Report

The Calibrated Bayes Approach to Sample Survey Inference Roderick Little Department of Biostatistics, University of Michigan Associate Director for Research & Methodology, Bureau of Census Learning Objectives 1. Understand basic features of alternative modes of inference for sample survey data. 2. Understand the mechanics of Bayesian inference for finite population quantitities under simple random sampling. 3. Understand the role of the sampling mechanism in sample surveys and how it is incorporated in a Calibrated Bayesian analysis. 4. More specifically, understand how survey design features, such as weighting, stratification, post-stratification and clustering, enter into a Bayesian analysis of sample survey data. 5. Introduction to Bayesian tools for computing posterior distributions of finite population quantities. Models for complex surveys 1: introduction 2 Acknowledgement and Disclaimer • These slides are based in part on a short course on Bayesian methods in surveys presented by Dr. Trivellore Raghunathan and I at the 2010 Joint Statistical Meetings. • While taking responsibility for errors, I’d like to acknowledge Dr. Raghunathan’s major contributions to this material • Opinions are my own and not the official position of the U.S. Census Bureau Models for complex surveys 1: introduction 3 Module 1: Introduction • Distinguishing features of survey sample inference • Alternative modes of survey inference – Design-based, superpopulation models, Bayes • Calibrated Bayes Models for complex surveys 1: introduction 4 Distinctive features of survey inference 1. Primary focus on descriptive finite population quantities, like overall or subgroup means or totals – Bayes – which naturally concerns predictive distributions -- is particularly suited to inference about such quantities, since they require predicting the values of variables for non-sampled items – This finite population perspective is useful even for analytic model parameters: = m odel param eter (m eaningful only in c ontext of the m odel) (Y ) = "estim ate" of from fitting m odel to w hole population Y (a finite population quantity, exists regardless of validity of m odel) A good estim ate of should be a good est im ate of (if not, then w hat's being estim ated?) Models for complex surveys 1: introduction 5 Distinctive features of survey inference 2. Analysis needs to account for "complex" sampling design features such as stratification, differential probabilities of selection, multistage sampling. • Samplers reject theoretical arguments suggesting such design features can be ignored if the model is correctly specified. • Models are always misspecified, and model answers are suspect even when model misspecification is not easily detected by model checks (Kish & Frankel 1974, Holt, Smith & Winter 1980, Hansen, Madow & Tepping 1983, Pfeffermann & Holmes (1985). • Design features like clustering and stratification can and should be explicitly incorporated in the model to avoid sensitivity of inference to model misspecification. Models for complex surveys 1: introduction 6 Distinctive features of survey inference 3. A production environment that precludes detailed modeling. • Careful modeling is often perceived as "too much work" in a production environment (e.g. Efron 1986). • Some attention to model fit is needed to do any good statistics • “Off-the-shelf" Bayesian models can be developed that incorporate survey sample design features, and for a given problem the computation of the posterior distribution is prescriptive, via Bayes Theorem. • This aspect would be aided by a Bayesian software package focused on survey applications. Models for complex surveys 1: introduction 7 Distinctive features of survey inference 4. Antipathy towards methods/models that involve strong subjective elements or assumptions. • Government agencies need to be viewed as objective and shielded from policy biases. • Addressed by using models that make relatively weak assumptions, and noninformative priors that are dominated by the likelihood. • The latter yields Bayesian inferences that are often similar to superpopulation modeling, with the usual differences of interpretation of probability statements. • Bayes provides superior inference in small samples (e.g. small area estimation) Models for complex surveys 1: introduction 8 Distinctive features of survey inference 5. Concern about repeated sampling (frequentist) properties of the inference. • Calibrated Bayes: models should be chosen to have good frequentist properties • This requires incorporating design features in the model (Little 2004, 2006). Models for complex surveys 1: introduction 9 Approaches to Survey Inference • Design-based (Randomization) inference • Superpopulation Modeling – Specifies model conditional on fixed parameters – Frequentist inference based on repeated samples from superpopulation and finite population (hybrid approach) • Bayesian modeling – Specifies full probability model (prior distributions on fixed parameters) – Bayesian inference based on posterior distribution of finite population quantities – argue that this is most satisfying approach Models for complex surveys 1: introduction 10 Design-Based Survey Inference Z ( Z1 ,..., Z N ) design variables, known for population I ( I1 ,..., I N ) = Sample Inclusion Indicators 1, unit included in sample Ii 0, otherwise I Z Y (Y1 , ..., Y N ) = population values, recorded only for sam ple Yinc Yinc ( I ) part of Y included in the survey N ote: here I is random variable, ( Y , Z ) are fixed Q Q(Y , Z ) = target finite population quantity qˆ qˆ ( I , Yinc , Z ) = sample estimate of Q Vˆ ( I , Y , Z ) = sample estimate of V 1 1 1 0 0 0 0 0 Y Yinc [Yexc ] inc qˆ 1.96 Vˆ , qˆ 1.96 Vˆ 95% confidence interval for Q Models for complex surveys 1: introduction 11 Random Sampling • Random (probability) sampling characterized by: – Every possible sample has known chance of being selected – Every unit in the sample has a non-zero chance of being selected – In particular, for simple random sampling with replacement: “All possible samples of size n have same chance of being selected” Z {1, ..., N } = set of units in the sam ple fram e N N N! 1/ , I i n , N P r( I | Z )= n i 1 ; n n !( N n ) ! 0, otherw ise E ( I i | Z ) Pr( I i 1 | Z ) n / N Models for complex surveys 1: introduction 12 Example 1:N Mean for Simple Random Sample 1 Q Y N N y , population mean i 1 i Random variable qˆ ( I ) y I i yi / n, the sample mean i 1 Fixed quantity, not modeled U nbiased for Y : E I I i y i / n i 1 N N E N I ( I i ) yi / n i 1 (n / N ) y i /n Y i 1 1 N 2 VarI ( y ) V (1 n / N ) S / n, S ( y Y ) i N 1 i 1 (1 n / N ) finite population correction N 1 2 2 2 Vˆ (1 n / N ) s / n, s = sample variance = I ( y y ) i i n 1 i 1 2 2 95% confidence interval for Y y 1.96 Vˆ , y 1.96 Vˆ Models for complex surveys 1: introduction 13 Example 2: Horvitz-Thompson estimator Q(Y ) T Y1 ...YN i E ( I i | Y ) = inclusion probability 0 N N N i 1 i 1 i 1 tˆHT I iYi / i , E I (tˆHT ) E ( I i )Yi / i = iYi / i T vˆHT Variance estimate, depends on sample design tˆ HT 1.96 vˆ H T , tˆH T 1.96 vˆ H T = 95% C I for T • Pro: unbiased under minimal assumptions • Cons: – variance estimator problematic for some designs (e.g. systematic sampling) – can have poor confidence coverage and inefficiency Models for complex surveys 1: introduction 14 Role of Models in Classical Approach • Inference not based on model, but models are often used to motivate the choice of estimator. E.g.: – Regression model regression estimator – Ratio model ratio estimator – Generalized Regression estimation: model estimates adjusted to protect against misspecification, e.g. HT estimation applied to residuals from the regression estimator (Cassel, Sarndal and Wretman book). • Estimates of standard error are then based on the randomization distribution • This approach is design-based, model-assisted Models for complex surveys 1: introduction 15 Model-Based Approaches • In our approach models are used as the basis for the entire inference: estimator, standard error, interval estimation • This approach is more unified, but models need to be carefully tailored to features of the sample design such as stratification, clustering. • One might call this model-based, design-assisted • Two variants: – Superpopulation Modeling – Bayesian (full probability) modeling • Common theme is “Infer” or “predict” about non-sampled portion of the population conditional on the sample and model Models for complex surveys 1: introduction 16 Superpopulation Modeling • Model distribution M: Y ~ f (Y | Z , ), Z = design variables, fixed parameters • Predict non-sampled values Yˆexc : yˆi E ( yi | zi , ˆ), ˆ model estimate of I R S T yi , if unit sampled; ~ ~ q Q(Y ), yi yi , if unit not sampled ( q), over distribution of I and M v mse aq 1.96 v , q 1.96 v f= 95% C I for Q 1 1 1 0 0 0 0 0 Z Y Yinc Yˆexc In the modeling approach, prediction of nonsampled values is central In the design-based approach, weighting is central: “sample represents … units in the population” Models for complex surveys 1: introduction 17 Bayesian Modeling • Bayesian model adds a prior distribution for the parameters: (Y , ) ~ ( | Z ) f (Y | Z , ), ( | Z ) prior distribution Inference about is based on posterior distribution from Bayes Theorem: I Z Y p ( | Z , Yinc ) ( | Z ) L( | Z , Yinc ), L = likelihood Inference about finite population quantitity Q(Y ) based on 1 Yinc 1 p (Q(Y ) | Yinc ) posterior predictive distribution 1 0 of Q given sample values Yinc p (Q(Y ) | Z , Yinc ) p(Q(Y ) | Z , Y inc , ) p( | Z , Yinc ) d 0 0 0 0 Yˆexc (Integrates out nuisance parameters ) In the super-population modeling approach, parameters are considered fixed and estimated In the Bayesian approach, parameters are random and integrated out of posterior distribution – leads to better small-sample inference Models for complex surveys 1: introduction 18 Bayesian Point Estimates • Point estimate is often used as a single summary “best” value for the unknown Q • Some choices are the mean, mode or the median of the posterior distribution of Q • For symmetrical distributions an intuitive choice is the center of symmetry • For asymmetrical distributions the choice is not clear. It depends upon the “loss” function. Models for complex surveys: simple random sampling 19 Bayesian Interval Estimation • Bayesian analog of confidence interval is posterior probability or credibility interval – Large sample: posterior mean +/- z * posterior se – Interval based on lower and upper percentiles of posterior distribution – 2.5% to 97.5% for 95% interval – Optimal: fix the coverage rate 1-a in advance and determine the highest posterior density region C to include most likely values of Q totaling 1-a posterior probability Models for complex surveys: simple random sampling 20 Bayes for population quantities Q • Inferences about Q are conveniently obtained by first conditioning on and then averaging over posterior of . In particular, the posterior mean is: E ( Q | Yinc ) E E ( Q | Yinc , ) | Yinc and the posterior variance is: V ar ( Q | Yinc ) E V ar ( Q | Yinc , ) | Yinc V ar E ( Q | Yinc , ) | Yinc • Value of this technique will become clear in applications • Finite population corrections are automatically obtained as differences in the posterior variances of Q and • Inferences based on full posterior distribution useful in small samples (e.g. provides “t corrections”) Models for complex surveys: simple random sampling 21 Simulating Draws from Posterior Distribution • For many problems, particularly with high-dimensional it is often easier to draw values from the posterior distribution, and base inferences on these draws (d ) • For example, if ( 1 : d 1,..., D ) is a set of draws from the posterior distribution for a scalar parameter 1 , then 1 D 1 d 1 1 D s ( D 1) 2 1 (d ) approxim ates posterior m ean D d 1 ( 1 (d ) 1 ) approxim ates posterior variance 2 ( 1 1.96 s ) or 2.5 th to 97.5th percentiles of draw s approxim ates 95% posterior credibility i nterva l for G iven a draw (d ) of , usually easy to draw non-sam pled values of data, and hence finite population quantities Models for complex surveys: simple random sampling 22 Calibrated Bayes • Any approach (including Bayes) has properties in repeated sampling • We can study the properties of Bayes credibility intervals in repeated sampling – do 95% credibility intervals have 95% coverage? • A Calibrated Bayes approach yields credibility intervals with close to nominal coverage • Frequentist methods are useful for forming and assessing model, but the inference remains Bayesian • See Little (2004) for more discussion Models for complex surveys 1: introduction 23 Summary of approaches • Design-based: – Avoids need for models for survey outcomes – Robust approach for large probability samples – Less suited to small samples – inference basically assumes large samples – Models needed for nonresponse, response errors, small areas – this leads to “inferential schizophrenia” Models for complex surveys 1: introduction 24 Summary of approaches • Superpopulation/Bayes models: – Familiar: similar to modeling approaches to statistics in general – Models needs to reflect the survey design – Unified approach for large and small samples, nonresponse and response errors. – Frequentist superpopulation modeling has the limitation that uncertainty in predicting parameters is not reflected in prediction inferences: – Bayes propagates uncertainty about parameters, making it preferable for small samples – but needs specification of a prior distribution Models for complex surveys 1: introduction 25 Module 2: Bayesian models for simple random samples 2.1 Continuous outcome: normal model 2.2 Difference of two means 2.3 Regression models 2.4 Binary outcome: beta-binomial model 2.5 Nonparametric Bayes Models for complex surveys 1: introduction 26 Models for simple random samples • Consider Bayesian predictive inference for population quantities • Focus here on the population mean, but other posterior distribution of more complex finite population quantities Q can be derived • Key is to compute the posterior distribution of Q conditional on the data and model – Summarize the posterior distribution using posterior mean, variance, HPD interval etc • Modern Bayesian analysis uses simulation technique to study the posterior distribution • Here consider simple random sampling: Module 3 considers complex design features Models for complex surveys: simple random sampling 27 Diffuse priors • In much practical analysis the prior information is diffuse, and the likelihood dominates the prior information. • Jeffreys (1961) developed “noninformative priors” based on the notion of very little prior information relative to the information provided by the data. • Jeffreys derived the noninformative prior requiring invariance under parameter transformation. • In general, ( ) | J ( ) | 1/ 2 w here 2 log f ( y | ) J ( ) E t Models for complex surveys: simple random sampling 28 Examples of noninformative priors Normal: ( , ) 2 B inom ial: ( ) P oisson: ( ) 1/ 2 2 (1 ) 1/ 2 1/ 2 N orm al regression w ith slopes : ( , ) 2 2 In simple cases these noninformative priors result in numerically same answers as standard frequentist procedures Models for complex surveys: simple random sampling 29 2.1 Normal simple random sample Yi ~ iid N ( , ); i 1, 2, ..., N 2 ( , ) 2 2 sim ple random sam ple results in Yinc ( y1 , ..., y n ) Q Y ny ( N n )Y exc N f y (1 f ) Y exc D erive posterior distribution of Q Models for complex surveys: simple random sampling 30 2.1 Normal Example Posterior distribution of (,2) p ( , ( ) 2 2 1 2 n / 2 1 | Yinc ) (2 ) exp 2 n / 2 1 1 2 exp ( y i y ) / 2 iinc 2 2 2 ( yi ) 2 iinc 2 2 n( y ) / The above expressions imply that (1) 2 | Yinc ~ ( y i y ) / n 1 2 2 i inc (2) | Yinc , 2 ~ N ( y , 2 / n) Models for complex surveys: simple random sampling 31 2.1 Posterior Distribution of Q Y exc | , 2 2 ~ N , N n Y exc | , Yinc 2 2 2 2 ~ N y, N n n (1 f ) n Q f y (1 f ) Y exc Q | , Yinc 2 Y exc | Yinc Q | Yinc (1 f ) ~ N y, n 2 2 s ~ t n 1 y , (1 f ) n 2 (1 f ) s ~ t n 1 y , n Models for complex surveys: simple random sampling 32 2.1 HPD Interval for Q Note the posterior t distribution of Q is symmetric and unimodal -- values in the center of the distribution are more likely than those in the tails. Thus a (1-a)100% HPD interval is: y t n 1,1 a / 2 (1 f ) s 2 n Like frequentist confidence interval, but recovers the t correction Models for complex surveys: simple random sampling 33 2.1 Some other Estimands • Suppose Q=Median or some other percentile • One is better off inferring about all non-sampled values • As we will see later, simulating values of Yexc adds enormous flexibility for drawing inferences about any finite population quantity • Modern Bayesian methods heavily rely on simulating values from the posterior distribution of the model parameters and predictive-posterior distribution of the nonsampled values • Computationally, if the population size, N, is too large then choose any arbitrary value K large relative to n, the sample size – National sample of size 2000 – US population size 306 million – For numerical approximation, we can choose K=2000/f, for some small f=0.01 or 0.001. Models for complex surveys: simple random sampling 34 2.1 Comments • Even in this simple normal problem, Bayes is useful: – t-inference is recovered for small samples by putting a prior distribution on the unknown variance – Inference for other quantities, like Q=Median or some other percentile, is achieved very easily by simulating the nonsampled values (more on this below) • Bayes is even more attractive for more complex problems, as discussed later. Models for complex surveys: simple random sampling 35 2.2 Comparison of Two Means • Population 1 • Population 2 P opulation size N 1 P opulation size N 2 Sam ple size n1 Sam ple size n 2 Y1i ind N ( 1 , ) Y 2 i ind N ( 2 , 2 ) 2 2 1 ( 1 , ) 2 1 2 ( 2 , 2 ) 2 2 2 1 2 2 Sam ple Statistics : ( y 1 , s1 ) Sam ple Statistics : ( y 2 , s 2 ) P osterior distributions : P osterior distributions : ( n1 1) s1 / 1 ~ n1 1 ( n 2 1) s 2 / 2 ~ n 2 1 1 ~ N ( y1 , 1 / n1 ) 2 ~ N ( y2 , 2 / n2 ) Y1i ~ N ( 1 , 1 ), i exc Y 2 i ~ N ( 2 , 2 ), i exc 2 2 2 2 2 2 2 2 2 2 Models for complex surveys: simple random sampling 36 2.2 Estimands • Examples – Y1 Y 2 (Finite sample version of Behrens-Fisher Problem) – Difference Pr(Y1 c ) Pr(Y2 c ) – Difference in the population medians – Ratio of the means or medians – Ratio of Variances • It is possible to analytically compute the posterior distribution of some these quantities • It is a whole lot easier to simulate values of nonsampled Y in Population 1 and Y in Population 2 's 1 's 2 Models for complex surveys: simple random sampling 37 2.3 Ratio and Regression Estimates • Population: (yi,xi; i=1,2,…N) • Sample: (yi, iinc, xi, i=1,2,…,N). For now assume SRS Objective: Infer about the population mean N Q yi i 1 Excluded Y’s are missing values y1 x1 y2 x2 . . . . . . yn xn xn 1 xn 2 . . . Models for complex surveys: simple random sampling xN 38 2.3 Model Specification (Yi | x i , , ) ~ ind N ( x i , x i ) 2 2 2g i 1, 2,..., N g know n 2 P rior distribution: ( , ) 2 g=1/2: Classical Ratio estimator. Posterior variance equals randomization variance for large samples g=0: Regression through origin. The posterior variance is nearly the same as the randomization variance. g=1: HT model. Posterior variance equals randomization variance for large samples. Note that, no asymptotic arguments have been used in deriving Bayesian inferences. Makes small sample corrections and uses tdistributions. Models for complex surveys: simple random sampling 39 2.3 Posterior Draws for Normal Linear Regression g = 0 ˆ ( , s ) ls estim ates of slopes and resid var iance 2 (d )2 (d ) ( n p 1) s / n p 1 2 T ˆ A z 2 (d ) n p 1 = chi-squared deviate w ith n p 1 df 2 z ( z1 , ..., z p 1 ) , z i ~ N (0,1) T A upper triangular C holesky factor of ( X X ) T A A (X X ) T T 1 : 1 N onsam pled values y i | (d ) , (d ) ~ N ( (d ) xi , (d )2 ) • Easily extends to weighted regression Models for complex surveys: simple random sampling 40 2.4 Binary outcome: consulting example • In India, any person possessing a radio, transistor or television has to pay a license fee. • In a densely populated area with mostly makeshift houses practically no one was paying these fees. • Target enforcement in areas where the proportion of households possessing one or more of these devices exceeds 0.3, with high probability. Models for complex surveys: simple random sampling 41 2.4 Consulting example (continued) N P opulation S ize in particular area 1, if household i has a device Yi 0, otherw ise N Q Y i / N P roportion of households w ith a devic e i 1 Q uestion of Interest: P r( Q 0.3) • Conduct a small scale survey to answer the question of interest • Note that question only makes sense under Bayes paradigm Models for complex surveys: simple random sampling 42 2.4 Consulting example srs of size n, Yinc {Y1 ,..., Yn }, Yexc {Yn1 ,..., YN } Yi | ~ iid Bernoulli( ) n x Yi i 1 f ( x | ) ( nx ) x (1 ) n x Model for observable ( ) 1 (0,1) Prior distribution N Q Yi / N x Yi / N i 1 i n 1 N Models for complex surveys: simple random sampling Estimand 43 2.4 Beta Binomial model The posterior distribution is p ( | x ) p ( | x ) f ( x | ) ( ) f ( x | ) ( ) d ( ) (1 ) n x x n x ( x ) (1 ) n x n x f ( x | ) ( ) 1 d | x ~ B eta ( x 1, n x 1) Models for complex surveys: simple random sampling 44 2.4 Infinite Population F or N , Y N P r(Y N 0.3 | x ) P r( 0.3 | x ) C om pute using cum ulative distribution fu nction of a beta distribution w hich is a standa rd function in m ost softw are such as S A S , R What is the maximum proportion of households in the population with devices that can be said with great certainty? P r( ? | x ) 0.9 Inverse C D F of B eta D istribution Models for complex surveys: simple random sampling 45 2.5 Bayesian Nonparametric Inference • • • • • Population: Y1 , Y2 , Y3 , ..., Y N All possible distinct values: d1 , d 2 ,..., d K Model: Pr(Yi d k ) k Prior: (1 , 2 ,..., k ) k1 if k 1 k k Mean and Variance: E (Yi | ) d k k k V ar (Yi | ) 2 dkk 2 2 k Models for complex surveys: simple random sampling 46 2.5 Bayesian Nonparametric Inference • SRS of size n with nk equal to number of dk in the sample • Objective is to draw inference about the population mean:Q f y (1 f ) Y • As before we need the posterior distribution of and 2 exc Models for complex surveys: simple random sampling 47 2.5 Nonparametric Inference • Posterior distribution of is Dirichlet: ( | Yinc ) k n k 1 if k 1 and n k k k n k • Posterior mean, variance and covariance of E ( k | Yinc ) nk n , V ar ( k | Yinc ) C ov ( k , l | Yinc ) nk (n nk ) n ( n 1) 2 n k nl n ( n 1) 2 Models for complex surveys: simple random sampling 48 2.5 Inference for Q E ( | Yinc ) dk k nk y n s n 1 2 V ar ( | Yinc ) E ( 2 | Yinc ) s n n 1 2 ;s 2 1 n 1 i inc ( yi y ) 2 n 1 n 1 Hence posterior mean and variance of Q are: E ( Q | Yinc ) f y (1 f ) E ( | Yinc ) y s n 1 2 V ar ( Q | Yinc ) (1 f ) n n 1 Models for complex surveys: simple random sampling 49 Module 3: complex sample designs • Considered Bayesian predictive inference for population quantities • Focused here on the population mean, but other posterior distribution of more complex finite population quantities Q can be derived • Key is to compute the posterior distribution of Q conditional on the data and model – Summarize the posterior distribution using posterior mean, variance, HPD interval etc • Modern Bayesian analysis uses simulation technique to study the posterior distribution • Models need to incorporate complex design features like unequal selection, stratification and clustering Models for complex surveys: simple random sampling 50 Modeling sample selection • Role of sample design in model-based (Bayesian) inference • Key to understanding the role is to include the sample selection process as part of the model • Modeling the sample selection process – Simple and stratified random sampling – Cluster sampling, other mechanisms – See Chapter 7 of Bayesian Data Analysis (Gelman, Carlin, Stern and Rubin 1995) Models for complex sample designs 51 Full model for Y and I p (Y , I | Z , , ) p (Y | Z , ) p ( I | Y , Z , ) Model for Population Model for Inclusion • Observed data: (Yinc , Z , I ) (N o m issing values) • Observed-data likelihood: L ( , | Yinc , Z , I ) p (Yinc , I | Z , , ) p (Y , I | Z , , ) dY exc • Posterior distribution of parameters: p ( , | Yinc , Z , I ) p ( , | Z ) L ( , | Yinc , Z , I ) Models for complex sample designs 52 Ignoring the data collection process • The likelihood ignoring the data-collection process is based on the model for Y alone with likelihood: L ( | Yinc , Z ) p (Yinc | Z , ) p (Y | Z , ) dY exc • The corresponding posteriors for and Yexc are: p ( | Yinc , Z ) p ( | Z ) L ( | Yinc , Z ) p (Y exc | Yinc , Z ) p (Y exc Posterior predictive distribution of Yexc | Yinc , Z , ) p ( | Yinc , Z ) d • When the full posterior reduces to this simpler posterior, the data collection mechanism is called ignorable for Bayesian inference about , Yexc . Models for complex sample designs 53 Bayes inference for probability samples • A sufficient condition for ignoring the selection mechanism is that selection does not depend on values of Y, that is: p ( I | Y , Z , ) p ( I | Z , ) for all Y . • This holds for probability sampling with design variables Z • But the model needs to appropriately account for relationship of survey outcomes Y with the design variables Z. • Consider how to do this for (a) unequal probability samples, and (b) clustered (multistage) samples Models for complex sample designs 54 Ex 1: stratified random sampling • Population divided into J strata • Z is set of stratum indicators: Sample Population Z Y Z 1, if unit i is in stratum j ; zi 0, otherw ise. • Stratified random sampling: simple random sample of n j units selected from population of N j units in stratum j. • This design is ignorable providing model for outcomes conditions on the stratum variables Z. • Same approach (conditioning on Z works for poststratification, with extensions to more than one margin. Models for complex sample designs 55 Inference for a mean from a stratified sample • Consider a model that includes stratum effects: [ y i | z i j ] ~ ind N ( j , j ) 2 • For simplicity assume 2 j is known and the flat prior: p ( j | Z ) const . • Standard Bayesian calculations lead to [Y | Yinc , Z , { j }] ~ N ( y st , st ) 2 where: 2 J y st P j y j , P j N j / N , y j sam ple m ean in stratum j , j 1 J st 2 P j (1 f j ) 2 2 j / nj, fj nj / N j j 1 Models for complex sample designs 56 Bayes for stratified normal model • Bayes inference for this model is equivalent to standard classical inference for the population mean from a stratified random sample • The posterior mean weights case by inverse of inclusion probability: y st N 1 J N j 1 j yj N 1 J yi / j , j 1 i : x i j w here j n j / N j selection probability in stratum j . • With unknown variances, Bayes’ for this model with flat prior on log(variances) yields useful t-like corrections for small samples Models for complex sample designs 57 Suppose we ignore stratum effects? • Suppose we assume instead that: [ y i | z i j ] ~ ind N ( , ), 2 the previous model with no stratum effects. • With a flat prior on the mean, the posterior mean of Y is then the unweighted mean J E (Y | Yinc , Z , ) y 2 pj yj, pj nj / n j 1 • This is potentially a very biased estimator if the selection rates j n j / N j vary across the strata – The problem is that results from this model are highly sensitive violations of the assumption of no stratum effects … and stratum effects are likely in most realistic settings. – Hence prudence dictates a model that allows for stratum effects, such as the model in the previous slide. Models for complex sample designs 58 Design consistency • Loosely speaking, an estimator is design-consistent if (irrespective of the truth of the model) it converges to the true population quantity as the sample size increases, holding design features constant. • For stratified sampling, the posterior mean y st based on the stratified normal model converges to Y , and hence is designconsistent • For the normal model that ignores stratum effects, the posterior mean y converges to Y J j 1 j N j Y j / j 1 j N J j and hence is not design consistent unless j const . • We generally advocate Bayesian models that yield designconsistent estimates, to limit effects of model misspecification Models for complex sample designs 59 Ex 2. A continuous (post)stratifier Z Consider PPS sampling, Z = measure of size Standard design-based estimator is weighted Horvitz-Thompson estimate yHT 1 n yi / i N i 1 Sample Population Z Y Z ; i selection prob (H T ) y H T m odel-based prediction estim ate for y i ~ N or ( i , i ) ("H T m odel") 2 2 When the relationship between Y and Z deviates a lot from the HT model, HT estimate is inefficient and CI’s can have poor coverage Models for complex sample designs 60 Ex 4. One continuous (post)stratifier Z y wt 1 n yi / i N i 1 ; i selection prob (H T ) Sample Population Z Y Z A m odeling alternative to the H T estim at or is create predictions from a m ore robust m odel relating Y to Z : 1 n y m od = yi N i 1 ˆ y i , yˆ i predictions from : i n 1 N y i ~ N or ( S ( i ), i ); S ( i ) = penalized s pline of Y on Z 2 k (Zheng and Little 2003, 2005) Models for complex sample designs 61 Ex 3. Two stage sampling • Most practical sample designs involve selecting a cluster of units and measure a subset of units within the selected cluster • Two stage sample is very efficient and cost effective • But outcome on subjects within a cluster may be correlated (typically, positively). • Models can easily incorporate the correlation among observations Models for complex sample designs 62 Two-stage samples • Sample design: – Stage 1: Sample c clusters from C clusters – Stage 2: Sample k i units from the selected cluster i=1,2,…,c K i P opulation size of cluster i C N K i i 1 • Estimand of interest: Population mean Q • Infer about excluded clusters and excluded units within the selected clusters Models for complex sample designs 63 Models for two-stage samples • Model for observables Yij ~ N ( i , ); i 1, ..., C ; j 1, 2, ..., K i 2 i ~ iid N ( , ) 2 A ssum e and are know n • Prior distribution ( ) 1 Models for complex sample designs 64 Estimand of interest and inference strategy • The population mean can be decomposed as c NQ [k C i y i ( K i k i )Yi ,exc ] i 1 K i Yi i c 1 • Posterior mean given Yinc c [k E ( N Q | Yinc , i , i 1, 2, ..., c ; ) C i yi ( K i ki ) i ] i 1 [k C i y i ( K i k i ) E ( i | Yinc )] i 1 w here E ( i | Yinc ) ˆ E ( | Yinc ) K i i c 1 c E ( N Q | Yinc ) K i E ( | Yinc ) i c 1 2 2 y i ( k i / ) ˆ (1 / ) ki / y i / ( 2 2 2 1/ 2 / ki ) i 1 / ( 2 2 / ki ) i Models for complex sample designs 65 Posterior Variance • Posterior variance can be easily computed c V ar ( N Q | Yinc ) C ( K i k i )( 2 ( K i k i ) ) 2 i 1 K i ( 2 K i ) 2 i c 1 V ar (Yi ,exc | Yinc ) E [V ar (Yi ,exc | Yinc , i ) | Yinc ] V ar [ E (Y i ,exc | Yinc , i ) | Yinc ] 2 , i 1, 2, 2 K i ki ,c V ar (Yi | Yinc ) E [V ar (Y i | Yinc , i ) | Yinc ] V ar [ E (Y i | Y inc , i ) | Y inc ] 2 / K i , i c 1, c 2, 2 ,C Models for complex sample designs 66 Inference with unknown and • For unknown and – Option 1: Plug in maximum likelihood estimates. These can be obtained using PROC MIXED in SAS. PROC MIXED actually gives estimates of ,, and E(i|Yinc) (Empirical Bayes) – Option 2: Fully Bayes with additional prior 2 ( , , ) 2 2 2 v exp b / (2 ) 2 where b and v are small positive numbers Models for complex sample designs 67 Extensions and Applications • Relaxing equal variance assumption Yil ~ N ( i , i ) 2 ( i , log i ) ~ iid B V N ( , ) • Incorporating covariates (generalization of ratio and regression estimates) Yil ~ N ( x il i , i ) 2 ( i , log i ) ~ iid M V N ( , ) • Small Area estimation. An application of the hierarchical model. Here the quantity of interest is E (Yi | Yinc ) ( k i y i ( K i k i ) E (Yi ,exc | Yinc )) / K i Models for complex sample designs 68 Extensions • Relaxing normal assumptions Yil | i ~ G lim ( i h ( x il i ), v ( i )) 2 v : a know n function i ~ iid M V N ( , ) • Incorporate design features such as stratification and weighting by modeling explicitly the sampling mechanism. Models for complex sample designs 69 Summary • Bayes inference for surveys must incorporate design features appropriately • Stratification and clustering can be incorporated in Bayes inference through design variables • Unlike design-based inference, Bayes inference is not asymptotic, and delivers good frequentist properties in small samples Models for complex sample designs 70 Module 4: Short introduction to Bayesian computation • A Bayesian analysis uses the entire posterior distribution of the parameter of interest. • Summaries of the posterior distribution are used for statistical inferences – Means, Median, Modes or measures of central tendency – Standard deviation, mean absolute deviation or measures of spread – Percentiles or intervals • Conceptually, all these quantities can be expressed analytically in terms of integrals of functions of parameter with respect to its posterior distribution • Computations – Numerical integration routines – Simulation techniques – outline here Models for Complex Surveys: Bayesian Computation 71 Types of Simulation • Direct simulation (as for normal sample, regression) • Approximate direct simulation – Discrete approximation of the posterior density – Rejection sampling – Sampling Importance Resampling • Iterative simulation techniques – Metropolis Algorithm – Gibbs sampler – Software: WINBUGS Models for Complex Surveys: Bayesian Computation 72 Approximate Direct Simulation • Approximating the posterior distribution by a normal distribution by matching the posterior mean and variance. – Posterior mean and variance computed using numerical integration techniques • An alternative is to use the mode and a measure of curvature at the mode – Mode and the curvature can be computed using many different methods • Approximate the posterior distribution using a grid of values of the parameter and compute the posterior density at each grid and then draw values from the grid with probability proportional to the posterior density Models for Complex Surveys: Bayesian Computation 73 Normal Approximation P osterior density : ( | x ) E asy to w ork w ith log-posterior density l ( ) log( ( | x )) A t the m ode, f ( ) l '( ) 0 C urvature : f '( ) l ''( ) F o r lo g arith m o f th e n o rm al d en sity M o d e is th e m ean an d th e cu rvatu re at th e m o d e is n eg ative o f th e p recisio n (P recisio n :recip ro cal o f varian ce) Models for Complex Surveys: Bayesian Computation 74 Rejection Sampling • Actual Density from which to draw from • Candidate density from which it is easy to draw • The importance ratio is bounded • Sample from g, accept with probability p otherwise redraw from g ( | data ) g ( ), w ith g ( ) 0 for all w ith ( | data ) 0 ( | data ) g ( ) p M ( | data ) M g ( ) Models for Complex Surveys: Bayesian Computation 75 Sampling Importance Resampling • Target density from which to ( | data ) draw g ( ), such that g ( ) 0 • Candidate density from which it is easy to draw for all w ith ( | data ) 0 • The importance ratio ( | data ) w ( ) • Sample M values of from g • Compute the M importance ratios and resample with probability proportional to the importance ratios. g ( ) 1* , *2 , ... , *M w ( i ); i 1,2 , ... , M * Models for Complex Surveys: Bayesian Computation 76 Markov Chain Simulation • In real problems it may be hard to apply direct or approximate direct simulation techniques. • The Markov chain methods involve a random walk in the parameter space which converges to a stationary distribution that is the target posterior distribution. – Metropolis-Hastings algorithms – Gibbs sampling Models for Complex Surveys: Bayesian Computation 77 Metropolis-Hastings algorithm • Try to find a Markov Chain whose stationary distribution is the desired posterior distribution. • Metropolis et al (1953) showed how and the procedure was later generalized by Hastings (1970). This is called Metropolis-Hastings algorithm. • Algorithm: – Step 1 At iteration t, draw y ~ p( y | x (t ) ) y : C andidate P oint p : C andidate D ensity Models for Complex Surveys: Bayesian Computation 78 – Step 2: Compute the ratio (t ) f ( y) / p( y | x ) w M in 1, (t ) (t ) f (x ) / p(x | y) – Step 3: Generate a uniform random number, u X X – – – – ( t 1) y if u w ( t 1) X (t ) o th erw ise This Markov Chain has stationary distribution f(x). Any p(y|x) that has the same support as f(x) will work If p(y|x)=f(x) then we have independent samples Closer the proposal density p(y|x) to the actual density f(x), faster will be the convergence. Models for Complex Surveys: Bayesian Computation 79 Gibbs sampling • Gibbs sampling a particular case of Markov Chain Monte Carlo method suitable for multivariate problems x ( x1 , x 2 , ..., x p ) ~ f ( x ) f ( x i | x1 , x 2 , ..., x i 1 , x i 1 , ..., x p ) G ibbs sequence : ( t 1) x1 ( t 1) x2 ( t 1) xi ( t 1) xp (t ) (t ) (t ) ~ f ( x1 | x 2 , x 3 , ..., x p ) ( t 1) (t ) ~ f ( x 2 | x1 ( t 1) ~ f ( x i | x1 ( t 1) (t ) (t ) , ..., x i 1 , x i 1 , ..., x p ) ( t 1) ~ f ( x p | x1 (t ) , x 3 , ..., x p ) ( t 1) 1. This is also a Markov Chain whose stationary Distribution is f(x) 2. This is an easier Algorithm, if the conditional densities are easy to work with 3. If the conditionals are harder to sample from, then use MH or Rejection technique within the Gibbs sequence , ..., x p 1 ) Models for Complex Surveys: Bayesian Computation 80 Conclusion • Design-based: limited, asymptotic • Bayesian inference for surveys: flexible, unified, now feasible using modern computational methods • Calibrated Bayes: build models that yield inferences with good frequentist properties – diffuse priors, strata and post-strata as covariates, clustering with mixed effects models • Software: Winbugs, but software targeted to surveys would help. • The future may be Calibrated Bayes! Models for complex surveys 1: introduction 81