Multiple Regression I 4/9/12 Transformations The model Individual coefficients R 2 ANOVA for regression Residual standard error Section 9.4, 9.5 Professor Kari Lock Morgan Duke University

Project 2 Proposal (due Wednesday, 4/11) Project 2 Proposal Homework 9 (due Monday, 4/16) Homework 9 Project 2 Presentation (Thursday, 4/19) Project 2 Presentation Project 2 Paper (Wednesday, 4/25) Project 2 Paper To Do

Non-Constant Variability

Non-Normal Residuals

Transformations If the conditions are not satisfied, there are some common transformations you can apply to the response variable You can take any function of y and use it as the response, but the most common are log(y) (natural logarithm - ln)  y (square root) y 2 (squared) e y (exponential))

log(y) Original Response, y : Logged Response, log(y) :

yy Original Response, y : Square root of Response,  y :

y2y2 Original Response, y : Squared response, y 2 :

eyey Original Response, y : Exponentiated Response, e y :

Multiple regression extends simple linear regression to include multiple explanatory variables: Multiple Regression

We’ll use your current grades to predict final exam scores, based on a model from last semester’s students Response: final exam score Explanatory: hw average, clicker average, exam 1, exam 2 Grade on Final

What variable is the most significant predictor of final exam score? a) Homework average b) Clicker average c) Exam 1 d) Exam 2 Grade on Final

The p-value for explanatory variable x i is associated with the hypotheses For intervals and p-values of coefficients in multiple regression, use a t-distribution with degrees of freedom n – k – 1, where k is the number of explanatory variables included in the model Inference for Coefficients

Estimate your score on the final exam. What type of interval do you want for this estimate? a) Confidence interval b) Prediction interval Grade on Final

Estimate your score on the final exam. (hw average is out of 10, clicker average is out of 2) Grade on Final

Is the clicker coefficient really negative?!? Give a 95% confidence interval for the clicker coefficient (okay to use t* = 2). Grade on Final

Is your score on exam 2 really not a significant predictor of your final exam score?!? Grade on Final

The coefficient (and significance) for each explanatory variable depend on the other variables in the model! In predicting final exam scores, if you know someone’s score on Exam 1, it doesn’t provide much additional information to know their score on Exam 2 (both of these explanatory variables are highly correlated) Coefficients

If you take Exam 1 out of the model… Grade on Final Model with Exam 1: Now Exam 2 is significant!

If you include Project 1 in the model… Grade on Final Model without Project 1:

Grades

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

Residuals Are the conditions satisfied? (a) Yes(b) No

Evaluating a Model How do we evaluate the success of a model? How we determine the overall significance of a model? How do we choose between two competing models?

Variability One way to evaluate a model is to partition variability A good model “explains” a lot of the variability in Y Total Variability Variability Explained by the Model Error Variability

Exam Scores Without knowing the explanatory variables, we can say that a person’s final exam score will probably be between 60 and 98 (the range of Y) Knowing hw average, clicker average, exam 1 and 2 grades, and project 1 grades, we can give a narrower prediction interval for final exam score We say the some of the variability in y is explained by the explanatory variables How do we quantify this?

Variability How do we quantify variability in Y? a)Standard deviation of Y b)Sum of squared deviations from the mean of Y c)(a) or (b) d)None of the above

Sums of Squares Total Variability Variability Explained by the model Error variability SSTSSMSSE

Variability If SSM is much higher than SSE, than the model explains a lot of the variability in Y

R2R2 R 2 is the proportion of the variability in Y that is explained by the model Total Variability Variability Explained by the Model

R2R2 For simple linear regression, R 2 is just the squared correlation between X and Y For multiple regression, R 2 is the squared correlation between the actual values and the predicted values

R2R2

Final Exam Grade

Is the model significant? If we want to test whether the model is significant (whether the model helps to predict y), we can test the hypotheses: We do this with ANOVA!

ANOVA for Regression k: number of explanatory variables n: sample size Source Model Error Total df k n-k-1 n-1 Sum of Squares SSM SSE SST Mean Square MSM = SSM/k MSE = SSE/(n-k-1) F MSM MSE p-value Use F k,n-k-1

Final Exam Grade For this model, do the explanatory variables significantly help to predict final exam score? (calculate a p-value). (a) Yes (b) No n = 69 SSM = SSE =

ANOVA for Regression Source Model Error Total df Sum of Squares Mean Square F p-value  0

Final Exam Grade

Simple Linear Regression For simple linear regression, the following tests will all give equivalent p-values: t-test for non-zero correlation t-test for non-zero slope ANOVA for regression

Mean Square Error (MSE) Mean square error (MSE) measures the average variability in the errors (residuals) The square root of MSE gives the standard deviation of the residuals (giving a typical distance of points from the line) This number is also given in the R output as the residual standard error, and is known as s  in the textbook

Final Exam Grade

Simple Linear Model Residual standard error =  MSE = s e estimates the standard deviation of the residuals (the spread of the normal distributions around the predicted values)

Residual Standard Error Use the fact that the residual standard error is and your predicted final exam score to compute an approximate 95% prediction interval for your final exam score NOTE: This calculation only takes into account errors around the line, not uncertainty in the line itself, so your true prediction interval will be slightly wider

How do we decide which explanatory variables to include in the model? How do we use categorical explanatory variables? What is the coefficient of one explanatory variable depends on the value of another explanatory variable? To Come…