Using Multivariate Statistics

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**1. Introduction.**

Multivariate Statistics: Why?The Domain of Multivariate Statistics:
Numbers of IVs and DVs. Experimental and Nonexperimental Research.
Computers and Multivariate Statistics. Why Not.Some Useful
Definitions.Continuous, Discrete, and Dichotomous Data. Samples and
Populations. Descriptive and Inferential Statistics. Orthogonality.
Standard and Sequential Analyses.Combining Variables.Number and
Nature of Variables to Include.Statistical Power.Data Appropriate
for Multivariate Statistics.The Data Matrix. The Correlation
Matrix. The Variance-Covariance Matrix. The Sum-of-Squares and
Cross-Products Matrix. Residuals.Organization of the Book.

**2. A Guide to Statistical Techniques: Using the Book.**

Research Questions and Associated Techniques.Degree of Relationship
among Variables. Significance of Group Differences. Prediction of
Group Membership. Structure. Time Course of Events.A Decision
Tree.Technique Chapters.Preliminary Check of the Data.

**3. Review of Univariate and Bivariate Statistics.**

Hypothesis Testing.One-Sample z Test as Prototype. Power.
Extensions of the Model.Analysis of Variance.One-Way
Between-Subjects ANOVA. Factorial Between-Subjects ANOVA.
Within-Subjects ANOVA. Mixed Between-Within-Subjects ANOVA. Design
Complexity. Specific Comparisons.Parameter Estimation.Strength of
Association.Bivariate Statistics: Correlation and
Regression.Correlation. Regression.Chi-Square Analysis.

**4. Cleaning Up Your Act: Screening Data Prior to
Analysis.**

Important Issues in Data Screening.Accuracy of Data File. Honest
Correlations. Missing Data. Outliers. Normality, Linearity, and
Homoscedasticity. Common Data Transformations. Multicollinearity
and Singularity. A Checklist and Some Practical
Recommendations.Complete Examples of Data Screening.Screening
Ungrouped Data. Screening Grouped Data.

**5. Multiple Regression.**

General Purpose and Description.Kinds of Research Questions.Degree
of Relationship. Importance of IVs. Adding IVs. Changing IVs.
Contingencies among IVs. Comparing Sets of IVs. Predicting DV
Scores for Members of a New Sample. Parameter Estimates.Limitations
to Regression Analyses.Theoretical Issues. Practical
Issues.Fundamental Equations for Multiple Regression.General Linear
Equations. Matrix Equations. Computer Analyses of Small Sample
Example.Major Types of Multiple Regression.Standard Multiple
Regression. Sequential Multiple Regression. Statistical (Stepwise)
Regression. Choosing among Regression Strategies.Some Important
Issues.Importance of IVs. Statistical Inference. Adjustment of R2.
Suppressor Variables. Regression Approach to ANOVA. Centering When
Interactions and Powers of IVs Are Included.Complete Examples of
Regression Analysis.Evaluation of Assumptions. Standard Multiple
Regression. Sequential Regression.Comparison of Programs.SPSS
Package. SAS System. SYSTAT System.

**6. Canonical Correlation.**

General Purpose and Description.Kinds of Research Questions.Number
of Canonical Variate Pairs. Interpretation of Canonical Variates.
Importance of Canonical Variates. Canonical Variate
Scores.Limitations.Theoretical Limitations. Practical
Issues.Fundamental Equations for Canonical Correlation.Eigenvalues
and Eigenvectors. Matrix Equations. Proportions of Variance
Extracted. Computer Analyses of Small Sample Example.Some Important
Issues.Importance of Canonical Variates. Interpretation of
Canonical Variates.Complete Example of Canonical
Correlation.Evaluation of Assumptions. Canonical
Correlation.Comparison of Programs.SAS System. SPSS Package. SYSTAT
System.

**7. Multiway Frequency Analysis.**

General Purpose and Description.Kinds of Research
Questions.Associations among Variables. Effect on a Dependent
Variable. Parameter Estimates. Importance of Effects. Strength of
Association. Specific Comparisons and Trend Analysis.Limitations to
Multiway Frequency Analysis.Theoretical Issues. Practical
Issues.Fundamental Equations for Multiway Frequency
Analysis.Screening for Effects. Modeling. Evaluation and
Interpretation. Computer Analyses of Small Sample Example.Some
Important Issues.Hierarchical and Nonhierarchical Models.
Statistical Criteria. Strategies for Choosing a Model.Complete
Example of Multiway Frequency Analysis.Evaluation of Assumptions:
Adequacy of Expected Frequencies. Hierarchical Loglinear
Analysis.Comparison of Programs.SPSS Package. SAS System. SYSTAT
System.

**8. Analysis of Covariance.**

General Purpose and Description.Kinds of Research Questions.Main
Effects of IVs. Interactions among IVs. Specific Comparisons and
Trend Analysis. Effects of Covariates. Strength of Association.
Parameter Estimates.Limitations to Analysis of
Covariance.Theoretical Issues. Practical Issues.Fundamental
Equations for Analysis of Covariance.Sums of Squares and Cross
Products. Significance Test and Strength of Association. Computer
Analyses of Small Sample Example.Some Important Issues.Test for
Homogeneity of Regression. Design Complexity. Evaluation of
Covariates. Choosing Covariates. Alternatives to ANCOVA.Complete
Example of Analysis of Covariance.Evaluation of Assumptions.
Analysis of Covariance.Comparison of Programs.SPSS Package. SYSTAT
System. SAS System.

**9. Multivariate Analysis of Variance and Covariance.**

General Purpose and Description.Kinds of Research Questions.Main
Effects of IVs. Interactions among IVs. Importance of DVs.
Parameter Estimates. Specific Comparisons and Trend Analysis.
Strength of Association. Effects of Covariates. Repeated-Measures
Analysis of Variance.Limitations to Multivariate Analysis of
Variance and Covariance.Theoretical Issues. Practical
Issues.Fundamental Equations for Multivariate Analysis of Variance
and Covariance.Multivariate Analysis of Variance. Computer Analyses
of Small Sample Example. Multivariate Analysis of Covariance.Some
Important Issues.Criteria for Statistical Inference. Assessing DVs.
Specific Comparisons and Trend Analysis. Design Complexity. MANOVA
vs. ANOVAs.Complete Examples of Multivariate Analysis of Variance
and Covariance.Evaluation of Assumptions. Multivariate Analysis of
Variance. Multivariate Analysis of Covariance.Comparison of
Programs.SPSS Package. SYSTAT System. SAS System.

**10. Profile Analysis: The Multivariate Approach to Repeated
Measures.**

General Purpose and Description.Kinds of Research
Questions.Parallelism of Profiles. Overall Difference among Groups.
Flatness of Profiles. Contrasts Following Profile Analysis.
Parameter Estimates. Strength of Association.Limitations to Profile
Analysis.Theoretical Issues. Practical Issues.Fundamental Equations
for Profile Analysis.Differences in Levels. Parallelism. Flatness.
Computer Analyses of Small Sample Example.Some Important
Issues.Contrasts in Profile Analysis. Univariate vs. Multivariate
Approach to Repeated Measures. Doubly Multivariate Designs.
Classifying Profiles. Imputation of Missing Values.Complete
Examples of Profile Analysis.Profile Analysis of Subscales of the
WISC. Doubly Multivariate Analysis of Reaction Time.Comparison of
Programs.SPSS Package. SAS System. SYSTAT System.

**11. Discriminant Function Analysis.**

General Purpose and Description.Kinds of Research
Questions.Significance of Prediction. Number of Significant
Discriminant Functions. Dimensions of Discrimination.
Classification Functions. Adequacy of Classification. Strength of
Association. Importance of Predictor Variables. Significance of
Prediction with Covariates. Estimation of Group Means.Limits to
Discriminant Function Analysis.Theoretical Issues. Practical
Issues.Fundamental Equations for Discriminant Function
Analysis.Derivation and Test of Discriminant Functions.
Classification. Computer Analyses of Small Sample Example.Types of
Discriminant Function Analysis.Direct Discriminant Function
Analysis. Sequential Discriminant Function Analysis. Stepwise
(Statistical) Discriminant Function Analysis.Some Important
Issues.Statistical Inference. Number of Discriminant Functions.
Interpreting Discriminant Functions. Evaluating Predictor
Variables. Design Complexity: Factorial Designs. Use of
Classification Procedures.Complete Example of Discriminant Function
Analysis.Evaluation of Assumptions. Direct Discriminant Function
Analysis.Comparison of Programs.SPSS Package. SYSTAT System. SAS
System.

**12. Logistic Regression.**

General Purpose and Description.Kinds of Research
Questions.Prediction of Group Membership or Outcome. Importance of
Predictors. Interactions among Predictors. Parameter Estimates.
Classification of Cases. Significance of Prediction with
Covariates. Strength of Association.Limitations to Logistic
Regression Analysis.Theoretical Issues. Practical
Issues.Fundamental Equations for Logistic Regression.Testing and
Interpreting Coefficients. Goodness-of-Fit. Comparing Models.
Interpretation and Analysis of Residuals. Computer Analyses of
Small Sample Example.Types of Logistic Regression.Direct Logistic
Regression. Sequential Logistic Regression. Stepwise (Statistical)
Logistic Regression. Probit and Other Analyses.Some Important
Issues.Statistical Inference. Number and Type of Outcome
Categories. Strength of Association for a Model. Coding Outcome and
Predictor Categories. Classification of Cases. Hierarchical and
Nonhierarchical Analysis. Interpretation of Coefficients Using
Odds. Logistic Regression for Matched Groups.Complete Examples of
Logistic Regression.Evaluation of Limitations. Direct Logistic
Regression with Two-Category Outcome. Sequential Logistic
Regression with Three Categories of Outcome.Comparison of
Programs.SPSS Package. SAS System. SYSTAT System.

**13. Principal Components and Factor Analysis.**

General Purpose and Description.Kinds of Research Questions.Number
of Factors. Nature of Factors. Importance of Solutions and Factors.
Testing Theory in FA. Estimating Scores on
Factors.Limitations.Theoretical Issues. Practical
Issues.Fundamental Equations for Factor Analysis.Extraction.
Orthogonal Rotation. Communalities, Variance, and Covariance.
Factor Scores. Oblique Rotation. Computer Analyses of Small Sample
Example.Major Types of Factor Analysis.Factor Extraction
Techniques. Rotation. Some Practical Recommendations.Some Important
Issues.Estimates of Communalities. Adequacy of Extraction and
Number of Factors. Adequacy of Rotation and Simple Structure.
Importance and Internal Consistency of Factors. Interpretation of
Factors. Factor Scores. Comparisons among Solutions and
Groups.Complete Example of FA.Evaluation of Limitations. Principal
Factors Extraction with Varimax Rotation.Comparison of
Programs.SPSS Package. SAS System. SYSTAT System.

**14. Structural Equation Modeling by Jodie B. Ullman.**

General Purpose and Description.Kinds of Research
Questions.Adequacy of the Model. Testing Theory. Amount of Variance
in the Variables Accounted for by the Factors. Reliability of the
Indicators. Parameter Estimates. Mediation. Group Differences.
Longitudinal Differences. Multilevel Modeling.Limitations to
Structural Equation Modeling.Theoretical Issues. Practical
Issues.Fundamental Equations for Structural Equations
Modeling.Covariance Algebra. Model Hypotheses. Model Specification.
Model Estimation. Model Evaluation. Computer Analysis of Small
Sample Example.Some Important Issues.Model Identification.
Estimation Techniques. Assessing the Fit of the Model. Model
Modification. Reliability and Proportion of Variance. Discrete and
Ordinal Data. Multiple Group Models. Mean and Covariance Structure
Models.Complete Examples of Structural Equation Modeling
Analysis.Model Specification for CFA. Evaluation of Assumptions for
CFA. Model Modification. SEM Model Specification. SEM Model
Estimation and Preliminary Evaluation. Model
Modification.Comparison of Programs.EQS. LISREL. SAS System.
AMOS.

**15. Survival/Failure Analysis.**

General Purpose and Description.Kinds of Research
Questions.Proportions Surviving at Various Times. Group Differences
in Survival. Survival Time with Covariates.Limitations to Survival
Analysis.Theoretical Issues. Practical Issues.Fundamental Equations
for Survival Analysis.Life Tables. Standard Error of Cumulative
Proportion Surviving. Hazard and Density Functions. Plot of Life
Tables. Test for Group Differences. Computer Analyses of Small
Sample Example.Types of Survival Analysis.Actuarial and
Product-Limit Life Tables and Survivor Functions. Prediction of
Group Survival Times from Covariates.Some Important
Issues.Proportionality of Hazards. Censored Data. Effect Size and
Power. Statistical Criteria. Odds Ratios.Complete Example of
Survival Analysis.Evaluation of Assumptions. Cox Regression
Survival Analysis.Comparison of Programs.SAS System. SYSTAT System.
SPSS Package.

**16. Time Series Analysis.**

General Purpose and Description.Kinds of Experimental
Questions.Pattern of Autocorrelation. Seasonal Cycles and Trends.
Forecasting. Effect of an Intervention. Comparing Time Series. Time
Series with Covariates. Effect Size and Power.Assumptions of Time
Series Analysis.Theoretical Issues. Practical Issues.Fundamental
Equations for Time Series ARIMA Models.Identification of ARIMA (p,
d, q) Models. Estimating Model Parameters. Diagnosing a Model.
Computer Analysis of Small Sample Time Series Example.Types of Time
Series Analysis.Models with Seasonal Components. Models with
Interventions. Adding Continuous Variables.Some Important
Issues.Patterns of ACFs and PACFs. Effect Size. Forecasting.
Statistical Methods for Comparing Two Models.Complete Example of a
Time Series Analysis.Evaluation of Assumptions. Baseline Model
Identification. Baseline Model Diagnosis. Intervention
Analysis.Comparison of Programs.SPSS Package. SAS System. SYSTAT
System.

**17. An Overview of the General Linear Model.**

Linearity and the General Linear Model.Bivariate to Multivariate
Statistics and Overview of TechniquesBivariate Form. Simple
Multivariate Form. Full Multivariate Form.Alternative Research
Strategies.

**Appendix A. A Skimpy Introduction to Matrix Algebra.**

The Trace of a Matrix.Addition or Subtraction of a Constant to a
Matrix.Multiplication or Division of a Matrix by a
Constant.Addition and Subtraction of Two Matrices.Multiplication,
Transposes, and Square Roots of Matrices.Matrix "Division"
(Inverses and Determinants).Eigenvalues and Eigenvectors:
Procedures for Consolidating Variance from a Matrix.

**Appendix B. Research Designs for Complete Examples.**

Women's Health and Drug Study.Sexual Attraction Study.Learning
Disabilities Data Bank.Reaction Time to Identify Figures.Clinical
Trial for Primary Biliary Cirrhosis.Impact of Seat Belt Law.

**Appendix C. Statistical Tables.**

Normal Curve Areas.Critical Values of the t Distribution for a =
.05 and .-1, Two-Tailed Test.Critical Values of the f
Distribution.Critical Values of Chi Square (c2).Critical Values for
Squares Multiple Correlation (R2) in Forward Stepwise
Selection.Critical Values for Fmax (S2max/S2min) Distribution for a
= .05 and .01.

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