Principal Components AnalysisPrincipal component analysisPrincipal components factor analysisStrategy for solving problems - 1Strategy for solving problems - 2Strategy for solving problems - 3Substituting components for variablesNotes - 1Notes - 2The Problem in BlackBoardStatement about Level of MeasurementMarking the Statement about Level of MeasurementStatement about Sample SizeRun the Principal Components Analysis - 1Run the Principal Components Analysis - 2Run the Principal Components Analysis - 3Run the Principal Components Analysis - 4Run the Principal Components Analysis - 5Run the Principal Components Analysis - 6Run the Principal Components Analysis - 7Run the Principal Components Analysis - 8Output for Sample Size RequirementMarking the Statement about Sample SizeThe Statement about Suitability for Factor Analysis: Sufficient CorrelationsSufficient Correlations in Correlation MatrixMarking the Statement about Sufficient CorrelationsThe Statement about Suitability for Factor Analysis: Test of SphericityBartlett’s Test of SphericityMarking the Statement about Bartlett’s Test of SphericityThe Statement about Suitability for Factor Analysis: Sampling AdequacyMeasures of Sampling Adequacy for Individual VariablesKaiser-Meyer-Olkin Measure of Sampling AdequacyMarking the Statement about Measures of Sampling AdequacyStatement about Initial Number of FactorsInitial Number of Factors: Eigenvalues Greater than OneInitial Number of Factors: Percentage of Variance ExplainedMarking the Statement about Initial Number of FactorsStatement about First Iteration of Factor ExtractionOutput for Communalities on First IterationMarking the Statement about First Iteration of Factor ExtractionRemoving a Variable from the Factor Analysis - 1Removing a Variable from the Factor Analysis - 2Removing a Variable from the Factor Analysis - 3Statement about Second Iteration of Factor ExtractionOutput for Communalities on Second IterationOutput for Factor Structure on Second IterationMarking the Statement about Second Iteration of Factor ExtractionSlide 48Slide 49Slide 50Statement about Third Iteration of Factor ExtractionOutput for Communalities on Third Iteration - 1Output for Factor Structure on Third Iteration - 2Output for Factor Structure on Third Iteration - 3Marking the Statement about Third Iteration of Factor ExtractionStatement about Variables Loading on the First ComponentOutput for Component OneMarking the Statement about Variables Loading on the First ComponentStatement about Variables Loading on the Second ComponentOutput for Component TwoMarking the Statement about Variables Loading on the Second ComponentStatement about Percentage of Variance Explained by FactorsOutput for Percentage of Variance Explained by FactorsMarking the Statement about Percentage of Variance Explained by FactorsPrincipal Components Analysis: Level of MeasurementPrincipal Components Analysis: Sample SizePrincipal Components Analysis: Suitability for Factor Analysis - 1Principal Components Analysis: Suitability for Factor Analysis - 2Logic Diagram for Solving Homework Problems: Stepwise RelationshipPrincipal Components Analysis: Individual RelationshipsPrincipal Components Analysis: Classification AccuracyPrincipal Components Analysis: Cross-validationSteps in answering questions about principal components analysis - 1Steps in answering questions about principal components analysis - 2Steps in answering questions about principal components analysis – 3aSteps in answering questions about principal components analysis – 3bSteps in answering questions about principal components analysis - 4Steps in answering questions about principal components analysis - 5Steps in answering questions about principal components analysis - 6Steps in answering questions about principal components analysis - 7Steps in answering questions about principal components analysis - 8Slide 1Principal Components AnalysisSlide 2Principal component analysisPrincipal component analysisStrategy for solving problemsSample problemSteps in principal component analysisHomework problemsSlide 3Principal components factor analysisObtaining a factor solution through principal components analysis is an iterative process that usually requires repeating the SPSS factor analysis procedure a number of times to reach a satisfactory solution.We begin by identifying a group of variables whose variance we believe can be represented more parsimoniously by a smaller set of factors, or components. The end result of the principal components analysis will tell us which variables can be represented by which components, and which variables should be retained as individual variables because the factor solution does not adequately represent their information.Slide 4Strategy for solving problems - 1A principal component factor analysis requires:The variables included must be metric level or dichotomous (dummy-coded) nominal levelThe sample size must be greater than 50 (preferably 100)The ratio of cases to variables must be 5 to 1 or largerThe correlation matrix for the variables must contain 2 or more correlations of 0.30 or greaterVariables with measures of sampling adequacy less than 0.50 must be removedThe overall measure of sampling adequacy is 0.50 or higherThe Bartlett test of sphericity is statistically significant.The first phase of a principal component analysis is devoted to verifying that we meet these requirements. If we do not meet these requirements, factor analysis is not appropriate.Slide 5Strategy for solving problems - 2The second phase of a principal component factor analysis focuses on deriving a factor model, or pattern of relationships between variables and components, that satisfies the following requirements:The derived components explain 50% or more of the variance in each of the variables, i.e. have a communality greater than 0.50None of the variables have loadings, or correlations, of 0.40 or higher for more than one component, i.e. do not have complex structureNone of the components has only one variable in itTo meet these requirements, we remove problematic variables from the analysis and repeat the principal component analysis procedure in SPSS.Slide 6Strategy for solving problems - 3If, at the conclusion of this process, we can substitute the components for the variables in further analyses if: the components have more than one variable loading on them, the components explain