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MULTIPLE REGRESSION After complete this chapter, you should be able to understand mold building using nonuple reversion analysis apply triune atavism analysis to business organisation decision-making situations analyze and interpret the computer output for a multiple lapse model test the significance of the separate proteans in a multiple regression toward the mean model use inconsistent transformations to model nonlinear kindreds bonk potential problems in multiple regression analysis and take the locomote to correct the problems. ncorporate qualitative variants into the regression model by using lacuna changeables. Multiple atavism Assumptions The errors be norm exclusivelyy distributed The mean of the errors is zero Errors come a constant variance The model errors be breakaway fashion model Specification Decide what you want to do and select the dependent variable Determine the potential independent variables for your model Gather sample info (observations ) for all variables The Correlation Matrix Correlation surrounded by the dependent variable and selected independent variables can be found using ExcelTools / entropy compendium / Correlation Can check for statistical significance of correlation with a t test Example A distributor of flash-frozen desert pies wants to evaluate factors thought to influence demand Dependent variable Pie sales ( building blocks per hebdomad) Independent variables Price (in $) advert ($100s) Data is composed for 15 calendar weeks Pie Sales archetype Sales = b0 + b1 (Price) + b2 (Advertising) Interpretation of portendd Coefficients Slope (bi) Estimates that the amount value of y changes by bi units for all(prenominal) 1 unit increase in Xi holding all other variables constantExample if b1 = -20, so sales (y) is expected to decrease by an dependd 20 pies per week for each $1 increase in selling price (x1), net of the cause of changes due to advertising (x2) y-intercept (b0) The estimated ave rage value of y when all xi = 0 (assuming all xi = 0 is within the range of notice values) Pie Sales Correlation Matrix Price vs. Sales r = -0. 44327 at that place is a negative association mingled with price and sales Advertising vs. Sales r = 0. 55632 There is a positive association surrounded by advertising and sales Scatter DiagramsComputer softwargon is generally used to soften the coefficients and measures of goodness of work for multiple regression Excel Tools / Data Analysis / relapsing Multiple reverting create The Multiple Regression equality Using The present to Make Predictions Input values Multiple Coefficient of finis Reports the proportion of total variation in y explained by all x variables taken together Multiple Coefficient of Determination Adjusted R2 R2 neer decreases when a cutting x variable is added to the model This can be a disadvantage when comparing modelsWhat is the net effect of adding a virgin variable? We lose a degree of emancipatio n when a innovative x variable is added Did the new x variable add lavish instructive power to offset the loss of wiz degree of freedom? Shows the proportion of variation in y explained by all x variables adjust for the turning of x variables used (where n = sample size, k = offspring of independent variables) Penalize excessive use of unimportant independent variables Smaller than R2 Useful in comparing among models Multiple Coefficient of Determination Is the Model Significant? F-Test for overall Significance of the ModelShows if there is a linear relationship between all of the x variables considered together and y Use F test statistic Hypotheses H0 ? 1 = ? 2 = = ? k = 0 (no linear relationship) HA at least one ? i ? 0 (at least one independent variable affects y) F-Test for Overall Significance Test statistic where F has (numerator) D1 = k and (denominator) D2 = (n k 1) degrees of freedom H0 ? 1 = ? 2 = 0 HA ? 1 and ? 2 not both zero ( = . 05 df1= 2 df2 = 12 ar Indi vidual Variables Significant? Use t-tests of individual variable slopes Shows if there is a linear relationship between the variable xi and yHypotheses H0 ? i = 0 (no linear relationship) HA ? i ? 0 (linear relationship does exist between xi and y) H0 ? i = 0 (no linear relationship) HA ? i ? 0 (linear relationship does exist between xi and y) t Test Statistic (df = n k 1) Inferences about the Slope t Test Example H0 ? i = 0 HA ? i ? 0 Confidence Interval Estimate for the Slope Standard Deviation of the Regression Model The estimate of the modular deviation of the regression model is Standard Deviation of the Regression Model The standard deviation of the regression model is 47. 46 A close together(p) prediction range for pie sales in a given week isPie sales in the sample were in the 300 to 500 per week range, so this range is probably too large to be acceptable. The analyst may want to look for additional variables that can explain to a greater extent of the variation in week ly sales OUTLIERS If an observation exceeds UP=Q3+1. 5*IQR or if an observation is smaller than LO=Q1-1. 5*IQR where Q1 and Q3 are quartiles and IQR=Q3-Q1 What to do if there are outliers? Sometimes it is appropriate to delete the entire observation containing the oulier. This get out generally increase the R2 and F test statistic values Multicollinearity Multicollinearity High correlation exists between two independent variablesThis means the two variables contribute unnecessary information to the multiple regression model Including two highly check independent variables can adversely affect the regression results No new information provided Can lead to unstable coefficients (large standard error and guerrillaary t-values) Coefficient signs may not match prior expectations Some Indications of Severe Multicollinearity infatuated signs on the coefficients Large change in the value of a foregoing coefficient when a new variable is added to the model A previously probatory vari able becomes insignificant when a new independent variable is addedThe estimate of the standard deviation of the model increases when a variable is added to the model Output for the pie sales example Since there are only two explanatory variables, only one VIF is reported VIF is 5 There is no severalize of collinearity between Price and Advertising Qualitative ( bosom) Variables Categorical explanatory variable (dummy variable) with two or more levels yes or no, on or off, staminate or female coded as 0 or 1 Regression intercepts are different if the variable is significant Assumes equal slopes for other variables The number of dummy variables needed is (number of levels 1)Dummy-Variable Model Example (with 2 Levels) Interpretation of the Dummy Variable Coefficient Dummy-Variable Models (more than 2 Levels) The number of dummy variables is one less than the number of levels Example y = nominate price x1 = square feet The style of the house is also thought to matter Style = ran ch, split level, condo Dummy-Variable Models (more than 2 Levels) Interpreting the Dummy Variable Coefficients (with 3 Levels) Nonlinear Relationships The relationship between the dependent variable and an independent variable may not be linear Useful when scatter diagram indicates non-linear relationshipExample Quadratic model The second independent variable is the square of the first variable Polynomial Regression Model where ?0 = macrocosm regression constant ?i = Population regression coefficient for variable xj j = 1, 2, k p = Order of the multinomial (i = Model error Linear vs. Nonlinear Fit Quadratic Regression Model Testing for Significance Quadratic Model Test for Overall Relationship F test statistic = Testing the Quadratic Effect equivalence quadratic model with the linear model Hypotheses (No 2nd order polynomial term) (2nd order polynomial term is needed) Higher Order Models interaction EffectsHypothesizes fundamental interaction between pairs of x variables Respon se to one x variable varies at different levels of another x variable Contains two-way cross output terms Effect of Interaction Without interaction term, effect of x1 on y is measured by ? 1 With interaction term, effect of x1 on y is measured by ? 1 + ? 3 x2 Effect changes as x2 increases Interaction Example Hypothesize interaction between pairs of independent variables Hypotheses H0 ? 3 = 0 (no interaction between x1 and x2) HA ? 3 ? 0 (x1 interacts with x2) Model building Goal is to develop a model with the best set of independent variablesEasier to interpret if unimportant variables are removed Lower probability of collinearity piecemeal regression procedure Provide evaluation of alternative models as variables are added Best-subset approach Try all combinations and select the best using the highest adjusted R2 and lowest s? Idea develop the least squares regression compare in steps, either with forward selection, backward elimination, or through standard stepwise regressio n The coefficient of partial determination is the measure of the peripheral contribution of each independent variable, given that other independent variables are in the modelBest Subsets Regression Idea estimate all possible regression equations using all possible combinations of independent variables Choose the best fit by looking for the highest adjusted R2 and lowest standard error s? Aptness of the Model Diagnostic checks on the model include confirm the assumptions of multiple regression Each xi is linearly related to y Errors have constant variance Errors are independent Error are normally distributed Residual Analysis The Normality Assumption Errors are fictional to be normally distributed Standardized residuals can be calculated by computerExamine a histogram or a normal probability plot of the regularise residuals to check for normality Chapter Summary Developed the multiple regression model Tested the significance of the multiple regression model Developed adjusted R2 Tested individual regression coefficients Used dummy variables Examined interaction in a multiple regression model Described nonlinear regression models Described multicollinearity Discussed model building Stepwise regression Best subsets regression Examined residual plots to check model assumptions