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Fienberg S., Holland P.W., Bishop Y.M. Discrete Multivariate Analysis
Fienberg S., Holland P.W., Bishop Y.M.  Discrete Multivariate Analysis


: Discrete Multivariate Analysis

: Fienberg S., Holland P.W., Bishop Y.M.


The scientist searching for structure in large systems of data finds inspiration in his own discipline, support from modern computing, and guidance from statistical models. Because large sets of data are likely to be complicated, and because so many approaches suggest themselves, a codification of techniques of analysis, regarded as attractive paths rather than as straitjackets, offers the scientist valuable directions to try. The literature on discrete multivariate analysis, although extensive, is widely scattered. This book brings that literature together in an organized way.

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ed2k: ed2k stats

: 2007

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: 14.04.2008

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$(FT)^2$      see Freeman Tukey
$G^2$      see Likelihood ratio statistic
$O_p,o_p$ notation      
see Stochastic order Stochastic
$\chi^2$      see Pearson's chi square
Additive models      2324; see also Partitioning chi square Lancaster's
Agreement, measures of      393400 501502; measures
AID      see Automatic interaction detection
Angular transformation for binomial      367369 491492; transformations variance-stabilizing
Anomalies in data      332337
Arcsin transformation      367369 491492
Association, measures of, asymptotic variances for      374 501502
Association, measures of, for $2 \times 2$ tables      376385
Association, measures of, for $I \times J$ tables      385393
Association, measures of, geometry of      383385
Association, measures of, interpretability of      374375
Association, measures of, norming      375376
Association, measures of, sensitivity to margins      375 392393; measures Correlation u-terms
Association, measures of, symmetry      376
Association, surface of constant      5253
Asymptotic theory      457530; see also Bayes estimators
Automatic interaction detection      360
BAN (best asymptotic normal)      58 26 349 352 518
Bayes estimators      404410
Bayes estimators, asymptotic results for      410416
Bayes estimators, risk function of      406407
Bayes estimators, small sample results for      416419 421424
Binomial distribution      see also Sampling distributions
Binomial distribution, angular transformation for      332337 367369 491492
Binomial distribution, asymptotic distribution of powers      478479 488
Binomial distribution, asymptotic normality of      464465
Binomial distribution, behavior of minimax estimators for      416417
Binomial distribution, estimation of proportion      437
Binomial distribution, estimation of sample size      437438
Binomial distribution, moments of      436437
Binomial distribution, negative      316317 339340 452454 500
Binomial distribution, orders of      477
Binomial distribution, variance of squared proportions      485
Binomial distribution, variance stabilization for      491492
Birch's results for MLEs, regularity conditions for      5095
Birch's results for MLEs, uniqueness of      69
Block-diagonal table, l8      2
Block-stairway table, l9      4 98
Block-triangular table l9      4 98
Bootstrap-Bayes estimators      see Pseudo-Bayes estimators
Breaking-apart      see Rearranging data
BSS (between-group sum of squares)      390
Capture-recapture models, general      246254
Capture-recapture models, incomplete tables and      233 237239 246247
Capture-recapture models, interpreting dependencies in      230 255
Capture-recapture models, three-sample      237245
Capture-recapture models, two-sample      231237
Cell values      see also Configuration Zero
Cell values, elementary      13 61
Cell values, nonelementary      61
Cell values, smoothing      123 401433
Cell values, standardized residual      136139
Cell values, sums of      61
Chains, Markov for different strata      277279
Chains, Markov, order of      269270
Chernoff Pratt theory      see Stochastic order Stochastic
Chi square distribution, percentage points of      527; see also Chi square statistic
Chi square statistic      see also Minimum logit $\chi^2$ Neyman's Pearson's
Chi square statistic, association measures based on      330332 385387
Chi square statistic, asymptotic distribution, under alternative hypothesis      329330 473474 518
Chi square statistic, asymptotic distribution, under null hypothesis      472473 516218
Chi square statistic, comparing several      330332
Closed-form estimates      see Direct estimates
Cochran's Q test      307309
Cochran's test for combined tables      146147
Cohen's measure of agreement, K      395397
Collapsing, general definition of      47
Collapsing, general theorem for      4748
Collapsing, illustration of      4143
Collapsing, theorem for      3
Collapsing, two-way tables      2728
Collapsing, way array      3940
Combination of $2 \times 2$ tables      365366; see also Mantel Haensel test Collapsing
Combining categories      2729; see also Collapsing
Complete data, fitting incomplete models to      111114; see also Incomplete arrays Comprehensive
Complete tables, definition of      58
Comprehensive model, definition      38 6667
Computer programs      7
Conditional distribution      see also Incomplete tables
Conditional distribution, exact theory for      364366
Conditional distribution, for Poisson variates      440441;
Conditional tests for interaction      148 364366
Conditional tests for marginal homogeneity      293296 307309
Confidence interval      see also Variance asymptotic
Confidence interval for measures of association      374 380 382
Confidence interval, asymptotic for cell probability, p      478
Confidence interval, use of asymptotic variances for      499500
Configuration of same order, degrees of freedom      119122
Configuration, correspondence of, with u-terms      66
Configuration, definition of      61
Configuration, empty cells in      90
Connectivity      see Separability
Constructing tables      2526
Contingency tables      57
Continuous variables      360
Contrasts      see Linear contrasts
Convergence in distribution      463465 467475
Convergence in distribution and $O_n (1)$      478479
Convergence in distribution and $O_p (1)$      474
Convergence in distribution and probability      477479
Convergence in probability      465467
Convergence of moments      484486
Correlation coefficient, association measures based on      380385
Correlation coefficient, chi square and      382
Correlation coefficient, partial, related to collapsing      41
Correlation coefficient, related to cross product      1415
Correlation coefficient, variance stabilization of      50050
Cramer's measure of association, V      386387
Cross-product ratio as loglinear contrast      15 2627
Cross-product ratio related to correlation coefficient      15
Cross-product ratio related to u-terms      1718
Cross-product ratio, association measures based on      377380 393
Cross-product ratio, asymptotic variance of      377 494497
Cross-product ratio, definition of      13
Cross-product ratio, functions of      379 601
Cross-product ratio, interpretation of      1415
Cross-product ratio, properties of      14 377
Cross-validation of models      319320
Cyclic descent, method of      see Fitting iterative proportional
D-association      see Separability
D-separability      see Separability
Degrees of freedom for closed-form models      121122
Degrees of freedom for complete tables      25 35 66 114122
Degrees of freedom for marginal homogeneity      287 294 307
Degrees of freedom for quasi-independence      187188
Degrees of freedom for quasi-loglinear models      216220
Degrees of freedom for quasi-symmetry      287 303
Degrees of freedom for symmetry      283 301302
Degrees of freedom, adjusting, for empty cells      115119
Delta method for binomial distribution      481482
Delta method for calculating asymptotic distributions      486502
Delta method for capture-recapture problem      249250
Delta method, multivariate theorem      492497
Delta method, one-dimensional theorem      487490
Deming Stephan procedure      see Fitting iterative
Direct estimates for multiway incomplete tables      217 221223 248
Direct estimates for quasi-independence      192201
Direct estimates in three or four dimensions      7475
Direct estimates, difference between models with      129130
Direct estimates, equivalence to iterative      74 201202
Direct estimates, partitioning $G^2$ for models with      170171
Direct estimates, rules for detecting      7677
Direct estimates, theorems for detecting      7882
Dirichlet distribution      405406
Disagreement      398399; see also Agreement
Distributions      see Sampling distributions
e, approximating      460462
Empirical-Bayes      see Pseudo-Bayes
Empty cells      see Zero entries
Entropy, maximum, theorem relating to linear models      346; see also Information theory
Error, prediction, measures of reduction in      387389
Estimability of parameters in incomplete table      219220
Estimability, compared with connectedness      2l5
Estimation see      BAN Bayes Information Maximum Minimum methods Minimum
Exact theory for $2 \times 2$ tables      364365
Exact theory for combined tables      365366
Exact theory, practicability of      366
Exponential distribution as limit of long waiting times      467468
Exponential distribution, distribution of minimum      477478
Fitting, iterative proportional      see also Maximum likelihood estimates
Fitting, iterative proportional for asymptotic variances      250251 253254
Fitting, iterative proportional for MLEs      83
Fitting, iterative proportional for quasi-symmetry      289
Fitting, iterative proportional in three dimensions      8485
Fitting, iterative proportional, classical usage      97102 337 392
Fitting, iterative proportional, convergence of      8587 9495 289
Fitting, iterative proportional, general algorithm      9197
Fitting, iterative proportional, incomplete tables      188192 217
Fitting, iterative proportional, initial estimates      9295
Fitting, iterative proportional, properties of      83
Fitting, iterative proportional, stopping rules      9596
Four-fold table ($2 \times 2$ table), definition of      1112
Four-fold table ($2 \times 2$ table), descriptive parameters ofl      1314
Four-fold table ($2 \times 2$ table), geometry of      4955 383385
Freeman Tukey, chi square      130 137 513516
Freeman Tukey, transformation for binomial      367368 492
Freeman Tukey, transformation for Poisson      137 492
Freeman Tukey, transformation, residuals for      130 137140 334337
Frequency of frequencies      337342; see also Sampling distributions
Geometry of Bayes estimators      408410
Geometry of measures for $2 \times 2$ tables      4955 383385
Gini's definition of variation      389390
Goodness-of-fit      see also Chi square Likelihood Neyman's Pearson's
Goodness-of-fit for sampling distributions      3l5317
Goodness-of-fit of internal cells      l36141
Goodness-of-fit when model is false      329332
Goodness-of-fit, asymptotic distribution of test statistics for      513529
Goodness-of-fit, asymptotic equivalence of 3 statistics      513514
Goodness-of-fit, fitting and testing for      317324
Goodness-of-fit, multinomial general theory of      507508
Goodness-of-fit, too good a fit      324329
Hansen frequencies, asymptotic distribution of      489
Hierarchy of models      320324
Hierarchy principle      33 38 6768
Homeogeneity of proportions      347
Homeogeneity, marginal      5455 282
Homeogeneity, test for      293296 306309
Homogeneous Markov chain      262
Hypergeometric distribution      see also Exact theory
Hypergeometric distribution, multivariate      450452
Hypergeometric distribution, univariate      488450
Incomplete arrays for subsets of complete arrays      111 206210 225228
Incomplete arrays, multiway      210225
Incomplete arrays, square tables and      283 290
Incomplete arrays, triangular      31
Incomplete arrays, two-way      178206
Independence as lack of association      374380
Independence in rectangular array      2829
Independence in three-way array      38
Independence, quasi      178182 287293
Independence, surface of      51
Indirect estimates      see Fitting iterative Partitioning
Information theory, minimum discrimination information (MDI)      344347
Information theory, model generation      345346
Information theory, modified MDI (MMDI)      295296 347
Interaction      see also u-terms
Interaction, assessing magnitude of      146155
Interaction, test for      146155 364366
Irregular arrays      31 48;
Isolates      l93
Iterative proportional fitting      83102 188191
Iterative scaling      see Iterative proportional fitting
Jackknifing      3l9
K, sum of Dirichlet parameters      401402 405408 420426; K
Kullback Liebler distance      345; see also Information theory Likelihood
Lambda as measure of association      388389
Lambda as weight for Dirichlet prior      405406 419426 429433
Latent structure analysis      344
Least squares, weighted      352357
Likelihood ratio statistic, $G^2$      125130
Likelihood ratio statistic, $G^2$ for direct models      129130 158160
Likelihood ratio statistic, $G^2$ for incomplete tables      191
Likelihood ratio statistic, $G^2$ for indirect models, bounds for      160161
Likelihood ratio statistic, $G^2$ for symmetry      283
Likelihood ratio statistic, $G^2$, asymptotic distribution of      5135 6
Likelihood ratio statistic, $G^2$, calculus for partitioning      169175
Likelihood ratio statistic, $G^2$, compared with Pearson's chi-square      125126
Likelihood ratio statistic, $G^2$, conditional breakdown of      120 127
Likelihood ratio statistic, $G^2$, conditional test for marginal homogeneity      293294 307
Likelihood ratio statistic, $G^2$, definition of      58
Likelihood ratio statistic, $G^2$, structural breakdown of      127130
Linear contrast      see also Log-linear model u-terms
Linear contrast, asymptotic variance of      494497
Linear contrast, interpretations of      15 16 2627 181 211222
Linear models      see Additive models
Log-linear model      see also Linear contrast Cross Direct
Log-linear model for $2 \times 2 \times 2$ table      3233
Log-linear model for $2 \times 2$ table      1617 79
Log-linear model for $I \times J \times K$ table      3541
Log-linear model for $I \times J \times K$ table, interpretation of      37 39
Log-linear model for $I \times J$ table      2426 179
Log-linear model with terms of uniform order      119121 156158
Log-linear model, general      4247
Log-linear model, general, interpretation of      4547
Log-linear model, random effects models      371
Logistic response function      see also Logit model
Logistic response function, multivariate      360
Logistic response function, univariate      357360
Logit model for $2 \times J$ table      30
Logit model, alternatives to      367371
Logit model, continuous variable      358
Logit model, definition of      2223
Logit model, discrete variable      357 361
Logit model, minimum c-hi square for      355357
Logit model, mixed variable      358
Loglinear contrasts      see Linear contrast
Loops, closed      76
Mantel Haenszel test      133 147148 151
Marginal homogeneity      see Homogeneity
margins      see Cell values Configuration
Margins, lines of constant      53
Margins, measures insensitive to      392393
Markov models for cross classified states      273279
Markov models for different strata      277279
Markov models, assessing order of      269270
Markov models, assessing order of, for aggregate data      260 261
Markov models, higher order      267270
Markov models, single sequence transitions of      270 273
Markov models, time-homogeneity      261267 269
Maximum entropy      345346; see also Information theory
Maximum likelihood estimates (MLEs)      see also Fitting Iterative proportional Birch's Direct
Maximum likelihood estimates (MLEs) for marginal homogeneity      294295 306
Maximum likelihood estimates (MLEs) for Markov chain transition probabilities      Maximum likelihood estimates (MLEs) 262263 267268
Maximum likelihood estimates (MLEs) for quasi-symmetry      289 306319
Maximum likelihood estimates (MLEs) for symmetry      28 330302
Maximum likelihood estimates (MLEs), advantages of      58
Maximum likelihood estimates (MLEs), asymptotic behavior of      509 513
Maximum likelihood estimates (MLEs), conditional for closed populations      231 238
Maximum likelihood estimates (MLEs), existence of, for incomplete tables      186 216
Maximum likelihood estimates (MLEs), expansion of, for multinorniat      511513
Maximum likelihood estimates (MLEs), methods of obtaining      71 83
Maximum likelihood estimates (MLEs), relative precision of      313315
Maximum likelihood estimates (MLEs), suitability for likelihood ratio statistic      125126
Maximum likelihood estimates (MLEs), uniqueness of in incomplete tables      185
Maximum likelihood estimates (MLEs), when model is incorrect      513
McNemar test      258 285
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