Textbook 754 sharma

INDEX 485 D compared with logistic regression. 332-333 Data analytic methods dependence methods. 4-10 discrete discriminant analysis. 8 interdependence merhods. 4. 10- )2 discrete multiple-group discriminant structural models. 13-14 analysis, 10 Data manipulations multiple-group discriminant analysis, 10 computer procedures for. 55-57 compared to multivariate analysis of degrees of freedom. 36-38 generalized variance. 39 variance (MANOVA), 350 group analysis, 40-42 with one dependent/more than one mean, 36 mean-corrected data. 36 independent variable, 7-8 standardization of data, 39 situations for use, 8. 237 sum of cross products. 39 stepwise discriminant analysis, 246. sum of squares, 38-39 variance. 38 264-273 Discriminant function. multiple-group Degrees of freedom, 36-38 computation of. 37-38 discriminant analysis. 294-303 assessment of imponance of, 303 Dendrogram, for clustering process, 190-191 computation options. 294, 297 Dependence methods, 4-10 estimate of, 297-299 group differences. examination of. 307 analysis of variance, 7 labeling of, 307 canonical correlation. 9 number needed. 299-303 conjoint analysis, 8-9 practical significance of, 302-303 discrete discriminant analysis. 8 statistical significance of, 299-302 discrete multiple-group discriminant Discriminant function. two-group discriminant analysis, 250-254 analysis. 10 bootstrap method for validation of. 274 discriminant analysis. 7-8 canonical discriminant function, 251 logistic regression. 8 computation options. 250-251 for more than one dependent/more than estimate of, 251-252 holdout method for validation of, 273 one independent variables, 9-1 0 linear discriminant function. 242 multiple-group discriminant analysis. 10 meaning of, 242 multivariate analysis of variance. 10 practical significance of, 253 standardized canonical discriminant for one dependent/more than one function. 253-254 independent variable. 5-9 statistical significance of. 252-253 U-method for validation of. 273-274 for one dependent/one independent Discriminan[ score, meaning of. 242 variable. 5. 6 Discriminant variables, 238 assessment of imponance of, 253 - 254 regression, 5 Distance measures, 218-220 situations for use of. 4- Euclidian distance. 19, 219 Detrended normal plot, 378-379 Mahalanobis distance, 44-45, 220 Dimensional reduction, principal Minkowski distance, 218 components analysis as. 64-65 statistical distance, 42-44 Direction cosines. 25 Distinguishability of observations, 45 Discordant pair, 325 Discrete discriminant analysis, with one E dependent/more ~an one independent variable, 8 Effec[ size Discrete multiple-group discriminant in MANOVA. 348-349 analysis multivariate. 349 with more than one dependent/one or univariate. 349 more independent variables, 10 Eigenstructure of covariance matrix, 84-85 situations for use, 10 computer analysis. 87-89 Discriminant analysis. see also Multiple- group discriminant analysis; Two-group discriminant analysis

486 INDEX Equality of covariance matrices Factor extraction methods, 141-142 assumption. 383-387 principal axis factoring, 142 principal components factoring, 141-142 errors and violation of, 383-384 tests for checking equality, 384-387 Factor rotation problem, 97, 100-102, 136 Equivalent vectors, 20 basis of. 97 Error sums of squares, 193 geometric view of, 100-102 Euclidian distance, 19 in cluster analysis, 219: Factor rotations, 137-141 in similarity matrix. 187-188 oblique rotation, 140-141 for standardized data, 219 orthogonal rotation, 137 and statistical distance, 43-44 quartimax rotation. 120-121. 137 Event. 325 varimax. rotation, 119-120, 138, Exploratory factor analysis, 128 139-140 situations for use, 144 Factor scores, 96, 142-143 F Farthest-neighbor method. hierarchical Factor analysis. see also Confinnatory clustering method. 192, 217 factor analysis F-distribution alpha factor analysis, 109 in multiple-group discriminant analysis. appropriateness of data for. 116, 123. 125 294 choosing technique for, 108 common factors, 96. ]08 table of, 460-465 communalities problem, estimation of, Fisher's linear discriminant function, 245, 100 277-278 and communality estimation problem, computation of, 277-278 Fisher's Z transfonnation, 383 136 Forward selection, stepwise discriminant computer estimation of, 109-115 analysis, 265 concepts/tenns related to. 90-93 F-ratio confinnatol)' factor analysis, 128 in mUltiple-group discriminant analysis. exploratory factor analysis, 128 factor extraction methods, 141-142 294, 301 factor inderenninacy. 97-98, 136 in multiple-group MANOVA, 351 factor rotation problem, 97. 100-102, 136 relationship to Wilks'A, 348 factor rotations, types of. 137-141 in stepwise discriminant analysis. 266, factor scores. %, 142-143 factor solution, 1] 7 -1 ] 8 271 fundamental factor analysis equation. 136 F-test, in multiple-group discriminant geometric view of, 99-102 image analysis. 109 analysis, 293, 294 interpretation of factor structure. 125 Fundamental factor analysis equation. 136 model with more than two factors. 96. G 102. 135-136 number of factors needed. ] 16-117 Generalized variance, 39, 5.0-51 objectives of, 99 equality to detenninant of covariance one-factor model. 93. 132-133 matrix, 54-55 principal axis factoring (PAF). 107. 142 geometric representation of. 50-51 compared to principal components Geometric concepts analysis, 125-128 Cartesian coordinate system, 17- 19 principal components factoring (PCF). vectors. 19-33 J03-107.141-142 Goodness-of-fit measures representation of factors. 118 -119 chi-square. 378 situations for use, 11. 90 LlSREL, 157-159 two-factor model. 93-96. 133 -135 in logical regression. 324 Gram-Schmidt orthononnalization procedure. 32 Group analysis. 40-42 between-group analysis. 41-42 within-group analysis. 40-41

INDEX 48'; H Laten[ factor. 91 Leptokurtic distributions. nature of. 375 Helmert contrasts, 360-361 Linear combination. vectors. 26-27 Heuristic measures. of model fi t. 157 - 160 Linear discriminant function. 242 Hierarchical clustering methods. 188-193 Fisher·s. 245. 277 -278 average-linkage method. 192-193 LlSREL. 148-177 centroid method. 188-191 chaining effect in, 211. 217 adjusted goodness-of-fit index. 159 complete-linkage or farthest-neighbor commands. 150-152 constrained analysis. 171, 173 method. 192 estimated model parameters. evaluation computer analysis. 193 -202 evaluation of, 211-217 of. 162-1M example of, 221-228 example of use. 174-178 single-linkage or nearest-neighbor goodness-of-fit indkes. 157-159 initial estimates. 152-157 method. 191 Ward's method. 193 maximum likelihood estimates. Holdout method, discriminant function 422-424 validation, 273 McDonald's transformation of the I noncentrality parameter (MDN). Image analysis, 109 159-160 Implied covariance matrix, 444-449 model fit. evaluation of. 157-162 matrix algebra, 445-446 model information and parameter specifications. 152 models with observable constructs, 444-446 model respecification, 164-165 models with unobservable constructs modification indices. 164 mu1tigroup analysis. 170-173 446-449 . null h~pothesis test, 157 null model. 160, 161 Independence assumption, 387-388 one-factor model. 153-156 lack of tests for, 388 parameter estimates. 162- 163 relative goodness-of-fit index, 159 Indicator, 91 relative noncentrality index. 160 Interdependence methods, 4, 10-12 rescaled noncentrality parameter. 159. cluster analysis, 12 160 correspondence analysis. 12 residual matrix. 160-162 factor analysis, 11 root mean square residual, 159 squared multiple correlation. 163-164 loglinear models. 12 structural model in. 421-434 principal components analysis, 11 terminology related to, 148-149 situations for use of. 4, 10 total coefficient of determination, 164 Interval scale, use of. 2-3 Tucker-Lewis index. 160 K two-factor model. 165-170 unconstrained analysis, 171 Kaiser-Meyer-Olkin measure, 116 Loadings. -W4 K-means clustering. 205 Kolmogorov-Smirnov test. 378. 379 Logistic regression Kurtosis classification, 326-327 with combination categorical/continuous and leptokurtic distribution. 375 independent variables. 328-332 normalization of. 375 of univariate normal distribution. 375 compared with discriminant analysis, 332-333 L computer analysis. 321-335 Latent constructs contingency table analysis, 327 -328 meaning of. 13 example as illustration of. 333-335 and structural models, 13. 14 example of use. 333-335 logistic regression model. 319-321

488 INDEX Logistic regression (continued) Minkowski distance, 218 maximum likelihood estimation . Modification indices. LISREL, 164 procedure in, 321, 324-325.339-341 Monotonic analysis of variance model fit. assessment of, 323-324 model information, 321 (MONANOVA), situations for use, multiple logistic regression, 320 8-9 with one categorical variable, 321-327 with one dependent/more than one Multicollinearity independent variable, 8 population-based, 273 parameter estimates. 324-325 sample-based, 273 predicted probabilities. association of. and stepwise discriminant analysis, 325-326 272-273 probability and odds in, 317-321 si tuations for use, 8. 317 Multigroup analysis, with LISREL, stepwise selection procedure, 329, 170-173 331-332 Multiple-group discriminant analysis Logit function. 320 analytical approach to, 293-294 Logit transfonnation. 383 classification. 293, 303-304, 311-312 LogIinear models, situations for use, 12 313 Loss of homogeneity. in cluster analysis. 200 computer analysis, 294-307 discrete multiple-group discriminant M analysis, 10 discriminant function, 294-303 Mahalanobis distance, 44-45 F-test in, 293, 294 as classification method, 258 geometric view of, 287-293 in cluster analysis, 220 with more than one dependent/one or definition of. 44 more independent variables, 10 formula for, 44 multivariate normal distribution, in MANOVA. 343 312-316 squared distance in stepwise new axes, identification of, 289, 293 discriminant analysis. 266 number of discriminant functions needed, 288-289 MANOVA, see Multivariate analysis of significance of variables, estimation of, variance (MANOVA) 294 situations for use, 10 Maximum likelihood estimation technique computation of, 181-185 Multiple-group MANOVA, 355-366 computer analysis, 148-173 computer analysis, 355-366 in LISREL, 422-424 correlated contrasts, 363-366 in logistic regression, 321, 324-325, mu1tivariate effects, 356 339-341 orthogonal contrasts, 356-363 univariate effects, 356 McDonald's transfonnation of the noncentrality parameter (MDN). Multiple regression 159-160 as canonical correlation. 9 discriminant analysis, 262-263 Mean. computation of, 36 with one dependent/more than one Mean-corrected data, nature of. 36 independent variable, 5-9 Measure of I, 91 Measurement model, 14 Multivariate analysis Measurement scales number of variables, 5 objectives of, 238 interval scale. 2-3 nominal scale. 2 Multivariate analysis of variance and number of variables, 3-4 (MANOVA) ordinal scale. 2 ratio scale. 3 see also MUltiple-group MANOVA; Minimum average partial correlation Two-group MANOVA (MAP). 117 analytic computations for. 346-350 computer analysis. 350-370 compared to discriminant analysis. 350 effect size. 348-350

INDEX 489 geometric view of, 342-346 K-means clustering. 205 with more than one dePendent/one or method to obtain initial seeds. 202-203 reassignment rules, 203 more independent variables, 10 steps in. 202 multiple-group, 355-366 Normality assumptions, 375 and multivariate effect size, 349-350 Norm of vector, 20 multivariate significance tests in, Null hypothesis and power of test, 349-350. 375 346-348 and Type I and Type II errors. 374-375 with one independenr/one dependent :i statistic for testing of. 157, 162 variable, 343 with one independent/p dependent Null model, 160 Null vector, 21 variables, 344-346 with one independenr/two or more o dependent variables, 343-344 Oblique basis, vectors, 31 power of test in, 349-350 Oblique factor rotation, 140-141 situations for use, 10, 342 Observation space, graphical representation two-group, 350-355 with two independent variables, of data in, 47-50 Odds, in logistic regression. 318-321 366-370 One-factor model and univariate effect size, 349 univariate significance tests in, 348-349 computation of, 132-133 Multivariate effect size, 349-350 with covariance matrix, 145 -147 Multivariate normal distributions with LISREL. l..i3-156 classification rules for, 281-283 situations for use, 93 multiple-group discriminant analysis. Ordinal scale, use of, 2 Origin. in Cartesian coordinate !iystem, 312-316 skewness of, 375 17-19 two-group discriminant analysis, Orthogonal contrasts 281-283 computer analysis, 360-363 Multivariate normality assumption, 8 multiple-group MANOVA, 356-363 multivariate significance tests for, and discriminant analysis, 263-264 Multivariate normality tests, 380-383 359-360 situations for use. 357 graphical test, 380-383 univariate significance tests for, transformations, 383 Multivariate significance tests, 252 357-359 . for contrasts. 359-360, 363 OnhogonaJ factor model, 94 in multivariate analysis of variance Onhogonal factor rotation. 137-140 Orthonormal vectors. 25, 31. 32 (MANOVA), 346-348 in two-group MANOVA. 351, 353 p N Parallel analysis, 77, 79 Parallelogram law of vector addition. 22 Naive prediction rule, 260 Pattern loadings. 91, 94 Nearest-neighbor method, hierarchical Pearson product moment correlation, 39 clustering method, 191 as similarity measure, 220 Newton-Raphson method, 340 Percent points of normal probability plot No-event, 325 Nominal scale. use of. 2, 7 correlation coefficient, table of, 466 Nonhierarchical clustering, 202-211 Perceptual map. purpose of, 307 Population-based multicollinearity, 273 algorithms in, 203-207 Power of test cluster solution, evaluation/interpretation in M.A..NOVA. 350 of,210 purpose of, 349-350, 375 computer analysis, 207-211 evaluation of, 217 example of, 228-232

490 INDEX Principal axis factoring, 107 Reliability, cluster analysis, 221 for factor exaaction, 141 Rescaled noncentraliry parameter, 159, 160 Residual matrix, LlSREL, 160-162 Principal components, 63, 66 Root-mean-square residual, 106-107, 118 Principal components analysis USREL.159 algebraic approach to, 67 - 71 Root-mean-square total-sample standard computer analysis. 67-71 deviation as dimensional reducing technique, of the cluster, 198.230 fonnula for, 197 64-65 R-squared, cluster analysis, 198, 200 eigenstructure of covariance matrix, s 84-85 compared to factor analysis, 125-128 Sample-based multicollinearity, 273 geometric view of, 59-66 SAS. see Statistical Analysis System goals of, 58. 66 interpretation of principal components, (SAS) Saturated models, 421 79-80 Scalar product, of two vectors, 20-21, issues related to use of, 71-81 number of components to extract, 76-79 27-28 and objective of study, 75-76 Scale invariant, meaning of, 46 singular value decomposition, 85-86 Schwartz's criterion. 324 situations for use, 11, 58 Scree plot, 76-77 spectral decomposition of matrix. 86-87 Scree plot test, 79 compared to two-group discriminant Semipartial R-squared, cluster analysis, analysis, 241-242 198.200 type of data, effect on analysis, 72-75 Shapiro-Wilk test, 378, 379 Principal components factoring. 103-107 Significance tests for factor extIaction, 141-142 Principal components scores, 63, 66 for main effects, 370 use of, 80 MANOVA for two independent Probabilities, in logistic regression, 317-321 variables, 367-370 Projection vector, 23. 27- 28 multivariate significance tests, 346- 348, Pythagorean theorem, Euclidian distance computation, 19 351. 353 univariate significance tests, 348-349, Q 353-355 Q-factor analysis. situations for use, 187 Similarity measures, 218 -220 Q-Q plot, 376-378 Quartimax factor rotation, 120-121, 137 association coefficients, 220 correlation coefficient, 220 R distance measures, 218-220 Simple regression, 5 Ratio scale, use of, 3 Simulation percentiles of b2, table of, 467 Ray's V, in stepwise discriminant analysis, Simulation probability points of jii;, 266 Rectangular Cartesian axes, 17 table of, 467 Reflection, of vectors, 21 Single-linkage method, hierarchical Regressio:· clustering method. 191 logistic regression, 8 Singular value decomposition, 85-86 multiple regression. 5-9 Skewness simple regression, 5 Relative goodness-of-fit index, USREL, of multivariate nonnal distribution. 375 of univariate nonnal distribution, 375 159 Space Relative noncentrality index, 160 observation space. 47-50 variable space, 45 -46 Spectral decomposition of matrix, 86-87 SPSS. see Statistical Package for the Social Sciences (SPSS)

C'IDEX 491 Squared multiple correlation Statistical significance computation of. 181 LISREL, 163-164 at\" discriminant function. 252-253, 299-302 Square-root transformation, 383 structural models. 430, 439 Standard basis vectors, 25 Scandardization of data. 39 tests for canonical correlations. 402-404 correlation coefficient. 39 Statistical tables Euclidian distance for s[.1ndardized data. F-<iistribution, 460-465 percent points of normi:U probability plot 219 correlation coefficient. 466 graphical representations in space, simulation percentiles of ~, 467 47-50 simulation probability points of jb;.467 Standardized canonical discriminant standard normal probabilities, 457 function. 253-254 student's t-discribution critical points. Standard normal probabilities. table of, r 458 457 critical points, 459 Statistical Analysis System (SAS) Statistical tests multivariate normality tests, 380-383 canonical correlation, 398-406 power of test. 375 chi-square plot, 389-390 cluster analysis. hierarchical, 193-202 Type I and Type II errors, 374-375 cluster analysis. nonhierarchical, univariate normality tests, 375-380 207-210 data manipulations. 55 - 57 Stepwise discriminant analysis. 264-273 factor analysis, 109-115 logistic regression, 321-335 backward selection, 265 maximum likelihood estimation computer analysis. 267 - 273 cutoff values for selection criteria. techI\"ique, 148 measurement models estimation, 14- 266-267 ordinary least squares estimation. 421 forward selection. 265 principal components analysis, F-ratio in. 266, '271 ~Iahalanobis squared distance in. 266 67-71 and multicollinearity, 272-273 structural model estimation. 426 Ray's V in, 266 Statistical decision theory, 256-257 selection criteria, 265-266 classification rules, development of, stepwise selection. 265 Wilks' .\\ test statistic in, 251. 266 279-281 Stepwise selection procedure. in logistic Statistical distance, ~2-44 regression, 329. 331-332 Structural correlations. 404 and Euclidian distance. 43-44 Structural models, 13-14.435-437 squared statistical distance. 43 assessment of, 438-439 Statistical Package for the Social Sciences direct effects, 425. 450. 452 effects among constructs. 430 (SPSS),67 effects among endogenous constructs. classification, 256-257, 261 confirmatory factor analysis. LlSREL, ~50-~5'2 148-177 effects of constructs on its indicators, discriminant analysis. 245-262 Helmel1 con[fasts. 360-361 452-~53 MANOVA, mUltiple group. 355-366 MANOVA. two-group, 350-355 effects of endogenous constructs on MANOVA, for two independent indicators, 434 variables. 366-371 effects of exogenous constructs on measurement models estimation. 14 endogenous constructs, 452 multiple-group discriminant analysis, effects of exogenous constructs on 294-308 indicators, 430 orthogonal contrasts, 360-363 stepwise discriminant analysis, 267-273 effects of exogenous constructs on univariate normality, 378-379 indicators of endogenous constructs, 435 estimation procedures in computer packages. 14

492 INDEX Structural models (continued) multivariate nonnality test, 383 examples of use, 13,435-439 square-root transformation. 383 and implied covariance matrix, Triangular inequality, and Mahalonobis 444-449 distance, 45 indirect effects, 425, 450-451, 452, T-test, in discriminant analysis, 244-245, 452-453 246,250 and latent constructs, 13, 14 TUcke~Le~s index, 160 LISREL estimation, 421-434 T-values, in structural model, 425 measurement model, assessment of, 437 Two-factor model model fit, assessment of, 435-437 computation of. 133-135 model respecification, 435-437 with correlated constructs. 147 with observable constructs, 420-426, with LISREL. 165-170 444-446 situations for use, 93-96 overall model fit, 425 Two-group discriminant analysis saturated models, 421 analytical approach to, 244- 245 standardized solution, 426, 435 and classification, 242-244, 278-284 statistical significance, 430, 439 computer analysis. 245-262 structural equations, 419-420 discriminant function, 242, 250-254 total coefficient of detennination for, discriminating variables, evaluation of 424 total effects, 425, 451 significance, 246, 250 t-values, 425 discriminator variables, selection of, with unobse.rvable constructs, 426-435. _ 446-449 244-245 equality of covariance matrices, 264 Structure coefficients, 254 Fisher's linear discriminant function, Structure loading. 92 Student'S t-distribution critical points, 245,277-278 geometric view of. 237-244 table of, 458 identification of set of variables. 238 Sum of cross products, 39 multiple regression approach to. computation of, 39 262-263 sum of squares and cross products muiuvariate normality assumption, matrix, 39 263-2(:4 Summary measures new axis, identification of. 239-242 objectives of, 237, 241, 242 computer procedures for, 55-57 compared to principal components data manipUlations for, 36-42 types of, 36 analysis, 241-242 Sum of squares, 38-39 stepwise discriminant analysis, 246, for correlated contrasts. 365-366 Sum of squares and cross products 264-273 matrix, 39 validation of discriminant function, and between-group analysis, 42 and within-group analysis, 40-41 273-274 Symmetry, and Mahalonobis distance. 45 Two-group MANOVA. 350-355 T cell means, 351 computer analysis, 350-355 Territorial map, purpose of. 303 homogeneity of variances, 351 ned pairs, 325-326 multivariate significance tests and Total coefficient of determination power, 351. 353 LISREL,I64 univariate significance tests and power, for structural equations. 424 Transfonnation 353-355 Fisher's Z transformation, 383 Two-stage least-squares approach, in logit transformation, 383 LISREL,152 Type I errors nature of, 374 and violation of equality of covariance mauices, 384 Type II errors, nature of, 375

u INDEX 493 U-method, discriminant function dimensionality of, 30-31 validation, 273-274 distance and angle between two vectors, Unconstrained analysis, USREL. 171 27 Univariate analysis equivalent vectors, 20 initial and tenninal point. 19 number of variables, 5 linear combination of. 26-17 objectives of, 238 multiplication by real number, 20-21 Univariate effect size, 349 multipliC3tion of [wo vectors. 22-23 Univariate nonnal distribution nonn of, 20 kurtosis. of, 375 null vector. 21 zero skewness of, 375 oblique basis. 31 Univariate nonnality tests, 375-380 orthononnal vectors, 25, 31. 32 analytical procedures, 378 projection into subspace, 28-29 computer analysis, 378-379 projection of one onto another, 23 graphical tests, 376-377 projection vector. 23, 27-28 Univariate significance tests, 250 reflection, 21 for contrasts, 357-359, 360, 362-363 representation of points with respect to in multivariate analysis of variance new axes. 32-33 (MANOVA), 348-349 scalar product of two vectors, 20-21, two-group MANOVA, 353 Unobservable construct. 91 27-28 signed length of. 28 v standard basis vectors, 25 subtraction of, 22 Validity of canonical coefficients, 409 Ward's method. hierarchical clustering cluster analysis, 221 method, 193. 217 Variables, number for measurement scales, Wilks'A test statistic in discriminant analysis, 246-250 3-4 relationship to F-ratio, 348 in stepwise discriminant analysis, 251, Variable space, graphical representation of 266 data in. 45 -46 testing for canonical correlations. 401-403 Variance computation of, 38 Within-group analysit:, 40-~1 generalized variance, 39, 50-51, 54-55 sum of squares and cross products situations for use, 38 matrices. 40-41 of standardized variables, 39 x Varimax factor rotation. 119-120, 138 ,r critical points. table of. 459 Vectors, 19-32 .r statistic. null hypothesis testing. 157. addition of, ~1-22 arithmetic operations on, 25-26 162 basis vectors, 25, 31 in Cartesian coordinate system, 23-25 centroid. 45-46 changing basis, 31-32