Principal component analysisPrincipal components factor analysisStrategy for solving problems - 1Strategy for solving problems - 2Strategy for solving problems - 3Notes - 1Notes - 2Problem 1Computing a principal component analysisAdd the variables to the analysisCompete the descriptives dialog boxSelect the extraction methodCompete the extraction dialog boxSelect the rotation methodCompete the rotation dialog boxComplete the request for the analysisLevel of measurement requirementSample size requirement: minimum number of casesSample size requirement: ratio of cases to variablesAppropriateness of factor analysis: Presence of substantial correlationsAppropriateness of factor analysis: Sampling adequacy of individual variablesSlide 22Excluding a variable from the factor analysisRepeating the factor analysisRemoving the variable from the list of variablesReplicating the factor analysisAppropriateness of factor analysis: Sample adequacy for revised factor analysisAppropriateness of factor analysis: Sample adequacy for set of variablesAppropriateness of factor analysis: Bartlett test of sphericityNumber of factors to extract: Latent root criterionNumber of factors to extract: Percentage of variance criterionEvaluating communalitiesCommunality requiring variable removalSlide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Communality satisfactory for all variablesIdentifying complex structureSlide 43Slide 44Slide 45Checking for single-variable componentsVariable loadings on componentsInterpreting the principal componentsVariance explained in individual variablesTotal variance explainedAnswering the problem questionProblem 2Slide 53Slide 54Slide 55Slide 56Slide 57Slide 58Slide 59Slide 60Slide 61Slide 62Slide 63Slide 64Steps in principal component analysis - 1Steps in principal component analysis - 2Steps in principal component analysis - 3Steps in principal component analysis - 4Steps in principal component analysis - 5SW388R7Data Analysis & Computers IISlide 1Principal component analysisPrincipal component analysisStrategy for solving problemsSample problemsSteps in principal component analysisSW388R7Data Analysis & Computers IISlide 2Principal 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.SW388R7Data Analysis & Computers IISlide 3Strategy 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.SW388R7Data Analysis & Computers IISlide 4Strategy 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.SW388R7Data Analysis & Computers IISlide 5Strategy for solving problems - 3If, at the conclusion of this process, we have components that have more than one variable loading on them, have components that explain at least 50% of the variance in the included variables, and have components that collectively explain more than 60% of the variance in the set of variables, we can substitute the components for the variables in further analyses.Variables that were removed in the analysis should be included individually in further analyses.Substitution of components for individual variables is accomplished by using only the highest loading variable, or by combining the variables loading on each component to create a new variable.SW388R7Data Analysis & Computers IISlide 6Notes - 1When evaluating measures of sampling adequacy, communalities, or factor loadings, we ignore the sign of the numeric value and base our decision on the size or magnitude of the value. The sign of the number indicates the direction of the relationship. A loading of -0.732 is just as strong as a loading of 0.732. The minus sign indicates an inverse or negative relationship; the absence of a sign is meant to imply a plus sign indicating a direct or positive relationship.SW388R7Data Analysis & Computers IISlide 7Notes - 2If there are two or more components in the component matrix, the pattern of loadings is based on the SPSS Rotated Component Matrix. If there is only one component in the solution, the Rotated Component Matrix is not computed, and the pattern of loadings is based on the Component Matrix.It is possible that the analysis will break down and we will have too few variables in the analysis to support the use of principal component analysis.SW388R7Data Analysis & Computers IISlide 8Problem 1In the dataset GSS2000.sav, is the following statement true, false, or an incorrect application of a statistic? Answer the question based on the results of a principal component analysis prior to testing for outliers, split sample validation, and a test of reliability. Assume that there is no problematic