##### Document Text Contents

Page 2

Statistics for Social and Behavioral Sciences

Advisors:

S.E. Fienberg

W.J. van der Linden

For further volumes:

http://www.springer.com/series/3463

Page 168

negative if the patient is not ill) both equal to 1. The result for each test is either

positive (+) or negative (�). For each subject, the outcome consists of a 4-tuple of

test results; for instance, ��++ would represent a negative result on the two first,

and a positive result on the two last, tests. There are 16 possible patterns, for which

Menten et al. specify a number of alternative models. For example, their Model 4

assumes correlated outcomes for tests 1 and 2, and for tests 3 and 4; and their

conditional independence model assumes (implausibly) that the outcomes of the

four tests are independent.

Figure 7.5 shows three posterior predictive check histograms; in each case a

vertical line indicates where the observed frequency lies. The left panel shows that

the observed frequency of ++�� patterns is far higher than the conditional inde-

pendence model would lead one to expect, while the center panel shows that it is in

line with what Model 4 would predict. On the other hand, the right panel in Fig. 7.5

shows that Model 4 does a relatively poor job of predicting the �++� pattern. If it

is important to use a model that gets the ++�� pattern right, then the diagnostic

results in Fig. 7.5. suggest that Model 4 would be appropriate, or at least certainly a

lot better than the conditional independence model.

The examples in this section involve the use of posterior predictive checking in

the context of data related to living standards. For a further discussion of this

method, and examples in other areas of application, we refer the reader to Gelman

et al. (2003), Sect. 6.3.

7.5 Combining Models: Bayesian Model Averaging

Suppose we are interested in finding the determinants of poverty, and we have at

hand a long list of K plausible explanatory variables. The classical econometric

approach assumes that we know the fundamental structure of the model that

Fig. 7.5 Posterior predictive histograms for selected outcomes of tests for visceral leishmaniasis

in East Africa. (Source: Menten et al. 2006)

7.5 Combining Models: Bayesian Model Averaging 145

Page 169

generates this poverty; the main task is then to use the data to measure the strength

of the effects of the different variables.

In contrast, the Bayesian Model Averaging approach treats the models them-

selves (within a set M of possible models) as random variables: we do not

necessarily know which variables to include in the analysis, and furthermore, we

often do not know how to combine these variables in an appropriate modelMj. We

therefore need to use the data to help resolve some of the ambiguity about model

selection; this will reduce the precision with which the size of effects are measured,

but this also implies that the classical approach, by ignoring model uncertainty,

generates statistically significant results too often.

Bayesian Model Averaging is a method for dealing with model uncertainty that

generates a posterior distribution for the effects or parameters of interest (y) given

data y – the left hand side of (7.22) – as a weighted average of the posterior

distributions of the parameters under each model Mj. More formally, if we have a

total of K covariates that could be used in linear combinations, we have

PðyjyÞ ¼

X2K

j¼1

Pðyjy;MjÞ � PðMjjyÞ; (7.23)

since there are 2

K

potential models, corresponding to the inclusion or not of a

covariate. The weights here are the posterior model probabilities, which equal the

posterior probabilities of the models, given the data, and are given by

PðMjjyÞ ¼

PðyjMjÞ � PðMjÞP2K

i¼1 PðyjMiÞ � PðMiÞ

: (7.24)

Here the P(Mi) terms are the prior probabilities of the models Mi belonging to

M; and P(y|Mi) is the marginal likelihood of modelMi, that is, the likelihood of the

data given the parameters and the modelMi, integrated with respect to the priors of

the parameters.

It appears, both in theory and in practice, that Bayesian averaging over all the

models has greater predictive ability than using any single modelM* (Hoeting et al.

1999), even a single model that has been chosen using, for instance, the Bayesian

Information Criterion (BIC). This provides a powerful argument for using this

technique, at least in principle.

7.5.1 Practical Issues

If the number of variables under consideration is small, then Bayesian Model

Averaging (BMA) is relatively straightforward: for each of the 2

K

models, find

146 7 Bayesian Analysis

Page 336

Shortcuts to measuring poverty. See Poverty

Simple random sample, 51–53, 56, 57, 129,

130, 132

Simultaneity, 24, 43–47

Slovakia, 112–114

Small area estimation, 151, 198, 273–286

Snowball sampling. See Sampling

Social capital, 46, 47, 260

Social Capital and Poverty Survey.

See Tanzania

Social network, 64

Social Weather Stations [in the Philippines],

198, 199

Son preference, 292, 293, 295, 300–302

South Africa, 19, 113–116, 204, 267

South Asia, 19, 145, 212

South Korea, 178, 268

SpaceStat, 173

Spatial autoregressive model. See Spatial

models

Spatial contiguity

contiguity matrix, 163, 164

queen contiguity, 163

rook contiguity, 163

Spatial dependence. See Spatial models

Spatial errors model. See Spatial models

Spatial expansion model. See Spatial models

Spatial heterogeneity. See Spatial models

Spatial lag model. See Spatial models

Spatial models

first order spatial autoregressive model, 164

general spatial model, 166

mixed autoregressive-regressivemodel, 164

spatial autoregressive model, 164, 166

spatial dependence, 159–160, 165, 166

spatial errors model, 165, 167

spatial expansion model, 169–170

spatial heterogeneity, 159–161, 170

spatial lag model, 164, 166, 167, 169

Spline, 68, 70, 71, 76–78, 83, 151

Squared poverty gap index (P2). See Poverty

measure

Statistical control.See Instrumental variables (IV)

Strata, 36, 51, 52, 54, 58, 61, 252, 291

Stratification, 10, 23, 55, 206

Stratified sampling, 51

Structuralists, 91, 93–94

Stunting. See Malnutrition

Subjective poverty line. See Poverty

Sub-Saharan Africa, 212, 213

Sudan, 145

Survey design, 52, 132

Survival function, 289–294, 296, 297, 301

“Synthetic” estimators, 274

“Synthetic regression” estimator, 275

T

Tableau, 18

Tanzania

Human Resource Development Survey, 46

Social Capita and Poverty survey, 46

Tetrad, 79, 97, 101–104, 106

Thailand, socio-economic survey, 9, 11, 42, 44,

183–185, 237, 246, 248, 253, 260, 266

Thailand Village Fund, 42–44, 183, 236–239,

245, 246, 249–251, 253, 256. See also

Microcredit

Time taken to exit. See Poverty measure

T€ornqvst index. See Price deflator [used in

measuring poverty]

Total survey error, 53

Trabajar II [Argentina], 255

Transfer axiom. See Poverty measure

Transient poor, 177

Transition matrix, 176, 177, 214

Treatment, 24, 43, 92, 94, 145, 183, 185, 186,

189, 235, 238–244, 246–252, 254,

256–262, 264, 267, 289

Trellis plot, 10

Tri-cube function, 74

Triple differences, 247, 263

t-test, 10, 231–233, 256

U

Uganda, 204

U-matrix. See Kohonen map

Unconfoundedness, 240, 249

United States, 50, 57, 84, 144, 192, 196, 212,

213, 250, 274

Unit non-response error, 54

Unobserved area heterogeneity, 245, 247

Unobserved heterogeneity [in panel data], 178

Unobserved household heterogeneity, 245,

247, 266

Unobserved individual heterogeneity, 245

Urban Poverty Survey. See Vietnam

U.S. National Longitudinal Survey of

Youth, 144

V

Variance inflation factor, 33

VCE. See Asymptotic variance–covariance

matrix of the estimator

Index 313

Page 337

Vietnam

Household Living Standards Survey, 17,

24, 42, 53, 54, 59, 60, 73–75, 118,

209, 276, 278, 280

Living Standards Survey, 3, 4, 7, 8,

11–14, 17, 24, 25, 29, 30, 36, 37,

42, 53, 54, 59, 60, 73–75, 79, 84,

118, 132, 135, 175, 176, 178, 179,

191, 197, 209, 214, 227, 231, 233,

275, 276, 278, 280, 291, 293, 295,

300, 302

Urban Poverty Survey, 61, 62

Vietnam Living Standards Survey, 3, 4, 7, 8,

11–14, 17, 24, 25, 29, 30, 36, 37, 42, 53,

54, 59, 60, 73–75, 79, 84, 118, 132, 135,

175, 176, 178, 179, 191, 197, 209, 214,

227, 231, 233, 275, 276, 278, 280, 291,

293, 295, 300, 302

Village Fund. See Thailand Village Fund

Violin plot, 10, 11

Visceral leishmaniasis, 144, 145

Vulnerability, 114–116, 189–218, 227–230

to poverty, 214–218, 227

W

Wasting. See Malnutrition

Watts index. See Poverty measure

Weibull model, 296, 299, 300, 303, 304

Weighted regression. See Regression

Weight for height. See Malnutrition

“Welfarist” approach [to poverty], 189

White’s robust estimator, 35, 36

White’s test, 37

Wild bootstrap, 231

WinBugs, 130, 134, 136, 137, 151–152

Women and Love, 49

Z

Zambia, 151

Zero-stage rule, 5

314 Index

Statistics for Social and Behavioral Sciences

Advisors:

S.E. Fienberg

W.J. van der Linden

For further volumes:

http://www.springer.com/series/3463

Page 168

negative if the patient is not ill) both equal to 1. The result for each test is either

positive (+) or negative (�). For each subject, the outcome consists of a 4-tuple of

test results; for instance, ��++ would represent a negative result on the two first,

and a positive result on the two last, tests. There are 16 possible patterns, for which

Menten et al. specify a number of alternative models. For example, their Model 4

assumes correlated outcomes for tests 1 and 2, and for tests 3 and 4; and their

conditional independence model assumes (implausibly) that the outcomes of the

four tests are independent.

Figure 7.5 shows three posterior predictive check histograms; in each case a

vertical line indicates where the observed frequency lies. The left panel shows that

the observed frequency of ++�� patterns is far higher than the conditional inde-

pendence model would lead one to expect, while the center panel shows that it is in

line with what Model 4 would predict. On the other hand, the right panel in Fig. 7.5

shows that Model 4 does a relatively poor job of predicting the �++� pattern. If it

is important to use a model that gets the ++�� pattern right, then the diagnostic

results in Fig. 7.5. suggest that Model 4 would be appropriate, or at least certainly a

lot better than the conditional independence model.

The examples in this section involve the use of posterior predictive checking in

the context of data related to living standards. For a further discussion of this

method, and examples in other areas of application, we refer the reader to Gelman

et al. (2003), Sect. 6.3.

7.5 Combining Models: Bayesian Model Averaging

Suppose we are interested in finding the determinants of poverty, and we have at

hand a long list of K plausible explanatory variables. The classical econometric

approach assumes that we know the fundamental structure of the model that

Fig. 7.5 Posterior predictive histograms for selected outcomes of tests for visceral leishmaniasis

in East Africa. (Source: Menten et al. 2006)

7.5 Combining Models: Bayesian Model Averaging 145

Page 169

generates this poverty; the main task is then to use the data to measure the strength

of the effects of the different variables.

In contrast, the Bayesian Model Averaging approach treats the models them-

selves (within a set M of possible models) as random variables: we do not

necessarily know which variables to include in the analysis, and furthermore, we

often do not know how to combine these variables in an appropriate modelMj. We

therefore need to use the data to help resolve some of the ambiguity about model

selection; this will reduce the precision with which the size of effects are measured,

but this also implies that the classical approach, by ignoring model uncertainty,

generates statistically significant results too often.

Bayesian Model Averaging is a method for dealing with model uncertainty that

generates a posterior distribution for the effects or parameters of interest (y) given

data y – the left hand side of (7.22) – as a weighted average of the posterior

distributions of the parameters under each model Mj. More formally, if we have a

total of K covariates that could be used in linear combinations, we have

PðyjyÞ ¼

X2K

j¼1

Pðyjy;MjÞ � PðMjjyÞ; (7.23)

since there are 2

K

potential models, corresponding to the inclusion or not of a

covariate. The weights here are the posterior model probabilities, which equal the

posterior probabilities of the models, given the data, and are given by

PðMjjyÞ ¼

PðyjMjÞ � PðMjÞP2K

i¼1 PðyjMiÞ � PðMiÞ

: (7.24)

Here the P(Mi) terms are the prior probabilities of the models Mi belonging to

M; and P(y|Mi) is the marginal likelihood of modelMi, that is, the likelihood of the

data given the parameters and the modelMi, integrated with respect to the priors of

the parameters.

It appears, both in theory and in practice, that Bayesian averaging over all the

models has greater predictive ability than using any single modelM* (Hoeting et al.

1999), even a single model that has been chosen using, for instance, the Bayesian

Information Criterion (BIC). This provides a powerful argument for using this

technique, at least in principle.

7.5.1 Practical Issues

If the number of variables under consideration is small, then Bayesian Model

Averaging (BMA) is relatively straightforward: for each of the 2

K

models, find

146 7 Bayesian Analysis

Page 336

Shortcuts to measuring poverty. See Poverty

Simple random sample, 51–53, 56, 57, 129,

130, 132

Simultaneity, 24, 43–47

Slovakia, 112–114

Small area estimation, 151, 198, 273–286

Snowball sampling. See Sampling

Social capital, 46, 47, 260

Social Capital and Poverty Survey.

See Tanzania

Social network, 64

Social Weather Stations [in the Philippines],

198, 199

Son preference, 292, 293, 295, 300–302

South Africa, 19, 113–116, 204, 267

South Asia, 19, 145, 212

South Korea, 178, 268

SpaceStat, 173

Spatial autoregressive model. See Spatial

models

Spatial contiguity

contiguity matrix, 163, 164

queen contiguity, 163

rook contiguity, 163

Spatial dependence. See Spatial models

Spatial errors model. See Spatial models

Spatial expansion model. See Spatial models

Spatial heterogeneity. See Spatial models

Spatial lag model. See Spatial models

Spatial models

first order spatial autoregressive model, 164

general spatial model, 166

mixed autoregressive-regressivemodel, 164

spatial autoregressive model, 164, 166

spatial dependence, 159–160, 165, 166

spatial errors model, 165, 167

spatial expansion model, 169–170

spatial heterogeneity, 159–161, 170

spatial lag model, 164, 166, 167, 169

Spline, 68, 70, 71, 76–78, 83, 151

Squared poverty gap index (P2). See Poverty

measure

Statistical control.See Instrumental variables (IV)

Strata, 36, 51, 52, 54, 58, 61, 252, 291

Stratification, 10, 23, 55, 206

Stratified sampling, 51

Structuralists, 91, 93–94

Stunting. See Malnutrition

Subjective poverty line. See Poverty

Sub-Saharan Africa, 212, 213

Sudan, 145

Survey design, 52, 132

Survival function, 289–294, 296, 297, 301

“Synthetic” estimators, 274

“Synthetic regression” estimator, 275

T

Tableau, 18

Tanzania

Human Resource Development Survey, 46

Social Capita and Poverty survey, 46

Tetrad, 79, 97, 101–104, 106

Thailand, socio-economic survey, 9, 11, 42, 44,

183–185, 237, 246, 248, 253, 260, 266

Thailand Village Fund, 42–44, 183, 236–239,

245, 246, 249–251, 253, 256. See also

Microcredit

Time taken to exit. See Poverty measure

T€ornqvst index. See Price deflator [used in

measuring poverty]

Total survey error, 53

Trabajar II [Argentina], 255

Transfer axiom. See Poverty measure

Transient poor, 177

Transition matrix, 176, 177, 214

Treatment, 24, 43, 92, 94, 145, 183, 185, 186,

189, 235, 238–244, 246–252, 254,

256–262, 264, 267, 289

Trellis plot, 10

Tri-cube function, 74

Triple differences, 247, 263

t-test, 10, 231–233, 256

U

Uganda, 204

U-matrix. See Kohonen map

Unconfoundedness, 240, 249

United States, 50, 57, 84, 144, 192, 196, 212,

213, 250, 274

Unit non-response error, 54

Unobserved area heterogeneity, 245, 247

Unobserved heterogeneity [in panel data], 178

Unobserved household heterogeneity, 245,

247, 266

Unobserved individual heterogeneity, 245

Urban Poverty Survey. See Vietnam

U.S. National Longitudinal Survey of

Youth, 144

V

Variance inflation factor, 33

VCE. See Asymptotic variance–covariance

matrix of the estimator

Index 313

Page 337

Vietnam

Household Living Standards Survey, 17,

24, 42, 53, 54, 59, 60, 73–75, 118,

209, 276, 278, 280

Living Standards Survey, 3, 4, 7, 8,

11–14, 17, 24, 25, 29, 30, 36, 37,

42, 53, 54, 59, 60, 73–75, 79, 84,

118, 132, 135, 175, 176, 178, 179,

191, 197, 209, 214, 227, 231, 233,

275, 276, 278, 280, 291, 293, 295,

300, 302

Urban Poverty Survey, 61, 62

Vietnam Living Standards Survey, 3, 4, 7, 8,

11–14, 17, 24, 25, 29, 30, 36, 37, 42, 53,

54, 59, 60, 73–75, 79, 84, 118, 132, 135,

175, 176, 178, 179, 191, 197, 209, 214,

227, 231, 233, 275, 276, 278, 280, 291,

293, 295, 300, 302

Village Fund. See Thailand Village Fund

Violin plot, 10, 11

Visceral leishmaniasis, 144, 145

Vulnerability, 114–116, 189–218, 227–230

to poverty, 214–218, 227

W

Wasting. See Malnutrition

Watts index. See Poverty measure

Weibull model, 296, 299, 300, 303, 304

Weighted regression. See Regression

Weight for height. See Malnutrition

“Welfarist” approach [to poverty], 189

White’s robust estimator, 35, 36

White’s test, 37

Wild bootstrap, 231

WinBugs, 130, 134, 136, 137, 151–152

Women and Love, 49

Z

Zambia, 151

Zero-stage rule, 5

314 Index