1Chapter 1Overview and Examples of Multivariate MethodsSection 1.1Introduction and Examples of Multivariate Statistics 3Objective Recognize appropriate analyses for a variety of multivariate research questions.4Univariate versus Multivariate StatisticsUnivariate statistics consider only one dependent variable (DV) at a time.– Examples: sample mean, t-test, ANOVAMultivariate statistics consider more than one dependent variable at a time.– Examples: vector of sample means, Hotelling’s T2, MANOVA5Advantages of Multivariate MethodsUnivariate statistics  increase the risk of type-I error with many DVs only demonstrate the relationships between independent variables (IVs) and a DV, but miss relationships among the DVs. Multivariate statistics  control type-I error by considering a set of dependent variables in multidimensional space account for relationships among the DVs as well as the relationships between IVs and DVs.6Applications of Multivariate StatisticsMultivariate statistics can be used to address a wide variety of research questions. Consider several examples of multivariate statistical applications in scientific research.27A Comparison of Advertising StrategiesEvaluate the effectiveness of three different commercials. What makes an effective commercial?  Product recognition Product recall Product liking Price willing to pay for product This example has three levels of the independent variable four response variables.Chapter 28The Effectiveness of a DrugA drug company wants to compare the effectiveness of two different drug formulations (Old, New) across different dosages (50, 100, 200 mg). How is effectiveness evaluated?  Score on a depression scale Scores on two different obsessive-compulsive behavior scales.2 × 3 factorial design, three responses.Chapter 29Multivariate Analysis of Variance: MANOVAMV analogy to ANOVA. Tests for significant differences between groups on two or more related dependent variables simultaneously while accounting for the correlation among the dependent variables.Research question: “Are there significant differences between two or more groups on a set of responses?”Chapter 210Corporate Training ExampleA company wants to compare the effectiveness of three employee training methods in a repeated measures study. Effectiveness is defined as:Score on a test of corporate policiesScore on a test of job-specific skills.Employees are tested at three time intervals (2 weeks, 4 weeks, and 6 weeks).Chapter 311Doubly Multivariate Repeated Measures Tests for significant group differences over time across a set of response variables measured at each time while accounting for correlation among the responses. Research question:“Do the factors have an effect on a set of responses over time?”“Do changes in an independent variable over time predict changes in the set of responses?”Chapter 312The Diagnostic Usefulness of an InstrumentHow well does a new psychological instrument perform compared to an established instrument? The established instrument, based on diagnostic criteria, contains twelve items and must be administered by a trained interviewer.  The test instrument contains twenty items and can be completed by the respondent with pen and paper. This example has twelve continuous predictors twenty continuous responses.Chapter 3313Multivariate Multiple RegressionTest for significant linear relationship between a set of predictors and a set of responses while accounting for the correlations among the responses. Research Question:“Does variation in a set of continuous independent variables adequately predict a set of continuous responses?”Chapter 314Canonical Correlation AnalysisCanonical correlation analysis tests the same hypotheses as multivariate regression, but also allows you to  interpret how the predictors are related to the responses interpret how the responses are related to the predictors examine how many dimensions the variable sets share in common.Chapter 315Pathological Gambling ExampleResearchers want to use responses to questionnaire items to classify people identified as steady gamblers, binge gamblers, and control/non-gamblers. A twelve-item questionnaire is administered to three groups of participants.Question: What linear combination of responses accounts for most of the variation in classification of gamblers?Chapter 416Customer Profiling and PredictionA credit card company is interested in using financial information to decide whether potential customers represent good or bad risk before offering a credit card. An analyst is interested in understanding what combination of demographic variables best predict whether a customer prefers one of several different marketing strategies.Chapter 417Discriminant Function Analysis Discriminant function analysis (DFA) is a dimension reduction method that can be used to identify a linear combination of variables that produces the greatest distance between categories. DFA is conceptually similar to logistic regression for multivariate data, and it is computationally similar to MANOVA. Chapter 418Bird Habitat ExampleA researcher is interested in understanding the habitat of a species of bird. Twenty characteristics are measured for each habitat. Many of these measures are associated. These variables will be used in regression, discriminant, and cluster analyses.The researcher wants to reduce the total number of variables from 20 to something smaller and eliminate potential collinearity problems.Chapter 5419Principal Components AnalysisA dimension reduction technique creates new variables that are linear combinations of a set of correlated variables  does not assume an underlying latent factor structure.Practical question:“How can I reduce the set of 20 correlated variables to a more manageable number of uncorrelated variables?”Chapter 520Perceptions of Mathematics in SchoolAn researcher wants to know whether students’ self-perceptions in math reflect several underlying latent factors or one single factor.Questionnaire items from several instruments intended to measure mathematics-related perceptions are administered to 4000 students.  Exploratory analysis identifies possible underlying factors.  Confirmatory analysis tests hypotheses about factors.Chapter 521Factor AnalysisExploratory factor analysis is a variable identification technique with superficial resemblance to principal components analysis,