2. Simple & Multiple Regression 1: Simple Regression - Prediction models 1

3. r =.81 68 ? ? ? Regression techniques allow us to do this Suppose we wanted to predict the weight of a person who was 68in tall? Let’s take our scatterplot as a start… 1

4. r =.81 We use a method of least squares estimation (cue statistical hocus pocus music)… And we generate a line through the data so that all deviations (vertical) between the line and the data points are minimized 1 2

5. r =.81 This line will have a certain slope… …brings a change in weight… A change in height… SLOPE And it will have a value on the y-axis for the zero value of the x-axis -234 INTERCEPT 1 2 3

6. The intercept can be seen more clearly if we redraw the graph with appropriate axes… -234lbs 1 2

7. 68 r =.81 From the line, we can predict that an increase in height of 1 inch should be accompanied by a rise in weight of 5.434lbs. We can also find the expected weight for a person of 68in height. 135lbs Using regression to make predictions… 1 2 3 4

8. From this data file Where is this in SPSS, and what is this going to look like elsewhere? 1

9. Choose this analysis Where is this in SPSS, and what is this going to look like elsewhere? 1

10. Specify dependent and independent variables Where is this in SPSS, and what is this going to look like elsewhere? 1

11. SPSS output: SLOPE INTERCEPT Where is this in SPSS, and what is this going to look like elsewhere? 1

12. And how about Excel?  Excel’s regression function can be accessed via the wizard, but it still needs some extra knowledge to get it to work, so I’m just going to show you the muggle (non-wizard) way 1

13. 1. Select a 2 (columns) by 5 (rows) array And how about Excel? 1

14. 2. Use the “linest” (linear estimate) function 3. The first array is the dependent variable 4. The second array is the independent variable 5. After 2 commas, “true” means you want all the stats Excel… 1

15. 6. Hit [CTRL_SHIFT_ENTER] at end of function – NOT enter… …and here’s all the stuff slope intercept R2R2 F Excel… 1

16. General form of equation: Y’ = a + bX SLOPE INTERCEPT Weight’ = -234 + 5.434 (Height) Predicted values of the d.v. values of the i.v. (predictor) The regression equation 1 2 3 4

17. A note on the equation and error  Here is another general form of the equation from a text book: Don’t be confused by this…it’s obvious really. It’s the error term. Note “actual” y, rather than predicted y, is on the left For an actual value y… 1

18. A note on the equation and error The least squares method used in regression just minimizes the sum of these squared vertical distances e1e1 e2e2 e3e3 e4e4 e5e5 e6e6 e7e7 1 2 3 4

19. How good, generally, is the fit?  R 2  Coefficient of determination  Standard error of the estimate  The average size of the error in predicting any value of Y  The standard deviation of actual Y’s about predicted Y’s  Or, the SD of the “e’s” (residuals)  Critically related to R 2 1 2 3 4 5 6

20. r =.81 More on the SE of estimate  At any point of X, the various Y’s are expected to be normally distributed about the regression line 1 2 3 Height = 63”

21. More on the SE of estimate  That means that you can set up expected margins of error of Y about Y’  E.G. What proportion of Y’ would fit within 2 standard errors of the estimate?  ??  All depends upon key assumptions…  Homoscedasticity  Linear relationship between X and Y  Y normally distributed about Y’ 1 2 3

22. Time for a break… KNR 445 Regression: Deep stuff - slide 21 1