1. 1 Regression Analysis Modeling Relationships

2. 2 Regression Analysis Regression Analysis is a study of the relationship between a set of independent variables and the dependent variable. Dependent Variable Independent Variables The Linear Equation representing the ‘true’ or population relationship:

3. 3 Variables Dependent Variable: Also called the predicted variable. Its value depends on, or can be predicted by the independent variables. Independent Variables: Also called the predictor variables. These can be measured directly, and are used to predict the dependent (or to simply understand it better).

4. 4 Modeling Process Define GoalTo study the impact of various factors on individual health Choose yLung Capacity, measured in cc. List possible XsMinutes of Exercise per day, # of days/week of exercise, ethnicity, gender, age, height, altitude at which lived. Collect DataPrimary, Secondary sources Preliminary AnalysesUnivariate, bivariate Build Regression ModelHow is y related to all the Xs? Evaluate ModelHow good is the model at predicting y? Implement/MonitorCreate DSS, monitor, update

5. 5 The Data YX1X2X3X4X5 Lung Capacity (cc)GenderHeightSmokerExerciseAge 5673169.502547 5632170.102467 5712168.202636 5723170.902668 5484171.912058 5308169.211519 5133171.91040 A portion of the data is shown below. See Spreadsheet for all data.

6. 6 Preliminary Analyses Lung Capacity (cc)GenderHeightSmokerExerciseAge Mean5325.600.5068.230.3921.3546.42 Stdev410.480.503.450.498.9113.98 Min4233.710.0058.930.00 19.00 Max6261.001.0076.611.0040.2982.14 The table below shows some descriptive statistics for each variable. What basic statements about our data can we make from this?

7. 7 Capacity by Gender, Smoking Gender SmokerDataFemaleMaleGrand Total Non-SmokerAverage of Lung Capacity (cc)5427.675662.225546.87 StdDev of Lung Capacity (cc)256.41284.71293.75 Count of Smoker30.0031.0061.00 SmokerAverage of Lung Capacity (cc)4837.455129.054979.51 StdDev of Lung Capacity (cc)273.74297.51318.12 Count of Smoker20.0019.0039.00 Total Average of Lung Capacity (cc)5191.585459.615325.60 Total StdDev of Lung Capacity (cc)391.51387.93410.48 Total Count of Smoker50.00 100.00 Does there appear to be a relationship between, Smoking, Gender, and Lung Capacity?

8. 8 Distributions

9. 9 Bivariate Analysis – Matrix Plot

10. 10 Capacity distribution by Gender, Smoking Men have a larger lung capacity than women, on average. Non-Smokers have a larger lung capacity than smokers on average. What about the variance?

11. 11 Simple Regression How well can exercise time alone predict the lung capacity?

12. 12 Multiple Regression How do all the Xs together help predict y? SUMMARY OUTPUT Regression Statistics Multiple R0.8798341 R Square0.7741081 Adjusted R Square0.7620926 Standard Error200.21 Observations100 Coefficients Standard Errort StatP-value Intercept1662.3965475.14566343.4987091920.000716253 Gender202.328241.868610424.8324568095.23607E-06 Height50.34687.082073357.1090589892.24959E-10 Smoker-278.971152.71395448-5.2921694927.88193E-07 Exercise11.29492.9911709723.7761126140.000279023 Age-0.11741.462303258-0.0803033670.936166702

13. 13 Final Model SUMMARY OUTPUT Regression Statistics Multiple R0.879825 R Square0.774093 Adjusted R Square0.764581 Standard Error199.164 Observations100 CoefficientsStandard Errort StatP-value Intercept1656.937467.79033.542050.000617 Gender202.10441.556954.863324.57E-06 Height50.3597.0430827.1502711.78E-10 Smoker-279.02552.43341-5.32156.85E-07 Exercise11.2592.9434943.8253420.000234 1656.937 + 202.104 * Gender + 50.359 * Height – 279.025 * Smoker + 11.259 * Exercise

14. 14 Prediction Exercise 1. Predict the lung capacity for a non- smoking female who does not exercise, and is 66 inches tall, based on the model above. 2. What would be the predicted value if she smoked? 3. What would it be for a male in both the above cases?