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Correlation
CJ 526 Statistical Analysis in Criminal Justice
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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
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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
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GPA
ACT
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Characteristics of a Relationship
1. Direction (sign)1. +: Positive
2. -: Negative
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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
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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
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Magnitude -- continued
Zero correlation1. Result of no systematic relationship between X
and Y
2. Knowing X would be of no value in predicting Y
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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
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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
1. Form:
Linear and non-linear relationships
Linear: every change in X is accompanied by a corresponding change in Y
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Nonlinear Relationship
1. No linear relationship1. A change in X does not correspond to any
predictable change in Y
Example: 0 correlation
Parabola
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Nonlinear Relationships
1. Exponential1. Time and retention
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Retention
Time
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Performance
Arousal
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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
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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)
1. divided by the variations in X and the variation in Y
2. See p. 462 for the computational formula
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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)
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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
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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
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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
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Coefficient of Nondetermination
Proportion of variance of Y that is not explained or accounted for by the variance of X
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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
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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, p < .01).