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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. - PowerPoint PPT Presentation
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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