# Bayesian inference for population prediction of individuals without health insurance in Florida

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1 Bayesian inference for population prediction of individuals without health insurance in Florida Neung Soo Ha 1 1 NISS 1 / 24

2 Outline Motivation Description of the Behavioral Risk Factor Surveillance System, BRFSS Description of posterior predictive distribution under informative/non-informative sampling Selection bias Inference for proportions of individuals without the health insurance Variation of maps 2 / 24

3 Motivation: Interest: 1. Inference for the county-level proportions of persons without health insurance in FL, using the 2010 Behavioral Risk Factor Surveillance System, BRFSS. 2. Display the variation of maps 3 / 24

4 2010 BRFSS: Survey description The largest on-going telephone based survey; administered by the Centers for Disease Control and Prevention. Collect data on individual risk behaviors and preventive health practices for the adult population(18 years of age and older); collects state-specific data ex: alcohol/cigarette consumption, general health status, health insurance, Uses a disproportionate stratified sample (DSS) design. In FL, 63 out of 67 counties were sampled. No cell phones for / 24

5 Examples of observed data Observed proportion of non insured persons Observed population proportion of non insured persons: Hisp Figure: Observed: All races Figure: Observed: Hispanics white=having insurance, blue=not having insurance 5 / 24

6 Complication 1. Small or non-existent number of observations for some sub-population groups in certain geographical areas. 2. Possible presence of selection bias in the sample 6 / 24

7 Model specification for the BRFSS data P(Y ikl = 1 θ ik ) = θ ik logit(θ ik ) = X k β + ν i ν i N(0, σ 2 ν) Y ikl = 1 : not having insurance, Y ikl = 0 : having insurance i = 1,..., M: county k = 1,..., K: population class (9 age groups 3 races 2 genders ) l = 1,..., N ik : units in county i and group k N ik = population size in ith county and kth group X k = vector of indicators for group k. p(β) = const on (, ) Gelman et al.(2004) p(σ 2 ν) = const on (0, ) 7 / 24

8 Model Evaluation Model fit Bayes residual plots QQ plots Bayesian tests: Partial posterior predictive p-values, Bayarri and Berger(2002) Predictions Cross-validation to simulate finite population inference Use 100(1-p)% of observed data to make predictions for the 100*p% that were held-out 8 / 24

9 Posterior predictive distribution with non-informative sampling Use posterior predictive distribution to make inference for the remaining non-sampled units Under the non-informative sampling, i.e. independence between the probability of selecting a person and the response: Y s = sampled units, Y ns = non-sampled units Y ns Y s θ, X f (Y ns Y s, X) = g(y ns Y s, X, θ)p(θ Y s, X)dθ = g(y ns θ, X)p(θ Y s, X)dθ p(θ Y s, X) L(θ Y s, X)p(θ β, σν)p(β, 2 σν): 2 posterior distribution θ = a vector of model parameters (β, σ 2 ν ) = a vector of hyperparameters 9 / 24

10 Under informative sampling Selection probabilities are related to the outcome variables. The observed outcomes may not be representative of the population outcomes. May cause bias for inferences about the finite population parameters. 10 / 24

11 Selection bias Let I = (I 1,..., I N ), I i = 1 if i s, I i = 0 if i / s, s = sample The posterior predictive distribution with I f (Y ns Y s, X, θ, I) g(y ns θ, X)p(I Y, X, θ) (1) Note: p(i X, Y, θ) p(i X, θ) i.e. selection bias. Selection information only comes from the sampled units. 11 / 24

12 Procedure: Ha and Sedransk(2015) 1. Pfeffermann and Sverchkov (2009): p(i Y, X, θ) = p(i Y, X) = k f 1(I k Y, X k ) 2. Using the Poisson sampling approximation, Chambers et. al.(1998) p(i Y, X) = M K { P(I ikl = 1 Y, X k )}{ (1 P(I ikl = 1 Y, X k ))} l s ik l/ s ik i=1 k=1 3. P(I ikl = 1 Y, X k ) = E U (π ikl Y, X k ) = {E s (w ikl Y, X k )} 1, Pfeffermann and Sverchkov (2009), πikl = probability of unit selected w ikl = 1/π ikl 12 / 24

13 Posterior predictive distribution with inclusion probability g(y ns Y s, X, θ, I) g(y ns θ, X) M i=1 K k=1 (1 a k) M ik1(1 b k ) M ik0, a k = 1 w k,y0, b k = 1 w k,y1 w k,y0 = average of inverse probability of selected units for class k with response Y ikl = 0 wk,y1 = average of inverse probability of selected units for class k with response Y ikl = 1 Mik1 : # non-sampled units with Y ikl = 1 M ik0 : # non-sampled units with Y ikl = 0 M ik1 + M ik0 = N ik n ik Use rejection sampling algorithm to obtain g(y ns ), Robert, C. and Casella, G. (2004) In our data set, 0.96 (1 a k )/(1 b k ) 1.01 with 43 of the 54 K groups. No substantial selection bias. 13 / 24

14 Comparison between prediction to observation County comparison Observed proportion of non insured persons Predicted population proportion of non insured persons Figure: Observed: All races Figure: Predicted: All races 14 / 24

15 Comparison between prediction to observation Observed population proportion of non insured persons: Hisp Predicted population proportion of non insured persons: Hisp Figure: Observed: Hispanics Figure: Predicted: Hispanics 15 / 24

16 Comparison between races Predicted population proportion of non insured persons: White Predicted population proportion of non insured persons: Hisp Figure: Predicted: Whites Figure: Predicted: Hispanics 16 / 24

17 Variation of maps Objective: Assess the variation of the entire map as a unit rather than expressing the variation separately for each geographic unit. Produce a map, based on each MCMC replicate. Analyze variations from a set of maps Illustrate variation between the counties from ν i, the county-level random effect. 17 / 24

18 random effect: mean Mean choropleth map of random effect quintiles [ 0.616, 0.236] ( 0.236, ] ( ,0.074] (0.074,0.246] (0.246,0.684] Posterior means of county-level random effects, ν i, are partitioned into quintiles Each quintile is color coded Figure: Mean map of ˆν i 18 / 24

19 Besag Method, Besag et al (1995) Provide 100(1 a)% regions for each of ν i : ν i (L) < ν i < ν i (U) Construct lower choropleth map of ν i (L) and ν i (U) 1. Denote the posterior MCMC sample by {ν (t) i : i = 1,..., M, t = 1,..., T } 2. Order {ν (t) i x [t] i and ranks r (t) i : t = 1,..., T } separately for each i to obtain order statistics 3. For fixed k {1,..., T }, let t be the smallest integer such that x [T +1 t ] i x (t) i x [t ] i, for all i, for at least k values of t 4. t = kth order statistic from the set a (t) = max{max i r (t) i, T + 1 min i r (t) i }, t = 1,..., T, i.e. t = a [k] 5. Then, {[x [T +1 t ] i, x [t ] i ] : i = 1,..., M} are a set of simultaneous credible regions containing at least 100k/T % of the distribution 19 / 24

20 Lower and Upper maps random effect: Besag L 90 random effect: mean quintiles random effect: Besag U 90 [ 1.12, 0.76] ( 0.76, 0.562] ( 0.562, 0.359] ( 0.359, 0.19] ( 0.19,0.264] [ 0.616, 0.236] ( 0.236, ] ( ,0.074] (0.074,0.246] (0.246,0.684] [ 0.172,0.203] (0.203,0.359] (0.359,0.519] (0.519,0.697] (0.697,1.07] Figure: 90 % Lower, Mean, 90% Upper 20 / 24

21 Variation in Choropleth map Goal: Illustrate the uncertainty of the choropleth maps. Use T = 1000 MCMC replications of ν i Each replication can be used to produce a choropleth map. Each quintile is color coded. For each replication, the value of ith county is color-coded according to its quintile. 21 / 24

22 Heat map Q1 Q2 Q3 Q4 Q5 Rows: replicates, columns: counties Variation in maps is evaluated by color changes 22 / 24

23 Summary Present a method for posterior predictive inference that includes selection information for sampled units Illustrate variation of maps 23 / 24

24 Reference 1. Chambers, R. L., Dorfman, A. and Wang, W. (1998). Limited Information Likelihood Analysis of Survey Data. Journal of the Royal Statistical Society, 60, Bayesian benchmarking with application to small area estimation Test Gelman, A., Carlin, J. Stern, H. and Rubin, D. (2004), Bayesian Data Analysis. Chapman and Hall/ CRC, New York, NY. 3. Pfeffermann, D. and Sverchkov, M. (2009) Inference under Informative Sampling Sample Surveys: Inference and Analysis, 29B, Robert, C. and Casella, G. (2004), Monte Carlo Statistical Methods, Springer, New York, NY. 24 / 24

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