Introduction to applied Bayesian statistics and estimation for social scientists:
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| Format: | Book |
| Language: | English |
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Berlin
Springer New York
2007
Berlin Springer Berlin |
| Edition: | 1. ed. |
| Series: | Statistics for Social Science and Behavorial Sciences
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | XXVIII, 357 S. graph. Darst. |
| ISBN: | 9780387712642 9781441924346 |
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| 100 | 1 | |a Lynch, Scott M. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Introduction to applied Bayesian statistics and estimation for social scientists |c Scott M. Lynch |
| 250 | |a 1. ed. | ||
| 264 | 1 | |a Berlin |b Springer New York |c 2007 | |
| 264 | 1 | |a Berlin |b Springer Berlin | |
| 300 | |a XXVIII, 357 S. |b graph. Darst. | ||
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| _version_ | 1804138360688607232 |
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| adam_text | Contents
Preface vii
Contents xiii
List of Figures xix
List of Tables xxvii
1 Introduction 1
1.1 Outline 3
1.2 A note on programming 5
1.3 Symbols used throughout the book 6
2 Probability Theory and Classical Statistics 9
2.1 Rules of probability 9
2.2 Probability distributions in general 12
2.2.1 Important quantities in distributions 17
2.2.2 Multivariate distributions 19
2.2.3 Marginal and conditional distributions 23
2.3 Some important distributions in social science 25
2.3.1 The binomial distribution 25
2.3.2 The multinomial distribution 27
2.3.3 The Poisson distribution 28
2.3.4 The normal distribution 29
2.3.5 The multivariate normal distribution 30
2.3.6 t and multivariate t distributions 33
2.4 Classical statistics in social science 33
2.5 Maximum likelihood estimation 35
2.5.1 Constructing a likelihood function 36
2.5.2 Maximizing a likelihood function 38
2.5.3 Obtaining standard errors 39
xiv Contents
2.5.4 A normal likelihood example 41
2.6 Conclusions 44
2.7 Exercises 44
2.7.1 Probability exercises 44
2.7.2 Classical inference exercises 45
3 Basics of Bayesian Statistics 47
3.1 Bayes Theorem for point probabilities 47
3.2 Bayes Theorem applied to probability distributions 50
3.2.1 Proportionality 51
3.3 Bayes Theorem with distributions: A voting example 53
3.3.1 Specification of a prior: The beta distribution 54
3.3.2 An alternative model for the polling data: A gamma
prior/ Poisson likelihood approach 60
3.4 A normal prior-normal likelihood example with a2 known .... 62
3.4.1 Extending the normal distribution example 65
3.5 Some useful prior distributions 68
3.5.1 The Dirichlet distribution 69
3.5.2 The inverse gamma distribution 69
3.5.3 Wishart and inverse Wishart distributions 70
3.6 Criticism against Bayesian statistics 70
3.7 Conclusions 73
3.8 Exercises 74
4 Modern Model Estimation Part 1: Gibbs Sampling 77
4.1 What Bayesians want and why 77
4.2 The logic of sampling from posterior densities 78
4.3 Two basic sampling methods 80
4.3.1 The inversion method of sampling 81
4.3.2 The rejection method of sampling 84
4.4 Introduction to MCMC sampling 88
4.4.1 Generic Gibbs sampling 88
4.4.2 Gibbs sampling example using the inversion method ... 89
4.4.3 Example repeated using rejection sampling 93
4.4.4 Gibbs sampling from a real bivariate density 96
4.4.5 Reversing the process: Sampling the parameters given
the data 100
4.5 Conclusions 103
4.6 Exercises 105
5 Modern Model Estimation Part 2: Metroplis—Hastings
Sampling 107
5.1 A generic MH algorithm 108
5.1.1 Relationship between Gibbs and MH sampling 113
Contents xv
5.2 Example: MH sampling when conditional densities are
difficult to derive 115
5.3 Example: MH sampling for a conditional density with an
unknown form 118
5.4 Extending the bivariate normal example: The full
multiparameter model 121
5.4.1 The conditionals for fix and fiy 122
5.4.2 The conditionals for o2x, a^, and p 123
5.4.3 The complete MH algorithm 124
5.4.4 A matrix approach to the bivariate normal distribution
problem 126
5.5 Conclusions 128
5.6 Exercises 129
6 Evaluating Markov Chain Monte Carlo Algorithms and
Model Fit 131
6.1 Why evaluate MCMC algorithm performance? 132
6.2 Some common problems and solutions 132
6.3 Recognizing poor performance 135
6.3.1 Trace plots 135
6.3.2 Acceptance rates of MH algorithms 141
6.3.3 Autocorrelation of parameters 146
6.3.4 K and other calculations 147
6.4 Evaluating model fit 153
6.4.1 Residual analysis 154
6.4.2 Posterior predictive distributions 155
6.5 Formal comparison and combining models 159
6.5.1 Bayes factors 159
6.5.2 Bayesian model averaging 161
6.6 Conclusions 163
6.7 Exercises 163
7 The Linear Regression Model 165
7.1 Development of the linear regression model 165
7.2 Sampling from the posterior distribution for the model
parameters 168
7.2.1 Sampling with an MH algorithm 168
7.2.2 Sampling the model parameters using Gibbs sampling.. 169
7.3 Example: Are people in the South nicer than others? 174
7.3.1 Results and comparison of the algorithms 175
7.3.2 Model evaluation 178
7.4 Incorporating missing data 182
7.4.1 Types of missingness 182
7.4.2 A generic Bayesian approach when data are MAR:
The niceness example revisited 186
xvi Contents
7.5 Conclusions 191
7.6 Exercises 192
8 Generalized Linear Models 193
8.1 The dichotomous probit model 195
8.1.1 Model development and parameter interpretation 195
8.1.2 Sampling from the posterior distribution for the model
parameters 198
8.1.3 Simulating from truncated normal distributions 200
8.1.4 Dichotomous probit model example: Black-white
differences in mortality 206
8.2 The ordinal probit model 217
8.2.1 Model development and parameter interpretation 218
8.2.2 Sampling from the posterior distribution for the
parameters 220
8.2.3 Ordinal probit model example: Black-white differences
in health 223
8.3 Conclusions 228
8.4 Exercises 229
9 Introduction to Hierarchical Models 231
9.1 Hierarchical models in general 232
9.1.1 The voting example redux 233
9.2 Hierarchical linear regression models 240
9.2.1 Random effects: The random intercept model 241
9.2.2 Random effects: The random coefficient model 251
9.2.3 Growth models 256
9.3 A note on fixed versus random effects models and other
terminology 264
9.4 Conclusions 268
9.5 Exercises 269
10 Introduction to Multivariate Regression Models 271
10.1 Multivariate linear regression 271
10.1.1 Model development 271
10.1.2 Implementing the algorithm 275
10.2 Multivariate probit models 277
10.2.1 Model development 278
10.2.2 Step 2: Simulating draws from truncated multivariate
normal distributions 283
10.2.3 Step 3: Simulation of thresholds in the multivariate
probit model 289
10.2.4 Step 5: Simulating the error covariance matrix 295
10.2.5 Implementing the algorithm 297
10.3 A multivariate probit model for generating distributions 303
Contents xvii
10.3.1 Model specification and simulation 307
10.3.2 Life table generation and other posterior inferences .... 310
10.4 Conclusions 315
10.5 Exercises 317
11 Conclusion 319
A Background Mathematics 323
A.I Summary of calculus 323
A.I.I Limits 323
A.1.2 Differential calculus 324
A.1.3 Integral calculus 326
A. 1.4 Finding a general rule for a derivative 329
A.2 Summary of matrix algebra 330
A.2.1 Matrix notation 330
A.2.2 Matrix operations 331
A.3 Exercises 335
A.3.1 Calculus exercises 335
A.3.2 Matrix algebra exercises 335
B The Central Limit Theorem, Confidence Intervals, and
Hypothesis Tests 337
B.I A simulation study 337
B.2 Classical inference 338
B.2.1 Hypothesis testing 339
B.2.2 Confidence intervals 342
B.2.3 Some final notes 344
References 345
Index 353
List of Figures
2.1 Sample Venn diagram: Outer box is sample space; and circles
are events A and B 11
2.2 Two uniform distributions 15
2.3 Histogram of the importance of being able to express
unpopular views in a free society (1 = Not very important...6
= One of the most important things) 16
2.4 Sample probability density function: A linear density. 17
2.5 Sample probability density function: A bivariate plane density. . 20
2.6 Three-dimensional bar chart for GSS data with best planar
density superimposed 22
2.7 Representation of bivariate cumulative distribution function:
Area under bivariate plane density from 0 to 1 in both
dimensions 23
2.8 Some binomial distributions (with parameter n = 10) 27
2.9 Some Poisson distributions 29
2.10 Some normal distributions 31
2.11 Two bivariate normal distributions 32
2.12 The t(0,1,1), t(0,1,10), and t(0,1,120) distributions (with an
N(0,1) distribution superimposed) 34
2.13 Binomial (top) and Bernoulli (bottom) likelihood functions
for the OH presidential poll data 37
2.14 Finding the MLE: Likelihood and log-likelihood functions for
the OH presidential poll data 39
3.1 Three beta distributions with mean a/(a + /3) = .5 56
3.2 Prior, likelihood, and posterior for polling data example: The
likelihood function has been normalized as a density for the
parameter K 58
xx List of Figures
3.3 Posterior for polling data example: A vertical line at K = .5 is
included to show the area needed to be computed to estimate
the probability that Kerry would win Ohio 59
3.4 Some examples of the gamma distribution 61
4.1 Convergence of sample means on the true beta distribution
mean across samples sizes: Vertical line shows sample size of
5,000; dashed horizontal lines show approximate confidence
band of sample estimates for samples of size n = 5,000; and
solid horizontal line shows the true mean 81
4.2 Example of the inversion method: Left-hand figures show the
sequence of draws from the [7(0,1) density (upper left) and
the sequence of draws from the density f{x) = (1/40) (2a; + 3)
density (lower left); and the right-hand figures show these
draws in histogram format, with true density functions
superimposed 83
4.3 The three-step process of rejection sampling 86
4.4 Sample of 1,000 draws from density using rejection sampling
with theoretical density superimposed 87
4.5 Results of Gibbs sampler using the inversion method for
sampling from conditional densities 92
4.6 Results of Gibbs sampler using the inversion method for
sampling from conditional densities: Two-dimensional view
after 5, 25, 100, and 2,000 iterations 93
4.7 Results of Gibbs sampler using rejection sampling to sample
from conditional densities 97
4.8 Results of Gibbs sampler using rejection sampling to sample
from conditional densities: Two-dimensional view after 5, 25,
100, and 2,000 iterations 98
4.9 Results of Gibbs sampler for standard bivariate normal
distribution with correlation r = .5: Two-dimensional view
after 10, 50, 200, and 2,000 iterations 100
4.10 Results of Gibbs sampler for standard bivariate normal
distribution: Upper left and right graphs show marginal
distributions for x and y (last 1,500 iterations); lower left
graph shows contour plot of true density; and lower right
graph shows contour plot of true density with Gibbs samples
superimposed 101
4.11 Samples from posterior densities for a mean and variance
parameter for NHIS years of schooling data under two Gibbs
sampling approaches: The solid lines are the results for the
marginal-for-(72-but conditional-for-/j approach; and the
dashed lines are the results for the full conditionals approach. .. 104
List of Figures xxi
5.1 Example of symmetric proposals centered over the previous
and candidate values of the parameters 110
5.2 Example of asymmetric proposals centered at the mode over
the previous and candidate values of the parameters Ill
5.3 Example of asymmetry in proposals due to a boundary
constraint on the parameter space 113
5.4 Trace plot and histogram of m parameter from linear density
model for the 2000 GSS free speech data 117
5.5 Trace plot and histogram of p parameter from bivariate
normal density model for the 2000 GSS free speech and
political participation data 120
5.6 Trace plot and histogram of p from bivariate normal model for
the 2000 GSS free speech and political participation data 126
6.1 Trace plot for first 1,000 iterations of an MH algorithm
sampling parameters from planar density for GSS free speech
and political participation data 136
6.2 Trace plot for first 4,000 iterations of an MH algorithm
sampling parameters from planar density for GSS free speech
and political participation data 137
6.3 Trace plot for all 50,000 iterations of an MH algorithm
sampling parameters from planar density for GSS free speech
and political participation data with means superimposed 138
6.4 Trace plot of cumulative and batch means for MH algorithm
sampling parameters from planar density for GSS free speech
and political participation data 139
6.5 Trace plot of cumulative standard deviation of mi from MH
algorithm sampling parameters from planar density for GSS
free speech and political participation data 140
6.6 Posterior density for mi parameter from planar density MH
algorithm and two proposal densities: [/(—.0003, .0003) and
t/(-.003, .003) 142
6.7 Trace plot of the first 1,000 iterations of an MH algorithm for
the planar density with a (relatively) broad C/(-.OO3, .003)
proposal density 143
6.8 Posterior density for mi parameter from planar density MH
algorithm and two proposal densities: U(—.003, .003) and
N(0, .00085) 144
6.9 Two-dimensional trace plot for initial run of MH algorithm
sampling parameters from planar density for GSS free speech
and political participation data: Two possible bivariate normal
proposal densities are superimposed 145
xxii List of Figures
6.10 Trace plot for 5000 iterations of an MH algorithm sampling
parameters from planar density for GSS free speech and
political participation data: Bivariate normal proposal density
with correlation —.9 146
6.11 Autocorrelation plots of parameter m,2 from MH algorithms
for the planar density: Upper figure is for the MH algorithm
with independent uniform proposals for mi and 7712; and
lower figure is for the MH algorithm with a bivariate normal
proposal with correlation —.9 148
6.12 Histogram of marginal posterior density for mi parameter:
Dashed reference line is the posterior mean 149
6.13 Trace plot of first 600 sampled values of p from MH algorithms
with three different starting values (GSS political participation
and free speech data) 151
6.14 Two-dimensional trace plot of sampled values of a and r^
from MH algorithms with three different starting values (GSS
political participation and free speech data) 152
6.15 Trace plot of the scale reduction factor R for p, a^., and Ty
across the first 500 iterations of the MH algorithms 153
6.16 Posterior predictive distributions for the ratio of the mean to
the median in the bivariate normal distribution and planar
distribution models: Vertical reference line is the observed
value in the original data 158
6.17 Correlations between observed cell counts and posterior
predictive distribution cell counts for the bivariate normal and
planar distribution models 160
7.1 Scale reduction factors by iteration for all regression
parameters 176
7.2 Trace plot of error variance parameter in three different
MCMC algorithms 177
7.3 Posterior predictive distributions: (1) The distribution of all
replicated samples and the distribution of the original data;
(2) the distribution of the ratio of the sample mean of y to
the median, with the observed ratio superimposed; and (3)
the distribution of the ranges of predicted values with the
observed range superimposed 180
7.4 Posterior predictive distributions (solid lines) and observed
values (dashed vertical lines) for persons 1,452 and 1,997 181
7.5 Distribution of age, education, and income for entire sample
and for 31 potential outliers identified from posterior
predictive simulation (reference lines at means) 183
7.6 Examples of types of missingness: No missing, missing are not
OAR, and missing are not MAR 185
List of Figures xxiii
8.1 Depiction of simulation from a truncated normal distribution... 201
8.2 Depiction of rejection sampling from the tail region of a
truncated normal distribution 203
8.3 Trace plot and autocorrelation function (ACF) plot for
intercept parameter in dichotomous probit model example
(upper plots show samples thinned to every 10 sample; lower
plot shows the ACF after thinning to every 24 post-burn-in
sample) 209
8.4 Sampled latent health scores for four sample members from
the probit model example 211
8.5 Model-predicted traits and actual latent traits for four
observations in probit model (solid line = model-predicted
latent traits from sample values of /3; dashed line = latent
traits simulated from the Gibbs sampler) 212
8.6 Latent residuals for four observations in probit model
(reference line at 0) 213
8.7 Distribution of black and age-by-black parameters (Cases 2
and 3 = both values are positive or negative, respectively;
Cases 1 and 4 = values are of opposite signs. Case 4 is the
typically-seen/expected pattern) 215
8.8 Distribution of black-white crossover ages for ages 0 and
200 (various summary measures superimposed as reference
lines) 216
8.9 Depiction of latent distribution for Y* and Y with thresholds
superimposed 220
8.10 Trace plot for threshold parameters in ordinal probit model. . .. 224
8.11 Distributions of latent health scores for five persons with
different observed values of y 225
8.12 Distributions of R2 and pseudo-i?2 from OLS and probit
models, respectively 228
9.1 Two-dimensional trace plot of Q(o) and a^) parameters
(dashed lines at posterior means for each parameter) 249
9.2 Scatterplots of four persons random intercepts and slopes
from growth curve model of health (posterior means
superimposed as horizontal and vertical dashed lines) 261
9.3 Predicted trajectories and observed health for four persons:
The solid lines are the predicted trajectories based on the
posterior means of the random intercepts and slopes from
Figure 9.2; and the dashed lines are the predicted trajectories
based on the individuals covariate profiles and posterior
means of the parameters in Table 9.4 265
xxiv List of Figures
10.1 Depiction of a bivariate outcome space for continuous latent
variables Z and Z2: The observed variables Y and Yi are
formed by the imposition of the vectors of thresholds in each
dimension; and the darkened area is the probability that an
individual s response falls in the [2,3] cell, conditional on the
distribution for Z 282
10.2 Comparison of truncated bivariate normal simulation using
naive simulation and conditional/decomposition simulation:
Upper plots show the true contours (solid lines) of the bivariate
normal distribution with sampled values superimposed (dots);
and lower plots show histograms of the values sampled from
the two dimensions under the two alternative sampling
approaches 287
10.3 Truncated bivariate normal distribution: Marginal distribution
for x when truncation of X2 is ignored 288
10.4 Comparison of truncated bivariate normal distribution
simulation using naive simulation and two iterations of Gibbs
sampling: Upper plots show the true contours (solid lines)
of the bivariate normal distribution with sampled values
superimposed (dots); and lower plots show histograms of
the values sampled from the two dimensions under the two
alternative approaches to sampling 290
10.5 Gibbs sampling versus Cowles algorithm for sampling
threshold parameters 293
10.6 Posterior predictive distributions for the probability a
30-year-old white male from the South self-identifies as a
conservative Republican across time 304
10.7 Representation of state space for a three state model 305
10.8 Two-dimensional outcome for capturing a three-state state
space 306
10.9 Trace plots of life table quantities computed from Gibbs
samples of bivariate probit model parameters 315
lO.lOHistograms of life table quantitities computed from Gibbs
samples of bivariate probit model parameters 316
A.I Generic depiction of a curve and a tangent line at an arbitrary
point 324
A.2 Finding successive approximations to the area under a curve
using rectangles 328
B.I Distributions of sample means for four sample sizes (n = 1, 2,
30, and 100) 339
B.2 Empirical versus theoretical standard deviations of sample
means under the CLT 340
B.3 Sampling distributions of variances in the simulation 341
List of Figures xxv
B.4 z-based confidence intervals for the mean: Top plot shows
intervals for samples of size n = 2; second plot shows intervals
for samples of size n = 30; third plot shows intervals for
samples of size n = 100 343
List of Tables
1.1 Some Symbols Used Throughout the Text 7
2.1 Cross-tabulation of importance of expressing unpopular views
with importance of political participation 21
3.1 CNN/USAToday/Gallup 2004 presidential election polls 57
4.1 Cell counts and marginals for a hypothetical bivariate
dichotomous distribution 94
6.1 Posterior predictive tests for bivariate normal and planar
distribution models 159
7.1 Descriptive statistics for variables in OLS regression example
(2002 and 2004 GSS data, n = 2,696) 175
7.2 Results of linear regression of measures of niceness on three
measures of region 178
7.3 Results of regression of empathy on region with and without
missing data: Missing data assumed to be MAR 190
8.1 Link functions and corrresponding generalized linear models
(individual subscripts omitted) 194
8.2 Descriptive statistics for NHANES/NHEFS data used in
dichotomous probit model example (baseline n = 3, 201) 207
8.3 Gibbs sampling results for dichotomous probit model
predicting mortality 214
8.4 Maximum likelihood and Gibbs sampling results for ordinal
probit model example 226
8.5 Various summary measures for the black-white health
crossover age 229
xxviii List of Tables
9.1 Results of hierarchical model for voting example under
different gamma hyperprior specifications 240
9.2 Results of hierarchical model for two-wave panel of income
and Internet use data 247
9.3 Results of growth model for two-wave panel of income and
Internet use data 257
9.4 Results of growth curve model of health across time 262
10.1 Results of multivariate regression of measures of niceness on
three measures of region 277
10.2 Political orientation and party affiliation 279
10.3 Multivariate probit regression model of political orientation
and party affiliation on covariates (GSS data, n = 37,028) 300
10.4 Results of multivariate probit regression of health and
mortality on covariates 310
B.I Percentage of confidence intervals capturing the true mean in
simulation 343
|
| adam_txt |
Contents
Preface vii
Contents xiii
List of Figures xix
List of Tables xxvii
1 Introduction 1
1.1 Outline 3
1.2 A note on programming 5
1.3 Symbols used throughout the book 6
2 Probability Theory and Classical Statistics 9
2.1 Rules of probability 9
2.2 Probability distributions in general 12
2.2.1 Important quantities in distributions 17
2.2.2 Multivariate distributions 19
2.2.3 Marginal and conditional distributions 23
2.3 Some important distributions in social science 25
2.3.1 The binomial distribution 25
2.3.2 The multinomial distribution 27
2.3.3 The Poisson distribution 28
2.3.4 The normal distribution 29
2.3.5 The multivariate normal distribution 30
2.3.6 t and multivariate t distributions 33
2.4 Classical statistics in social science 33
2.5 Maximum likelihood estimation 35
2.5.1 Constructing a likelihood function 36
2.5.2 Maximizing a likelihood function 38
2.5.3 Obtaining standard errors 39
xiv Contents
2.5.4 A normal likelihood example 41
2.6 Conclusions 44
2.7 Exercises 44
2.7.1 Probability exercises 44
2.7.2 Classical inference exercises 45
3 Basics of Bayesian Statistics 47
3.1 Bayes' Theorem for point probabilities 47
3.2 Bayes' Theorem applied to probability distributions 50
3.2.1 Proportionality 51
3.3 Bayes' Theorem with distributions: A voting example 53
3.3.1 Specification of a prior: The beta distribution 54
3.3.2 An alternative model for the polling data: A gamma
prior/ Poisson likelihood approach 60
3.4 A normal prior-normal likelihood example with a2 known . 62
3.4.1 Extending the normal distribution example 65
3.5 Some useful prior distributions 68
3.5.1 The Dirichlet distribution 69
3.5.2 The inverse gamma distribution 69
3.5.3 Wishart and inverse Wishart distributions 70
3.6 Criticism against Bayesian statistics 70
3.7 Conclusions 73
3.8 Exercises 74
4 Modern Model Estimation Part 1: Gibbs Sampling 77
4.1 What Bayesians want and why 77
4.2 The logic of sampling from posterior densities 78
4.3 Two basic sampling methods 80
4.3.1 The inversion method of sampling 81
4.3.2 The rejection method of sampling 84
4.4 Introduction to MCMC sampling 88
4.4.1 Generic Gibbs sampling 88
4.4.2 Gibbs sampling example using the inversion method . 89
4.4.3 Example repeated using rejection sampling 93
4.4.4 Gibbs sampling from a real bivariate density 96
4.4.5 Reversing the process: Sampling the parameters given
the data 100
4.5 Conclusions 103
4.6 Exercises 105
5 Modern Model Estimation Part 2: Metroplis—Hastings
Sampling 107
5.1 A generic MH algorithm 108
5.1.1 Relationship between Gibbs and MH sampling 113
Contents xv
5.2 Example: MH sampling when conditional densities are
difficult to derive 115
5.3 Example: MH sampling for a conditional density with an
unknown form 118
5.4 Extending the bivariate normal example: The full
multiparameter model 121
5.4.1 The conditionals for fix and fiy 122
5.4.2 The conditionals for o2x, a^, and p 123
5.4.3 The complete MH algorithm 124
5.4.4 A matrix approach to the bivariate normal distribution
problem 126
5.5 Conclusions 128
5.6 Exercises 129
6 Evaluating Markov Chain Monte Carlo Algorithms and
Model Fit 131
6.1 Why evaluate MCMC algorithm performance? 132
6.2 Some common problems and solutions 132
6.3 Recognizing poor performance 135
6.3.1 Trace plots 135
6.3.2 Acceptance rates of MH algorithms 141
6.3.3 Autocorrelation of parameters 146
6.3.4 "K" and other calculations 147
6.4 Evaluating model fit 153
6.4.1 Residual analysis 154
6.4.2 Posterior predictive distributions 155
6.5 Formal comparison and combining models 159
6.5.1 Bayes factors 159
6.5.2 Bayesian model averaging 161
6.6 Conclusions 163
6.7 Exercises 163
7 The Linear Regression Model 165
7.1 Development of the linear regression model 165
7.2 Sampling from the posterior distribution for the model
parameters 168
7.2.1 Sampling with an MH algorithm 168
7.2.2 Sampling the model parameters using Gibbs sampling. 169
7.3 Example: Are people in the South "nicer" than others? 174
7.3.1 Results and comparison of the algorithms 175
7.3.2 Model evaluation 178
7.4 Incorporating missing data 182
7.4.1 Types of missingness 182
7.4.2 A generic Bayesian approach when data are MAR:
The "niceness" example revisited 186
xvi Contents
7.5 Conclusions 191
7.6 Exercises 192
8 Generalized Linear Models 193
8.1 The dichotomous probit model 195
8.1.1 Model development and parameter interpretation 195
8.1.2 Sampling from the posterior distribution for the model
parameters 198
8.1.3 Simulating from truncated normal distributions 200
8.1.4 Dichotomous probit model example: Black-white
differences in mortality 206
8.2 The ordinal probit model 217
8.2.1 Model development and parameter interpretation 218
8.2.2 Sampling from the posterior distribution for the
parameters 220
8.2.3 Ordinal probit model example: Black-white differences
in health 223
8.3 Conclusions 228
8.4 Exercises 229
9 Introduction to Hierarchical Models 231
9.1 Hierarchical models in general 232
9.1.1 The voting example redux 233
9.2 Hierarchical linear regression models 240
9.2.1 Random effects: The random intercept model 241
9.2.2 Random effects: The random coefficient model 251
9.2.3 Growth models 256
9.3 A note on fixed versus random effects models and other
terminology 264
9.4 Conclusions 268
9.5 Exercises 269
10 Introduction to Multivariate Regression Models 271
10.1 Multivariate linear regression 271
10.1.1 Model development 271
10.1.2 Implementing the algorithm 275
10.2 Multivariate probit models 277
10.2.1 Model development 278
10.2.2 Step 2: Simulating draws from truncated multivariate
normal distributions 283
10.2.3 Step 3: Simulation of thresholds in the multivariate
probit model 289
10.2.4 Step 5: Simulating the error covariance matrix 295
10.2.5 Implementing the algorithm 297
10.3 A multivariate probit model for generating distributions 303
Contents xvii
10.3.1 Model specification and simulation 307
10.3.2 Life table generation and other posterior inferences . 310
10.4 Conclusions 315
10.5 Exercises 317
11 Conclusion 319
A Background Mathematics 323
A.I Summary of calculus 323
A.I.I Limits 323
A.1.2 Differential calculus 324
A.1.3 Integral calculus 326
A. 1.4 Finding a general rule for a derivative 329
A.2 Summary of matrix algebra 330
A.2.1 Matrix notation 330
A.2.2 Matrix operations 331
A.3 Exercises 335
A.3.1 Calculus exercises 335
A.3.2 Matrix algebra exercises 335
B The Central Limit Theorem, Confidence Intervals, and
Hypothesis Tests 337
B.I A simulation study 337
B.2 Classical inference 338
B.2.1 Hypothesis testing 339
B.2.2 Confidence intervals 342
B.2.3 Some final notes 344
References 345
Index 353
List of Figures
2.1 Sample Venn diagram: Outer box is sample space; and circles
are events A and B 11
2.2 Two uniform distributions 15
2.3 Histogram of the importance of being able to express
unpopular views in a free society (1 = Not very important.6
= One of the most important things) 16
2.4 Sample probability density function: A linear density. 17
2.5 Sample probability density function: A bivariate plane density. . 20
2.6 Three-dimensional bar chart for GSS data with "best" planar
density superimposed 22
2.7 Representation of bivariate cumulative distribution function:
Area under bivariate plane density from 0 to 1 in both
dimensions 23
2.8 Some binomial distributions (with parameter n = 10) 27
2.9 Some Poisson distributions 29
2.10 Some normal distributions 31
2.11 Two bivariate normal distributions 32
2.12 The t(0,1,1), t(0,1,10), and t(0,1,120) distributions (with an
N(0,1) distribution superimposed) 34
2.13 Binomial (top) and Bernoulli (bottom) likelihood functions
for the OH presidential poll data 37
2.14 Finding the MLE: Likelihood and log-likelihood functions for
the OH presidential poll data 39
3.1 Three beta distributions with mean a/(a + /3) = .5 56
3.2 Prior, likelihood, and posterior for polling data example: The
likelihood function has been normalized as a density for the
parameter K 58
xx List of Figures
3.3 Posterior for polling data example: A vertical line at K = .5 is
included to show the area needed to be computed to estimate
the probability that Kerry would win Ohio 59
3.4 Some examples of the gamma distribution 61
4.1 Convergence of sample means on the true beta distribution
mean across samples sizes: Vertical line shows sample size of
5,000; dashed horizontal lines show approximate confidence
band of sample estimates for samples of size n = 5,000; and
solid horizontal line shows the true mean 81
4.2 Example of the inversion method: Left-hand figures show the
sequence of draws from the [7(0,1) density (upper left) and
the sequence of draws from the density f{x) = (1/40) (2a; + 3)
density (lower left); and the right-hand figures show these
draws in histogram format, with true density functions
superimposed 83
4.3 The three-step process of rejection sampling 86
4.4 Sample of 1,000 draws from density using rejection sampling
with theoretical density superimposed 87
4.5 Results of Gibbs sampler using the inversion method for
sampling from conditional densities 92
4.6 Results of Gibbs sampler using the inversion method for
sampling from conditional densities: Two-dimensional view
after 5, 25, 100, and 2,000 iterations 93
4.7 Results of Gibbs sampler using rejection sampling to sample
from conditional densities 97
4.8 Results of Gibbs sampler using rejection sampling to sample
from conditional densities: Two-dimensional view after 5, 25,
100, and 2,000 iterations 98
4.9 Results of Gibbs sampler for standard bivariate normal
distribution with correlation r = .5: Two-dimensional view
after 10, 50, 200, and 2,000 iterations 100
4.10 Results of Gibbs sampler for standard bivariate normal
distribution: Upper left and right graphs show marginal
distributions for x and y (last 1,500 iterations); lower left
graph shows contour plot of true density; and lower right
graph shows contour plot of true density with Gibbs samples
superimposed 101
4.11 Samples from posterior densities for a mean and variance
parameter for NHIS years of schooling data under two Gibbs
sampling approaches: The solid lines are the results for the
marginal-for-(72-but conditional-for-/j approach; and the
dashed lines are the results for the full conditionals approach. . 104
List of Figures xxi
5.1 Example of symmetric proposals centered over the previous
and candidate values of the parameters 110
5.2 Example of asymmetric proposals centered at the mode over
the previous and candidate values of the parameters Ill
5.3 Example of asymmetry in proposals due to a boundary
constraint on the parameter space 113
5.4 Trace plot and histogram of m parameter from linear density
model for the 2000 GSS free speech data 117
5.5 Trace plot and histogram of p parameter from bivariate
normal density model for the 2000 GSS free speech and
political participation data 120
5.6 Trace plot and histogram of p from bivariate normal model for
the 2000 GSS free speech and political participation data 126
6.1 Trace plot for first 1,000 iterations of an MH algorithm
sampling parameters from planar density for GSS free speech
and political participation data 136
6.2 Trace plot for first 4,000 iterations of an MH algorithm
sampling parameters from planar density for GSS free speech
and political participation data 137
6.3 Trace plot for all 50,000 iterations of an MH algorithm
sampling parameters from planar density for GSS free speech
and political participation data with means superimposed 138
6.4 Trace plot of cumulative and batch means for MH algorithm
sampling parameters from planar density for GSS free speech
and political participation data 139
6.5 Trace plot of cumulative standard deviation of mi from MH
algorithm sampling parameters from planar density for GSS
free speech and political participation data 140
6.6 Posterior density for mi parameter from planar density MH
algorithm and two proposal densities: [/(—.0003, .0003) and
t/(-.003, .003) 142
6.7 Trace plot of the first 1,000 iterations of an MH algorithm for
the planar density with a (relatively) broad C/(-.OO3, .003)
proposal density 143
6.8 Posterior density for mi parameter from planar density MH
algorithm and two proposal densities: U(—.003, .003) and
N(0, .00085) 144
6.9 Two-dimensional trace plot for initial run of MH algorithm
sampling parameters from planar density for GSS free speech
and political participation data: Two possible bivariate normal
proposal densities are superimposed 145
xxii List of Figures
6.10 Trace plot for 5000 iterations of an MH algorithm sampling
parameters from planar density for GSS free speech and
political participation data: Bivariate normal proposal density
with correlation —.9 146
6.11 Autocorrelation plots of parameter m,2 from MH algorithms
for the planar density: Upper figure is for the MH algorithm
with independent uniform proposals for mi and 7712; and
lower figure is for the MH algorithm with a bivariate normal
proposal with correlation —.9 148
6.12 Histogram of marginal posterior density for mi parameter:
Dashed reference line is the posterior mean 149
6.13 Trace plot of first 600 sampled values of p from MH algorithms
with three different starting values (GSS political participation
and free speech data) 151
6.14 Two-dimensional trace plot of sampled values of a\ and r^
from MH algorithms with three different starting values (GSS
political participation and free speech data) 152
6.15 Trace plot of the scale reduction factor R for p, a^., and Ty
across the first 500 iterations of the MH algorithms 153
6.16 Posterior predictive distributions for the ratio of the mean to
the median in the bivariate normal distribution and planar
distribution models: Vertical reference line is the observed
value in the original data 158
6.17 Correlations between observed cell counts and posterior
predictive distribution cell counts for the bivariate normal and
planar distribution models 160
7.1 Scale reduction factors by iteration for all regression
parameters 176
7.2 Trace plot of error variance parameter in three different
MCMC algorithms 177
7.3 Posterior predictive distributions: (1) The distribution of all
replicated samples and the distribution of the original data;
(2) the distribution of the ratio of the sample mean of y to
the median, with the observed ratio superimposed; and (3)
the distribution of the ranges of predicted values with the
observed range superimposed 180
7.4 Posterior predictive distributions (solid lines) and observed
values (dashed vertical lines) for persons 1,452 and 1,997 181
7.5 Distribution of age, education, and income for entire sample
and for 31 potential outliers identified from posterior
predictive simulation (reference lines at means) 183
7.6 Examples of types of missingness: No missing, missing are not
OAR, and missing are not MAR 185
List of Figures xxiii
8.1 Depiction of simulation from a truncated normal distribution. 201
8.2 Depiction of rejection sampling from the tail region of a
truncated normal distribution 203
8.3 Trace plot and autocorrelation function (ACF) plot for
intercept parameter in dichotomous probit model example
(upper plots show samples thinned to every 10 sample; lower
plot shows the ACF after thinning to every 24 post-burn-in
sample) 209
8.4 Sampled latent health scores for four sample members from
the probit model example 211
8.5 Model-predicted traits and actual latent traits for four
observations in probit model (solid line = model-predicted
latent traits from sample values of /3; dashed line = latent
traits simulated from the Gibbs sampler) 212
8.6 Latent residuals for four observations in probit model
(reference line at 0) 213
8.7 Distribution of black and age-by-black parameters (Cases 2
and 3 = both values are positive or negative, respectively;
Cases 1 and 4 = values are of opposite signs. Case 4 is the
typically-seen/expected pattern) 215
8.8 Distribution of black-white crossover ages for ages 0 and
200 (various summary measures superimposed as reference
lines) 216
8.9 Depiction of latent distribution for Y* and Y with thresholds
superimposed 220
8.10 Trace plot for threshold parameters in ordinal probit model. . . 224
8.11 Distributions of latent health scores for five persons with
different observed values of y 225
8.12 Distributions of R2 and pseudo-i?2 from OLS and probit
models, respectively 228
9.1 Two-dimensional trace plot of Q(o) and a^) parameters
(dashed lines at posterior means for each parameter) 249
9.2 Scatterplots of four persons' random intercepts and slopes
from growth curve model of health (posterior means
superimposed as horizontal and vertical dashed lines) 261
9.3 Predicted trajectories and observed health for four persons:
The solid lines are the predicted trajectories based on the
posterior means of the random intercepts and slopes from
Figure 9.2; and the dashed lines are the predicted trajectories
based on the individuals' covariate profiles and posterior
means of the parameters in Table 9.4 265
xxiv List of Figures
10.1 Depiction of a bivariate outcome space for continuous latent
variables Z\ and Z2: The observed variables Y\ and Yi are
formed by the imposition of the vectors of thresholds in each
dimension; and the darkened area is the probability that an
individual's response falls in the [2,3] cell, conditional on the
distribution for Z 282
10.2 Comparison of truncated bivariate normal simulation using
naive simulation and conditional/decomposition simulation:
Upper plots show the true contours (solid lines) of the bivariate
normal distribution with sampled values superimposed (dots);
and lower plots show histograms of the values sampled from
the two dimensions under the two alternative sampling
approaches 287
10.3 Truncated bivariate normal distribution: Marginal distribution
for x\ when truncation of X2 is ignored 288
10.4 Comparison of truncated bivariate normal distribution
simulation using naive simulation and two iterations of Gibbs
sampling: Upper plots show the true contours (solid lines)
of the bivariate normal distribution with sampled values
superimposed (dots); and lower plots show histograms of
the values sampled from the two dimensions under the two
alternative approaches to sampling 290
10.5 Gibbs sampling versus Cowles' algorithm for sampling
threshold parameters 293
10.6 Posterior predictive distributions for the probability a
30-year-old white male from the South self-identifies as a
conservative Republican across time 304
10.7 Representation of state space for a three state model 305
10.8 Two-dimensional outcome for capturing a three-state state
space 306
10.9 Trace plots of life table quantities computed from Gibbs
samples of bivariate probit model parameters 315
lO.lOHistograms of life table quantitities computed from Gibbs
samples of bivariate probit model parameters 316
A.I Generic depiction of a curve and a tangent line at an arbitrary
point 324
A.2 Finding successive approximations to the area under a curve
using rectangles 328
B.I Distributions of sample means for four sample sizes (n = 1, 2,
30, and 100) 339
B.2 Empirical versus theoretical standard deviations of sample
means under the CLT 340
B.3 Sampling distributions of variances in the simulation 341
List of Figures xxv
B.4 z-based confidence intervals for the mean: Top plot shows
intervals for samples of size n = 2; second plot shows intervals
for samples of size n = 30; third plot shows intervals for
samples of size n = 100 343
List of Tables
1.1 Some Symbols Used Throughout the Text 7
2.1 Cross-tabulation of importance of expressing unpopular views
with importance of political participation 21
3.1 CNN/USAToday/Gallup 2004 presidential election polls 57
4.1 Cell counts and marginals for a hypothetical bivariate
dichotomous distribution 94
6.1 Posterior predictive tests for bivariate normal and planar
distribution models 159
7.1 Descriptive statistics for variables in OLS regression example
(2002 and 2004 GSS data, n = 2,696) 175
7.2 Results of linear regression of measures of "niceness" on three
measures of region 178
7.3 Results of regression of empathy on region with and without
missing data: Missing data assumed to be MAR 190
8.1 Link functions and corrresponding generalized linear models
(individual subscripts omitted) 194
8.2 Descriptive statistics for NHANES/NHEFS data used in
dichotomous probit model example (baseline n = 3, 201) 207
8.3 Gibbs sampling results for dichotomous probit model
predicting mortality 214
8.4 Maximum likelihood and Gibbs sampling results for ordinal
probit model example 226
8.5 Various summary measures for the black-white health
crossover age 229
xxviii List of Tables
9.1 Results of hierarchical model for voting example under
different gamma hyperprior specifications 240
9.2 Results of hierarchical model for two-wave panel of income
and Internet use data 247
9.3 Results of "growth" model for two-wave panel of income and
Internet use data 257
9.4 Results of growth curve model of health across time 262
10.1 Results of multivariate regression of measures of "niceness" on
three measures of region 277
10.2 Political orientation and party affiliation 279
10.3 Multivariate probit regression model of political orientation
and party affiliation on covariates (GSS data, n = 37,028) 300
10.4 Results of multivariate probit regression of health and
mortality on covariates 310
B.I Percentage of confidence intervals capturing the true mean in
simulation 343 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Lynch, Scott M. |
| author_facet | Lynch, Scott M. |
| author_role | aut |
| author_sort | Lynch, Scott M. |
| author_variant | s m l sm sml |
| building | Verbundindex |
| bvnumber | BV035185379 |
| classification_rvk | CM 4000 MR 2100 |
| ctrlnum | (OCoLC)634046070 (DE-599)BVBBV035185379 |
| discipline | Soziologie Psychologie Mathematik |
| discipline_str_mv | Soziologie Psychologie Mathematik |
| edition | 1. ed. |
| format | Book |
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| id | DE-604.BV035185379 |
| illustrated | Illustrated |
| index_date | 2024-07-02T22:59:25Z |
| indexdate | 2024-07-09T21:26:57Z |
| institution | BVB |
| isbn | 9780387712642 9781441924346 |
| language | English |
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| series2 | Statistics for Social Science and Behavorial Sciences |
| spelling | Lynch, Scott M. Verfasser aut Introduction to applied Bayesian statistics and estimation for social scientists Scott M. Lynch 1. ed. Berlin Springer New York 2007 Berlin Springer Berlin XXVIII, 357 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Statistics for Social Science and Behavorial Sciences Hier auch später erschienene, unveränderte Nachdrucke Bayes-Verfahren (DE-588)4204326-8 gnd rswk-swf Statistik (DE-588)4056995-0 gnd rswk-swf Soziologe (DE-588)4055930-0 gnd rswk-swf Soziologe (DE-588)4055930-0 s Statistik (DE-588)4056995-0 s Bayes-Verfahren (DE-588)4204326-8 s 1\p DE-604 Erscheint auch als Online-Ausgabe 978-0-387-71265-9 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016992074&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Lynch, Scott M. Introduction to applied Bayesian statistics and estimation for social scientists Bayes-Verfahren (DE-588)4204326-8 gnd Statistik (DE-588)4056995-0 gnd Soziologe (DE-588)4055930-0 gnd |
| subject_GND | (DE-588)4204326-8 (DE-588)4056995-0 (DE-588)4055930-0 |
| title | Introduction to applied Bayesian statistics and estimation for social scientists |
| title_auth | Introduction to applied Bayesian statistics and estimation for social scientists |
| title_exact_search | Introduction to applied Bayesian statistics and estimation for social scientists |
| title_exact_search_txtP | Introduction to applied Bayesian statistics and estimation for social scientists |
| title_full | Introduction to applied Bayesian statistics and estimation for social scientists Scott M. Lynch |
| title_fullStr | Introduction to applied Bayesian statistics and estimation for social scientists Scott M. Lynch |
| title_full_unstemmed | Introduction to applied Bayesian statistics and estimation for social scientists Scott M. Lynch |
| title_short | Introduction to applied Bayesian statistics and estimation for social scientists |
| title_sort | introduction to applied bayesian statistics and estimation for social scientists |
| topic | Bayes-Verfahren (DE-588)4204326-8 gnd Statistik (DE-588)4056995-0 gnd Soziologe (DE-588)4055930-0 gnd |
| topic_facet | Bayes-Verfahren Statistik Soziologe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016992074&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT lynchscottm introductiontoappliedbayesianstatisticsandestimationforsocialscientists |