LINEAR REGRESSION: Evaluating Regression Models

Overview

• Assumptions for Linear Regression• Evaluating a Regression Model

Assumptions for Bivariate Linear Regression

• Quantitative data (or dichotomous)• Independent observations• Predict for same population that was

sampled

Assumptions for Bivariate Linear Regression

• Linear relationship– Examine scatterplot

• Homoscedasticity – equal spread of residuals at different values of predictor– Examine ZRESID vs ZPRED plot

Checking for Homoscedasticity

Assumptions for Bivariate Linear Regression

• Independent errors– Durbin Watson should be close to 2

• Normality of errors– Examine frequency distribution of residuals

Checking for Normality of Errors

Influential Cases

• Influential cases have greater impact on the slope and y-intercept

• Select casewise diagnostics and look for cases with large residuals

Standard Error of the Estimate

• Index of how far off predictions are expected to be

• Larger r means smaller standard error• Standard deviation of y scores around

predicted y scores

Sums of Squares

• Total SS – total squared differences of Y scores from the mean of Y

• Model SS – total squared differences of predicted Y scores from the mean of Y

• Residual SS – total squared differences of Y scores from predicted Y scores

Coefficient of Determination

• r2 is the proportion of variance in Y explained by X

• Adjusted r2 corrects for the fact that the r2 often overestimates the true relationship. Adjusted r2 will be lower when there are fewer subjects.

Goodness of Fit

• Dividing the Model SS by the Total SS produces r2

• The ANOVA F-test determines whether the regression equation accounted for a significant proportion of variance in Y

• F is the Model Mean Square divided by the Residual Mean Square

Coefficients

• The Constant B under “unstandardized” is the y-intercept b0

• The B listed for the X variable is the slope b1

• The t test is the coefficient divided by its standard error

• The standardized slope is the same as the correlation

Example of Reporting a Regression Analysis

The linear regression for predicting quiz enjoyment from level of statistics anxiety did not account of a significant portion of variance, F(1, 24) = 1.75, p = .20, r2 = .07.

Take-Home Points

• The validity of a regression procedure depends on multiple assumptions.

• A regression model can be evaluated based on whether and how well it predicts an outcome variable.