Linear Models المحاضرة 11

1. Linear Modelsالمحاضرة 11

2. Multiple Regression Models • A general additive multiple regression model, which relates a dependent variable y to k predictor variables x1, x2,…, xk is given by the model equation • y = a + b1x1 + b2x2 + … + bkxk + e • The random deviation e is assumed to be normally distributed with mean value 0 and variance s2 for any particular values of x1, x2,…, xk. This implies that for fixed x1, x2,…, xk values, y has a normal distribution with variance s2 and • (mean y value for • fixed x1, x2,…, xk values) = a + b1x1 + b2x2 + … + bkxk

3. Multiple Regression Models • The bi’s are called population regression coefficients; each bi can be interpreted as the true average change in y when the predictor xi increases by 1 unit and the values of all the other predictors remain fixed. • The deterministic portion a + b1x1 + b2x2 + … + bkxk is called the population regression function.

4. Polynomial Regression Models • The kth degree polynomial regression model • y = a + b1x + b2x2 + … + bkxk + e • Is a special case of the general multiple regression model with x1 = x, x2 = x2, … , xk = xk. • The population regression function (mean value of y for fixed values of the predictors) is a+ b1x + b2x2 + … + bkxk . The most important special case other than simple linear regression (k = 1) is the quadratic regression model y = a+ b1x + b2x2. This model replaces the line y = a+ bx with a parabolic cure of mean values a+ b1x + b2x2. If b2 > 0, the curve opens upward, whereas if b2 < 0, the curve opens downward.

5. Interaction • If the change in the mean y value associated with a 1-unit increase in one independent variable depends on the value of a second independent variable, there is interaction between these two variables. When the variables are denoted by x1 and x2, such interaction can be modeled by including x1x2, the product of the variables that interact, as a predictor variable.

6. Qualitative Predictor Variables. • Up to now, we have only considered the inclusion of quantitative (numerical) predictor variables in a multiple regression model. • Two types are very common: • Dichotomous variable: One with just two possible categories coded 0 and 1 Example • Gender {male, female} • Marriage status {married, not-married} • Ordinal variables: Categorical variables that have a natural ordering • Activity level {light, moderate, heavy} coded respectively as 1, 2 and 3 • Education level {none, elementary, secondary, college, graduate} coded respectively 1, 2, 3, 4, 5 (or for that matter any 5 consecutive integers}