Multiple Regression Models: Interactions and Indicator VariablesToday’s Data SetFirst order modelThe regression equation is…Tests of Individual β ParametersWhat if the relationship between E(y) and either of the independent variables depends on the other?For our exampleAre these slopes parallel, or do they change with the number of bidders?Caution!Another Example: Graph and interpret the following findingsConclusionsBasic premises up to this pointDummy VariablesCreating Dummy VariablesReturning to the Clocks at Auction Data SetLet’s Create a Dummy Variable in MinitabBefore we run the analysis…Thinking Through the VariablesRun the AnalysisLet’s look at your Journal ApplicationNested ModelsWhich is better? How do we decide?Error is Always Greater for the Reduced ModelTest statistic: FConclusions?Question 3 (Journal)Multiple Regression Models: Interactions and Indicator VariablesToday’s Data SetA collector of antique grandfather clocks knows that the price received for the clocks increases linearly with the age of the clocks. Moreover, the collector hypothesizes that the auction price of the clocks will increase linearly as the number of bidders increases.(Let’s hypothesize a first order MLR model.)First order modely = β0 + β1x1 + β2x2 + εy = auction pricex1= age of clock (years)x2 = number of biddersThe regression equation is…AuctionPrice = - 1339 + 12.7 AgeOfClock + 86.0 BiddersAnalysis of VarianceSource DF SS MS F PRegression 2 4283063 2141531 120.19 0.000Residual Error 29 516727 17818Total 31 4799790Tests of Individual β ParametersPredictor Coef SE Coef T PConstant -1339.0 173.8 -7.70 0.000AgeOfClock 12.7406 0.9047 14.08 0.000Bidders 85.953 8.729 9.85 0.000What if the relationship between E(y) and either of the independent variables depends on the other?In this case, the two independent variables interact, and we model this as a cross-product of the independent variables.For our exampleDo age and the number of bidders interact? In other words, is the rate of increase of the auction price with age driven upward by a large number of bidders? In this case, as the number of bidders increases, the slope of the price versus age line increases.To facilitate investigation, number of bidders has been separated into:A: 0-6 biddersB: 7-10 biddersC: 11-15 biddersAre these slopes parallel, or do they change with the number of bidders?AgeOfClockAuctionPrice190180170160150140130120110100225020001750150012501000750500Bidders3CatCABScatterplot of AuctionPrice vs AgeOfClockCaution!Once an interaction has been deemed important in a model, all associated first-order terms should be kept in the model, regardless of the magnitude of their p-values.Another Example: Graph and interpret the following findingsLet’s say we want to study how hard students work on tests. We have some achievement-oriented students and some achievement-avoiders. We create two random halves in each sample, and give half of each sample a challenging test, the other an easy test. We measure how hard the students work on the test. The means of this study are: Achievement-oriented (n=100)Achievement –avoiders (n=100)Challenging test 10 5Easy test 5 10ConclusionsE(y)= β0 + β1x1 + β2x2+ β3x1x2The effect of test difficulty (x1) on effort (y) depends on a student’s achievement orientation (x2).Thus, the type of achievement orientation and test difficulty interact in their effect on effort.This is an example of a two-way interaction between achievement orientation and test difficulty.Basic premises up to this pointWe have used continuous variables (we can assume that having a value of 2 on a variable means having twice as much of it as a 1.) We often work with categorical variables in which the different values have no real numerical relationship with each other (race, political affiliation, sex, marital status…) Democrat(1), Independent(2), Republican(3)Is a Republican three times as politically affiliated as a Democrat?How do we resolve this problem?Dummy VariablesA dummy variable is a numerical variable used in regression analysis to represent subgroups of the sample in your study.Dummy variables have two values: 0, 1 "Republican" variable: someone assigned a 1 on this variable is Republican and someone with an 0 is not. They act like 'switches' that turn various parameters on and off in an equation.Creating Dummy VariablesIn Minitab, we can recode the categorical variable into a set of dummy variables, each of which has two levels. In the regression model, we will use all but one of the original levels. The level which is not included in the analysis is the category to which all other categories will be compared (base level.) You decide this.The coefficient on the variable in your regression will show the effect that being that variable has on your dependent variable.Returning to the Clocks at Auction Data SetThe collector of antique grandfather clocks knows that the price received for the clocks increases linearly with the age of the clocks and he hypothesized that the auction price of the clocks will increase linearly as the number of bidders increases. But let’s say he doesn’t have the exact number of bidders, only knows if there was a high number of bidders (we’ll say 9 and above) or a low number (below 9.)Let’s Create a Dummy Variable in MinitabWe’ll use the Bidders2Cat columnCalc Make Indicator VariablesIn top box, specify that you want to make indicator variables for Bidders2CatLet’s store results in C10 - C11Once you have created the variables, name columns 10 and 11 ManyBidders and FewBiddersWhich is which? Why?Before we run the analysis…Let’s say we decide to include ManyBidders (FewBidders is the base level.)Because FewBidders is not included, we can determine if ManyBidders predicts a different Auction Price than FewBidders. If ManyBidders is significant in our regression, with a positive β coefficient, we conclude that ManyBidders has a significant effect on the price of the clocks at auction.Thinking Through the VariablesWhat is x1?Let’s hypothesize the model in plain English… just looking at high/low bidders.)What’s the Null Hypothesis?Run the AnalysisResults of the t-test?What would