Introduction to Bivariate Regression

Purpose of regression line:

1. To estimate the average value of the dependent variable at any specified value of the independent variable

2. To predict the value of the dependent variable, given that individual's value for the independent variable

Slope interpretation: the slope of the regression line tells us the amount on average that the predicted dependent variable changes when the independent variable changes by 1 unit.

Intercept interpretation: the intercept tells us the average value of the predicted dependent variable when the independent variable equals 0.

Using SPSS to obtain the Regression Equation

1. Analyse --> regression --> linear

Output

We can find the slope and intercept of the regression equation in the table Coefficients.

intercept (constant) = -42.35

slope = 0.59

Predicted Weight =

-42.35 + 0.59 * Height

We can also see the Pearson's r for the relationship between height (IV) and weight (DV) at the Standardised Coefficient column. The Pearson's r is 0.84

Bivariate regression/Multiple r (other words 1 IV and 1 DV) is also given. This is found in the Model Summary.

Multiple R: correlation between the observed value of the dependent variable and the value predicted by the regression equation.

Multiple R is used in multiple regression where we have more than one independent variable as a predictor. As such multiple R is a gauge of how well the overall model predicts the observed data. Hence we find it in the Model Summary.

Predictions and Residuals

• The regression equation can be used to predict the value of the dependent variable.

• Prediction error is referred to as "residual".

• Residual = observed value - predicted value

• The least squares regression line minimises the sum of squared residuals for the observed data set. In SPSS this basically just means that it will make a straight line with the smallest vertical residuals.

• This means that if we switch the DV and IV, there will be a new regression equation.