1. Statistics: Unlocking the Power of Data Lock 5 STAT 250 Dr. Kari Lock Morgan Multiple Regression SECTIONS 10.1, 10.3 (?) Multiple explanatory variables (10.1, 10.3)

2. Statistics: Unlocking the Power of Data Lock 5 Today we’ll finally learn a way to handle more than 2 variables! More than 2 variables!

3. Statistics: Unlocking the Power of Data Lock 5

4. Multiple regression extends simple linear regression to include multiple explanatory variables: Each x is a different explanatory variable k is the number of explanatory variables Multiple Regression

5. Statistics: Unlocking the Power of Data Lock 5 Predicting Body Fat Percentage The percentage of a person’s weight that is made up of body fat is often used as an indicator of health and fitness Accurate measures of percent body fat at difficult to implement For example, you can immerse the body in water to estimate density, then apply a formula Another option: build a model predicting % body fat based on easy to obtain measurements

6. Statistics: Unlocking the Power of Data Lock 5 Body Fat Data Measurements were collected on 100 men Response variable: percent body fat Explanatory variables:  Age (in years)  Weight (in pounds)  Height (in inches)  Neck circumference (in cm)  Chest circumference (in cm)  Abdomen circumference (in cm)  Ankle circumference (in cm)  Biceps circumference (in cm)  Wrist circumference (in cm) A sample taken from data provided by Johnson R., "Fitting Percentage of Body Fat to Simple Body Measurements," Journal of Statistics Education, 1996,

7. Statistics: Unlocking the Power of Data Lock 5 Predicting Percent Body Fat We’ll start with just three explanatory variables and fit the model: Bodyfat = 49.6 + 0.1653 Age + 0.2264 Weight - 1.117 Height What can we do with this?  Make predictions  Interpret coefficients  Inference  Interpret R 2  and much, more more!

8. Statistics: Unlocking the Power of Data Lock 5 Making Predictions Bodyfat = 49.6 + 0.1653 Age + 0.2264 Weight - 1.117 Height If you are male, you can use this to predict your percent body fat! Age: years, weight: pounds, height: inches

9. Statistics: Unlocking the Power of Data Lock 5 Percent Body Fat

10. Statistics: Unlocking the Power of Data Lock 5 Interpreting Coefficients Bodyfat = 49.6 + 0.1653 Age + 0.2264 Weight - 1.117 Height Intercept: a man 0 years old, weighs 0 lbs, and is 0 inches tall would have 49.6% body fat Slope: Keeping weight and height constant, percent body fat increases by 0.1653 for every additional year Keeping age and height constant, percent body fat increases by 0.2264 for every additional pound

11. Statistics: Unlocking the Power of Data Lock 5 Interpreting Coefficients Bodyfat = 49.6 + 0.1653 Age + 0.2264 Weight - 1.117 Height Which of the following is a correct interpretation? a) Keeping age and weight constant, height decreases by 1.117 for every additional percent of body fat b) Keeping age and weight constant, percent body fat decreases by 1.117 for every additional inch c) Predicted body fat decreases by 1.117 for every additional inch

12. Statistics: Unlocking the Power of Data Lock 5 Inference Are our explanatory variables significant predictors? All of the p-values corresponding to the explanatory variables are very small Age, weight, and height are all significant predictors of percent body fat (given the other variables in the model)

13. Statistics: Unlocking the Power of Data Lock 5 Explaining Variability How much of the variability in percent body fat is explained by this model? Which of the following would tell us this? a) p-value b) correlation c) slope coefficients d) R 2 e) confidence interval

14. Statistics: Unlocking the Power of Data Lock 5 Explaining Variability About 55% of the variability in percent body fat is explained by age, weight, and height Can we do better?

15. Statistics: Unlocking the Power of Data Lock 5 Comparing with BMI BMI is used more commonly than percent body fat because it is easy to calculate Currently, our predicted percent body fat is not using much more information than BMI (just age as an extra predictor) What’s wrong with body mass index (BMI) as a indicator of health and fitness? How might we improve our model to fix this problem?

16. Statistics: Unlocking the Power of Data Lock 5 New Model Bodyfat = -55.9 + 0.0067 Age - 0.1724 Weight + 0.099 Height + 1.066 Abdomen Anything look odd about this equation??? Model without Abdomen: Bodyfat = 49.6 + 0.1653 Age + 0.2264 Weight - 1.117 Height

17. Statistics: Unlocking the Power of Data Lock 5 Significance Which explanatory variable(s) are significant? a) All of them – age, weight, height, abdomen b) Weight and height c) Weight, height, abdomen d) Weight and abdomen e) Abdomen only

18. Statistics: Unlocking the Power of Data Lock 5 Multiple Regression The coefficient for each explanatory variable is the predicted change in y for one unit change in x, given the other explanatory variables in the model! The p-value for each coefficient indicates whether it is a significant predictor of y, given the other explanatory variables in the model! If explanatory variables are associated with each other, coefficients and p-values will change depending on what else is included in the model

19. Statistics: Unlocking the Power of Data Lock 5 Full Model

20. Statistics: Unlocking the Power of Data Lock 5 Which explanatory variable(s) are significant? a) All of them b) Weight and abdomen c) Neck only d) Abdomen and wrist

21. Statistics: Unlocking the Power of Data Lock 5 Insignificant Terms What should we do with the insignificant variables? Keep them in the model? Take them out? ??? Deciding which variables to keep in the model (variable selection) is an entire subfield of statistics, and beyond the scope of this class Want to learn more about it? Take STAT 462!

22. Statistics: Unlocking the Power of Data Lock 5 Electricity and Life Expectancy Cases: countries of the world Response variable: life expectancy Explanatory variable: electricity use (kWh per capita) Is a country’s electricity use helpful in predicting life expectancy?

23. Statistics: Unlocking the Power of Data Lock 5 Electricity and Life Expectancy

24. Statistics: Unlocking the Power of Data Lock 5 Electricity and Life Expectancy

25. Statistics: Unlocking the Power of Data Lock 5 Electricity and Life Expectancy

26. Statistics: Unlocking the Power of Data Lock 5 Electricity and Life Expectancy Is a country’s electricity use helpful in predicting life expectancy? (a) Yes (b) No

27. Statistics: Unlocking the Power of Data Lock 5 Electricity and Life Expectancy If we increased electricity use in a country, would life expectancy increase? (a) Yes (b) No (c) Impossible to tell

28. Statistics: Unlocking the Power of Data Lock 5 Confounding Variables Wealth is an obvious confounding variable that could explain the relationship between electricity use and life expectancy Multiple regression is a powerful tool that allows us to account for confounding variables We can see whether an explanatory variable is still significant, even after including potential confounding variables in the model

29. Statistics: Unlocking the Power of Data Lock 5 Electricity and Life Expectancy Is a country’s electricity use helpful in predicting life expectancy, even after including GDP in the model? (a) Yes(b) No

30. Statistics: Unlocking the Power of Data Lock 5 Cases: countries of the world Response variable: life expectancy Explanatory variable: number of mobile cellular subscriptions per 100 people Is a country’s cell phone subscription rate helpful in predicting life expectancy? Cell Phones and Life Expectancy

31. Statistics: Unlocking the Power of Data Lock 5 Cell Phones and Life Expectancy

32. Statistics: Unlocking the Power of Data Lock 5 Cell Phones and Life Expectancy

33. Statistics: Unlocking the Power of Data Lock 5 Is a country’s number of cell phone subscriptions per capita helpful in predicting life expectancy? (a) Yes (b) No Cell Phones and Life Expectancy

34. Statistics: Unlocking the Power of Data Lock 5 If we gave everyone in a country a cell phone and a cell phone subscription, would life expectancy in that country increase? (a) Yes (b) No (c) Impossible to tell Cell Phones and Life Expectancy

35. Statistics: Unlocking the Power of Data Lock 5 Is a country’s cell phone subscription rate helpful in predicting life expectancy, even after including GDP in the model? (a) Yes(b) No Cell Phones and Life Expectancy

36. Statistics: Unlocking the Power of Data Lock 5 Cell Phones and Life Expectancy This says that wealth alone can not explain the association between cell phone subscriptions and life expectancy This suggests that either cell phones actually do something to increase life expectancy (causal) OR there is another confounding variable besides wealth of the country

37. Statistics: Unlocking the Power of Data Lock 5 Confounding Variables Multiple regression is one potential way to account for confounding variables This is most commonly used in practice across a wide variety of fields, but is quite sensitive to the conditions for the linear model (particularly linearity) You can only “rule out” confounding variables that you have data on, so it is still very hard to make true causal conclusions without a randomized experiment

38. Statistics: Unlocking the Power of Data Lock 5 To Do Read 10.1 Do SRTEs (by Sunday, 5/3) Do HW 10.1 (due Friday, 5/1) Do online assessments (due Friday, 5/1)