Bivariate Regression

CJ 526 Statistical Analysis in Criminal Justice

Regression Towards the Mean -- continued

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

Sir Francis Galton

Regression

1. Prediction

Prediction Equation

1. Karl Pearson, Pearson r correlation coefficient

Bivariate Prediction

1. Uses one variable to make predictions about another variable

Multivariate Prediction

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

Y = a +b1x1+b2x2+b3x3+e

Criterion Variable

1. The variable who 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 Variable

1. The variable whose values are used to make predictions

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

1. Regression line that minimizes the size of errors that are made when using it to make predictions

Predicted Value (Y’)

1. Value of Y that is predicted by the regression line

Error of Prediction

1. e = Y - Y’

Least-Squares Criterion

1. 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

Example

1. Y’ = 3 X + 2

2. X = 4, Y’ = 3(4) + 2 = 14

3. X = 2, Y’ = 3(2) + 2 = 8

Standard Error of Estimate

1. Measure of the average amount of variability of the predictive error

Standard Error of Estimate

21 rSS YYX

Range of Predictive Error

1. 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