19-1 Chapter Nineteen MULTIVARIATE ANALYSIS: An Overview

19-2 Two Types of Multivariate Techniques Dependency –dependent (criterion) variables and independent (predictor) variables are present Interdependency –variables are interrelated without designating some dependent and others independent

19-3 Dependency Techniques Multiple regression Discriminant analysis Multivariate analysis of variance (MANOVA) Linear structural relationships (LISREL) Conjoint analysis

19-4 Multiple Regression Extension of bivariate linear regression to include more than one independent variable. –Y = βo + β 1 X1 + β 2 X2 + β 3 X3 + …..+ ε Use of multiple regression –Predict values for a criterion variable (dependent variable) by developing a self- weighting estimating equation.

19-5 Multiple Regression –Control for confounding variables to better evaluate the contribution of other variables –Test and explain causal theories Path analysis –Method of least squares (minimizing the sum of squared error terms) are used as in bivariate regression –Coefficients (B) vs. standardized coefficients (beta weights)

19-6 Multiple Regression Estimation Method –Enter method includes all the variables in the order of variables entered. –Forward selection starts with the constant and adds variables that results in the largest R 2. –Backward selection include all the variables and remove variable that change R 2 the least.

19-7 Multiple Regression –Stepwise selection The variable with the greatest explanatory power is added first. Subsequent variables are included according to their marginal (or incremental) contribution. A variable entered can be removed later if it becomes insignificant at a given alpha. This method which combines both forward and backward methods is the most popular method.

19-8 Tests –T- test for individual coefficients Ho : β i = 0, d.f. for t : n-k-1 –F-test for the overall model Ho : R 2 = 0 d.f. for F : (k, n-k-1) –As R 2 increases, standard error (of the estimate) decreases. The smaller standard error, the better model. Multiple Regression

19-9 Multiple Regression Collinearity (or Multicollinearity) problem –What is it? Situation where two or more independent variables are highly correlated. –What is consequence? Unreliable regression coefficients –How to detect? High correlation coefficients among independent variables (r >.8 requires attention)

19-10 Multiple Regression Collinearity problem continued Collinearity statistics (VIF): –If VIF>10, then multicollinearity suspicion –How to fix? Choose one and delete another when two independent variables are highly correlated. Create a new variable that is a composite of the two.

19-11 Multiple Regression Autocorrelation problem –Commonly found in time series data –What is it?: Error terms are correlated –What is consequence?: Unreliable coefficients –How to detect?: Visual detection, DW statistics –How to fix? Taking the first difference Taking logarithm Lagged dependent variable as an additional independent variable

19-12 Multiple Regression Use of dummy variables –Dummy variables are used when a nominal scale variable is to be included in the regression –When there are two categories of the variable, then one dummy variable is used. –When there are n categories, then n-1 dummy variables are used.

19-13 Discriminant Analysis Use –Classify persons or objects into various groups. –Analyze known groups to determine the relative influence of specific factors (or variables) Model –Similar to the multiple regression –Dependent variable: nominal One equation for two groups, two equations for three groups, and so on. –Independent variables: interval or ratio

19-14 MANOVA Assess relationship between two or more dependent variables and classificatory variables (or factors). Examples: measuring differences between –employees –customers –manufactured items –production parts

19-15 Uses of LISREL Explains causality among constructs not directly measured Two parts –Measurement model –Structural Equation model

19-16 Conjoint Analysis Mainly used for market research and product development. Evaluate a set of attributes to choose the product that best meets their needs

19-17 Interdependency Techniques Factor analysis: techniques to reduce many independent variables into a few manageable number. Cluster analysis: a set of techniques for grouping similar objects or people Multidimensional Scaling (MDS): a special description of a participant’s perception about a product, service, or other object of interest

19-18 Factor Analysis Computational techniques that reduce variables to a manageable number of factors that are not correlated with each other. Principal components analysis is most popular: construction of new set of variables (which are called “factors”) based on relationships in the correlation matrix.

19-19 Factor Analysis continued Loading and communalities(h 2 ) –Loading: correlation between a variable and a factor –Communalities: variance in each variable explained by all the factors Eigenvalue –A measure of explanatory power of each factor –Eignevalue/# of variables: % of total variance explained by each factor

19-20 Factor Analysis continued Rotation –To make pure constructs of each factor by focusing on a few major determinants of each factor. –To improve representations of variables by factors and to differentiate between factors. –Methods: Orthogonal vs. oblique

19-21 Steps in Cluster Analysis Select sample to be clustered Define measurement variables (e.g. market segment characteristics) Compute similarities among the entities through correlation, Euclidean distances, and other techniques Select mutually exclusive clusters Compare and validate the clusters

19-22 Multidimensional Scaling a special description of a participant’s perception about a product, service, or other object of interest Used in conjunction with cluster analysis or conjoint analysis. Used to understand difficult-to-measure constructs