The Multiple Regression Model

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1 Multiple Regression

2 The Multiple Regression Model Idea: Examine the linear relationship between 1 dependent (Y) & or more independent variables (X i ) Multiple Regression Model with k Independent Variables: Y-intercept Population slopes Random Error Y = β + β X + β X + K+ β X k k ε

3 Multiple Regression Equation The coefficients of the multiple regression model are estimated using sample data Multiple regression equation with k independent variables: Estimated (or predicted) value of y Estimated intercept Estimated slope coefficients y ˆ = b + b x + b x + K+ i 0 1 1i i b k x ki In this chapter we will always use a computer to obtain the regression slope coefficients and other regression summary measures.

4 x 1 Multiple Regression Equation Two variable model y y ˆ = b + (continued) 0 + b 1 x 1 b x x Slope for variable x1 Slope for variable x

5 Standard Multiple Regression Assumptions The values x i and the error terms ε i are independent The error terms are random variables with mean 0 and a constant variance, σ. E[ε i ] = 0 and E[ε i ] = σ for (i = 1, K,n) (The constant variance property is called homoscedasticity)

6 Standard Multiple Regression Assumptions (continued) The random error terms, ε i, are not correlated with one another, so that E[ε i ε j ] = 0 for all i j It is not possible to find a set of numbers, c 0, c 1,..., c k, such that c0 + c1x1i+ cxi+ K+ ckxki = 0 (This is the property of no linear relation for the X j s)

7 Example: Independent Variables A distributor of frozen desert pies wants to evaluate factors thought to influence demand Dependent variable: Pie sales (units per week) Independent variables: Price (in $) Data are collected for 15 weeks Advertising ($100 s)

8 Pie Sales Example Week Pie Sales Price ($) Advertising ($100s) Multiple regression equation: Sales = b 0 + b 1 (Price) + b (Advertising)

9 Estimating a Multiple Linear Regression Equation Excel will be used to generate the coefficients and measures of goodness of fit for multiple regression Excel: Tools / Data Analysis... / Regression PHStat: PHStat / Regression / Multiple Regression

10 Multiple Regression Output Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 15 Sales = (Price) (Advertising) ANOVA df SS MS F Significance F Regression Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Price Advertising

11 The Multiple Regression Equation Sales = (Price) + where Sales is in number of pies per week Price is in $ Advertising is in $100 s. b 1 = : sales will decrease, on average, by pies per week for each $1 increase in selling price, net of the effects of changes due to advertising (Advertising) b = : sales will increase, on average, by pies per week for each $100 increase in advertising, net of the effects of changes due to price

12 Coefficient of Determination, R Reports the proportion of total variation in y explained by all x variables taken together SSR R = = SST regression sum of squares total sum of squares This is the ratio of the explained variability to total sample variability

13 Coefficient of Determination, R (continued) Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 15 SSR R = = =.5148 SST % of the variation in pie sales is explained by the variation in price and advertising ANOVA df SS MS F Significance F Regression Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Price Advertising

14 Estimation of Error Variance Consider the population regression model The unbiased estimate of the variance of the errors is where The square root of the variance, s e, is called the standard error of the estimate 1 K n SSE 1 K n e s n 1 i i e = = = i Ki K i 1i 1 0 i ε x β x β x β β Y = L i i i y y e ˆ =

15 Standard Error, s e Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 15 s e = The magnitude of this value can be compared to the average y value ANOVA df SS MS F Significance F Regression Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Price Advertising

16 Adjusted Coefficient of Determination, R never decreases when a new X variable is added to the model, even if the new variable is not an important predictor variable This can be a disadvantage when comparing models What is the net effect of adding a new variable? We lose a degree of freedom when a new X variable is added Did the new X variable add enough explanatory power to offset the loss of one degree of freedom? R

17 Adjusted Coefficient of Determination, R Used to correct for the fact that adding non-relevant independent variables will still reduce the error sum of squares SSE /(n K 1) R = 1 SST /(n 1) (where n = sample size, K = number of independent variables) Adjusted R provides a better comparison between multiple regression models with different numbers of independent variables Penalize excessive use of unimportant independent variables Smaller than R (continued)

18 R Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 15 R = % of the variation in pie sales is explained by the variation in price and advertising, taking into account the sample size and number of independent variables ANOVA df SS MS F Significance F Regression Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Price Advertising

19 Coefficient of Multiple Correlation The coefficient of multiple correlation is the correlation between the predicted value and the observed value of the dependent variable R = r(y,y) ˆ = R Is the square root of the multiple coefficient of determination Used as another measure of the strength of the linear relationship between the dependent variable and the independent variables Comparable to the correlation between Y and X in simple regression

20 Test on All Coefficients F-Test for Overall Significance of the Model Shows if there is a linear relationship between all of the X variables considered together and Y Use F test statistic Hypotheses: H 0 : β 1 = β = = β k = 0 (no linear relationship) H 1 : at least one β i 0 (at least one independent variable affects Y)

21 F-Test for Overall Significance Test statistic: MSR s F= e = SSR/K SSE/(n K 1) where F has k (numerator) and (n K 1) (denominator) degrees of freedom The decision rule is Reject H if F> 0 F k,n K 1,α

22 F-Test for Overall Significance (continued) Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 15 MSR F = = = MSE 5.8 With and 1 degrees of freedom P-value for the F-Test ANOVA df SS MS F Significance F Regression Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Price Advertising

23 0 H 0 : β 1 = β = 0 H 1 : β 1 and β not both zero α =.05 df 1 = df = 1 Do not reject H 0 F-Test for Overall Significance Critical Value: F α = α =.05 Reject H 0 F.05 = F Test Statistic: MSR F = = MSE Decision: Since F test statistic is in the rejection region (pvalue <.05), reject H 0 Conclusion: (continued) There is evidence that at least one independent variable affects Y

24 Prediction Given a population regression model y i = β 0 + β 1 x 1i + β x i + L + β x + ε (i= 1,, K,n) K Ki i then given a new observation of a data point (x 1,n+1, x,n+1,..., x K,n+1 ) the best linear unbiased forecast of y^ n+1 is ˆ y n+ 1= b0+ b1x1,n+ 1+ bx,n+ 1+ L+ bkxk,n+ 1 It is risky to forecast for new X values outside the range of the data used to estimate the model coefficients, because we do not have data to support that the linear model extends beyond the observed range.

25 Using The Equation to Make Predictions Predict sales for a week in which the selling price is $5.50 and advertising is $350: Sales = (Price) (Advertising) = (5.50) (3.5) = 48.6 Predicted sales is 48.6 pies Note that Advertising is in $100 s, so $350 means that X = 3.5

26 Residuals in Multiple Regression Two variable model Residual = e i = (y i y i ) < y i y i y < Sample observation y ˆ = b + 0+ b1x1 bx x i x x 1i x 1

27 Linear vs. Nonlinear Fit Y Y X X residuals X residuals X Linear fit does not give random residuals random Nonlinear fit gives residuals

28 Multiple Regression Assumptions Errors (residuals) from the regression model: e i = (y i y i ) < Assumptions: The errors are normally distributed Errors have a constant variance The model errors are independent

29 Analysis of Residuals in Multiple Regression These residual plots are used in multiple regression: Residuals vs. y i < Residuals vs. x 1i Residuals vs. x i Residuals vs. time (if time series data) Use the residual plots to check for violations of regression assumptions

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