商业应用统计英文版ch14

1、Applied Business Statistics, 7th ed.by Ken Black,Chapter 14 Advanced Multiple Regression Models,Learning Objectives,Analyze and interpret nonlinear variables in multiple regression analysis. Understand the role of qualitative variables and how to use them in multiple regression analysis. Learn how t

2、o build and evaluate multiple regression models. Learn how to detect influential observations in regression analysis. Explain when to use logistic regression and interpret results.,Regression models presented thus far are based on thegeneral linear regression model, which has the form Y = 0 + 1X1 +

3、2X2 + 3X3 + . . . + kXk+ Y = the value of the dependent (response) variable 0 = the regression constant 1 = the partial regression coefficient of independent variable 1 2 = the partial regression coefficient of independent variable 2 k = the partial regression coefficient of independent variable k k

4、 = the number of independent variables = the error of prediction,General Linear Regression Model,In the general linear model, the parameters, i,are linear. However, dependent variable, y, is not necessarilylinearly related to the predictor variables. Multiple regression response surfaces are not res

5、tricted to linear surfaces and may be curvilinear. Regression models can be developed for more than two predictors.,General Linear Regression Model,Polynomial Regression,Regression models in which the highest power of any predictor variable is 1 and in which there are no interaction terms are referr

6、ed to as first-order models. If a second independent variable is added, the model is referred to as a first-order model with two independent variables. Polynomial regression models are regression models that are second- or higher-order models. Contain squared, cubed, or higher powers of thepredictor

7、 variable(s),Non Linear Models:Mathematical Transformation,Consider the table in the next slide. The table contains sales for 13 mfg companies along with the number of mfg representatives associated with each firm. A simple regression analysis to predict sales by the number of manufacturers represen

8、tatives resultsin the Excel output.,Sales Data and Scatter Plot for 13 Manufacturing Companies,Sales Data and Scatter Plot for 13 Manufacturing Companies,Excel Simple Linear Regression Outputfor the Manufacturing Example,Sales Data and Scatter Plot for 13 Manufacturing Companies,Researcher creates a

9、 second predictor variable, (number of manufacturers representatives2) to use in the regression analysis to predict sales along with number of manufacturers representatives. This variable can be created to explore second-order parabolic relationships by squaring the data from the independent variabl

10、e of the linear model and entering it into the analysis. With the new data, a multiple regression modelcan be developed.,Manufacturing Datawith Newly Created Variable,Computer Output for QuadraticModel to Predict Sales,Tukeys Ladder of Transformations,Tukeys ladder of expressions can be used to stra

11、ightenout a plot of x and y. Tukey used a four-quadrant approach to show which expressions on the ladder are more appropriate for agiven situation. If the scatter plot of x and y indicates a shape like that shown in the upper left quadrant, recoding should move “down the ladder” for the x variable t

12、oward or “up the ladder” for the y variable toward. If the scatter plot of x and y indicates a shape like that of the lower right quadrant, the recoding should move “upthe ladder” for the x variable toward or “down the ladder” for the y variable toward.,Tukeys Four Quadrant Approach,Regression Model

13、s with Interaction,When two different independent variables are usedin a regression analysis, an interaction occurs between the two variables. Interaction can be examined as a separate independent variable: An interaction predictor variable can be designed by multiplying the data values of one varia

14、ble by the valuesof another variable, thereby creating a new variable.,Example Three Stocks,Suppose the data in the following table represent the closing stock prices for three corporations over a period of 15 months. An investment firm wants to use the prices for stocks 2 and 3 to develop a regress

15、ion model to predict the price of stock 1.,Prices of Three Stocks overa 15-Month Period,First-order with Two Independent Variables,Second-order with an Interaction Term,Regression Models for the Three Stocks,The regression equation is Stock 1 = 50.9 - 0.119 Stock 2 - 0.071 Stock 3 Predictor Coef StD

16、ev T P Constant 50.855 3.791 13.41 0.000 Stock 2 -0.1190 0.1931 -0.62 0.549 Stock 3 -0.0708 0.1990 -0.36 0.728 S = 4.570 R-Sq = 47.2% R-Sq(adj) = 38.4% Analysis of Variance Source DF SS MS F P Regression 2 224.29 112.15 5.37 0.022 Error 12 250.64 20.89 Total 14 474.93,Regression for Three Stocks:Fir

17、st-order, Two Independent Variables,The regression equation is Stock 1 = 12.0 - 0.879 Stock 2 - 0.220 Stock 3 0.00998 Inter Predictor Coef StDev T P Constant 12.046 9.312 1.29 0.222 Stock 2 0.8788 0.2619 3.36 0.006 Stock 3 0.2205 0.1435 1.54 0.153 Inter -0.009985 0.002314 -4.31 0.001 S = 2.909 R-Sq

18、= 80.4% R-Sq(adj) = 25.1% Analysis of Variance Source DF SS MS F P Regression 3 381.85 127.28 15.04 0.000 Error 11 93.09 8.46 Total 14 474.93,Regression for Three Stocks:Second-order With an Interaction Term,Nonlinear Regression Models:Model Transformation,Data Set for Model Transformation Example t

19、o Predict Sales by Adv. Expenditure,Regression Output for Model Transformation Example,Prediction with the Transformed Model,Indicator (Dummy) Variables,Some variables are referred to as qualitative variables Qualitative variables do not yield quantifiable outcomes Qualitative variables yield nomina

20、l- or ordinal-level information; used more to categorize items. Qualitative variables are referred to as indicator,or dummy variables,As an example, consider the issue of sex discrimination in the salary earnings of workers in some industries. In examining this issue, suppose a random sample of 15 w

21、orkers is drawn from a pool of employed laborers in a particular industry and the workers average monthly salaries are determined, along with their age and gender. The data are shown in the following table. As sex can be only male or female, this variable is coded as a dummy variable with 0 = female

22、, 1 = male.,Monthly Salary Example,Data for the Monthly Salary Example,The regression equation is Salary = 1.732 + 0.111 Age + 0.459 Gender Predictor Coef StDev T P Constant 1.7321 0.2356 7.35 0.000 Age 0.11122 0.07208 1.54 0.149 Gender 0.45868 0.05346 8.58 0.000 S = 0.09679 R-Sq = 89.0% R-Sq(adj) =

23、 87.2% Analysis of Variance Source DF SS MS F P Regression 2 0.90949 0.45474 48.54 0.000 Error 12 0.11242 0.00937 Total 14 1.02191,Regression Output for the Monthly Salary Example,Regression Output for the Monthly Salary Example,MODEL-BUILDING,Suppose a researcher wants to develop a multiple regress

24、ion model to predict the world production of crude oil. The researcher decides to use as predictors the following five independent variables. U.S. energy consumption Gross U.S. nuclear electricity generation U.S. coal production Total U.S. dry gas (natural gas) production Fuel rate of U.S.-owned aut

25、omobiles,YWorld Crude Oil Production X1U.S. Energy Consumption X2U.S. Nuclear Generation X3U.S. Coal Production X4U.S. Dry Gas Production X5 U.S. Fuel Rate for Autos,Data for Multiple Regressionto Predict Crude Oil Production,Regression Analysis forCrude Oil Production,All Possible Regressions with

26、Five Independent Variables,Model-Building: Search Procedures,Search procedures are processes whereby more than one multiple regression model is developed for a given database, and the models are compared and sorted by different criteria, depending on the given procedure: All Possible Regressions Ste

27、pwise Regression Forward Selection Backward Elimination,Stepwise Regression,Stepwise regression is a step-by-step process that begins by developing a regression model with a single predictor variable and adds and deletes predictors one step at a time. Perform k simple regressions; and select the bes

28、t as the initial model. Evaluate each variable not in the model If none meet the criterion, stop Add the best variable to the model; reevaluate previous variables, and drop any which are not significant Return to previous step.,Stepwise: Step 1 - Simple RegressionResults for Each Independent Variabl

29、e,Stepwise: Regression,Step 2: Two Predictors,Step 3: Three Predictors,Minitab Stepwise Regression Output,Forward Selection,Forward selection is like stepwise regression, but once a variable is entered into the process, it isnever dropped out. Forward selection begins by finding the independent vari

30、able that will produce the largest absolute value of t (and largest R2) in predicting y.,Backward Elimination,Start with the “full model” (all k predictors). If all predictors are significant, stop. Otherwise, eliminate the most non-significant predictor; return to previous step.,Backward Elimination: Oil Production,Step 1:,Step 2:,Backward