© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Chapter 14 Using Multivariate Design and Analysis

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Multivariate Design and Analysis Multivariate design has multiple dependent variables Multiple dependent variables will be analyzed in a single multivariate analysis Correlational Multivariate Design Used when you have multiple measures in a correlational study Techniques available Multiple regression Discriminant analysis Canonical correlation Factor analysis

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Experimental Multivariate Design Multiple dependent variables in an experiment are analyzed in one analysis rather than separate univariate analyses Multivariate Analysis of Variance (MANOVA) is used to analyze data from multiple dependent variables MANOVA is more powerful than multiple univariate analyses of variance MANOVA is better than a univariate ANOVA for analyzing data from repeated measures designs

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Assumptions and Requirements of Multivariate Statistics Linearity The relationship between continuously measured variables must be linear Absence of Outliers The presence of extreme scores, or outliers, changes the slope of a regression line. Both univariate and multivariate outliers must be detected and corrected. Normality The population distribution underlying the sample population is assumed to be normal

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Homoscedasticity On a scatterplot, data points should form an elliptical pattern A conical pattern indicates heteroscedasticity Absence of Multicollinearity If variables are highly correlated, multicollinearity will exist and can affect the results of a multivariate analysis Sample Size Large samples are needed for multivariate analysis

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Factor Analysis Used to reduce a large number of variables to smaller sets comprising related variables A factor is a set of related variables representing a common underlying dimension The strength of the relationship between a variable and a factor is indicted by the factor loading Factor loadings below.3 are usually not interpreted Factor rotation is used to make factors more distinct and easier to interpret

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Two types of factor analysis Principle components analysis is used to reduce a large set of variables and to obtain an empirical summary of the data Principle factors analysis is used when your research is driven by theoretical or empirical predictions Exploratory factor analysis is used to describe a large set of variables in simpler terms and you have no a priori ideas about variable clustering Confrimatory factor analysis is used when you specify how variables should cluster

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Partial and Part Correlation Used when two variables are simultaneously influenced by a third variable Partial correlation removes the effect of a third variable from both other variables Not limited to the three variable case Part correlation removes the effect of a third variable from only ONE of the other variables

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Multiple Regression Multivariate analogue of bivariate regression Regression analysis for multiple dependent variables Types of multiple regression analysis Simple regression: All variables entered into regression equation at the same time Hierarchical regression order of variable entry determined by a theory or model Stepwise regression order of variable entry determined statistically

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Type of multiple regression to use depends on research needs Use hierarchical regression when you are working from a theoretical model Simple regression should be used in all other instances Stepwise regression should be avoided because it may capitalize on chance relationships Multiple R and R-square Multiple R is the correlation between predicted and observed values of Y R-square is an index of the amount of variability in the dependent variable accounted for by the predictor variables

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Regression weights Regression weights are used to interpret results Raw score regression weights are not used Standardized regression weights are used Standardized regression weights do not give an estimate of the unique contribution of a predictor to variability in the dependent variable The squared semipartial correlations should be used to determine the unique contribution of each predictor

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Discriminant Analysis Special case of multiple regression when you have a categorical dependent variable Can predict category membership based on a set of predictor variables The analysis works by forming discriminant functions More than one discriminant function can link variables The first one calculated is the strongest, with others being progressively weaker

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Discriminant analysis can be used in two ways Evaluate the amount of variability accounted for by each discriminant function Evaluate the degree of contribution of each predictor to the separation of groups

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Canonical Correlation Used to evaluate the relationship between a set of predictor variables and a set of dependent variables A canonical variate is calculated for each set The canonical correlation is the correlation between canonical variates Canonical correlation is not used much in psychological research

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Multiway Frequency Analysis Multivariate statistical analysis for categorical variables in an experiment Loglinear analysis is one form of multiway frequency analysis Similar to Chi-square Can easily be applied to designs with more than two independent variables

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Path Analysis Application of multiple regression to causal modeling Path coefficients provide estimates of the strength of causal connections between variables in a path model Causal paths can be decomposed into direct and indirect effects using Wright’s Rules