Chapter 12 Multiple Regression and Model Building

2 Multiple Regression Models The General Multiple Regression Model is the dependent variable are the independent variables is the deterministic portion of the model determines the contribution of the independent variable

3 Multiple Regression Models Analyzing a Multiple Regression Model 1.Hypothesize the deterministic component of the model 2.Use sample data to estimate β 0,β 1,β 2,… β k 3.Specify probability distribution of ε and estimate σ 4.Check that assumptions on ε are satisfied 5.Statistically evaluate model usefulness 6.Useful model used for prediction, estimation, other purposes

4 Multiple Regression Models Assumptions about Random Error ε 1.For any given set of values of x 1, x 2,…..x k, the random error has a normal probability distribution with mean 0 and variance σ 2 2.The random errors are independent

5 The First-Order Model: Estimating and Interpreting the  -Parameters For the chosen fitted model minimizes

6 The First-Order Model: Estimating and Interpreting the  -Parameters y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 + ε where Y = Sales price (dollars) X 1 = Appraised land value (dollars) X 2 = Appraised improvements (dollars) X 3 = Area (square feet )

7 The First-Order Model: Estimating and Interpreting the  -Parameters Plot of data for sample size n=20

8 The First-Order Model: Estimating and Interpreting the  -Parameters Fit model to data

9 The First-Order Model: Estimating and Interpreting the  -Parameters Interpret β estimates E(y), the mean sale price of the property is estimated to increase dollars for additional square foot of living area, holding other variables constant E(y), the mean sale price of the property is estimated to increase.8204 dollars for every $1 increase in appraised improvements, holding other variables constant E(y), the mean sale price of the property is estimated to increase.8145 dollars for every $1 increase in appraised land value, holding other variables constant

10 The First-Order Model: Estimating and Interpreting the  -Parameters Given a model E(y) = 1 +2x 1 +x 2, the effect of x 2 on E(y), holding x 1 constant is

11 The First-Order Model: Estimating and Interpreting the  -Parameters The plane E(y) = 1 +2x 1 +x 2

12 Inferences about the  -Parameters and the Overall Model Utility 2 types of inferences can be made, using either confidence intervals or hypothesis testing For any inferences to be made, the assumptions made about the random error term ε (normal distribution with mean 0 and variance σ 2, independence of errors) must be met

13 Inferences about the  -Parameters and the Overall Model Utility A Test of an Individual Parameter Coefficient One-Tailed Test Two-Tailed Test H 0 : β i =0 H a : β i 0) H 0 : β i =0 H a : β i ≠0 Rejection region: t< -t α (or t 0) Rejection region: |t|> t α/2 Where t α and t α/2 are based on n-(k+1) degrees of freedom

14 Inferences about the  -Parameters and the Overall Model Utility A 100(1-α)% Confidence Interval for a  -Parameter where t α/2 is based on n-(k+1) degrees of freedom and n = Number of observations k+1 = Number of  parameters in the model

15 Inferences about the  -Parameters and the Overall Model Utility A Minitab Analysis Use for confidence Intervals Use for hypotheses about parameter coefficients

16 Inferences about the  -Parameters and the Overall Model Utility 3 tests of overall model utility: 1.Multiple coefficient of determination R 2 2.Adjusted multiple coefficient of determination 3.Global F-test

17 Inferences about the  -Parameters and the Overall Model Utility Testing Global Usefulness of the Model: The Analysis of Variance F-test H 0 : β 1 = β 2=.... β k =0 H a : At least one β i ≠ 0 where n is the sample size and k is number of terms in the model Rejection region: F>F α, with k numerator degrees of freedom and [n- (k+1)] denominator degrees of freedom

18 Inferences about the  -Parameters and the Overall Model Utility Checking the Utility of a Multiple Regression Model 1.Conduct a test of overall model adequacy using the F-test. If H 0 is rejected, proceed to step 2 2.Conduct t-tests on β parameters of particular interest 3.Examine values of and 2s to evaluate how well the model fits the data

19 Using the Model for Estimation and Prediction As in Simple Linear Regression, intervals around a predicted value will be wider than intervals around an estimated value Most statistics packages will print out both confidence and prediction intervals

20 Model Building: Interaction Models An Interaction Model relating E(y) to Two Quantitative Independent Variables where represents the change in E(y) for every 1-unit increase in x 1, holding x 2 fixed represents the change in E(y) for every 1-unit increase in x 2, holding x 1 fixed

21 Model Building: Interaction Models When the relationship between two y and x i is not impacted by a second x (no interaction) When the linear relationship between y and x i depends on another x

22 Model Building: Interaction Models

23 Model Building: Quadratic and other Higher-Order Models A Quadratic (Second-Order) Model where is the y-intercept of the curve is a shift parameter is the rate of curvature

24 Model Building: Quadratic and other Higher-Order Models Home Size-Electrical Usage Data Size of Home, x (sq. ft.) Monthly Usage, y (kilowatt-hours) 1,2901,182 1,3501,172 1,4701,264 1,6001,493 1,7101,571 1,8401,711 1,9801,804 2,2301,840 2,4001,95 2,9301,954

25 Model Building: Quadratic and other Higher-Order Models

26 Model Building: Quadratic and other Higher-Order Models A Complete Second-Order Model with Two Quantitative Independent Variables where is the y-intercept, value of E(y) when x 1 = x 2 =0 changes cause the surface to shift along the x 1 and x 2 axes controls the rotation of the surface control the type of surface, rates of curvature

27 Model Building: Quadratic and other Higher-Order Models

28 Model Building: Qualitative (Dummy) Variable Models Dummy variables – coded, qualitative variables Codes are in the form of (1, 0), 1 being the presence of a condition, 0 the absence Create Dummy variables so that there is one less dummy variable than categories of the qualitative variable of interest Gender dummy variable coded as x = 1 if male, x=0 if female If model is E(y)=β 0 +β 1 x, β 1 captures the effect of being male on the dependent variable

29 Model Building: Models with both Quantitative and Qualitative Variables Start with a first order model with one quantitative variable, E(y)=β 0 +β 1 x 1 Adding a qualitative variable with no interaction, E(y)=β 0 +β 1 x 1 + β 2 x 2 + β 3 x 3

30 Model Building: Models with both Quantitative and Qualitative Variables Adding an interaction term, E(y)=β 0 +β 1 x 1 + β 2 x 2 + β 3 x 3 + β 4 x 1 x 2 + β 5 x 1 x 3 Main effect, Main effect Interaction x 1 x 2 and x 3

31 Model Building: Comparing Nested Models Models are nested if one model contains all the terms of the other model and at least one additional term. Complete (full) model – the more complex model Reduced model – the simpler model

32 Model Building: Comparing Nested Models Models are nested if one model contains all the terms of the other model and at least one additional term. Complete (full) model – the more complex model Reduced model – the simpler model

33 Model Building: Comparing Nested Models F-Test for comparing nested models: F-Test for Comparing Nested Models Reduced model Complete Model H 0 : β g+1 = β g+2=.... β k =0 H a : At least one β under test is nonzero. Rejection region: F>F α, with k-g numerator degrees of freedom and [n-(k+1)] denominator degrees of freedom

34 Model Building: Stepwise Regression Used when a large set of independent variables Software packages will add in variables in order of explanatory value. Decisions based on largest t-values at each step Procedure is best used as a screening procedure only

35 Residual Analysis: Checking the Regression Assumptions Regression Residual – the difference between an observed y value and its corresponding predicted value Properties of Regression Residuals The mean of the residuals equals zero The standard deviation of the residuals is equal to the standard deviation of the fitted regression model

36 Residual Analysis: Checking the Regression Assumptions Analyzing Residuals Top plot of residuals reveals non-random pattern, curved shape Second plot, based on second-order term being added to model, results in random pattern, better model

37 Residual Analysis: Checking the Regression Assumptions Identifying Outliers Residual plots can reveal outliers Outliers need to be checked to try to determine if error is involved If error is involved, or observation is not representative, analysis can be rerun after deleting data point to assess the effect. Outlier

38 Residual Analysis: Checking the Regression Assumptions With OutlierWithout Outlier Checking for Normal Errors

39 Residual Analysis: Checking the Regression Assumptions Checking for Equal Variances Pattern in residuals indicate violation of equal variance assumption Can point to use of transformation on the dependent variable to stabilize variance

40 Residual Analysis: Checking the Regression Assumptions Steps in Residual Analysis 1.Check for misspecified model by plotting residuals against quantitative independent variables 2.Examine residual plots for outliers 3.Check for non-normal error using frequency distribution of residuals 4.Check for unequal error variances using plots of residuals against predicted values

41 Some Pitfalls: Estimability, Multicollinearity, and Extrapolation Estimability – the number of levels of observed x-values must be one more than the order of the polynomial in x that you want to fit Multicollinearity – when two or more independent variables are correlated

42 Some Pitfalls: Estimability, Multicollinearity, and Extrapolation Multicollinearity – when two or more independent variables are correlated Leads to confusing, misleading results, incorrect parameter estimate signs. Can be identified by –checking correlations among x’s –non-significant for most/all x’s –signs opposite from expected in the estimated β parameters Can be addressed by –Dropping one or more of the correlated variables in the model –Restricting inferences to range of sample data, not making inferences about individual β parameters based on t-tests.

43 Some Pitfalls: Estimability, Multicollinearity, and Extrapolation Extrapolation – use of model to predict outside of range of sample data is dangerous Correlated Errors – most common when working with time series data, values of y and x’s observed over a period of time. Solution is to develop a time series model.