Alfred DeMaris - Regression with Social Data, Modeling Continuous and Limited Response Variables
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521
Regression with Social Data: Modeling Continuous and Limited Response Variables,
By Alfred DeMaris
ISBN 0-471-22337-9 Copyright © 2004 John Wiley & Sons, Inc.
Absolute value, 453–454
Accelerated failure-time (AFT) model, 393–397
Additive exponential model, 188–190
Additive model, 106, 133
Adjusted alpha level, 141–142
Adjusted mean difference, 138, 140
Adjusted means, 138–142
Adjusted R
2
, see R
2
, adjusted
Algebra, basic, 447–457
Algebra, covariance, see Covariance algebra
Algebra, matrix, see Matrix algebra
Algebra, scalar, 473
Alias matrix, 214
Alpha level, 34
Alternative estimator, see Ordinary least squares
(OLS), alternatives to, under multicollinearity
Alternative hypothesis, 34
Analysis of covariance (ANCOVA), 126, 136–142
Analysis of variance (ANOVA), 126
Antilog, 453
Approximately continuous variable, 40, 43
Area under the curve (AUC), 272–273, 293–294
Arithmetic average, 26
Associative property, 448, 481
Assumptions for linear regression, 4, 43–45, 86–87
Asymmetric distribution, 260–263
Asymptotic
efficiency, 32
normality, 24, 32, 197
properties, of MLE’s, 31–32
results, 37
unbiasedness, 31–32
Attenuation factor, probit and logit regression, 260
Authenticity, 7, 56, 69–70, 89, 95, 103–104
Average causal effect, 10
Averaging slopes, 84
Bartlett’s test, 153–154
Baseline category, of a multinomial variable, 295
Baseline hazard function, 429
absorption of unmeasured heterogeneity into,
427
in Cox model, 397, 399, 401, 408
multiepisode model, 425
Baseline survival function, 398, 402–403
Bell-shaped graph, 23
Bernoulli density, 5, 21–22
Best-fitting straight line, 40–41, 46–48
Betas, interpretation of, 85–86
Bias
due to an included-variable,
see Included-variable bias
due to an omitted-variable,
see Omitted-variable bias
in b
1
, due to excluded regressors, 214
of an estimator, 31
of OLS estimate of V(b) under heteroscedasticity,
202
of principal components estimator, 241
of ridge estimator, 233–234
Biased estimator, 231–232
Binary response, 86, 247
Bivariate normal distribution, 320–321, 334
Bonferroni multiple comparison procedure, 141–142
Index
bindex.qxd 30.8.04 12:00 Page 521
Breusch-Pagan test, 203–205
Burr-10 distribution, 260–261
Capitalization on chance, 141
Categorical explanatory variable, 126
Causal assumption, see Orthogonality condition
Causal inference(s), 9–13
Cell means, 135–136
Censored normal random variable, moments of,
318–319
Censored observations
in censored regression model, 327, 331, 333
in discrete-time analysis, 431, 433–434
in multiepisode models, 425–426
in survival analysis, 383–411
Censored regression model, 324–333
Censored sample, 318
Censoring
in PRM, 364
indicator, 394, 431–432
noninformative, 383
threshold, 318, 324–325
Centering
and scaling, 200
in logistic regression, 290
in MULR, 106, 110, 138, 147, 175, 179
in SLR, 73
targeted, see Targeted centering
Central limit theorem (CLT), 32–34, 59, 352
Centroid of the x’s, 219
Chain rule, 469
Chance classification rule, 272
Change in probability, see Probability, change in
Characteristic root, see Eigenvalue(s)
Characteristic vector, see Eigenvector(s)
Chi-squared density, 366
Chi-squared test, see Likelihood-ratio chi-squared
test; Model chi-squared
Chow test, 111–112, 148–154, 209, 283–284
Cities dataset, 74–76
Classification table, 271–273
Closed-form function, 23
Closed-form solution, 188, 253, 366
Coefficient changes, test for, 97–98, 103–104,
291–293
Coefficient differences, test for, 94–96, 125, 462
Coefficient of determination, 52, 54, 85. See also
Discriminatory Power; R
2
Cohabiting transitions dataset, 15, 421–423,
439–440
Collapsibility, 299
Collapsing interval variables, 18
Collinearity, 87, 175. See also Multicollinearity
Column vector, see Vector, column
Commutative property, 447–448, 481
Comparing models, see Chow test
Competing risks, 419–424, 439–440. See also
Multistate model
Complementary log-log model, 261–263.
See also Interval-censored Cox model
Complete separation, 268–269
Complex sampling design, 211, 213
Composition of functions, 468–469
Compound path, 102
Computer programs, see LIMDEP; MATLAB;
SAS; SPSS; STATA
Condition number, 228–230
Conditional
errors in GLM, 4
mean, NBRM, 366
mean, PRM, 353
mean, zero-truncated PRM, 362
mean, ZINB model, 373
mean, ZIP model, 372
mean in GLM, 4
mean of y, censored observations, 325–326
mean of y, linear regression, 44, 71, 84–85
mean of y, truncated population, 322
probability, 19, 390, 424, 430
random assignment, 10
variance, NBRM, 366
variance, PRM, 353
variance, zero-truncated PRM, 362
variance, ZINB model, 373
variance, ZIP model, 372
variance of y, 71
Conditioning, on the independent variable, 70
Confidence interval, 33–34
Cox model, 400–401
for β
0
, β
1
, 61
for β
k
,93
for odds ratio, 266–267
for population mean, 33–34
for probit and logit coefficients, 257
Conformability, of matrices, 478
Confounding, 98–100
Consistency, 11–12, 32, 34, 56, 99–100
Constant slope, see Slope, constant
Constant variance assumption, 63–64, 185–186,
250–251. See also Homoscedasticity
assumption
Constrained-intercept model, 149–152, 283–284
Constraints, on parameters, 93–96
Constrast category, 78
Contagion, 353
Continuous, treating ordinal variable as, 171, 217
Continuous density function, 21
Continuous variable, 18–19, 43, 45
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Continuous-time survival model, 384
Contrast, linear, 460
Contrast group, 127
Convergence in distribution, 32
Cook’s D, 221–222, 243
Correlated regressors, 80
Correlation coefficient, 26, 57, 60, 463
Correlation matrix, 199, 224
Count variable, 6, 348
Count-data models, vs. censored regression, 361
Counterfactual, 10
Counterintuitive sign, of regression coefficient,
229–230
Couples dataset, 14, 40, 203–208, 330–332,
367–369, 373–375
Covariance, 26
Covariance, of residuals with x, 64–65
Covariance, sample, see Sample covariance
Covariance algebra, 8, 96, 232, 460–464
Covariance structure model, 7
Covariate pattern, 4, 85, 112, 127, 143, 219
Covariate patterns, grouping, 269
Covariate(s), 39, 136
Covratio, 222
Cox regression model, 397–398, 430, 433
empirical consistency of, see Empirical
consistency in Cox model
in two-step approach, 423
interval-censored, see Interval-censored
Cox model
shared-frailty model, 428
stratified, see Stratified Cox model
vs. discrete-time model, 436–442
Cragg model, 329–330, 332–333
Criterion, for R
2
analogs, 273–275
Criterion value, for π
ˆ
, 271
Criterion variable, see Dependent variable
Critical value, 35–36
Crossproduct term(s)
for follow-up to slope homogeneity test, 210
for model comparison, 148, 151–152
for modeling interaction in MULR, 104–105
for modeling nonlinear interaction, 178–184
in logistic regression, 285–287
interpretation of, 104–105, 107–108
omitted-variable bias in, 215–218
problems with, 108–109
Cube-root transformation, 182
Cubic equation, 169–170, 175
Cumulative baseline hazard, 402–403
Cumulative distribution function (CDF),
see Distribution function
Cumulative logit model, see Ordered logit model;
Proportional odds model
Cumulative logits, 303
Cumulative odds, 303
Current-partner victims dataset, 15, 339–340,
342–343
Curvilinear model, see Quadratic model
Curvilinearity, 168–169, 171–172, 179–180
Cutpoint, 303–305
Data, 17, 43, 86
Datasets, used in book, 13–16
Deciles of distribution, 171–172
Deciles of risk, 269–270, 411
Declining hazard, see Hazard, declining
Decomposition
of X′X, 229
of population variability in y, 52–53
of sample variability in y, 53–54
of variation, binary response, 274
Decomposition, spectral, see Spectral decomposition
Degrees of freedom, 136
Delayed entry, see Left truncation
Deleting observations, justification for, 218–219
Density function, 20–21, 71, 252, 261–263.
See also Distribution function
Departure from linearity, 168–169, 172–175, 178
Dependence of events, 424
Dependent variable, 39
Derivative(s), 464–473
Design matrix, 224, 484, 490
Design variable, see Dummy variable(s)
Destination states, 419–424
Determinant, see Matrix, determinant of
Deterministic relationship, 43
Deviance R
2
, see Discriminatory power in
count-data models
Dfbetas, 221–223
Dffits, 221–222
Diagnostics, see Influence diagnostics; Outliers;
Regression diagnostics
Diagonal
elements, of Z′Z, 199
matrix, see Matrix, diagonal
of a square matrix, 477
of design matrix, 232–233
of hat matrix, 198
of R
-1
xx
, 227, 232–233
of σ
2
(X′X)
−1
, 198
of variance-covariance matrix of regression
coefficients, 226–227
Dichotomous independent variable, 78
Difference in means, 127, 143–144
Difference in regression coefficients, test for,
see Coefficient differences, test for
Differential calculus, 48
Direct effect, 101, 218
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Discrete density function, 20–21, 255
Discrete variable, 18
Discrete-time hazard model, 430–442
Discrete-time survival model, 384
Discrimination, of categories of y, 297
Discriminatory power, see also Coefficient of
determination; R
2
defined, 7
in censored regression, 328
in competing risks model, 421–422
in count-data models, 358–360, 363
in Cox model, 410–411
in logistic regression, 271–276, 293–294
in MULR, 80, 85, 88, 90, 174, 187–188
in NBRM, 369–370
in selection models, 343
in SLR, 52–57
in logy model, 186
in multinomial models, 307–308
in ZIP model, 374
vs. empirical consistency, 69
Disordinal interaction, see Interaction, disordinal
Distribution function, 20, 253–254, 260–262,
387, 413. See also Density function
Distributive property, 448, 482
Disturbance, 3, 43, 84–85
Disturbance variance, see Error variance(s)
Domain size, 353
Domain, of observation, 349
Dummy variable(s), 2, 43, 86, 127
coding, 78, 127–131
coefficient(s), 140, 289
exact linear combination of, 128–129
model, for departure from linearity, 173
naming convention, 128
Duration of risk, 384–385
Effect coding, 131–133, 139, 141
Effect estimators, 24
Effect size, 118
Efficiency, see Asymptotic efficiency
Eigenvalue(s), 224–231, 233, 240–241, 485–486
Eigenvector(s), 224–225, 227–228, 240, 485–486
Elasticity, 91–92
Empirical consistency
defined, 7
in count-data models, 357–358
in Cox model, 411
in logistic regression, 269–270, 293
in MULR, 112–118
in multinomial logistic regression, 307
in SLR, 54, 63–69
Episode, 384
Equal error variance assumption, see Error
variance(s)
Equality, of coefficients, test for, see Coefficient
differences, test for
Equality, of matrices, 476
Equality, of vectors, 476
Equation disturbance, see Disturbance
Equation error, see Disturbance
Equation parameters, see Parameter(s)
Equidispersion, 350, 360–361. See also
Overdispersion
Error variance(s), 84, 127, 148–149, 153–154, 205
Error-variance homogeneity, 208–209
Essential ill-conditioning, 109
Essentially nonlinear model, 164–165, 184
Estimation, see also Maximum likelihood
estimation; Ordinary least squares (OLS);
Partial likelihood estimation
of censored regression model under alternative
distributions, 327
of hazard function, interval-censored Cox
model, 438
of sample selection model, 335–336
of standard error of sample intercept in SLR, 61
of standard error of sample slope in SLR, 58
of survival function, interval-censored Cox
model, 438–439
Event, defined, 383
Event, nonrepeatable, defined, 384
Event count, defined, 349
Event history analysis, defined, 382
Event indicator, 433–434, 442
Event occurrence, rate of, 353–378
Exact linear dependence, 224, 230
Exact linear function, 43
Exactly identified model, see Just-identified
model
Exogenous selection, 336
Expectation, 25–26, 44
Expected value, 3, 61, 486–488, 492
Explained risk, 274–275
Explained variance, see Coefficient of
determination; Discriminatory power; R
2
Explanatory variable, 39
Exponential
AFT model, 393–397
density, 22, 30–31, 386–387, 392–393
family of density functions, 5–6
function for mean of event count, 353–378
function for squared errors in WLS, 206
hazard model, 438
model with additive error term, 164, 185
model with multiplicative error term, 164, 185
survival function, 387, 392–393
Exponents, 451–452
Exposure, 353, 363, 377
Externally studentized residuals, 220–221
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