1. 1 Chapter 14 Multiple Regression Models

2. 2 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 = + 1x1 + 2x2 + + kxk + e Multiple Regression Models 3. 3 The random deviation e is assumed to be normally distributed with mean value 0 and variance 2 for any particular values of x1, x2, , xk. This implies that for fixed x1, x2,, xk values, y has a normal distribution with variance 2 and Multiple Regression Models (mean y value for fixed x1, x2,, xk values) = + 1x1 + 2x2 + + kxk 4. 4 The is are called population regression coefficients; each i 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. Multiple Regression Models The deterministic portion + 1x1 + 2x2 + + kxk is called the population regression function. 5. 5 The kth degree polynomial regression model y = + 1x + 2x2 + + kxk + 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 + 1x + 2x2 + + kxk . Polynomial Regression Models 6. 6 The most important special case other than simple linear regression (k = 1) is the quadratic regression model y = + 1x + 2x2 . This model replaces the line y = + x with a parabolic cure of mean values + 1x + 2x2 . If 2 > 0, the curve opens upward, whereas if 2 < 0, the curve opens downward. Polynomial Regression Models 7. 7 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. Interaction 8. 8 Up to now, we have only considered the inclusion of quantitative (numerical) predictor variables in a multiple regression model. Two other types are very common: Dichotomous variable: One with just two possible categories coded 0 and 1 Examples Gender {male, female} Marriage status {married, not- married} Qualitative Predictor Variables. 9. 9 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} Qualitative Predictor Variables. 10. 10 According to the principle of least squares, the fit of a particular estimated regression function a + b1x1 + b2x2 + + bkxk to the observed data is measured by the sum of squared deviations between the observed y values and the y values predicted by the estimated function: [y (a + b1x1 + b2x2 + + bkxk )]2 Least Square Estimates The least squares estimates of , 1, 2,, k are those values of a, b1, b2, , bk that make this sum of squared deviations as small as possible. 11. 11 Predicted Values & Residuals Doing this successively for the remaining observations yields the predicted values (sometimes referred to as the fitted values or fits). L2 3 ky ,y , ,y The first predicted value is obtained by taking the values of the predictor variables x1, x2,, xk for the first sample observation and substituting these values into the estimated regression function. 1y 12. 12 Predicted Values & Residuals The residuals are then the differences between the observed and predicted y values. L1 1 2 2 k ky y ,y y , ,y y 13. 13 Sums of Squares The number of degrees of freedom associated with SSResid is n - (k + 1), because k + 1 df are lost in estimating the k + 1 coefficients , 1, 2,,k. The residual (or error) sum of sqyares, SSResid, and total sum of squares, SSTo, are given by where is the mean of the y observations in the sample. ( ) ( ) 2 2 SSResid= y-y SSTo= y-y y 14. 14 Estimate for 2 An estimate of the random deviation variance 2 is given by and is the estimate of . 2 e SSResid s n - (k + 1) = 2 e es s= 15. 15 Coefficient of Multiple Determination, R2 The coefficient of multiple determination, R2 , interpreted as the proportion of variation in observed y values that is explained by the fitted model, is 2 SSResid R 1 SSTo = 16. 16 Adjusted R2 Generally, a model with large R2 and small se are desirable. If a large number of variables (relative to the number of data points) is used those conditions may be satisfied but the model will be unrealistic and difficult to interpret. 17. 17 Adjusted R2 To sort out this problem, sometimes computer packages compute a quantity called the adjusted R2 , 2 n 1 SSResid adjusted R 1 n (k 1) SSTo = Notice that when a large number of variables are used to build the model, this value will be substantially lower than R2 and give a better indication of usability of the model. 18. F Distributions F distributions are similar to a Chi-Square distributions, but have two parameters, dfden and dfnum. 19. 19 The F Test for Model Utility The regression sum of squares denoted by SSReg is defined by SSREG = SSTo - SSresid 20. 20 The F Test for Model Utility When all k is are zero in the model y = + 1x1 + 2x2 + + kxk + e And when the distribution of e is normal with mean 0 and variance 2 for any particular values of x1, x2,, xk, the statistic has an F probability distribution based on k numerator df and n - (K+ 1) denominator df SSRegr kF SSResid n (k 1) = + 21. 21 The F Test for Utility of the Model y = + 1x1 + 2x2 + + kxk + e Null hypothesis: H0: 1 = 2 = = k =0 (There is no useful linear relationship between y and any of the predictors.) Alternate hypothesis: Ha: At least one among 1, 2, , k is not zero (There is a useful linear relationship between y and at least one of the predictors.) 22. 22 The F Test for Utility of the Model y = + 1x1 + 2x2 + + kxk + e Test statistic: SSRegr kF SSResid n (k 1) where SSreg = SST0 - SSresid. = + An alternate formula: 2 2 R kF (1 R ) n (k 1) where SSreg = SST0 - SSresid. = + 23. 23 The F Test Utility of the Model y = + 1x1 + 2x2 + + kxk + e The test is upper-tailed, and the information in the Table of Values that capture specified upper-tail F curve areas is used to obtain a bound or bounds on the P-value using numerator df = k and denominator df = n - (k + 1). Assumptions: For any particular combination of predictor variable values, the distribution of e, the random deviation, is normal with mean 0 and constant variance. 24. 24 Example A number of years ago, a group of college professors teaching statistics met at an NSF program and put together a sample student research project. They attempted to create a model to explain lung capacity in terms of a number of variables. Specifically, Numerical variables: height, age, weight, waist Categorical variables: gender, activity level and smoking status. 25. 25 Example They managed to sample 41 subjects and obtain/measure the variables. There was some discussion and many felt that the calculated variable (height)(waist)2 would be useful since it would likely be proportional to the volume of the individual. The initial regression analysis performed with Minitab appears on the next slide. 26. 26 Example Linear Model with All Numerical Variables The regression equation is Capacity = - 13.0 - 0.0158 Age + 0.232 Height - 0.00064 Weight - 0.0029 Chest + 0.101 Waist -0.000018 hw2 40 cases used 1 cases contain missing values Predictor Coef SE Coef T P Constant -13.016 2.865 -4.54 0.000 Age -0.015801 0.007847 -2.01 0.052 Height 0.23215 0.02895 8.02 0.000 Weight -0.000639 0.006542 -0.10 0.923 Chest -0.00294 0.06491 -0.05 0.964 Waist 0.10068 0.09427 1.07 0.293 hw2 -0.00001814 0.00001761 -1.03 0.310 S = 0.5260 R-Sq = 78.2% R-Sq(adj) = 74.2% 27. 27 Example The only coefficient that appeared to be significant and the 5% level was the height. Since the P-value for the coefficient on the age was very close to 5% (5.2%) it was decided that a linear model with the two independent variables height and age would be calculated. The resulting model is on the next slide. 28. 28 Example Linear Model with variables: Height & Age The regression equation is Capacity = - 10.2 + 0.215 Height - 0.0133 Age 40 cases used 1 cases contain missing values Predictor Coef SE Coef T P Constant -10.217 1.272 -8.03 0.000 Height 0.21481 0.01921 11.18 0.000 Age -0.013322 0.005861 -2.27 0.029 S = 0.5073 R-Sq = 77.2% R-Sq(adj) = 76.0% Notice that even though the R2 value decreases slightly, the adjusted R2 value actually increases. Also note that the coefficient on Age is now significant at 5%. 29. 29 Example In an attempt to determine if incorporating the categorical variables into the model would significantly enhance the it. Gender was coded as an indicator variable (male = 0 and female = 1), Smoking was coded as an indicator variable (No = 0 and Yes = 1), and Activity level (light, moderate, heavy) was coded respectively as 1, 2 and 3. The resulting Minitab output is given on the next slide. 30. 30 Example Linear Model with categorical variables added The regression equation is Capacity = - 7.58 + 0.171 Height - 0.0113 Age - 0.383 C-Gender + 0.260 C-Activity - 0.289 C-Smoke 37 cases used 4 cases contain missing values Predictor Coef SE Coef T P Constant -7.584 2.005 -3.78 0.001 Height 0.17076 0.02919 5.85 0.000 Age -0.011261 0.005908 -1.91 0.066 C-Gender -0.3827 0.2505 -1.53 0.137 C-Activi 0.2600 0.1210 2.15 0.040 C-Smoke -0.2885 0.2126 -1.36 0.185 S = 0.4596 R-Sq = 84.2% R-Sq(adj) = 81.7% 31. 31 Example It was noted that coefficient for the coded indicator variables gender and smoking were not significant, but after considerable discussion, the group felt that a number of the variables were related. This, the group felt, was confounding the study. In an attempt to determine a reasonable optimal subgroup of the variables to keep in the study, it was noted that a number of the variables were highly related. Since the study was small, a stepwise regression was run and the variables, Height, Age, Coded Activity, Coded Gender were kept and the following model was obtained. 32. 32 Example Linear Model with Height, Age & Coded Activity and Gender The regression equation is Capacity = - 6.93 + 0.161 Height - 0.0137 Age + 0.302 C-Activity - 0.466 C-Gender 40 cases used 1 cases contain missing values Predictor Coef SE Coef T P Constant -6.929 1.708 -4.06 0.000 Height 0.16079 0.02454 6.55 0.000 Age -0.013744 0.005404 -2.54 0.016 C-Activi 0.3025 0.1133 2.67 0.011 C-Gender -0.4658 0.2082 -2.24 0.032 S = 0.4477 R-Sq = 83.2% R-Sq(adj) = 81.3% 33. 33 Example Linear Model with Height, Age & Coded Activity and Gender Analysis of Variance Source DF SS MS F P Regression 4 34.8249 8.7062 43.44 0.000 Residual Error 35 7.0151 0.2004 Total 39 41.8399 Source DF Seq SS Height 1 30.9878 Age 1 1.3296 C-Activi 1 1.5041 C-Gender 1 1.0034 Unusual Observations Obs Height Capacity Fit SE Fit Residual St Resid 4 66.0 2.2000 3.2039 0.1352 -1.0039 -2.35R 23 74.0 5.7000 4.7635 0.2048 0.9365 2.35R 39 70.0 5.4000 4.4228 0.1064 0.9772 2.25R R denotes an observation with a large standardized residual The rest of the Minitab output is given below. 34. 34 Example Linear Model with Height, Age & Coded Activity and Gender All of the coefficients in this model were significant at the 5% level and the R2 and adjusted R2 were both fairly large. This appeared to be a reasonable model for describing lung capacity even though the study was limited by sample size, and measurement limitations due to antique equipment. Minitab identified 3 outliers (because the standardized residuals were unusually large. Various plots of the standardized residuals are produced on the next few slides with comments 35. 35 Example Linear Model with Height, Age & Coded Activity and Gender The histogram of the residuals appears to be consistent with the assumption that the residuals are a sample from a normal distribution. 1.00.80.60.40.20.0-0.2-0.4-0.6-0.8-1.0 10 5 0 Residual Frequency Histogram of the Residuals (response is Capacity) 36. 36 Example Linear Model with Height, Age & Coded Activity and Gender The normality plot also tends to indicate the residuals can reasonably be thought to be a sample from a normal distribution. 10-1 2 1 0 -1 -2 NormalScore Residual Normal Probability Plot of the Residuals (response is Capacity) 37. 37 Example Linear Model with Height, Age & Coded Activity and Gender The residual plot also tends to indicate that the model assumptions are not unreasonable, although there would be some concern that the residuals are predominantly positive for smaller fitted lung capacities. 65432 1 0 -1 Fitted Value Residual Residuals Versus the Fitted Values (response is Capacity)