11.4 Multivariate analysis techniques

Multivariate analysis techniques are powerful tools for understanding complex relationships in survey data. They allow researchers to examine multiple variables simultaneously, uncovering patterns and connections that might be missed with simpler methods.

From regression and classification to dimension reduction and clustering, these techniques offer a comprehensive toolkit for statistical analysis. They help researchers make sense of large datasets, test hypotheses, and draw meaningful conclusions from survey responses.

Regression and Classification Techniques

Multiple and Logistic Regression

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• Multiple regression analyzes relationships between multiple independent variables and one dependent variable
• Extends simple linear regression to include more than one predictor variable
• Uses least squares method to minimize the sum of squared residuals
• Regression equation takes the form: $Y = β₀ + β₁X₁ + β₂X₂ + ... + βₖXₖ + ε$
  • Y represents the dependent variable
  • X₁, X₂, ..., Xₖ represent independent variables
  • β₀, β₁, β₂, ..., βₖ represent regression coefficients
  • ε represents the error term
• Assumptions include linearity, independence of errors, homoscedasticity, and normality of residuals
• Logistic regression predicts binary outcomes (success/failure, yes/no)
• Uses maximum likelihood estimation to fit the model
• Logistic function transforms linear combination of predictors into probability between 0 and 1
• Equation for logistic regression: $P(Y=1) = \frac{1}{1 + e^{-(β₀ + β₁X₁ + β₂X₂ + ... + βₖXₖ)}}$
• Interprets results using odds ratios and log-odds

Discriminant Analysis

• Discriminant analysis classifies observations into predefined groups based on multiple predictor variables
• Aims to find linear combinations of variables that best separate groups
• Linear discriminant analysis (LDA) assumes equal covariance matrices for all groups
• Quadratic discriminant analysis (QDA) allows for different covariance matrices
• Discriminant function maximizes between-group variance relative to within-group variance
• Can be used for dimensionality reduction and visualization of multivariate data
• Evaluates classification accuracy using confusion matrices and cross-validation

Dimension Reduction Methods

Factor Analysis

• Factor analysis identifies underlying latent variables (factors) that explain correlations among observed variables
• Reduces large number of variables to smaller set of factors
• Exploratory factor analysis (EFA) discovers factor structure without prior hypotheses
• Confirmatory factor analysis (CFA) tests specific factor structure based on theory
• Steps include correlation matrix calculation, factor extraction, rotation, and interpretation
• Common factor extraction methods include principal axis factoring and maximum likelihood
• Factor rotation techniques (varimax, oblimin) improve interpretability of factor loadings
• Scree plot and eigenvalues guide decision on number of factors to retain
• Factor scores can be used in subsequent analyses or as composite variables

Principal Component Analysis and Multidimensional Scaling

• Principal component analysis (PCA) transforms correlated variables into uncorrelated components
• Maximizes variance explained by each successive component
• First principal component accounts for most variance, followed by second, third, and so on
• Eigenvalues and eigenvectors of covariance or correlation matrix determine principal components
• Scree plot helps determine number of components to retain (elbow method)
• Can be used for data compression, feature selection, and visualization
• Multidimensional scaling (MDS) visualizes similarities or dissimilarities between objects in lower-dimensional space
• Classical MDS uses Euclidean distances between objects
• Non-metric MDS preserves ordinal relationships between distances
• Stress value measures goodness of fit for MDS solutions
• Applications include market research, psychological scaling, and gene expression analysis

Clustering and Structural Analysis

Cluster Analysis Techniques

• Cluster analysis groups similar objects together based on multiple variables
• Hierarchical clustering builds nested clusters (dendrogram representation)
  • Agglomerative (bottom-up) starts with individual objects and merges clusters
  • Divisive (top-down) starts with one cluster and splits into smaller clusters
• K-means clustering partitions data into k predefined clusters
  • Iteratively assigns objects to nearest centroid and updates centroids
  • Requires specifying number of clusters in advance
• Density-based clustering (DBSCAN) identifies clusters of arbitrary shape based on density
• Evaluates cluster quality using silhouette coefficient, Calinski-Harabasz index, or Davies-Bouldin index
• Applications include customer segmentation, image segmentation, and document classification

Path Analysis and Structural Equation Modeling

• Path analysis examines direct and indirect relationships among variables
• Represents causal relationships using path diagrams
• Calculates path coefficients using multiple regression or correlation analysis
• Decomposes total effects into direct and indirect effects
• Structural equation modeling (SEM) combines factor analysis and path analysis
• Tests complex relationships between observed and latent variables
• Consists of measurement model (factor analysis) and structural model (path analysis)
• Evaluates model fit using chi-square test, CFI, RMSEA, and other fit indices
• Allows for testing and comparison of alternative models
• Used in psychology, sociology, and marketing research to test theoretical frameworks

Canonical Correlation Analysis

• Canonical correlation analysis (CCA) explores relationships between two sets of variables
• Finds linear combinations of variables in each set that maximize correlation between sets
• Produces canonical variates (linear combinations) and canonical correlations
• Number of canonical correlations equals minimum number of variables in either set
• Tests significance of canonical correlations using Wilks' lambda or other multivariate tests
• Interprets results using canonical loadings and cross-loadings
• Applications include relating personality traits to job performance measures or relating environmental factors to species abundance

Multivariate Hypothesis Testing

Multivariate Analysis of Variance (MANOVA)

• MANOVA extends univariate ANOVA to multiple dependent variables
• Tests for differences in means across groups on multiple outcome variables simultaneously
• Accounts for correlations among dependent variables
• Null hypothesis states no difference in population mean vectors across groups
• Test statistics include Wilks' lambda, Pillai's trace, Hotelling's trace, and Roy's largest root
• Assumes multivariate normality, homogeneity of covariance matrices, and independence of observations
• Post-hoc tests (discriminant analysis, univariate ANOVAs) follow significant MANOVA results
• Advantages over multiple ANOVAs include control of Type I error rate and increased power
• Used in psychology, education, and biology to compare groups on multiple outcomes
• Can be extended to multivariate analysis of covariance (MANCOVA) to include covariates