Multiple Regression ModelsExploring an example: “Chapter 4: Multiple Regression II” dataExplaining Assets with each predictor variableWhat is Multiple Regression?Strategy for Multiple RegressionThe Multiple Linear Regression ModelThe Prediction EquationBasic IdeaDoing the CalculationsLet’s Return to the ExampleAssessing the Utility of the Model: Hypothesis tests (see MLR handout)ANOVA: ANalysis Of VArianceSums of squaresSST = SSM + SSEF statisticProceed only if F and corresponding p-value indicate sufficient evidence that the overall model is usefulInterpreting coefficientsConfidence IntervalsReturning to our example…What if the Relationship is Curvilinear?Basic Quadratic ModelInterpreting the Coefficient (β) EstimatesAssessing Model UtilityWhat if I have a Qualitative Independent Variable?What if the relationship between E(y) and any one IV depends on the value of another IV?Example: Graph and interpret the following findingsCaution!ConclusionsMultiple Regression ModelsExploring an example: “Chapter 4: Multiple Regression II” dataOnline stock trading through the Internet has increased dramatically during the past several years. An article discussing this new method of investing provided data on the major Internet stock brokerages who provide this service. Here we have some data for the top 10 Internet brokerages. The variables are Mshare, the market share of the firm; Accts, the number of Internet accounts in thousands; and Assets, the total assets in billions of dollars. Describe the data:How many variables does the data set contain? How would you describe them in terms of levels of measurement?Explaining Assets with each predictor variableFind the correlation between Assets, and the explanatory variables Mshare and Accts.Use a Simple Linear Regression to predict Assets content using the number of accounts. What is the regression equation? What are the results of the significance test for the regression coefficient?Do the same using Mshare.What is Multiple Regression?What is Multiple Regression?Predicting an outcome (dependent variable) based upon several independent variables simultaneously. Why is this important? Why is this important? Behavior is rarely a function of just one variable, but is instead influenced by many variables. So the idea is that we should be able to obtain a more accurate predicted score if using multiple variables to predict our outcome.Strategy for Multiple RegressionHypothesize form of the model (choose which independent variables to include) Conduct exploratory data analysis Develop one or more tentative models Identify most suitable model Make inferences based on model Stop…StartThe Multiple Linear Regression ModelRegression applications in which there are several independent variables, x1, x2, … , xk . A multiple linear regression model with p independent variables has the equation  βi is the intercept and βi determines the contribution of the independent variable xi The ε is a random variable with mean 0 and variance σ2. y o 1 1 p px x      KThe Prediction EquationThe equation for this model fitted to data is Where denotes the “predicted” value computed from the equation, and bi denotes an estimate of βi. As with Simple Linear Regression, they’re obtained by the method of least squares Among the set of all possible values for the parameter estimates, I find the ones which minimize the sum of squared residuals.o 1 1 p pˆy b b x b x   KˆyBasic IdeaWith multiple regression, we form a 'linear combination' of multiple variables to best predict an outcome, and then we assess the contribution that each predictor variable makes to the equation. My research question might be: “How much does an independent variable contribute to explaining dependent variable after the effect of another independent variable is taken into account?”Doing the CalculationsComputation of the estimates by hand is tedious. They are ordinarily obtained using a regression computer program. Standard errors also are usually part of output from a regression program.Let’s Return to the ExampleConstruct a 3-D plot.ACome up with a prediction equation for the multiple regression model.Assessing the Utility of the Model: Hypothesis tests (see MLR handout) Test if all of the slope parameters are zero: F –test.Test if a particular slope parameter is zero given that all other x's remain in the model: t –test.ANOVA: ANOVA: ANalysis Of VAriance This is a test of the null hypothesis that Multiple R in the population = 0.0. If this is .05 or less, reject the null hypothesis.For a multiple linear regression model with p independent variables fitted to a data set with n observations is, the ANOVA is:Source ofVariation DF SS MSModel p SSM MSMErrorn-p-1 SSE MSETotal n-1 SSTSums of squaresThe sums of squares SSM, SSE, and SST have the same definitions in relation to the model as in simple linear regression:   222ˆSSR y yˆSSE y ySST y y   MSST = SSM + SSEThe value of SST does not change with the model.It depends only on the values of the dependent variable y. SSE decreases as variables are added to a model, and SSM increases by the same amount. This amount of increase in SSM is the amount of variation due to variables in the larger model that was not accounted for by variables in the smaller model.F statisticF is the statistic to test if ALL the slope parameters are zero. •ANOVA gives F statistic and p-value (be sure to set the α level)•Under the null hypothesis the F statistic has an F(p, n-p-1) distribution and the p-value is ___. According to this distribution, the chance of obtaining an F statistic of __ or larger is _(p-value). We conclude that the model is useful/not useful for predicting… 1 2: ... 0o pH     MSMFMSEProceed only if F and corresponding p-value indicate sufficient evidence that the overall model is useful If so, look to the individual variables to determine their contributionWe do this with t-testsp = .05 or less than each variable indicates a significant contributionInterpreting coefficientsConstant = slopeOther coefficients are the regression coefficients, interpreted as the change in the mean