply causality

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Criteria for Causality

1. Relationship between X (presumed cause) and Y
   (effect)

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Poverty and Crime

1. Poverty and crime are related

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Factors Affecting Pearson Correlation

1. Restricted range
2. Could overestimate or underestimate

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Interpreting Correlation in Terms of Variance

• Coefficient of Determination
• Proportion of variance of Y that is explained or
  accounted for by the variance of X
• R squared

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Coefficient of Nondetermination

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

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SPSS Procedure Graphs

• Use to generate scatterplot
• Determine whether the relationship is linear
• Graphs, Scatter
• Simple
• Define

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SPSS Procedure Correlate

• Analyze, Correlate, Bivariate
• Move variables over
• Options
• Statistics
• Means and standard deviations

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SPSS Procedure Correlate Output

• Descriptive Statistics
• Variables
• Mean
• Standard Deviation
• N
• Correlations
• Pearson Correlation
• Sig (2-tailed)
• N

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Hypothesis Tests With Pearson Correlations

• H0 The population correlation is zero
• H1 The population correlation is non-zero
• ? (rho)
• df N - 2

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Report Writing

• A correlation for the data revealed that
  population and crime rate were significantly
  related, r .97, n 32, p lt .01, two tails.