1. Multiple Regression

2. Regression: A Simplified Example
Let’s find the best-fitting equation for predicting new, as yet unknown scores on Y from scores on X. The regression equation takes the form Y = a + bX + e where Y is the dependent or criterion variable we’re trying to predict, a is the intercept or point where the regression line crosses the Y axis, X is the independent or predictor variable, b is the weight by which we multiply the value of X (it is the slope of the regression line, and is how many units Y increases (decreases) for every unit change in X), and e is an error term (basically an estimate of how much our prediction is “off”). a and b are often called “regression coefficients. When Y is an estimated value it is usually symbolized as Y’
X (predictor)
Y (criterion) 3 14 4 18 2 10 1 6 5 22 26

3. Finding the Regression Line with SPSS
First let’s use a scatterplot to visualize the relationship between X and Y. The first thing we notice is that the points appear to form a straight line and that that as X gets larger, Y gets larger, so it would appear that we have a strong, positive relationship between X and Y. Based on the way the points seem to fall, what do you think the value of Y would be for a person who obtained a score of 7 on X?

4. Fitting a Line to the Scatterplot
Next let’s fit a line to the scatterplot. Note that the points appear to be fit well by the straight line, and that the line crosses the Y axis (at the point called the intercept, or the constant a in our regression equation) at about the point y = 2. So it’s a good guess that our regression equation will be something like y = 2 + some positive multiple of X, since the values of Y look to be about 4-5 times the size of X

5. The Least Squares Solution to Finding the Regression Equation
Mathematically, the regression equation is that combination of constant and weights b on the predictors (the X’s) which minimizes the sum, across all subjects, of the squared differences between their predicted scores (e.g. the scores they would get if the regression equation were doing the predicting) and the obtained scores (their actual scores) on the criterion Y (that is, it minimizes the error sum of squares or residuals). This is known as the least squares solution
*SPSS uses R in the regression output even if there is only one predictor

6. Using SPSS to Calculate the Regression Equation
Download the Data File simpleregressionexample.sav and open it in SPSS
In Data Editor, we will go to Analyze/ Regression / Linear and move X into the Independent box (in regression the Independent variables are the predictor variables) and move Y into the dependent box and click OK. The dependent variable, Y, is the one for which we are trying to find an equation that will predict new cases of Y given than we know X

7. Obtaining the Regression Equation from the SPSS Output
This table gives us the regression coefficients. Look in the column called unstandardized coefficients. There are two values of β provided. The first one, labeled the constant, is the intercept a, or the point at which the regression line crosses the y axis. The second one, X, is the unstandardized regression weight or the b from our regression equation. So this output tells us that the best-fitting equation for predicting Y from X is Y = 2 + (4)X. Let’s check that out with a known value of X and Y. According to the equation, if X is 3, Y should be 2 + 4(3), or 14. How about when X = 5? X Y 3 14 4 18 2 10 1 6 5 22 26
The constant representing the intercept is the value that the dependent variable would take when all the predictors are at a value of zero. In some treatments this is called B0 instead of a

8. What is the Regression Equation when the Scores are in Standard (Z) Units?
When the scores on X and Y have been converted to Z scores, then the intercept disappears (because the two sets of scores are expressed on the same scale) and the equation for predicting Y from X just becomes Y = BetaX, where Beta is the standardized coefficient reported in your SPSS regression procedure output
In the bivariate case, where there is only one X and one Y, the
standardized beta weight will equal the correlation coefficient. Let’s confirm this by seeing what would happen if we convert our raw scores to Z scores

9. SPSS: 1) analyze, 2) regression, 3) linear

11. SPSS Screen

12. Interpret the coefficients
SPSS Output
Interpret the coefficients

13. What does the ANOVA result mean?
SPSS Output
Interpret the r square
What does the ANOVA result mean?

14. Finding the Regression Equation
Go to Analyze/ Regression/ Linear
Move the Average Female Life Expectancy variable into the dependent box and the Daily Calorie Intake variable into the independent box
Under Options, make sure “include constant in equation” is checked and click Continue
Under Statistics, Check Estimates, Confidence intervals, and Model Fit. Click Continue and then OK
Compare your output to the next slide