Graybill Conference Poster Session Introductions

Transcription

1 Graybill Conference Poster Session Introductions 2013 Graybill Conference in Modern Survey Statistics Colorado State University Fort Collins, CO June 10, 2013

2 Small Area Estimation with Incomplete Auxiliary Information Andreea L. Erciulescu and Wayne A. Fuller Department of Statistics, Iowa State University June 10, 2013

3 Background and Motivation Surveys are often designed to achieve specific information about totals and means, but direct estimates for small areas may not be reliable because of small sample sizes Procedures based on models have been used to construct estimates for small areas, by exploiting auxiliary information We fit nested models with a binary response and random area effects E(y ij b i ) = p ij (x ij, b i ) = exp(x ij β + b i) 1 + exp(x ij β + b i) The goal is to construct small area predictions for the mean of a binomial variable, using different amounts of auxiliary information The true small area mean of y is θ i = p ij (x ij, b i )df xi (x)

4 Results and Conclusions We consider three cases of auxiliary information Fxi (µ xi, Σ xx ) known µ xi unknown fixed, estimated using ω ij such that n i j=1 n i ω ij = 1 and E( ω ij x ij ) = µ xi j=1 µxi unknown random, estimated using µ xi (µ x, Σ δδ ) and x i µ xi (µ xi, Σ xx ) We construct small area predictions for the area means by integrating over the covariate distribution and over the random area effects distribution We compare the prediction error biases and mean squared errors using a simulation study We conclude that, generally, it is better to include auxiliary information in the model and estimate its distribution than to ignore the auxiliary information.

5 Variance Estimation after Multiple Imputation Jiwei Zhao Department of Biostatistics, Yale University School of Public Health June 10, 2013

6 Background and Motivation Consider (R, Y, X ), R = 1 is Y is observed, X is fully observed Missing data mechanism p(r = 1 Y, X ) Regression model p(y X ; θ) Problem of interest: comparison of variance estimation of estimates of θ before and after multiple imputation (MI) Under MAR assumption, the estimator after MI is less efficient, and more general results are obtained (Wang and Robins 1998, Kim and Shao 2013) Before MI, Vp = I 1 obs After MI, Vp,MI = I 1 obs + M 1 Ic 1 I mis Ic 1 What is the situation under nonignorable missingness?

7 Results and Conclusions How to propose preliminary estimator? Assume p(r = 1 Y, X ) = p(r = 1 Y ) (Tang, Little and Raghunathan, 2003) p(x Y, R = 1) = p(x Y ) = p(y X ;θ)p(x ) p(y X ;θ)p(x )dx How to conduct MI? p(y X, R = 0) = p(r=0 Y )p(y X ;θ) p(r=0 Y )p(y X ;θ)dy Simulation studies show that MI could improve the efficiency of estimator of θ Ongoing project: general results are still under investigation Any comments/suggestions are appreciated!

8 A Semiparametric Approach to Modeling Survey Data in the Presence of Informative Sampling Wade W. Herndon, Jean Opsomer, and F. Jay Breidt Department of Statistics, Colorado State University June 10, 2013

9 Background and Motivation Under an informative sampling design, the model that holds at the sample level does not hold at the population level We want to estimate f (y k x k ) Due to the informative sampling we must include additional design variables to account for the design information yielding f 1 (y k x k, z k ) We can recover the original regression relationship of interest via f (y k x k ) = f 1 (y k x k, z k )f 2 (z k x k )dz k The goal is to use model covariates to integrate out the design effects from model

10 Results and Conclusions For many applications, sample weights can be included as model covariates to account for the design bias, and then subsequently estimated by a nonparametric estimator using model covariates The full regression model is y = x T β + wx T γ + ɛ A semiparametric model is proposed where (ˆβ T, ˆγ T ) come from the regression of y on x and w E [w x] is estimated by a nonparametric, design-based estimator The nonparametric estimator is combined with the parametric regression to form an estimator for y that is a smooth function of x nonparametric methods are used here to integrate out the design effects from the model

11 The use of followups for propensity score adjustment with nonignorable nonresponse Jongho Im and Jae-Kwang Kim Department of Statistics, Iowa State University June 10, 2013

12 Background and Motivation Nonignorable nonreponse bias can be corrected with followups. Our goal is to provide a propensity score adjusted estimator, Ŷ = n d i δ i,t 1 y i + i=1 n i=1 (1 δ i,t 1 )δ it d i y i ˆp it for t = 1,, T with δ i0 = 0. d i is sampling weight. A t is a set of all respondents up to the t-th contact; A 1 A T. δ it is equal to 1 if i A t and 0 otherwise. p it is the conditional response probability at the t-th contact, p it P(δ it = 1 δ i,t 1 = 0, y i ) = {1 + exp(α t + φy i )} 1 Alho (1990) considered a conditional likelihood based approach to estimate ˆp it by assuming the multinomial likelihood on p it = P(δ it = 1 δ i,t 1 = 0, y i, δ it = 1) instead of p it.

13 Results and Conclusions Since E [δ it δ i,t 1, y i ] = p it, given the set of respondents A 1 and A 2, we can write δ i1 d i (1, y i ) = (N, Y ) & d i = N (1) p i1 i=a i A i A d i δ i1 (1, y i ) + i A d i (1 δ i1 )δ i2 p i2 (1, y i ) = (N, Y ) (2) We have 3 equations and 3 parameters in (1) and (2). We can apply the generalized method of moment (GMM) for the general followup cases that we have more equations than the number of parameters. Relatively easy to get variance estimation (GMM estimator). More robust rather than other likelihood based methods. Auxiliary variable information can be augmented as additional calibration equations.

14 Varying Coefficient Models in Finite Population Sampling Luis Fernando Contreras Cruz COLPOS Mexico June 10, 2013

15 Background and Motivation A model-assisted semiparametric method of estimating population totals is investigated to improve the precision of survey estimators by incorporating multivariate auxiliary information. The proposed superpopulation model is a varying coefficient model. The varying coefficient models (Hastie and Tibshirani,1993) and many of their variations (e.g. Hoover,1998) have gained much attention in the literature. The applications are found in various scientific areas, such as economics, business, medical science, etc. (see Fan, 2008 for a nice review). Both simulated and real data examples are given to illustrate the model and the proposed estimation methodology, which have provided strong evidence that corroborates with the asymptotic theory.

16 Results and Conclusions A way to obtain the smoothing parameters was proposed using cross-validation. The VCM identifies relations non linear between the variables. The VCM assisted-models contributes to semiparametric regression in survey sampling. The Variance estimation using cross-validation and g-weights work well in simulation studies and application. Use cross-validation to avoid overfitting problem.

17 Application of Z-estimation Theory to Calibrated Estimators for Semiparametric Models with Two-phase Stratified Sampling Jie Kate Hu, Gary Chan, Norman Breslow Department of Biotatistics University of Washington, Seattle, WA June 10, 2013

18 Motivation In epidemiology studies, we are usually interested in parameters specified in a (semi)parametric model describing an association between an exposure and an outcome. For example, λ(t Z) = λ 0 (t) + θ T Z. To improve the efficiency, we consider two-phase stratified sampling design and calibration estimators using auxiliary variables available for all cohort members. Our goal it to estimate both Euclidean and infinite dimensional parameters simultaneously in semiparametric models using inverse probability weighted estimating equation (IPW-EE) with calibration.

19 Results Let X be the variable of interest. Motivated by the semiparametric model, α 0 is defined as the unique solution to the map Ψ(a) = Eψ α (X ) = 0. Let vector Ṽ = Ṽ (V ) be the calibration variable. Calibrated estimator ˆα is obtained by solving the calibrated IPW-EE: N Ψ ψα,γ(x, V, R) = 0, ψ α,γ(x, V, R) = N (α, γ) = 1 N ( ψ 1,α,γ ψ 2,γ Asymptotic distribution of ˆα : i=1 (X, V, R) = R π 0(V ) exp( γt Ṽ )ψ α (X ) R (V, R) = π 0(V ) exp( γt Ṽ )Ṽ Ṽ N(ˆα α 0 ) = Ψ c 1 11 G N ψ1,α 0,0+ Ψ c 1 11 Ψ c c 1 12 Ψ 22 G Nψ2,0+o p (1). ).

20 Estimation of Cluster-level Regression Model under Nonresponse within Clusters Nuanpan Nangsue Social Sciences, University of Southampton, UK June 10, 2013

21 Background and Motivation Aim: Look at new methods for analysis which incorporate information on non-response in the model The model of interest is a cluster level regression model relating the cluster mean Ȳ i of y ij Ȳ i = x i β + ɛ i (3) We suppose that underlying (3) we may write y ij = x i β + ɛ ij (4) To model the response outcome R ij, we introduce a variable u ij so that R ij = 1 if u ij > 0 and R ij = 0, otherwise. We assume that u ij = z i γ + δ ij (5) The inferential problem is how to use observed data on y ij, x i and z i to make inference about β.

22 Results and Conclusions To develop an estimator following the approach of Heckman (1976), we may write ( z E(y ij R ij = 1) = x i ) β + cλ i γ, (6) ( ) ( ) ( where c = σ ɛδ σ 1 z δ, λ i γ z σ δ = φ i γ z σ δ /Φ i γ σ δ ). A simpler version of this estimator is obtained by noting that for large m i, the response rate p i = r i m i may be expressed approximately as ( z ) p i E(R ij ) = Φ i γ = Φ(Ψ i ) (7) ( Now set ˆΨ i = Φ 1 z (p i ) and replace λ i ˆγ by λ( ˆΨ i ) in the Heckman two-step approach. An approximate Heckman maximum likelihood estimator is also obtained in order to estimate the regression coefficients β and c. σ δ ˆσ δ ) σ δ

23 Proportion estimators in dual frame surveys with auxiliary information Hemilio Coelho 1, Camila Silva 1 and Cristiano Ferraz 2 1. Department of Statistics, Federal University of Paraiba 2. Department of Statistics, Federal University of Pernambuco June 10, 2013

24 Background and Motivation In dual frame surveys, probability samples are independently drawn from two overlapping frames, denoted by A and B, with A B The simultaneous use of both frames, in a dual frame design generate three domains mutually exclusive: a = A B c, b = B A c and ab = A B. Based on results proposed by Hartley (1962), we proposed three estimators to estimate the populational proportion assisted by regression models, denoted by ˆP 1, ˆP 2 and ˆP 3, where the model used in the third estimator was based on logistic regression; The goal is to evaluate the performance of these estimators through Monte Carlo Experiments. All estimators were evaluated on their replicates mean, standard deviation, mean squared error and relative bias.

25 Results and Conclusions The results show that estimators ˆP 1 and ˆP 2 presented less relative bias than the estimator ˆP 3 ; When we look for the standard deviation for all sample sizes, it is possible to note that the estimator ˆP 3 presented better performance; The results show that the relative bias of estimator ˆP 3 not changed for all sample sizes considered, which suggests a further study to correct this bias. The correct specification of the model or the number of auxiliary information present in study can improve the performance of estimator ˆP 3.

26 Impacts of Nonsampling Errors on Estimates for the Conservation Effects Assessment Project Andreea Erciulescu and Emily Berg Department of Statistics, Iowa State University June 10, 2013

27 Background and Objectives Conservation Effects Assessment Project (CEAP) Environmental impacts of conservation practices Population: cultivated cropland Estimation domains: watersheds (8-digits nested in 4-digits) Boone/Raccoon River Watershed (Iowa) Sample of locations classified as cultivated cropland according to the National Resources Inventory Computer model converts collected data to analysis variables Soil erosion (RUSLE2), wind erosion, nitrogen run-off Nonsampling errors in CEAP Nonresponse - refusals Frame undercoverage - limited information on land use at sample design stage

28 Methods and Results Auxiliary information to evaluate bias due to nonsampling errors Slope, soil erodibility index from Soil Survey (known for full population) Soil erosion based on Universal Soil Loss Equation from NRI (known for NRI sample) Compare means using t-tests and locations using nonparametric tests Little evidence of nonresponse bias Evidence of bias due to frame undercoverage Especially in southern watersheds, where slopes are steeper and changes between non-cultivated and cultivated cropland are more common On-going work Calibration to adjust for bias due to frame undercoverage Small area estimation, 8-digit watersheds

29 Jackknife Empirical Likelihood for Regression Imputation Estimation Sixia Chen and Pingshou Zhong Westat and Michigan State University June 10, 2013

30 Item Nonresponse in Auxiliary Variables Used in Weighting Adjustments for Survey Sample Data Raphael Nishimura Michigan Program in Survey Methodology, Institute for Social Research, University of Michigan June 10, 2013

31 Background and Motivation Auxiliary variables in weighting on survey sampling: Population aggregates (control totals) known for auxiliary variables: t x = x i U Adjust design-weights to match population totals: w i x i = t x Improve estimates precision Calibration (Deville and Sarndal, 1992) Special case: Linear GREG (Generalized REGression) estimator Requirement: auxiliary variables observed for all sampled elements However, some important auxiliary variables may not be completely observed In practice: auxiliary variables imputed when missing or not used in weighting What are the impact of such procedure in the survey estimates? s

32 Results and Conclusions Missing values in auxiliary variables used in weighting adjustments: Never use complete cases only Larger variance (reduced sample size) Potential bias Calibration using auxiliary variable with imputed values, worthwhile when MAR (correctly specified imputation model) High correlation with survey variable Missing rate is not high Otherwise, using other auxiliary variables with lower missing rates and/or higher correlation with survey variables might be better alternative