1. Statistics 350 Lecture 15

2. Today Last Day: More matrix results and Chapter 5 Today: Start Chapter 6

3. Multiple Regression So far, have consider only the simple linear regression model The simple linear regression model has only one predictor variable (X) It is rare to have a single explanatory variable in a study A more complex model can frequently help in prediction and explaining the system variability

4. Multiple Regression Suppose observe Y and also two explanatory variables: Model: Goal:

5. Multiple Regression Example (Stolen from Tom Loughin): One problem with any long surgery is a tendency for the patient's hemoglobin levels to fall. Since hemoglobin carries oxygen around the bloodstream, it is important that hemoglobin levels be maintained to aid the patient's recovery. Eight male patients with thyroid cancer were entered into a study in which their thyroids were to be removed. The length of surgery and total blood loss were recorded immediately following the procedure. Their hemoglobin levels were recorded 24 hours before and 24 hours after the operation, and the percent change was recorded.

6. Multiple Regression The goal of the study was to develop a regression equation which could be used to predict hemoglobin losses of future thyroidectomy patients. (Indeed, shortly following the study there were two new patients on whom the regression equation could be used.) Model:

7. Multiple Regression Notation:

8. Multiple Regression Estimation of model parameters:

9. Multiple Regression Fitted Values and Residuals: