Multiple Regression Methods

Transcription

1 Chapter 1: Multiple Regression Methods Hildebrand, Ott and Gray Basic Statistical Ideas for Managers Second Edition 1 Learning Objectives for Ch. 1 The Multiple Linear Regression Model How to interpret a slope coefficient in the multiple regression model The reason for using the adjusted coefficient of determination in multiple regression The meaning of multicollinearity and how to detect it How to test the overall utility of the predictor How to test the additional value of a single predictor How to test the significance of a subset of predictors in multiple regression The meaning of extrapolation Section 1.1 The Multiple Regression Model 3

2 1.1 The Multiple Regression Model Example (with two predictors): Y = Sales revenue per region (tens of thousands of dollars) x 1 = advertising expenditures (thousands of dollars) x = median household income (thousands of dollars) The data follow: The Multiple Regression Model The data are: Region Sales Adv Exp Income A B 1 38 C 1 4 D 3 35 E 3 41 F G H I J A graphical representation follows The Multiple Regression Model 3D Scatterplot of Sales vs Income vs Adv Exp 5.5 Sales Adv Exp Income Objective: Fit a plane through the points 6

3 1.1 The Multiple Regression Model Population Model: E Y ) = β + β x β where ( 1 1 or Y β + β x β + ε = 1 1 k xk ε is the error term. Interpretation of any β j Change in Y per unit change in x j, when all other independent variables are held constant. is called the partial slope, j = 1,,,k β j k x k The Multiple Regression Model First-order model No higher-order terms or interaction terms An interaction term is the product of two predictors: x 1 x Change in E(Y) per unit change in x 1 depends on the value of x. 8 Section 1. Estimating Multiple Regression 9

4 1. Estimating Multiple Regression Criterion used to estimate β 's Method of Least Squares minimize the sum of squared residuals Symbolically: min y i y i We will use software to do the calculations 1 1. Estimating Multiple Regression Example (Y = Sales, x 1 = Advertising Expenditures, x = Median Household Income): The Minitab output follows. Regression Analysis: Sales versus Adv Exp and Income The regression equation is Sales = Adv Exp Income Predictor Coef SE Coef T P VIF Constant Adv Exp Income S = R-Sq = 89.8% R-Sq(adj) = 86.8% Estimating Multiple Regression The fitted model is: Y vs. = x x Y = x 1 and Y = x The coefficient of an independent variable x j in a multiple regression equation does not, in general, equal the coefficient that would apply to that variable in a simple linear regression. In multiple regression, the coefficient refers to the effect of changing that x j variable while other independent variables stay constant. In simple linear regression, all other potential independent variables are ignored. (Hildebrand, Ott and Gray) 1

5 1. Estimating Multiple Regression Interpretation of.416: An additional unit (or an increase of $1,) of Advertising Expenditures leads to.416 increase in Sales when Median Household Income is fixed, i.e., regardless of whether x is 3 or 48. Does this seem reasonable? If Advertising Expenditures are increased by 1 unit, do you expect Sales to increase by.416 units regardless of whether the region has income of $3, or $48,? Estimating Multiple Regression The output also gives the estimate of σ ε, both directly and indirectly. Indirectly, σ ε can be estimated as follows: = sε = (Sum of Squared Residuals) /df MS (Residual Error) where df Error = n - (k + 1) = n k - 1 The estimate of σ ε can also be read directly from s on the output Estimating Multiple Regression Example (Sales vs. Adv. Exp. And Income): The Minitab output follows. Regression Analysis: Sales versus Adv Exp and Income The regression equation is Sales = Adv Exp Income S = R-Sq = 89.8% R-Sq(adj) = 86.8% Analysis of Variance Source DF SS MS F P Regression Residual Error Total 9. 15

6 1. Estimating Multiple Regression Use the output to locate the estimate of σ ε From the output, s ε =.5411 Or, s ε = MS( Error) =.98 = Estimating Multiple Regression Coefficient of Determination R or R y x x x 1 k Concept: we define the coefficient of determination as the proportional reduction in the squared error of Y, which we obtain by knowing the values of x1, x,..., xk. (Hildebrand, Ott and Gray) SSR SSE As in simple regression, R = = 1- SST SST Estimating Multiple Regression Example (Sales vs. Adv. Exp. And Income): From the output, R-Sq = 89.8% Interpretation: 89.8% of the variation in Sales is explained by a multiple regression model with Adv. Exp. and Income as predictors. 18

7 1. Estimating Multiple Regression Adjusted Coefficient of Determination ( = Ra SSE / (n - (k + 1)) 1- SST / (n -1) n -1 SSE = 1- n - (k + 1) SST R a SSE and SST are each divided by their degrees of freedom. Since (n - 1) / (n - (k + 1)) > 1 R < R a ) Estimating Multiple Regression R a Why use? SST is fixed, regardless of the number of predictors. SSE decreases when more predictors are used. R increases when more predictors are used. R a However, can decrease when another predictor is added to the fitted model, even though R increases. Why? The decrease in SSE is offset by the loss of a degree of freedom in [n (k +1)] for SSE. 1. Estimating Multiple Regression The following example illustrates this. Example: For a fitted model with 1 observations, suppose SST = 5. When k =, SSE = 5. When k = 3, SSE = 4.5. R R α K=: 1 =.9 1 = K=3: =.91 1 = Even though there has been a modest increase in R, R has decreased. α 1

8 1. Estimating Multiple Regression Sequential Sum of Squares (SEQ SS) Concept: The incremental contributions to SS (Regression) when the predictors enter the model in the order specified by the user. Example (Sales vs. Adv. Exp. And Income): The Minitab output follows for when Adv.Exp. is entered first. Analysis of Variance Source DF SS MS F P Regression Source DF Seq SS Adv Exp Income Estimating Multiple Regression SS (Regression Using x1 and x) SSR ( x 1, x ) = SS (Regression Using x 1 only) SSR ( x ) 1 = SS (Regression for xwhen x1 is already in the model) SSR ( x x1 ) = Estimating Multiple Regression Example (Sales vs. Adv. Exp. And Income): The Minitab output follows for when Income is entered first. Analysis of Variance Source DF SS MS F P Regression Source DF Seq SS Income Adv Exp

9 1. Estimating Multiple Regression x 1 SS (Regression Using and ) SSR ( x 1, x ) = {Unchanged SS (Regression Using x only) SSR ( x ) = SS (Regression for x 1 when x is already in the model) SSR ( x 1 x ) =.793 Regardless of which predictor is entered first, the sequential sums of squares, when added, equal SS (Regression). x 5 Section 1.3 Inferences in Multiple Regression Inferences in Multiple Regression Objective: Build a parsimonious model as few predictors as necessary Must now assume errors in population model are normally distributed F-test for overall model H : β1 = β =... β k = vs. : at least one β H a j 7

10 1.3 Inferences in Multiple Regression Test Statistic: F = MS (Regression)/MS (Residual Error) Concept: If SS (Regression) is large relative to SS (Residual), the indication is that there is real predictive value in [some of] the independent variables x1, x,..., x k. (Hildebrand, Ott and Gray) Decision Rule: Reject H if F > Fα, k, n k 1 or reject if p value < α H Inferences in Multiple Regression Example (Sales vs. Adv. Exp. and Income): The Minitab output follows: Analysis of Variance Source DF SS MS F P Regression Residual Error Total Inferences in Multiple Regression H β vs. Test : 1 = β = at the 5% level. H a : At least one β j Since F = 3.65 > F.5,, 7 = 4.74, reject H : at the 5% level. β1 = β = Or since p-value =. <.5, reject H : at the 5% level. β1 = β = Implication: At least one of the x s has some predictive power. 3

11 1.3 Inferences in Multiple Regression t-test for Significance of an Individual Predictor H : β j = vs. H a : β j, j = 1,,, k H implies that x j has no additional predictive value as the last predictor in to a model that contains all the other predictors ( ˆ β )/ Test Statistic: t = j s ˆ β j where s is the estimated standard error of β ˆ j β ˆj Inferences in Multiple Regression In Minitab notation, T = (Coef) / (SE Coef) Decision Rule: Reject H if t > tα /, n k 1 H or reject if p-value < α. Warning: Limit the number of t-tests to avoid a high overall Type 1 error rate Inferences in Multiple Regression Example (Sales vs. Adv. Exp. and Income): The Minitab output follows: Predictor Coef SE Coef T P VIF Constant Adv Exp Income Test H : β1 = vs. H a : β1 at the 5% level. 33

12 1.3 Inferences in Multiple Regression Since t =.4 > t reject H 3. 5, 7 = : β1 =.365, at the 5% level. Or since p-value =.19 <.5, reject H : β1 = at the 5% level. Implication: Advertising Expenditures provides additional predictive value to a model having Income as a predictor Inferences in Multiple Regression Multicollinearity Concept: High correlation between at least one pair of predictors, e.g., x 1 and x Correlated x's provide no new information. Example: In predicting heights of adults using the length of the right leg, the length of the left leg would be of little value. Symptoms of Multicollinearity Wrong signs for ˆ β s t-test isn't significant even though you believe the predictor is useful and should be in the fitted model Inferences in Multiple Regression Detection of Multicollinearity R x j x1 x j - 1 x j + 1 xk is the coefficient of determination obtained by regressing x j on the remaining (k - 1) predictors, denoted by R j. If R j >.9, this is a signal that multicollinearity is present. This criterion can be expressed in a different way. 36

13 1.3 Inferences in Multiple Regression Let VIF j denote the Variance Inflation Factor of the j th predictor: 1 VIFj = 1 R If VIF j > 1, this is a signal that multicollinearity is present. j Inferences in Multiple Regression Why is VIF j called the variance inflation factor for the j th predictor? The estimated standard error of β j in a multiple regression is: or s ˆ β j = s ε ( x ij 1 x ) (1 R ) VIF = j ˆ s β ε j s ( xij x j ) j j Inferences in Multiple Regression If VIF j is large, so is, which leads to a t-test sβ j that is not statistically significant. The VIF measures how much the variance (square of the standard error) of a coefficient is increased because of collinearity. (Hildebrand, Ott and Gray) 39

14 1.3 Inferences in Multiple Regression Example (Sales vs. Adv. Exp. and Income): The Minitab output follows. The regression equation is Sales = Adv Exp Income Predictor Coef SE Coef T P VIF Constant Adv Exp Income Since both VIFs = 1.8 < 1, multicollinearity between Advertising Expenditures and Median Household Income is not a problem Inferences in Multiple Regression The Minitab output for regressing Adv Exp on Income follows: The regression equation is Adv Exp = Income S = R-Sq = 43.% Since R =.43, VIF = 1/(1 -.43) = 1.8, as shown Inferences in Multiple Regression To illustrate multicollinearity, consider Exercise 1.19 Exercise 1.19: A study of demand for imported subcompact cars consists of data from 1 metropolitan areas. The variables are: Demand: Imported subcompact car sales as a percentage of total sales Educ: Average number of years of schooling completed by adults Income: Per capita income Popn: Area population Famsize: Average size of intact families 4

15 1.3 Inferences in Multiple Regression The Minitab output follows: The regression equation is Demand = Educ +.89 Income Popn Famsize Predictor Coef SE Coef T P VIF Constant Educ Income Popn Famsize S =.6868 R-Sq = 96.% R-Sq(adj) = 94.1% Inferences in Multiple Regression Is there a multicollinearity (MC) problem? Since the VIF = 1.3 > 1 for the variable Famsize, there is a MC problem. Note that the p-value for the F test =., indicating that at least one of the x s has predictive value. However, the smallest p-value for any t-test is.79, indicating that not one of the individual x s has predictive value. What is the source of the MC problem? A matrix plot, in conjunction with the correlations, could be useful. This exercise will be revisited in Section Inferences in Multiple Regression Remedies if multicollinearity is a problem. Eliminate one or more of the collinear predictors. Form a new predictor that is a surrogate of collinear predictor Multicollinearity could occur if one of the predictors is x. This can be eliminated by using ( x x) as the predictor. 45

16 Section 1.4 Testing a Subset of the Regression Testing a Subset of the Regression To illustrate the concept, consider Exercise Exercise 13.55: A bank that offers charge cards to customers studies the yearly purchase amount (in thousands of dollars) on the card as related to the age, income (in thousands of dollars), whether the cardholder owns or rents a home and years of education of the cardholder. The variable owner equals 1 if the cardholder owns a home and if the cardholder rents a home. The other variables are selfexplanatory. The original data set has information on 16 cardholders. Upon further examination of the data, you decide to remove the data for cardholder 19 because this is an older individual who has a high income from having saved early in life and having invested successfully. This cardholder travels extensively and frequently uses her/his charge card Testing a Subset of the Regression Problem to be investigated: The income and education predictors measure the economic well-being of a cardholder. Do these predictors have any predictive value given the age and home ownership variables? The null hypothesis is that the β s corresponding to these predictors are simultaneously equal to. 48

17 1.4 Testing a Subset of the Regression General Case Complete Model: E( Y ) β + β x + + β x + β x + + β x = 1 1 g g g + 1 g+ 1 Null hypothesis: H β = = β : g + 1 k = Reduced Model: E ( Y ) = β + β x + + β 1 1 g x g k k Testing a Subset of the Regression Exercise Complete Model: E( Y ) = β + β1( Age) + β( Owner) + β3( Income) + β4( Educn) Null hypothesis: H : β β Income = Educn = Reduced Model: E( Y) = β + β1( Age) + β( Owner) Testing a Subset of the Regression The test statistic is called the Partial F statistic Partial F Statistic [ SSEreduced SSEcomplete]/[ dfreduced df F = [ SSE ]/[ df ] Rationale complete complete complete SSE decreases as new terms are added to the model. If the x s from (g + 1) to k have predictive ability, then SSE complete should be much smaller than SSE reduced Their difference [SSE reduced SSE complete ] should be large ] 51

18 1.4 Testing a Subset of the Regression Note: df reduced -df complete = k g; df complete = n (k + 1) Note: Note: SSE complete /df complete = MSE complete Other versions of the partial F-test are in H,O&G. Decision criterion: Reject H if Partial F > F α,k-g,n-k Testing a Subset of the Regression Exercise 13.55: [ SSE F = reduced SSE [ SSE H : β Income = β Educn = complete complete ]/[ df ]/[ df reduced complete [ ]/[4 ] = = df (from the Minitab output that follows) Since 6.45 > F.5,,154 =3.55, reject H. Either Income or Education add predictive value to a model that contains Age and Owner ] complete ] Testing a Subset of the Regression Regression Analysis: Purch_1 versus Age_1, Income_1, Owner_1, Educn_1 The regression equation is Purch_1 = Age_ Income_ Owner_ Educn_1 S =.884 R-Sq = 95.% R-Sq(adj) = 94.8% Analysis of Variance Source DF SS MS F P Regression Residual Error Total

19 1.4 Testing a Subset of the Regression Regression Analysis: PURCH_1 versus AGE_1, OWNER_1 The regression equation is PURCH_1 = AGE_1 +. OWNER_1 S =.985 R-Sq = 94.6% R-Sq(adj) = 94.5% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Section 1.5 Forecasting Using Multiple Regression Forecasting Using Multiple Regression A major purpose of regression is to make predictions using the fitted model. In simple regression, we could obtain a confidence interval for E(Y) or a prediction interval for an individual Y. In both cases, the danger of extrapolation must be considered. Extrapolation occurs when using values of x far outside the range of x-values used to build the fitted model. 57

20 1.5 Forecasting Using Multiple Regression In regressing Sales on Advertising Expenditures, Advertising Expenditures ranged from 1 to 6. It would be incorrect to obtain a Confidence Interval for E(Y) or a Prediction Interval for Y far outside this range. We don t know if the fitted model is valid outside this range. In multiple regression, one must consider not only the range of each predictor but the set of values of the predictors taken together Forecasting Using Multiple Regression Consider the following example: Example: Y = sales revenue per region (tens of thousands of dollars) x 1 = advertising expenditures (thousands of dollars) x = median household income (thousands of dollars) The values for x 1 and x are: Region A B C D E F G H I J x x Forecasting Using Multiple Regression The scatterplot for x 1 vs. x follows. 6

21 1.5 Forecasting Using Multiple Regression Extrapolation occurs when using the fitted model to predict outside the elbow-shaped region. This would occur when Advertising Expenditures is 5 and income is 35. The Minitab output follows Forecasting Using Multiple Regression Regression Analysis: Sales versus Adv Exp, Income The regression equation is Sales = Adv Exp Income Predicted Values for New Observations New Obs Fit SE Fit 95% CI 95% PI (1.451, 3.956) (.913, 4.494) X X denotes a point that is an outlier in the predictors. Values of Predictors for New Observations New Obs Adv Exp Income Minitab indicates that this set of values for x 1 and x is an outlier 6 Keywords: Chapter 1 Multiple regression model Partial slopes First order model Adjusted Coefficient of Determination, R a Multicollinearity Variance Inflation Factor Overall F test t-test Complete model Reduced model Partial F test Extrapolation 63

22 Summary of Chapter 1 The Multiple Linear Regression Model Interpreting the slope coefficient of a single predictor in a multiple regression model Understanding the difference between the coefficient of determination (R a ) and the adjusted coefficient of determination (R ) The detection of multicollinearity and its impact Using the F statistic to test the overall utility of the predictors Using the t-test to test the additional value of a single predictors Using the partial F test for assessing the significance of a subset of predictors The meaning of extrapolation in multiple regression 64