1. Copyright © 1999-2008 Leland Stanford Junior University. All rights reserved. Warning: This presentation is protected by copyright law and international treaties. Unauthorized reproduction of this presentation, or any portion of it, may result in severe civil and criminal penalties and will be prosecuted to maximum extent possible under the law. HRP 223 - 2008 Topic 9 - Regression

2. HRP223 2008 Height and Resting Pulse  The spreadsheet RESTING.xls has height and pulse measures on 50 people. On average, does pulse go up or down with height?

5. Look before you leap!

7. HRP223 2008

15.  Root MSE = Estimated standard deviation of the error in the model (eta)  Dependent Mean = Mean of the outcome  CV = ratio of above * 100  In general r2 is interpreted as: –.1 small effect, 3. medium effect,.5 large effect  Adjusted R-square =1- ( (1- rsquare) * ((n-1)/n-m-1)) ) n=subjects m=variables – It penalizes you for putting extra terms in the model. – R-squared is typically reported if you have a single predictor variable. – Adjusted R-square is typically reported if you have several predictors.

16. HRP223 2008 Oxygen  The next set of data looks at the relationship between oxygen inhaled and exhaled. You would hope that there would be close to a perfect relationship between the two factors.

17. Add the library to a new flowchart. Add the SAS data set to the project.

18. Look at the Data  This is bad news…. At least it is symmetric.

19.  Simple correlation is questionable.

21. HRP223 2008

27. Are the residuals about normal?

29. HRP223 2008 Leave yourself a note on how to interpret the output.  Right click on the flowchart and choose New > Note.  Leave yourself some notes.  Right click on the Note icon > Link Note to > Quadratic

30. HRP223 2008 Ice cream!  In this example you will predict ice cream sales based on factors like price and temperature.  Start by making a library (or copy and paste the existing one) in a new flowchart.  The data is in a text file. Import the data.

31. Load the Data

32. Add Celsius Celsius is ( (5/9) * (Fahr-32) ) 1 1 2 2

38. HRP223 2008

42. Some people say VIF > 10 is a problem but that is arbitrary. If VIF is > 1/(1 - R- squared) then the factors are more related to other predictors than outcome.

43. HRP223 2008 Severely Dehydrated Children

44. HRP223 2008 A Look  Do univariate descriptive statistics. – Things look reasonable.  Do bivariate correlations. – Age and weight are correlated  Do univariate modeling. – There is a weak but statistically significant association.  Build a model with all 3 predictors and check variance inflation.

45. A Simpler Model  It explains a fair amount of the variability (45%). How can I check to make sure the model is working well and is not being driven by outliers?

46. HRP223 2008 Outliers Images from: Statistics I: Introduction to ANOVA, Regression, and Logistic Regression Course Notes (2005) and Categorical Data Analysis Using Logistic Regression Course Notes (2005), SAS Press.

47. First Check Residuals

49. HRP223 2008 What is influential?  Freund and Littell SAS System for Regression 3rd edition, page 70;  Variance inflation: – vifcheck = 1 /(1 – r2)  Leverage greater than this value: – leverageCheck = 2 * (predictors + 1) / records  Covariance more extreme than: – cov1Check = 1 + 3 * (predictors+1) / records – cov1Check = 1 - 3 * (predictors+1) / records  Dfits values with absolute value bigger than: – dffitsCheck = 2 * ((predictors + 1)/records) **.5

50. Influence Code