Using Multivariate Statistics - GBV

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FOURTH EDITIONUsing Multivariate StatisticsBarbara G. TabachnickCalifornia State University, NorthridgeLinda S. FidellCalifornia State University, NorthridgeTechnlsche Universitat DarmstadtFACHBEREICH 10 — BIOLOGIE— Bibliothek-— "SchnittspahnstraBe tOD-6 4 28 7 D a r m s t a d tAllyn and BaconBoston London Toronto Sydney Tokyo Singapore

CONTENTSPrefaceXxxvIntroduction11.1 Multivariate Statistics: Why?11.1.1 The Domain of Multivariate Statistics: Numbers of IVsand DVs11.1.2 Experimental and Nonexperimental Research21.1.2.1 Multivariate Statistics in Nonexperimental Research1.1.2.2 Multivariate Statistics in Experimental Research31.1.3 Computers and Multivariate Statistics41.1.3.1 Program Updates4V1.1.3.2 Garbage In, Roses Out?51.1.4 Why Not?51.2 Some Useful Definitions51.2.1 Continuous, Discrete, and Dichotomous Data1.2.2 Samples and Populations71.2.3 Descriptive and Inferential Statistics71.2.4 Orthogonality81.2.5 Standard and Sequential Analyses91.3 Combining Variables101.4 Number and Nature of Variables to Include1.5 Statistical Power511111.6 Data Appropriate for Multivariate Statistics121.6.1 The Data Matrix121.6.2 The Correlation Matrix131.6.3 The Variance-Covariance Matrix141.6.4 The Sum-of-Squares and Cross-Products Matrix1.6.5 Residuals16j1.7 Organization of the Book1416A Guide to Statistical Techniques: Using the Book2.1 Research Questions and Associated Techniques2.1.1 Degree of Relationship among Variables2.1.1.1 Bivariater172.1.1.2 Multiple fl 18171717ill

IVCONTENTS2.1.22.1.3/2.1.42.1.52.1.1.3 Sequential/?182.1.1.4 Canonical/?182.1.1.5 Multiway Frequency Analysis19Significance of Group Differences192.1.2.1 One-Way ANOVA and t Test192.1.2.2 One-Way ANCOVA192.1.2.3 Factorial ANOVA202.1.2.4 Factorial ANCOVA202.1.2.5 Hotelling's T2202.1.2.6 One-Way MANOVA212.1.2.7 One-Way MANCOVA212.1.2.8 Factorial MANOVA212.1.2.9 Factorial MANCOVA222.1.2.10 Profile Analysis22Prediction of Group Membership232.1.3.1 One-Way Discriminant Function232.1.3.2 Sequential One-Way Discriminant Function2.1.3.3 Multiway Frequency Analysis (Logit)242.1.3.4 Logistic Regression242.1.3.5 Sequential Logistic Regression242.1.3.6 Factorial Discriminant Function242.1.3.7 Sequential Factorial Discriminant FunctionStructure252.1.4.1 Principal Components252.1.4.2 Factor Analysis252.1.4.3 Structural Equation Modeling26Time Course of Events262.1.5.1 Survival/Failure Analysis262.1.5.2 Time-Series Analysis262.2 A Decision Tree2.3 Technique Chapters26292.4 Preliminary Check of the Data30Review of Univariate and Bivariate Statistics3.1 Hypothesis Testing313.1.1 One-Sample z Test as Prototype3.1.2 Power343.1.3 Extensions of the Model3531313.2 Analysis of Variance353.2.1 One-Way Between-Subjects ANOVA363.2.2 Factorial Between-Subjects ANOVA403.2.3 Within-Subjects ANOVA413.2.4 Mixed Between-Within-Subjects ANOVA442325

CONTENTS3.2.53.2.6Design Complexity453.2.5.1 Nesting453.2.5.2 Latin-Square Designs463.2.5.3 Unequal n and Nonorthogonality463.2.5.4 Fixed and Random Effects47Specific Comparisons473.2.6.1 Weighting Coefficients for Comparisons483.2.6.2 Orthogonality of Weighting Coefficients483.2.6.3 Obtained F for Comparisons493.2.6.4 Critical F for Planned Comparisons503.2.6.5 Critical F for Post Hoc Comparisons503.3 Parameter Estimation513.4 Strength of Association523.5 Bivariate Statistics: Correlation and Regression3.5.1 Correlation533.5.2 Regression543.6 Chi-Square Analysis5355Cleaning Up Your Act: Screening DataPrior to Analysis564.1 Important Issues in Data Screeningj?r574.1.1 Accuracy of Data File574.1.2 Honest Correlations574.1.2.1 Inflated Correlation57"V.4.1.2.2 Deflated Correlation57 4.1.3 Missing Data584.1.3.1 Deleting Cases or Variables '"594.1.3.2 Estimating Missing Data604.1.3.3 Using a Missing Data Correlation Matrix644.1.3.4 Treating Missing Data as Data654.1.3.5 Repeating Analyses with and without Missing Data654.1.3.6 Choosing among Methods for Dealingywith Missing Data654.1.4 Outliers664.1.4.1 Detecting Univariate and Multivariate Outliers674.1.4.2 Describing Outliers704.1.4.3 Reducing the Influence of Outliers714.1.4.4 Outliers in a Solution714.1.5 Normality, Linearity, and Homoscedasticity724.1.5.1 Normality734.1.5.2 Linearity774.1.5.3 Homoscedasticity, Homogeneity of Variance, andHomogeneity of Variance-Covariance Matrices79V

VICONTENTS4.1.64.1.74.1.8Common Data Transformations80Multicollinearity and Singularity82A Checklist and Some Practical Recommendations4.2 Complete Examples of Data Screening864.2.1 Screening Ungrouped Data864.2.1.1 Accuracy of Input, Missing Data, Distributions,and Univariate Outliers874.2.1.2 Linearity and Homoscedasticity904.2.1.3 Transformation924.2.1.4 Detecting Multivariate Outliers924.2.1.5 Variables Causing Cases to be Outliers944.2.1.6 Multicollinearity984.2.2 Screening Grouped Data994.2.2.1 Accuracy of Input, Missing Data, Distributions,Homogeneity of Variance, and Univariate Outliers4.2.2.2 Linearity1024.2.2.3 Multivariate Outliers104. 4.2.2.4 Variables Causing Cases to be Outliers1074.2.2.5 Multicollinearity108Multiple Regression85991115.1 General Purpose and Description1115.2 Kinds of Research Questions1125.2.1 Degree of Relationship113 5.2.2 Importance of IVs1135.2.3 Adding IVs1135.2.4 ' Changing IVs1135.2.5 Contingencies among IVs11415.2.6 Comparing Sets of IVs1145.2.7 Predicting DV Scores for Members of a New Sample5.2.8 Parameter Estimates115114' 5.3 Limitations to Regression Analyses1155.3.1 Theoretical Issues1155.3.2 Practical Issues1165.3.2.1 Ratio of Cases to IVs1175.3.2.2 Absence of Outliers among the IVs and on the DV1175.3.2.3 Absence of Multicollinearity and Singularity1185.3.2.4 Normality, Linearity, Homoscedasticity of Residuals1195.3.2.5 Independence of Errors1215.3.2.6 Outliers in the Solution1225.4 Fundamental Equations for Multiple Regression5.4.1 General Linear Equations1235.4.2 Matrix Equations124122

CONTENTS5.4.3Computer Analyses of Small-Sample Example5.5 Major Types of Multiple Regression1315.5.1 Standard Multiple Regression1315.5.2 Sequential Multiple Regression1315.5.3 Statistical (Stepwise) Regression1335.5.4 Choosing among Regression Strategies1385.6 Some Important Issues1395.6.1 Importance of IVs1395.6.1.1 Standard Multiple Regression1405.6.1.2 Sequential or Statistical Regression1425.6.2 Statistical Inference1425.6.2.1 Test for Multiple R1425.6.2.2 Test of Regression Components1435.6.2.3 Test of Added Subsefof IVs1445.6.2.4 Confidence Limits around B1455.6.2.5 Comparing Two Sets of Predictors1455.6.3 Adjustment of/?21475.6.4 Suppressor Variables1485.6.5 Regression Approach to ANOVA1495.6.6 Centering when Interactions and Powers of IVsAre Included1515.7 Complete Examples of Regression Analysis1535.7.1 'Evaluation of Assumptions1545.7.1.1 Ratio of Cases to IVs1545.7.1.2 Normality, Linearity, Homoscedasticity,and Independence of Residuals1545.7.1.3 Outliers1575.7.1.4 Multicollinearity and Singularity157- " 5.7.2 Standard Multiple Regression1595.7.3 Sequential Regression1655.8 Comparison of Programs1705.8.1 SPSS Package1705.8.2 SAS System1755.8.3 SYSTAT System176Canonical Correlation1776.1 General Purpose and Description1776.2 Kinds of Research Questions1786.2.1 Number of Canonical Variate Pairs6.2.2 Interpretation of Canonical Variates6.2.3 Importance of Canonical Variates6.2.4 Canonical Variate Scores178178178178128Vll

VlllCONTENTS6.3 Limitations1786.3.1 Theoretical Limitations1786.3.2 Practical Issues1806.3.2.1 Ratio of Cases to IVs1806.3.2.2 Normality, Linearity, and Homoscedasticity6.3.2.3 Missing Data181 J 6.3.2.4 Absence of Outliers1816.3.2.5 Absence of Multicollinearity and Singularity6.4 Fundamental Equations for Canonical Correlation6.4.1 Eigenvalues and Eigenvectors1836.4.2 Matrix Equations " 1856.4.3 Proportions of Variance Extracted1896.4.4 Computer Analyses of Small-Sample Example1801811821906.5 Some Important Issues1986.5.1 Importance of Canonical Variates1986.5.2 Interpretation of Canonical Variates1996.6 Complete Example of Canonical Correlation1996.6.1 Evaluation of Assumptions2006.6.1.1 Missing Data2006.6.1.2 Normality, Linearity, and Homoscedasticity6.6.1.3 Outliers2036.6.C1.4 Multicollinearity and Singularity2076.6.2 Canonical Correlation2166.7 Comparison of Programs2166.7.1 SAS System2166.7.2 SPSS Package2166.7.3' SYSTAT System218Multiway Frequency Analysis2197.1 General Purpose and Description219f7.2 Kinds of Research Questions2207.2.1 Associations among Variables2207.2.2 Effect on a Dependent Variable2217.2.3 Parameter Estimates2217.2.4 Importance of Effects2217.2.5 Strength of Association2217.2.6 Specific Comparisons and Trend Analysis2227.3 Limitations to Multiway Frequency Analysis2227.3.1 Theoretical Issues2227.3.2 Practical Issues2227.3.2.1 Independence2227.3.2.2 Ratio of Cases to Variables223200

CONTENTST7.3.2.3 Adequacy of Expected Frequencies7.3.2.4 Outliers in the Solution2242237.4 Fundamental Equations for Multiway Frequency Analysis7.4.1 Screening for Effects2257.4.1.1 Total Effect2267.4.1.2 First-Order Effects2277.4.1.3 Second-Order Effects2287:4.1.4 Third-Order Effect2327.4.2 Modeling2337.4.3 Evaluation and Interpretation2357,4.3.1 Residuals23577.4.3.2 Parameter Estimates2367.4.4 Computer Analyses of Small-Sample Example2417.5 Some Important Issues2507.5.1 Hierarchical and Nonhierarchical Models2507.5.2 Statistical Criteria2517.5.2.1 Tests of Models2517.5.2.2 Tests of Individual Effects2517.5.3 Strategies for Choosing a Model2527.5.3.1 SPSS HILOGLINEAR (Hierarchical)2527.5.3.2 SPSS GENLOG (General Log-linear)2537.5.3.3 SAS CATMOD, SYSTAT LOGLINEAR,and SYSTAT LOGLIN (General Log-linear)2537.6 Complete Example of Multiway Frequency Analysis253 7.6.1, Evaluation of Assumptions: Adequacyof Expected Frequencies2537.6.2 Hierarchical Log-linear Analysis254i .7.6.2.1 Preliminary Model Screening2547.6.2.2 Stepwise Model Selection256 7.6.2.3 Adequacy of Fit ,2587.6.2.4 Interpretation of the Selected Model2647.7 Comparison of Programs2707.7.1 SPSS Package2737.7.2 SAS System2747.7.3 SYSTAT System274cOAnalysis of Covariance2758:1 General Purpose and Description2758.2 Kinds of Research Questions2778.2.1 Main Effects of IVs2788.2.2 Interactions among IVs2788.2.3 Specific Comparisons and Trend Analysis8.2.4 Effects of Covariates278IX278224

CONTENTS8.2.58.2.6Strength of Association279Parameter Estimates2798.3 Limitations to Analysis of Covariance2798.3.1 Theoretical Issues2798.3.2 Practical Issues2808.3.2.1 Unequal Sample Sizes, Missing Data, and Ratio of Casesto IVs2808.3.2.2- Absence of Outliers2818.3.2.3 Absence of Multicollinearity and Singularity2818.3.2.4 Normality of Sampling Distributions2818.3.2.5 Homogeneity of Variance2818.3.2.6 Linearity2828.3.2.7 Homogeneity of Regression2828.3.2.8- Reliability of Covariates2838.4 Fundamental Equations for Analysis of Covariance8.4.1 Sums of Squares and Cross-Products2848.4.2 Significance Test and Strength of Association8.4.3 Computer Analyses of Small-Sample Example283288289.8.5 Some Important Issues2918.5.1 Test for Homogeneity of Regression2918.5.2 Design Complexity2938.5.2.1 Withih-Subjects and Mixed Within-Between Designs, 8.5.2.2 Unequal Sample Sizes2968.5.2.3 Specific Comparisons and Trend Analysis2988.5.2.4 Strength of Association3018.5.3 Evaluation of Covariates3028.5.4 Choosing Covariates3028.5.5 Alternatives to ANCOVA3038.6 Complete Example of Analysis of Covariance3048.6.1 Evaluation of Assumptions3058.6.1.1 Unequal n and Missing Data3058.6.1.2 Normality3053058.6. 1.3 Linearity3058.6. 1.4 Outliers8.6. 1.5 Multicollineartyand Singularity3098.6. .6 Homogeneity of Variance3093108.6. 1.7 Homogeneity of Regression3108.6. 1.8 Reliability of Covariates8.6.2 Analysis of Covariance3108.6.2.1 Main Analysis3108.6.2.2 Evaluation of Covariates3138.6.2.3 Homogeneity of Regression Run3158.7 Comparison of Programs3198.7.1 SPSS Package3198.7.2 SYSTAT System3198.7.3 SAS System321293

CONTENTSMultivariate Analysis of Variance and Covariance9.1 General Purpose and Description'XI3223229.2 Kinds of Research Questions3259.2.1 Main Effects of IVs3259.2.2 Interactions among IVs3269.2.3 Importance of DVs326, 9.2.4 -'Parameter Estimates3269.2.5' Specific Comparisons and Trend Analysis9:2.6 Strength of Association3279.2.7. - Effects of Covariates3279.2.8 Repeated-Measures Analysis of Variance3273279.3 Limitations to Multivariate Analysis of Varianceand Covariance3289.11 Theoretical Issues328- 9.3.2 Practical Issues3289.3.2.1 Unequal Sample Sizes, Missing Data, and Power329 9.3.2.2 Multivariate Normality3299.3.2.3 Absence of Outliers3309.3.2.4 Homogeneity of Variance-Covariance Matrices3309.3.2.5 Linearity3309.3.2.6 Homogeneity of Regression331,/ \9.3.2.7 Reliability of Covariates3319.3.2.8 Absence of Multicollinearity and Singularity3319.4 Fundamental Equations for Multivariate Analysisof Variance and Covariance3329.4.1 Multivariate Analysis of Variance3329.4.2 Computer Analyses of Small-Sample Example9.4.3 Multivariate Analysis of Covariance3403399.5 Some Important Issues3479.5.1 Criteria for Statistical Inference3479.5.2 Assessing DVs3489.5.2.1 Univariate F3489.5.2.2 Roy-Bargmann Stepdown Analysis3509.5.2.3 Using Discriminant Function Analysis3519.5.2.4 Choosing among Strategies for Assessing DVs9.5.3 Specific Comparisons and Trend Analysis3529.5.4 Design Complexity356 9.5.4.1 Within-Subjects and Between-Within Designs9.5.4.2 Unequal Sample Sizes3569.5.5 MANOVA vs. ANOVAs3579.6 Complete Examples of Multivariate Analysis of Varianceand Covariance3579.6.1 Evaluation of Assumptions3589.6.1.1 Unequal Sample Sizes and Missing Data358,351356

XUCONTENTS9.6.1.2 Multivariate Normality3609.6.1.3 Linearity3609.6.1.4 Outliers3609.6.1.5 Homogeneity of Variance-Covariance Matrices9.6.1.6 Homogeneity of Regression362 9.6.1.7 Reliability of Covariates3659.6.1.8 Multicollinearity and Singularity3659.6.2 Multivariate Analysis of Variance3659.6.3 Multivariate Analysis of Covariance376' 9.6.3.i Assessing Covariates377j9.6.3.2 Assessing DVs377—3619.7 Comparison of Programs3869.7.1 SPSS Package3899.7.2 SYSTAT System3899.7.3 SAS System390- XUProfile Analysis: The Multivariate Approachto Repeated Measures39110.1 General Purpose and Descriptioni39110.2 Kinds of Research Questions39210.2.1 Parallelism of Profiles39210.2.2 Overall Difference among Groups39310.2.3 Flatness of Profiles39310.2.4 Contrasts Following Profile Analysis39310.2.5 Parameter Estimates39310.2.6 Strength of Association39410.3 Limitations to Profile Analysis39410.3.1 Theoretical Issues39410.3.2 Practical Issues39410.3.2.1 Sample Size, Missing Data, and Power39410.3.2.2 Multivariate Normality395 10.3.2.3 Absence of Outliers39510.3.2.4 Homogeneity of Variance-Covariance Matrices39510.3.2.5 Linearity395' 10.3.2.6 Absence of Multicollinearity and Singularity39610.4 -Fundamental Equations for Profile Analysis39610.4.1 Differences in Levels39610.4.2 Parallelism39810.4 .3 Flatness40110.4.4 Computer Analyses of Small-Sample Example10.5 Some Important Issues41010.5.1 Contrasts in Profile Analysis410403

CONTENTS;Xlll10.5.1.1 Parallelism and Flatness Significant, Levels Not Significant(Simple-Effects Analysis) 413., 10.5.1.2 Parallelism and Levels Significant, Flatness Not Significant(Simple-Effects Analysis) 414U0.5/1.3 Parallelism, Levels, and Flatness Significant(Interaction Contrasts) 41610.5.1.4 Only Parallelism Significant42110.5:2 Univariate vs. Multivariate Approachto Repeated Measures42110.5.3 Doubly-Multivariate Designs42310.5.4,Classifying Profiles42910.5.5 Imputation of Missing Values42910.6 Complete Examples of Profile Analysis43010.6.1 Profile Analysis of Subscales of the WISC43010.6.1.1 Evaluation of Assumptions43110.6.1.2 Profile Analysis 43510.6.2 Doubly-Multivariate Analysis of Reaction Time442J10.6.2.1 Evaluation of Assumptions 44210.6.2.2 Doubly-Multivariate Analysis of Sloperand Intercept 44610.7J , Comparison of Programs45310.7.1 SPSS Package453 10.7.2 SAS System45510.7.3 SYSTAT System -455X'XDiscriminant Function Analysis45611.1 General Purpose and Description45611.2 Kinds of Research Questions45811.2.1 Significance of Prediction45811.2.2 Number of Significant Discriminant Functions45811.2.3 Dimensions of Discrimination45911.2.4 Classification Functions45911.2.5 Adequacy of Classification45911.2.6 Strength of Association46011.2.7 Importance of Predictor Variables46011.2.8 Significance of Prediction with Covariates46011.2.9 Estimation of Group Means460vS11.3 Limits to Discriminant Function Analysis46111.3.1 Theoretical Issues46111.3.2 Practical Issues46111.3.2.1 Unequal Sample Sizes, Missing Data, and Power11.3.2.2 Multivariate Normality 46211.3.2.3 Absence of Outliers 462461

XIVCONTENTS11.3.2.4 Homogeneity of Variance-Covariance Matrices11.3.2.5 Linearity46311.3.2.6 Absence of Multicollinearity and Singularity46246311.4 Fundamental Equations for Discriminant Function Analysis11.4.1 Derivation and Test of Discriminant Functions46411.4.2 Classification46711.4.3 Computer Analyses of Small-Sample Example46911.5 Types of Discriminant Function Analysis47711.5.1 Direct Discriminant Function Analysis47811.5.2 Sequential Discriminant Function Analysis47811.5.3 Stepwise (Statistical) Discriminant Function Analysis11.6 Some Important Issues48111 6.1 Statistical Inference48141.6.1.1 Criteria for Overall Statistical Significance' 11.671.2 Stepping Methods48211.6.2 Number of Discriminant Functions48211.6.3 Interpreting Discriminant Functions48311.6.3.1 Discriminant Function Plots483'11.6.3.2 Loading Matrices48411.6.4 Evaluating Predictor Variables48511.6.5 Design Complexity: Factorial Designs48811.6.6 Use of Classification Procedures48911.6.6.1 Cross-Validation and New Cases48911.6.6.2 Jackknifed Classification49011.6.6.3 Evaluating Improvement in Classificationlf48148149011.7 Complete Example of Discriminant Function Analysis49211.7.1 Evaluation of Assumptions49211.7.1.1 Unequal Sample Sizes and Missing Data49211.7.1.2 Multivariate Normality49211.7.1.3 Linearity49311.7.1.4 Outliers49311.7.1.5 Homogeneity of Variance-Covariance Matrices49311.7.1.6 Multicollinearity and Singularity49311.7.2 Direct Discriminant Function Analysis497C-X.Z46311.8 Comparison of Programs509c 11.8.1 SPSS Package51511.8.2 SYSTAT System51611.8.3 SAS System516Logistic Regression51712.1 General Purpose and Description517

CONTENTSXV12.2 Kinds of Research Questions51812.2.1- Prediction of Group Membership or Outcome51812.2:2 Importance of Predictors51812.2-.3 Interactions among Predictors51812.2.4 ParameterEstirhates52012.2.5 Classification of Cases52012.2.6 Significance of Prediction with Covariates52012.2.7 Strength of Association52012.3 Limitations to Logistic Regression Analysis52112.3.1 Theoretical Issues52112.3.2 Practical Issues52112.3.2.1 Ratio of Cases to Variables52112.3.2.2 Adequacy of Expected Frequencies and Power, ' * 12.3.2.3 Linearity in the Logit52212.3.2.4 Absence of Multicollinearity52212.3.2.5 Absence of Outliers in the Solution52312.3.2.6 Independence of Errors52352212.4 Fundamental Equations for Logistic Regression52312.4.1 Testing and Interpreting Coefficients52412.4.2 Goodness-of-Fit52512.4.3 Comparing Models52712.4.4 Interpretation and Analysis of Residuals52712.4.5 Computer Analyses of Small-Sample Example52712.5 Types of Logistic Regression533- 12.5.1 Direct Logistic Regression533'12.5.2 Sequential Logistic Regression53312.5.3 Stepwise (Statistical) Logistic Regression12.5.4 Probit and Other Analyses53553512.6 Some Important Issues53612.6.1 Statistical Inference53612.6.1.1 Assessing Goodness-of-Fit of Models53712.6.1.2 Tests of Individual Variables53912.6.2 Number and Type of Outcome Categories53912.6.2.1 Unordered Response Categorieswith SYSTAT LOGIT54012.6.2.2 Ordered Response Categories with SAS LOGISTIC12.6.3 Strength of Association for a Model54512.6.4 Coding Outcome and Predictor Categories54612.6.5 Classification of Cases54712.6.6 Hierarchical and Nonhierarchical Analysis54812.6.7 Interpretation of Coefficients using Odds54812.6.8 Importance of Predictors54912.6.9 Logistic Regression for Matc

1.6 Data Appropriate for Multivariate Statistics 12 1.6.1 The Data Matrix 12 1.6.2 The Correlation Matrix 13 1.6.3 The Variance-Covariance Matrix 14 1.6.4 The Sum-of-Squares and Cross-Products Matrix 14 1.6.5 Residuals 16 j 1.7 Organization of the Book 16 A G

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