Comparing latent inequality with ordinal health data

Transcription

1 Comparing latent inequality with ordinal health data David M. Kaplan University of Missouri Longhao Zhuo University of Missouri Midwest Econometrics Group October 2018 Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 0 / 32

2 Outline 1 Motivation 2 Results 3 Bayesian and frequentist inference 4 Empirical illustrations 5 Simulations 6 Conclusion Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 0 / 32

3 Outline 1 Motivation 2 Results 3 Bayesian and frequentist inference 4 Empirical illustrations 5 Simulations 6 Conclusion Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 0 / 32

4 Motivation: health inequality, SRHS From Deaton and Paxson (1998, pp ): Our interest in health inequality stems from a more general interest in the distribution of welfare. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 1 / 32

5 Motivation: health inequality, SRHS From Deaton and Paxson (1998, pp ): Our interest in health inequality stems from a more general interest in the distribution of welfare. SRHS: self-reported health status SRHS scale: excellent, very good, good, fair, poor SRHS benefits: 1 Useful over the complete adult life cycle 2 Strongly correlated with objective measures 3 Widely available (PSID, NHIS, etc.) 4 Synthesizes all dimensions of health Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 1 / 32

6 Inequality #1: within-group (dispersion) Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 2 / 32

7 Inequality #2: between-group (better/worse) Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 3 / 32

8 SRHS: latent model Why isn t SRHS distribution of ultimate interest? 1 No cardinal meaning (e.g., no mean/variance) 2 Ignores within-category variation (e.g., all good identical) Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 4 / 32

9 SRHS: latent model Why isn t SRHS distribution of ultimate interest? 1 No cardinal meaning (e.g., no mean/variance) 2 Ignores within-category variation (e.g., all good identical) Latent health H : cardinal, continuous, of interest, but censored SRHS H, fixed thresholds γ j Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 4 / 32

10 Latent model H=1 H=2 H=3 H=4 H=5 γ 1 γ 2 γ 3 γ 4 Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 5 / 32

11 Literature: SRHS inequality methodology Goal: compare latent health distributions using ordinal data Literature: parametric/mle or discrete latent distribution Literature: statistical inference rare Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 6 / 32

12 Literature: SRHS inequality methodology Goal: compare latent health distributions using ordinal data Literature: parametric/mle or discrete latent distribution Literature: statistical inference rare Allison and Foster (2004): median-preserving spread, called the breakthrough in analyzing inequality with [SRHS] data by Madden (2014, p. 206) Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 6 / 32

13 Literature: SRHS inequality methodology Goal: compare latent health distributions using ordinal data Literature: parametric/mle or discrete latent distribution Literature: statistical inference rare Allison and Foster (2004): median-preserving spread, called the breakthrough in analyzing inequality with [SRHS] data by Madden (2014, p. 206) SRHS-based inequality indexes: Good: complete ordering of distributions Bad: many possible indexes/weights Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 6 / 32

14 Our contribution Identification: characterize ordinal conditions informative about latent inequality Semi/nonparametric restrictions (not parametric) Continuous latent distribution (not discrete) Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 7 / 32

15 Our contribution Identification: characterize ordinal conditions informative about latent inequality Semi/nonparametric restrictions (not parametric) Continuous latent distribution (not discrete) Inference: frequentist and Bayesian (example code) Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 7 / 32

16 Outline 1 Motivation 2 Results 3 Bayesian and frequentist inference 4 Empirical illustrations 5 Simulations 6 Conclusion Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 7 / 32

17 Partial Identification Given H distributions, what can we learn about H distributions? Latent CDFs partially identified, given γ j Related: Stoye (2010) Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 8 / 32

18 Partial Identification γ 1 γ 2 γ 3 γ 4 Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 9 / 32

19 Thresholds: assumptions Do all populations have same γ j thresholds? Evidence of yes or constant shift : Lindeboom and van Doorslaer (2004), Hernández-Quevedo, Jones, and Rice (2005). Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 10 / 32

20 SD1: Proposition 2 Between-group inequality Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 11 / 32

21 SD1: Proposition 2 Between-group inequality Healthier if latent first-order stochastic dominance (SD1). X SD 1 Y F X ( ) F Y ( ) Ordinal SD1 = latent SD1? Latent SD1 = ordinal SD1? Proposition 2(i vi) in paper. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 11 / 32

22 Latent model (again) H=1 H=2 H=3 H=4 H=5 γ 1 γ 2 γ 3 γ 4 Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 12 / 32

23 SD1: Proposition 2 Same γ j Prop 2(i): latent SD1 = ordinal SD1, but not =. Reject ordinal SD1 = reject latent SD1. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 13 / 32

24 SD1: Proposition 2 Same γ j Prop 2(i): latent SD1 = ordinal SD1, but not =. Reject ordinal SD1 = reject latent SD1. Prop 2(v,vi): ordinal SD1 = latent restricted SD1 (Atkinson, 1987). Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 13 / 32

25 SD1: Prop 2(i) γ 1 γ 2 γ 3 γ 4 Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 14 / 32

26 SD1: Prop 2(v) Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 15 / 32

27 SD1: Prop 2(vi) Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 16 / 32

28 Dispersion: Proposition 3 (CDF crossing) Within-group inequality Pure location shift has zero effect on dispersion. Similarly, γ j can all shift by a constant (unlike for SD1). Prop 3(i): can learn about latent interquantile range (IQR) differences. Prop 3(ii): location scale = extrapolate to all IQR differences. Prop 3(iii): even stronger assumptions = latent SD2. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 17 / 32

29 Dispersion: Prop 3(i,ii) Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 18 / 32

30 Dispersion: Proposition 4 (fanning out) Can we ever infer dispersion changes without a CDF crossing? Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 19 / 32

31 Dispersion: Proposition 4 (fanning out) Can we ever infer dispersion changes without a CDF crossing? Yes, if symmetric, unimodal latent distributions Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 19 / 32

32 Dispersion: Proposition 4 (fanning out) Can we ever infer dispersion changes without a CDF crossing? Yes, if symmetric, unimodal latent distributions Location scale = extrapolation Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 19 / 32

33 Dispersion: Prop 4(i) 0.5 Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 20 / 32

34 Dispersion: Prop 4(i) τ 2 max IQR τ 1 min IQR max < min Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 21 / 32

35 Outline 1 Motivation 2 Results 3 Bayesian and frequentist inference 4 Empirical illustrations 5 Simulations 6 Conclusion Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 21 / 32

36 Ordinal SD1 inference Let θ j F 2 (j) F 1 (j) = E[1{H 2 j} 1{H 1 j}] Ordinal SD1: H 2 SD 1 H 1 θ j 0, j = 1, 2, 3, 4 Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 22 / 32

37 Ordinal SD1 inference Let θ j F 2 (j) F 1 (j) = E[1{H 2 j} 1{H 1 j}] Ordinal SD1: H 2 SD 1 H 1 θ j 0, j = 1, 2, 3, 4 Frequentist: recent moment inequality tests from Andrews and Barwick (2012), Romano, Shaikh, and Wolf (2014), McCloskey (2015), et al. Bayesian: nonparametric posterior for category probabilities = posterior probabilities for all relationships Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 22 / 32

38 Other relationships Unions and/or intersections of inequalities Bayes: just compute posteriors Frequentist: intersection union test (sometimes) Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 23 / 32

39 Outline 1 Motivation 2 Results 3 Bayesian and frequentist inference 4 Empirical illustrations 5 Simulations 6 Conclusion Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 23 / 32

40 Treatment effects Jones, Molitor, and Reif (2018): randomized workplace wellness program; measure SRHS Can compare treated/untreated latent health distributions: healthier (SD1)? inequality? Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 24 / 32

41 Treatment effects Jones, Molitor, and Reif (2018): randomized workplace wellness program; measure SRHS Can compare treated/untreated latent health distributions: healthier (SD1)? inequality? Can examine selection effects, too. Stay tuned... Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 24 / 32

42 Comparisons of U.S. states Goal: compare latent health across states Data: 2011 PSID, observations per state Posteriors: Bayesian bootstrap of Dong, Elliott, and Raghunathan (2014) (stratification, clustering, weights) Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 25 / 32

43 PSID 2011 posterior probabilities (%) SD 1 SC fans out X Y AZ MO NY UT IL NY MN NY IA MO Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 26 / 32

44 PSID 2011 posterior probabilities (%) X SD 1 Y ; Y is: X SC Y ; Y is: X MO KS NE IA IL MO KS NE IA IL MO * KS 34* 20* NE IA * 6 IL 40* 4 18* 43* Asterisk (*): satisfied in-sample Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 27 / 32

45 Outline 1 Motivation 2 Results 3 Bayesian and frequentist inference 4 Empirical illustrations 5 Simulations 6 Conclusion Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 27 / 32

46 Purpose and setup Goal: compare type I error rates of different frequentist and Bayesian tests. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 28 / 32

47 Purpose and setup Goal: compare type I error rates of different frequentist and Bayesian tests. DGP 1: P(X = j) = P(Y = j) = 1/5, j = 1,..., 5; all SD1 inequalities binding. DGP 2: change to P(X = j) = 1/10 for j = 1, 2, 3 and P(X = 4) = 1/2; only one binding inequality. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 28 / 32

48 Methods KS: Kolmogorov Smirnov. RMS: refined moment selection of Andrews and Barwick (2012). Bayes: Dirichlet multinomial, uninformative prior on parameters. Reject if posterior below α. adj: adjust prior to P(H 0 ) = 1/2. (Goutis, Casella, and Wells, 1996) Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 29 / 32

49 Results: α = 0.1 H 0 : X SD 1 Y H 0 : X SC Y DGP n KS RMS Bayes adj RMS Bayes adj Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 30 / 32

50 Results: α = 0.1 H 0 : X SD 1 Y H 0 : X SC Y DGP n KS RMS Bayes adj RMS Bayes adj RMS better than KS Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 30 / 32

51 Results: α = 0.1 H 0 : X SD 1 Y H 0 : X SC Y DGP n KS RMS Bayes adj RMS Bayes adj RMS better than KS Even with Bayes (adj), cannot treat posterior as p-value, or vice-versa Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 30 / 32

52 Outline 1 Motivation 2 Results 3 Bayesian and frequentist inference 4 Empirical illustrations 5 Simulations 6 Conclusion Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 30 / 32

53 Conclusion Summary: link latent and ordinal inequality, without parametric or discrete assumptions Summary: relationships defined by inequalities; frequentist and Bayesian inference (code on website) Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 31 / 32

54 Conclusion Summary: link latent and ordinal inequality, without parametric or discrete assumptions Summary: relationships defined by inequalities; frequentist and Bayesian inference (code on website) Future work: other shape restrictions? multivariate? regression? health, happiness,...? Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 31 / 32

55 Conclusion Summary: link latent and ordinal inequality, without parametric or discrete assumptions Summary: relationships defined by inequalities; frequentist and Bayesian inference (code on website) Future work: other shape restrictions? multivariate? regression? health, happiness,...? Thank you! (And further questions or comments are welcome) Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 31 / 32

56 References I Allison, R. A., Foster, J. E., Measuring health inequality using qualitative data. Journal of Health Economics 23 (3), URL Andrews, D. W. K., Barwick, P. J., Inference for parameters defined by moment inequalities: A recommended moment selection procedure. Econometrica 80 (6), URL Atkinson, A. B., On the measurement of poverty. Econometrica 55 (4), URL Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 32 / 32

57 References II Deaton, A., Paxson, C., Aging and inequality in income and health. American Economic Review (Papers and Proceedings) 88 (2), URL Dong, Q., Elliott, M. R., Raghunathan, T. E., A nonparametric method to generate synthetic populations to adjust for complex sampling design features. Survey Methodology 40 (1), 29. URL X Goutis, C., Casella, G., Wells, M. T., Assessing evidence in multiple hypotheses. Journal of the American Statistical Association 91 (435), URL Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 32 / 32

58 References III Hernández-Quevedo, C., Jones, A. M., Rice, N., Reporting bias and heterogeneity in self-assessed health. evidence from the British Household Panel Survey. HEDG Working Paper 05/04, Health, Econometrics and Data Group, The University of York. URL Jones, D., Molitor, D., Reif, J., What do workplace wellness programs do? evidence from the Illinois Workplace Wellness Study. NBER Working Paper 24229, National Bureau of Economic Research. URL Lindeboom, M., van Doorslaer, E., Cut-point shift and index shift in self-reported health. Journal of Health Economics 23 (6), URL Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 32 / 32

59 References IV Madden, D., Dominance and the measurement of inequality. In: Culyer, A. J. (Ed.), Encyclopedia of Health Economics. Vol. 1. Elsevier, pp URL McCloskey, A., On the computation of size-correct power-directed tests with null hypotheses characterized by inequalities, working paper, available at Adam_McCloskey/Research.html. Romano, J. P., Shaikh, A. M., Wolf, M., A practical two-step method for testing moment inequalities. Econometrica 82 (5), URL Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 32 / 32

60 References V Stoye, J., Partial identification of spread parameters. Quantitative Economics 1 (2), URL Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 32 / 32