Bivariate Regression

CJ 526 Statistical Analysis in Criminal Justice

Regression Towards the Mean

Measure tend to “fall toward” the mean

Tall parents have tall children, but not as tall as themselves

Sir Francis Galton

Regression

1. Prediction: predicting a variable from one or more variables

2. Karl Pearson, Pearson r correlation coefficient, uses one variable to make predictions about another variable (bivariate prediction)

Multivariate Prediction

Uses two or more variables (considered independent variables) to make predictions about another variable

Y = a +b1x1+b2x2+b3x3+e

Criterion Variable

Criterion variable: The variable whose value is predicted

A = a constant, x (1, 2, etc) the independent variables, and b(1,2,) are the slopes. They are standardized and referred to as beta weights

Predictor Variables

1. The variable(s) whose values are used to make predictions

2. Predictions are made based on independent variables which are weighted (by the beta weights) that BEST predict the predictor variable

Regression Line

1. A straight line that an be used to predict the value of the criterion variable from the value of the predictor variable

Line of Best Fit

2. Graphically, the regression line is the line that minimizes the size of errors that are made when using it to make predictions

Predicted Value (Y’)

1. Values of Y that are predicted by the regression line

2. The regression line is the line of best fit, that makes the prediction

3. There will be error

4. Error, or e = Y –Y’

Least-Squares Criterion

The regression line is determined such that the sum of the squared prediction errors for all observations is as small as possible

Regression Equation

1. The equation of a straight line (bivariate, one predictor and one predicted variable)

2. Y’ = 3 X + 2

3. X = 4, Y’ = 3(4) + 2 = 14

4. X = 2, Y’ = 3(2) + 2 = 8

Regression equation

Multiple regression equations an expansion of the equation example above to 2 or more predictor variables to predict a predicted variable

Standard Error of Estimate

Measure of the average amount of variability of the predictive error

Standard Error of Estimate

21 rSS YYX

Range of Predictive Error

SYX becomes smaller as r increases

Multiple regression

Multiple regression can tell us how much variance in a dependent variable is explained by independent variables that are combined into a predictor equation

Collinearity

Very often independent variables are intercorrelated, related to one another

i.e., lung cancer can be predicted from smoking, but smoking is intercorrelated with other factors such as diet, exercise, social class, medical care, etc.

Multiple Regression

One purpose of multiple regression is to determine how much prediction in variability is uniquely due to each IV

Proportion of variance

R squared

The F test can be used to determine the statistical significance of R squared.

SPSS Procedure Regression

Analyze, Regression, Linear Move DV into Dependent Move IV into Independent Method

Enter

Statistics Estimate Model fit R squared change Descriptives

SPSS Procedure Regression Output

Descriptive Statistics Variables Mean Standard Deviation N

Correlations Pearson Correlation Sig (1-tailed) N

SPSS Procedure Regression Output -- continued

Variables Entered/Removed

Model SummaryRR SquareAdjusted R SquareStandard Error of the Estimate

SPSS Procedure Regression Output -- continued

Change StatisticsR Square ChangeF ChangeDf1Df2Sig F Change

SPSS Procedure Regression Output -- continued

ANOVASum of SquaresDfMean SquaresFSig

SPSS Procedure Regression Output -- continued

Coefficients Model

Constant (Y-Intercept) IV

Unstandardized Coefficients B Standard Error of B

Standardized Coefficients Beta

t sig