CJ 526 Statistical Analysis in Criminal Justice Correlation CJ 526 Statistical Analysis in Criminal Justice

Introduction Correlation:

Correlation and Prediction If a relationship exists between two variables

Correlation and Ex Post Facto Designs Usually used with ex post facto designs No manipulation of independent variable by the researcher

Requirements for Correlation Requires two scores for each unit of analysis: X Y

Scatterplot Graphical representation of relationship between the two variables

GPA ACT

Characteristics of a Relationship Direction (sign) +: Positive -: Negative

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

Direction -- continued Negative As one variable increases, the other decreases Scatterplot goes to the left

Magnitude Strength

Magnitude -- continued Closer to 1, stronger the relationship Less predictive error

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

Magnitude -- continued Perfect correlations can be positive or negative

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 Form: Linear and non-linear relationships

Linear Relationship Linear relationship Every change in X is accompanied by a corresponding change in Y

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

Nonlinear Relationships Exponential Time and retention

Retention Time

Performance Arousal

Use of Correlation Reliability Test-retest and split-half

Pearson Product-Moment Correlation 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) divided by

Correlation and Causality Correlation does not imply causality

Criteria for Causality Relationship between X (presumed cause) and Y (effect)

Poverty and Crime Poverty and crime are related

Factors Affecting Pearson Correlation Restricted range Could overestimate or underestimate

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

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

SPSS Procedure Graphs Use to generate scatterplot Graphs, Scatter 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)

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, n = 32, p < .01, two tails.