2. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. Chapter 13 Multiple Regression Section 13.3 Using Multiple Regression to Make Inferences

3. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 3 Inferences about the Population Assumptions required when using a multiple regression model to make inferences about the population:  The regression equation truly holds for the population means. This implies that there is a straight-line relationship between the mean of y and each explanatory variable, with the same slope at each value of the other predictors.

4. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 4 Assumptions required when using a multiple regression model to make inferences about the population:  The data were gathered using randomization.  The response variable y has a normal distribution at each combination of values of the explanatory variables, with the same standard deviation. Inferences about the Population

5. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 5 Inferences about Individual Regression Parameters Consider a particular parameter,  If, the mean of y is identical for all values of, at fixed values of the other explanatory variables.  So, states that y and are statistically independent, controlling for the other variables.  This means that once the other explanatory variables are in the model, it doesn’t help to have in the model.

6. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 6 SUMMARY: Significance Test about a Multiple Regression Parameter 1. Assumptions:  Each explanatory variable has a straight-line relation with with the same slope for all combinations of values of other predictors in the model.  Data gathered using randomization.  Normal distribution for y with same standard deviation at each combination of values of other predictors in model.

7. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 7 2. Hypotheses:  When is true, y is independent of, controlling for the other predictors. SUMMARY: Significance Test about a Multiple Regression Parameter

8. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 8 3. Test Statistic: where se is the standard error for b 1 SUMMARY: Significance Test about a Multiple Regression Parameter

9. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 9 4. P-value: Two-tail probability from t-distribution of values larger than observed t test statistic (in absolute value). The t-distribution has: df = n – number of parameters in the regression equation SUMMARY: Significance Test about a Multiple Regression Parameter

10. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 10 5. Conclusion: Interpret P-value in context; compare to significance level if decision needed. SUMMARY: Significance Test about a Multiple Regression Parameter

11. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 11 Example: What Helps Predict a Female Athlete’s Weight? The “College Athletes” data set comes from a study of 64 University of Georgia female athletes. The study measured several physical characteristics, including total body weight in pounds (TBW), height in inches (HGT), the percent of body fat (%BF) and age.

12. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 12 The results of fitting a multiple regression model for predicting weight using the other variables: Table 13.10 Multiple Regression Analysis for Predicting Weight Predictors are HGT = height, %BF = body fat, and age of subject. Example: What Helps Predict a Female Athlete’s Weight?

13. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 13 Interpret the effect of age on weight in the multiple regression equation: Example: What Helps Predict a Female Athlete’s Weight?

14. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 14 The slope coefficient of age is -0.96. For athletes having fixed values for and, the predicted weight decreases by 0.96 pounds for a 1-year increase in age, and the ages vary only between 17 and 23. Example: What Helps Predict a Female Athlete’s Weight?

15. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 15 Run a hypothesis test to determine whether age helps to predict weight, if you already know height and percent body fat. Here are the steps: 1. Assumptions Met:  The 64 female athletes were a convenience sample, not a random sample.  Caution should be taken when making inferences about all female college athletes. Example: What Helps Predict a Female Athlete’s Weight?

16. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 16 2. Hypotheses:  3. Test statistic: Example: What Helps Predict a Female Athlete’s Weight?

17. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 17 4. P-value: This value is reported in the output as 0.144. 5. Conclusion: The P-value of 0.144 does not give much evidence against the null hypothesis that.  Age may not significantly predict weight if we already know height and % body fat. Example: What Helps Predict a Female Athlete’s Weight?

18. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 18 Confidence Interval for a Multiple Regression Parameter A 95% confidence interval for a slope parameter in multiple regression equals: The t-score has df = (n - # of parameters in the model). The assumptions are the same as for the t test.

19. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 19 Construct and interpret a 95% CI for, the effect of age while controlling for height and % body fat Confidence Interval for a Multiple Regression Parameter

20. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 20 At fixed values of and, we infer that the population mean of weight changes very little (and maybe not at all) for a 1 year increase in age.  The confidence interval contains 0.  Age may have no effect on weight, once we control for height and % body fat. Confidence Interval for a Multiple Regression Parameter

21. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 21 Estimating Variability Around the Regression Equation A standard deviation parameter,, describes variability of the observations around the regression equation. Its sample estimate is:

22. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 22  Anova Table for the “college athletes” data set: Example: What Helps Predict a Female Athletes’ Weight? Table 13.7 ANOVA Table for Multiple Regression Analysis of Athlete Weights

23. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 23 For female athletes at particular values of height, % of body fat, and age, estimate the standard deviation of their weights.  Begin by finding the Mean Square Error: Notice that this value (102.2) appears in the MS column in the ANOVA table. Example: What Helps Predict a Female Athletes’ Weight?

24. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 24  The standard deviation is:  This value is also displayed in the ANOVA table.  For athletes with certain fixed values of height, % body fat, and age, the weights vary with a standard deviation of about 10 pounds. Example: What Helps Predict a Female Athletes’ Weight?

25. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 25 Insight: The conditional distributions of weight are approximately bell-shaped, about 95% of the weight values fall within about 2s = 20 pounds of the true regression equation. Example: What Helps Predict a Female Athletes’ Weight?

26. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 26 The Collective Effect of Explanatory Variables Example: With 3 predictors in a model, we can check this by testing:

27. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 27 The test statistic for is denoted by F. It equals the ratio of the mean squares from the ANOVA table, The Collective Effect of Explanatory Variables

28. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 28  When is true, the expected value of the F test statistic is approximately 1.  When is false, F tends to be larger than 1.  The larger the F test statistic, the stronger the evidence against. The Collective Effect of Explanatory Variables

29. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 29 SUMMARY: F Test That All Beta Parameters = 0 1. Assumptions:  Multiple regression equation holds  Data gathered randomly  Normal distribution for y with same standard deviation at each combination of predictors

30. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 30 2. Hypothesis: 3. Test statistic: SUMMARY: F Test That All Beta Parameters = 0

31. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 31 4. P-value: Right-tail probability above observed F-test statistic value from F distribution with:  df1 = number of explanatory variables  df2 = n – (number of parameters in regression equation) SUMMARY: F Test That All Beta Parameters = 0

32. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 32 5. Conclusion: The smaller the P-value, the stronger the evidence that at least one explanatory variable has an effect on y.  If a decision is needed, reject if P-value significance level, such as 0.05. SUMMARY: F Test That All Beta Parameters = 0

33. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 33 Example: What Helps Predict a Female Athlete’s Weight? For the 64 female college athletes, the regression model for predicting y = weight using = height, = % body fat and = age is summarized in the ANOVA table on the next slide.

34. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 34 Anova table for the “college athletes” data set: Table 13.7 ANOVA Table for Multiple Regression Analysis of Athlete Weights Example: What Helps Predict a Female Athlete’s Weight?

35. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 35 Use the output in the ANOVA table to test the hypothesis: Example: What Helps Predict a Female Athlete’s Weight?

36. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 36  The observed F statistic is 40.5  The corresponding P-value is 0.000  We can reject H 0 at the 0.05 significance level In summary, we conclude that at least one predictor has an effect on weight. Example: What Helps Predict a Female Athlete’s Weight?

37. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 37 Insight:  The F-test tells us that at least one explanatory variable has an effect.  If the explanatory variables are chosen sensibly, at least one should have some predictive power.  The F-test result tells us whether there is sufficient evidence to make it worthwhile to consider the individual effects, using t-tests. Example: What Helps Predict a Female Athlete’s Weight?

38. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 38 The individual t-tests identify which of the variables are significant (controlling for the other variables) Table 13.10 Multiple Regression Analysis for Predicting Weight Predictors are HGT = height, %BF = body fat, and age of subject. Example: What Helps Predict a Female Athlete’s Weight?

39. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 39  If a variable turns out not to be significant, it can be removed from the model.  In this example, ‘age’ can be removed from the model. Example: What Helps Predict a Female Athlete’s Weight?