Multiple Regression – Assumptions and OutliersMultiple Regression and AssumptionsMultiple Regression and OutliersRelationship between assumptions and outliersOrder of analysis is importantStrategy for solving problemsTransforming dependent variablesTransforming independent variables - 1Transforming independent variables - 2Transforming independent variables - 3Impact of transformations and omitting outliersNotesProblem 1Dissecting problem 1 - 1Dissecting problem 1 - 2Dissecting problem 1 - 3R² before transformations or removing outliersSlide 18Slide 19Normality of the dependent variable: total family incomeSlide 21Linearity and independent variable: how many in family earned moneyLinearity and independent variable: how many in family earned moneyNormality of independent variable: how many in family earned moneySlide 25Slide 26Transformation for how many in family earned moneyNormality of independent variable: respondent’s incomeNormality of independent variable: respondent’s incomeLinearity and independent variable: respondent’s incomeSlide 31Homoscedasticity: sexSlide 33Adding a transformed variableThe transformed variable in the data editorThe regression to identify outliersSaving the measures of outliersThe variables for identifying outliersComputing the probability for Mahalanobis D²Formula for probability for Mahalanobis D²Multivariate outliersUnivariate outliersOmitting the outliersSpecifying the condition to omit outliersThe formula for omitting outliersCompleting the request for the selectionThe omitted multivariate outlierRunning the regression without outliersOpening the save options dialogClearing the request to save outlier dataOpening the statistics options dialogRequesting descriptive statisticsRequesting the outputSample size requirementSignificance of regression relationshipIncrease in proportion of varianceProblem 2Dissecting problem 2 - 1Dissecting problem 2 - 2Dissecting problem 2 - 3Slide 61Slide 62Normality of the dependent variableSlide 64Normality of independent variable: AgeSlide 66Linearity and independent variable: AgeSlide 68Transformation for AgeLinearity and independent variable: Highest year of school completedSlide 71Normality of independent variable: Highest year of school completedSlide 73Transformation for highest year of schoolSlide 75Slide 76Slide 77Slide 78Slide 79Slide 80Slide 81Slide 82Slide 83The multivariate outlierThe univariate outlierSlide 86Slide 87Slide 88Slide 89Slide 90Slide 91Slide 92Slide 93Slide 94Slide 95Slide 96Slide 97Slide 98Slide 99Impact of assumptions and outliers - 1Impact of assumptions and outliers - 2Impact of assumptions and outliers - 3Impact of assumptions and outliers - 4Impact of assumptions and outliers - 5SW388R7Data Analysis & Computers IISlide 1Multiple Regression – Assumptions and OutliersMultiple Regression and AssumptionsMultiple Regression and OutliersStrategy for Solving ProblemsPractice ProblemsSW388R7Data Analysis & Computers IISlide 2Multiple Regression and AssumptionsMultiple regression is most effect at identifying relationship between a dependent variable and a combination of independent variables when its underlying assumptions are satisfied: each of the metric variables are normally distributed, the relationships between metric variables are linear, and the relationship between metric and dichotomous variables is homoscedastic.Failing to satisfy the assumptions does not mean that our answer is wrong. It means that our solution may under-report the strength of the relationships.SW388R7Data Analysis & Computers IISlide 3Multiple Regression and OutliersOutliers can distort the regression results. When an outlier is included in the analysis, it pulls the regression line towards itself. This can result in a solution that is more accurate for the outlier, but less accurate for all of the other cases in the data set.We will check for univariate outliers on the dependent variable and multivariate outliers on the independent variables.SW388R7Data Analysis & Computers IISlide 4Relationship between assumptions and outliersThe problems of satisfying assumptions and detecting outliers are intertwined. For example, if a case has a value on the dependent variable that is an outlier, it will affect the skew, and hence, the normality of the distribution.Removing an outlier may improve the distribution of a variable.Transforming a variable may reduce the likelihood that the value for a case will be characterized as an outlier.SW388R7Data Analysis & Computers IISlide 5Order of analysis is importantThe order in which we check assumptions and detect outliers will affect our results because we may get a different subset of cases in the final analysis.In order to maximize the number of cases available to the analysis, we will evaluate assumptions first. We will substitute any transformations of variable that enable us to satisfy the assumptions. We will use any transformed variables that are required in our analysis to detect outliers.SW388R7Data Analysis & Computers IISlide 6Strategy for solving problems1. Run type of regression specified in problem statement on variables using full data set.2. Test the dependent variable for normality. If it does not satisfy the criteria for normality unless transformed, substitute the transformed variable in the remaining tests that call for the use of the dependent variable.3. Test for normality, linearity, homoscedasticity using scripts. Decide which transformations should be used.4. Substitute transformations and run regression entering all independent variables, saving studentized residuals and Mahalanobis distance scores. Compute probabilities for D².5. Remove the outliers (studentized residual greater than 3 or Mahalanobis D² with p <= 0.001), and run regression with the method and variables specified in the problem.6. Compare R² for analysis using transformed variables and omitting outliers (step 5) to R² obtained for model using all data and original variables (step 1).Our strategy for solving problems about violations of assumptions and outliers will include the following steps:SW388R7Data Analysis & Computers IISlide 7Transforming dependent variablesIf dependent variable is not normally distributed:Try log, square root, and inverse transformation. Use first transformed variable that satisfies normality criteria.If no transformation satisfies normality criteria, use untransformed variable and add caution for