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MULTIPLE arrested development After completing this chapter, you should be able to understand precedent building utilise double lapsing analysis apply eightfoldx retroflexion analysis to business decision-making situations analyze and interpret the computer make for a multiple infantile fixation model test the significance of the independent variants in a multiple turnabout model use multivariate transformations to model non bi running(a) kinds recognize potential problems in multiple regression analysis and take the steps to correct the problems. ncorporate qualitative variables into the regression model by using dummy variables. septuple Regression Assumptions The errors ar norm eithery distributed The mean of the errors is zero Errors have a unending variance The model errors are independent Model Specification Decide what you want to do and train the dependent variable Determine the potential independent variables for your model Gather sample data (observation s) for on the whole variables The Correlation Matrix Correlation between the dependent variable and selected independent variables can be outletuate using ExcelTools / entropy Analysis / Correlation Can visualise for statistical significance of correlation coefficient with a t test instance A distributor of frozen desert pies wants to evaluate factors betoken to influence demand Dependent variable Pie sales (units per week) Independent variables bell (in $) Advertising ($100s) Data is collected for 15 weeks Pie Sales Model Sales = b0 + b1 (Price) + b2 (Advertising) Interpretation of Estimated Coefficients Slope (bi) Estimates that the average look upon of y changes by bi units for each 1 unit increase in Xi holding all other variables constantExample if b1 = -20, then sales (y) is expected to decrease by an cipherd 20 pies per week for each $1 increase in selling price (x1), net of the effects of changes due to advertising (x2) y-intercept (b0) The estimated average 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 There is a negative knowledge between price and sales Advertising vs. Sales r = 0. 55632 There is a positive railroad tie between advertising and sales Scatter DiagramsComputer software is generally used to generate the coefficients and measures of goodness of mark off for multiple regression Excel Tools / Data Analysis / Regression nine-fold Regression Output The Multiple Regression Equation Using The Model to Make Predictions Input values Multiple Coefficient of Determination Reports the residuum of total sport in y explained by all x variables taken together Multiple Coefficient of Determination Adjusted R2 R2 never decreases when a new x variable is transmited to the model This can be a disadvantage when comparing modelsWhat is the net effect of adding a new variable? We lose a degree of freedom when a new x variable is added Did t he new x variable add enough explanatory power to offset the loss of 1 degree of freedom? Shows the proportion of variation in y explained by all x variables adjusted for the payoff of x variables used (where n = sample size, k = number of independent variables) Penalize excessive use of unimportant independent variables little than R2 utilitarian in comparing among models Multiple Coefficient of Determination Is the Model Significant? F-Test for overall Significance of the ModelShows if in that location is a running(a) 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 Are Individual Variables Significant? Us e 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 standard going of the regression model is Standard Deviation of the Regression Model The standard deviation of the regression model is 47. 46 A rough 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 spare variables that can explain more of the variation in weekly 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 hold to delete the entire observation containing the oulier. This will generally increase the R2 and F test statistic values Multicollinearity Multicollinearity advanced correlation exists between dickens independent variablesThis means the two variables contribute redundant information to the multiple regression model Including two highly cor link up independent variables can adversely affect the regression results No new information provided Can lead to unstable coefficients (large standard error and low t-values) Coefficient signs may not match prior expectations Some Indications of Severe Multicollinearity Incorrect signs on the coefficients Large change in the value of a previous coefficient when a new variable is added to the model A previously real variable becomes insignificant when a new independent variable is addedThe estimate of th e 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 evidence of collinearity between Price and Advertising Qualitative (Dummy) Variables Categorical explanatory variable (dummy variable) with two or more levels yes or no, on or off, male 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 = house price x1 = square feet The style of the house is also thought to matter Style = ranch, split level, condo Dummy-Variable Models (more than 2 Levels) Interpreting the Dummy Variable C oefficients (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 = Population regression constant ?i = Population regression coefficient for variable xj j = 1, 2, k p = Order of the polynomial (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 Compare quadratic model with the linear model Hypotheses (No 2nd order polynomial term) (2nd order polynomial term is needed) Higher Order Models interaction EffectsHypothesizes interaction between pairs of x variables Response to one x variable varies at different levels of other x variable Contains two-way cross product terms Effect of In teraction Without interaction term, effect of x1 on y is calculated 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 edifice Goal is to develop a model with the best set of independent variablesEasier to interpret if unimportant variables are removed Lower probability of collinearity Stepwise 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 equation in steps, either through forward selection, slow-witted elimination, or through standard stepwise regression The coefficient of partial determination is the measure of the marginal division 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 contain by looking for the highest adjusted R2 and lowest standard error s? Aptness of the Model diagnostic checks on the model include verifying the assumptions of multiple regression Each xi is linearly related to y Errors have constant variance Errors are independent Error are normally distributed counterweight Analysis The Normality Assumption Errors are assumed to be normally distributed Standardized residuals can be calculated by computerExamine a histogram or a normal probability plot of the standardized 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