Fundamental Statistics for the Behavioral Sciences, 4th edition Chapter 10 Regression Fundamental Statistics for the Behavioral Sciences, 4th edition David C. Howell ©1999 Brooks/Cole Publishing Company/ITP     

Major Points The basic problem An example The regression line Chapter 10 Regression Major Points The basic problem An example The regression line Accuracy of prediction Hypothesis testing Review questions

The Basic Problem How do we predict one variable from another? Chapter 10 Regression The Basic Problem How do we predict one variable from another? How does one variable change as the other changes? Cause and effect

An Example Cigarettes and CHD Mortality from Chapter 9 Chapter 10 Regression An Example Cigarettes and CHD Mortality from Chapter 9 Data repeated on next slide We want to predict level of CHD mortality in a country averaging 20 cigarettes per day. Landwehr, J.M. & Watkins, A.E. (1987) Exploring Data: Teacher’s Edition. Palo Alto, CA: Dale Seymour Publications.

Chapter 10 Regression The Data

Chapter 10 Regression

Why Do We Care? We may want to make a prediction. Chapter 10 Regression Why Do We Care? We may want to make a prediction. More likely, we want to understand the relationship. How fast does CHD mortality rise with a one unit increase in smoking? Note we speak about predicting, but often don’t actually predict.

Regression Line Formula  = the predicted value of Y (CHD mortality) Chapter 10 Regression Regression Line Formula  = the predicted value of Y (CHD mortality) X = smoking incidence for that country

Regression Coefficients Chapter 10 Regression Regression Coefficients “Coefficients” are a and b b = slope Change in predicted Y for one unit change in X a = intercept value of  when X = 0

Chapter 10 Regression Calculation Slope Intercept

For Our Data CovXY = 11.13 s2X = 2.332 = 5.43 b = 11.13/5.43 = 2.04 Chapter 10 Regression For Our Data CovXY = 11.13 s2X = 2.332 = 5.43 b = 11.13/5.43 = 2.04 a = 14.52 - 2.04*5.95 = 2.37 See SPSS printout on next slide Answers are not exact due to rounding error and desire to match SPSS.

Chapter 10 Regression SPSS Printout

Note: The values we obtained are shown on printout. Chapter 10 Regression Note: The values we obtained are shown on printout. The intercept is labeled “constant.” Slope is labeled by name of predictor variable.

Chapter 10 Regression Making a Prediction Assume that we want CHD mortality when cigarette consumption of 6. We predict 14.61 people/10,000 in that country will die of coronary heart disease.

Accuracy of Prediction Chapter 10 Regression Accuracy of Prediction Finnish smokers smoke 6 cigarettes/adult/day We predict 14.61 deaths/10,000 They actually have 23 deaths/10,000 Our error (“residual”) = 23 - 14.61 = 8.39 a large error

Chapter 10 Regression Residual Prediction

Errors of Prediction Residual variance Standard error of estimate Chapter 10 Regression Errors of Prediction Residual variance The variability of predicted values Standard error of estimate The standard deviation of predicted values

Standard Error of Estimate Chapter 10 Regression Standard Error of Estimate A common measure of the accuracy of our predictions We want it to be as small as possible.

r 2 as % Predictable Variability Chapter 10 Regression r 2 as % Predictable Variability Define Sum of Squares The remaining error divided by the original error

Chapter 10 Regression For Our Data r = .713 r 2 = .7132 =.508 Approximately 50% in variability of incidence of CHD mortality is associated with variability in smoking. Elaborate on what this means.

Hypothesis Testing Null hypotheses Chapter 10 Regression Hypothesis Testing Null hypotheses b* = 0 a* = 0 Define b* and a* population correlation () = 0 We saw how to test the last one in Chapter 9.

Testing Slope and Intercept Chapter 10 Regression Testing Slope and Intercept These are given in computer printout as a t test.

Chapter 10 Regression Testing The t values in the second from right column are tests on slope and intercept. The associated p values are next to them. The slope is significantly different from zero, but not the intercept. Why do we care? Cont.

Testing--cont. What does it mean if slope is not significant? Chapter 10 Regression Testing--cont. What does it mean if slope is not significant? How does that relate to test on r? What if the intercept is not significant? Does significant slope mean we predict quite well?

Review Questions How does regression differ from correlation? Chapter 10 Regression Review Questions How does regression differ from correlation? If the slope is negative, what does that tell us about the sign of r? Why do we call Y-  a residual? What does it mean to say that r 2 = % variation accounted for? Cont.

Review Questions--cont. Chapter 10 Regression Review Questions--cont. If the slope is significant, what does that tell us about the significance of r? What does a nonsignificant slope tell us?