MultipleLinearRegression-Statistics.pdfVIP

Multiple Linear Regression The population model ? In a simple linear regression model, a single response measurement Y is related to a single predictor (covariate, regressor) X for each observation. The critical assumption of the model is that the conditional mean function is linear: E(Y |X) = α + βX. In most problems, more than one predictor variable will be available. This leads to the following “multiple regression” mean function: E(Y |X) = α + β1X1 + · · · + βpXp, where α is caled the intercept and the βj are called slopes or coe?cients. ? For example, if Y is annual income ($1000/year), X1 is educational level (number of years of schooling), X2 is number of years of work experience, and X3 is gender (X3 = 0 is male, X3 = 1 is female), then the population mean function may be E(Y |X) = 15 + 0.8 · X1 + 0.5 · X2 ? 3 · X3. Based on this mean function, we can determine the expected income for any person as long as we know his or her educational level, work experience, and gender. For example, according to this mean function, a female with 12 years of schooling and 10 years of work experience would expect to earn $26,600 annually. A male with 16 years of schooling and 5 years of work experience would expect to earn $30,300 annually. ? Going one step further, we can specify how the responses vary around their mean values. This leads to a model of the form Yi = α + β1Xi,1 + · · · + βpXi,p + i. which is equivalent to writing Yi = E(Y |Xi) + i. We write Xi,j for the jth predictor variable measured for the ith observation. The main assumptions for the errors i is that E i = 0 and var( i) = σ2 (all variances are equal). Also the i should be independent of each other. For small sample sizes, it is also important that the i approximately have a normal distribution. 1 ? For example if we have the population model Y = 15 + 0.8 · X1 + 0.5 · X2 ? 3 · X3 + . as above, and we know that σ = 9, we can answer questions like: “what is the probability that a female with 16 years educat