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 / 28

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 / 28

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 / 28

4 Motivation: self-reported health status (SRHS) From Deaton and Paxson (1998a, 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 / 28

5 Motivation: self-reported health status (SRHS) From Deaton and Paxson (1998a, pp ): Our interest in health inequality stems from a more general interest in the distribution of welfare. SRHS scale: excellent, very good, good, fair, poor. SRHS is useful over the complete adult life cycle and strongly correlated with more objective measures (mortality, activities of daily living, etc.). Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 1 / 28

6 Motivation: self-reported health status (SRHS) From Deaton and Paxson (1998a, pp ): Our interest in health inequality stems from a more general interest in the distribution of welfare. SRHS scale: excellent, very good, good, fair, poor. SRHS is useful over the complete adult life cycle and strongly correlated with more objective measures (mortality, activities of daily living, etc.). Interested in whether inequality in health status... increases with age as well as across socioeconomic groups. 2 types of inequality: within-distribution (large dispersion), between distributions (more/less healthy). Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 1 / 28

7 SRHS: latent model Why isn t SRHS distribution of ultimate interest? 2 reasons. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 2 / 28

8 SRHS: latent model Why isn t SRHS distribution of ultimate interest? 2 reasons. 1) no cardinal meaning; no mean, variance, etc., although Deaton and Paxson (1998a,b) try. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 2 / 28

9 SRHS: latent model Why isn t SRHS distribution of ultimate interest? 2 reasons. 1) no cardinal meaning; no mean, variance, etc., although Deaton and Paxson (1998a,b) try. 2) ignores within-category variation; e.g., treats all good as identical. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 2 / 28

10 SRHS: latent model Why isn t SRHS distribution of ultimate interest? 2 reasons. 1) no cardinal meaning; no mean, variance, etc., although Deaton and Paxson (1998a,b) try. 2) ignores within-category variation; e.g., treats all good as identical. Instead: imagine cardinal, continuous, latent (unobserved) health. Latent health is of interest, but only observe censored SRHS. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 2 / 28

11 SRHS: latent model Why isn t SRHS distribution of ultimate interest? 2 reasons. 1) no cardinal meaning; no mean, variance, etc., although Deaton and Paxson (1998a,b) try. 2) ignores within-category variation; e.g., treats all good as identical. Instead: imagine cardinal, continuous, latent (unobserved) health. Latent health is of interest, but only observe censored SRHS. Assume latent health H, SRHS H, fixed thresholds γ j : H = 1 H = 2 H = 3 H = 4 H = 5 γ 1 γ 2 γ 3 γ 4 H Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 2 / 28

12 Literature: SRHS inequality methodology Goal: compare latent health distributions using ordinal data. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 3 / 28

13 Literature: SRHS inequality methodology Goal: compare latent health distributions using ordinal data. Literature: either parametric/mle or discrete latent distribution. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 3 / 28

14 Literature: SRHS inequality methodology Goal: compare latent health distributions using ordinal data. Literature: either parametric/mle or discrete latent distribution. E.g., Allison and Foster (2004): median-preserving spread (MPS), 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 3 / 28

15 Literature: SRHS inequality methodology Goal: compare latent health distributions using ordinal data. Literature: either parametric/mle or discrete latent distribution. E.g., Allison and Foster (2004): median-preserving spread (MPS), called the breakthrough in analyzing inequality with [SRHS] data by Madden (2014, p. 206). SRHS-based inequality indexes (scalar summary of ordinal probabilities): Abul Naga and Yalcin (2008), Reardon (2009), Silber and Yalonetzky (2011), Lazar and Silber (2013), Lv, Wang, and Xu (2015), and Yalonetzky (2016). Provides complete ordering of distributions, but many possible indexes and weighting parameters/functions, implicit assumptions. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 3 / 28

16 Literature: SRHS inequality methodology Goal: compare latent health distributions using ordinal data. Literature: either parametric/mle or discrete latent distribution. E.g., Allison and Foster (2004): median-preserving spread (MPS), called the breakthrough in analyzing inequality with [SRHS] data by Madden (2014, p. 206). SRHS-based inequality indexes (scalar summary of ordinal probabilities): Abul Naga and Yalcin (2008), Reardon (2009), Silber and Yalonetzky (2011), Lazar and Silber (2013), Lv, Wang, and Xu (2015), and Yalonetzky (2016). Provides complete ordering of distributions, but many possible indexes and weighting parameters/functions, implicit assumptions. Only Lazar and Silber (2013) mention statistical inference. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 3 / 28

17 Our contribution Semi/nonparametric restrictions (not parametric). Happiness data: Bond and Lang (2018) Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 4 / 28

18 Our contribution Semi/nonparametric restrictions (not parametric). Happiness data: Bond and Lang (2018) Continuous latent distribution (not discrete). Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 4 / 28

19 Our contribution Semi/nonparametric restrictions (not parametric). Happiness data: Bond and Lang (2018) Continuous latent distribution (not discrete). Results: characterize various ordinal conditions that are informative about latent inequality. Unions and intersections of parameter inequalities. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 4 / 28

20 Our contribution Semi/nonparametric restrictions (not parametric). Happiness data: Bond and Lang (2018) Continuous latent distribution (not discrete). Results: characterize various ordinal conditions that are informative about latent inequality. Unions and intersections of parameter inequalities. Statistical inference: frequentist and Bayesian. (Code on website.) Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 4 / 28

21 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 4 / 28

22 (Partial) Identification Given (observable) population ordinal distribution, what can be learned about (unobservable) population latent distribution? Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 5 / 28

23 (Partial) Identification Given (observable) population ordinal distribution, what can be learned about (unobservable) population latent distribution? One view: latent CDFs are partially identified given thresholds γ j. Related: Stoye (2010). Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 5 / 28

24 (Partial) Identification Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 6 / 28

25 Thresholds: assumptions Do all populations have the same γ j thresholds? Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 7 / 28

26 Thresholds: assumptions Do all populations have the same γ j thresholds? Evidence of yes or constant shift : (Hernández-Quevedo, Jones, and Rice, 2005; Lindeboom and van Doorslaer, 2004). Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 7 / 28

27 Inequality: SD1 One type of inequality : one group healthier than another. Healthier : latent first-order stochastic dominance (SD1). X SD 1 Y F X ( ) F Y ( ) Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 8 / 28

28 Inequality: SD1 One type of inequality : one group healthier than another. Healthier : latent first-order stochastic dominance (SD1). X SD 1 Y F X ( ) F Y ( ) Ordinal SD1 = latent SD1? ( verifiable ) Latent SD1 = ordinal SD1? (testable implication; refutable ) Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 8 / 28

29 Inequality: SD1 One type of inequality : one group healthier than another. Healthier : latent first-order stochastic dominance (SD1). X SD 1 Y F X ( ) F Y ( ) Ordinal SD1 = latent SD1? ( verifiable ) Latent SD1 = ordinal SD1? (testable implication; refutable ) Proposition 2(i vi) in paper. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 8 / 28

30 SD1: Proposition 2 H = 1 H = 2 H = 3 H = 4 H = 5 γ 1 γ 2 γ 3 γ 4 H Pictures to follow. Assume constant γ j. Helpful: F (j) = F (γ j ). Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 9 / 28

31 SD1: Proposition 2 H = 1 H = 2 H = 3 H = 4 H = 5 γ 1 γ 2 γ 3 γ 4 H Pictures to follow. Assume constant γ j. Helpful: F (j) = F (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 9 / 28

32 SD1: Proposition 2 H = 1 H = 2 H = 3 H = 4 H = 5 γ 1 γ 2 γ 3 γ 4 H Pictures to follow. Assume constant γ j. Helpful: F (j) = F (γ j ). Prop 2(i): latent SD1 = ordinal SD1, but not =. Reject ordinal SD1 = reject latent SD1. Lemma 1: latent location scale model = (at most) single latent CDF crossing. Prop 2(vi): ordinal SD1 = latent restricted SD1 (Atkinson, 1987) if latent location scale model. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 9 / 28

33 SD1: Proposition 2 H = 1 H = 2 H = 3 H = 4 H = 5 γ 1 γ 2 γ 3 γ 4 H Pictures to follow. Assume constant γ j. Helpful: F (j) = F (γ j ). Prop 2(i): latent SD1 = ordinal SD1, but not =. Reject ordinal SD1 = reject latent SD1. Lemma 1: latent location scale model = (at most) single latent CDF crossing. Prop 2(vi): ordinal SD1 = latent restricted SD1 (Atkinson, 1987) if latent location scale model. Prop 2(v): ordinal SD1 = latent restricted SD1 if ordinal super SD1. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 9 / 28

34 SD1: Prop 2(i,vi) Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 10 / 28

35 SD1: Prop 2(v) Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 11 / 28

36 Dispersion: Proposition 3 (CDF crossing) Inequality within a population. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 12 / 28

37 Dispersion: Proposition 3 (CDF crossing) Inequality within a population. Pure location shift has zero effect on dispersion. Similarly, γ j can all shift by a constant (unlike for SD1). Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 12 / 28

38 Dispersion: Proposition 3 (CDF crossing) Inequality within a population. 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. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 12 / 28

39 Dispersion: Proposition 3 (CDF crossing) Inequality within a population. 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. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 12 / 28

40 Dispersion: Proposition 3 (CDF crossing) Inequality within a population. 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 12 / 28

41 Dispersion: Prop 3(i,ii) Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 13 / 28

42 Dispersion: Prop 3(i) If first ordinal CDF crosses the second from below, then know some IQRs larger in second latent distribution. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 14 / 28

43 Dispersion: Prop 3(i) If first ordinal CDF crosses the second from below, then know some IQRs larger in second latent distribution. Ordinal crossing = not ordinal SD1. But, possibly latent SD1 if γ j shift. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 14 / 28

44 Dispersion: Prop 3(i) If first ordinal CDF crosses the second from below, then know some IQRs larger in second latent distribution. Ordinal crossing = not ordinal SD1. But, possibly latent SD1 if γ j shift. Median-preserving spread of Allison and Foster (2004) is special case of single CDF crossing. (But, they interpret differently, since they assume discrete latent + same γ j.) Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 14 / 28

45 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 15 / 28

46 Dispersion: Proposition 4 (fanning out) Can we ever infer dispersion changes without a CDF crossing? Yes, with stronger assumptions: symmetric, unimodal latent distributions. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 15 / 28

47 Dispersion: Proposition 4 (fanning out) Can we ever infer dispersion changes without a CDF crossing? Yes, with stronger assumptions: symmetric, unimodal latent distributions. Again, location scale assumption allows extrapolation. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 15 / 28

48 Dispersion: Prop 4(i) 0.5 Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 16 / 28

49 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 17 / 28

50 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 17 / 28

51 Ordinal SD1 inference Let θ j F 2 (j) F 1 (j) = E[1{H 2 j} 1{H 1 j}]. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 18 / 28

52 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 18 / 28

53 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. Can use recent moment inequality tests: Andrews and Barwick (2012), Romano, Shaikh, and Wolf (2014), McCloskey (2015), et al. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 18 / 28

54 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. Can use recent moment inequality tests: Andrews and Barwick (2012), Romano, Shaikh, and Wolf (2014), McCloskey (2015), et al. Bayesian: nonparametric posterior for category probabilities = posterior probabilities for all relationships. (E.g., iid: Dirichlet multinomial model.) Coherent : e.g., posterior probabilities of SD1 and non-sd1 sum to 100%. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 18 / 28

55 Other relationships Other relationships are 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 19 / 28

56 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 19 / 28

57 Treatment effects Jones, Molitor, and Reif (2018): large, randomized workplace wellness program. Measure SRHS. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 20 / 28

58 Treatment effects Jones, Molitor, and Reif (2018): large, randomized workplace wellness program. Measure SRHS. Can compare treated/untreated latent health distributions using SRHS: does treatment improve health (SD1)? increase inequality? etc. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 20 / 28

59 Treatment effects Jones, Molitor, and Reif (2018): large, randomized workplace wellness program. Measure SRHS. Can compare treated/untreated latent health distributions using SRHS: does treatment improve health (SD1)? increase inequality? etc. Can examine selection effects, too. Stay tuned... Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 20 / 28

60 Comparisons of U.S. states Goal: compare latent health across different states in U.S. Data: 2011 PSID. Around observations per state. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 21 / 28

61 Comparisons of U.S. states Goal: compare latent health across different states in U.S. Data: 2011 PSID. Around observations per state. Posteriors: Bayesian bootstrap of Dong, Elliott, and Raghunathan (2014) to incorporate stratification, clustering, weights. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 21 / 28

62 PSID 2011 posterior probabilities (%) SD 1 SC fans out X Y AZ MO IL NY IA MO LA NY MN NY NY UT Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 22 / 28

63 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 23 / 28

64 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 / 28

65 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 24 / 28

66 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 24 / 28

67 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 25 / 28

68 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 26 / 28

69 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 26 / 28

70 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 26 / 28

71 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 26 / 28

72 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 27 / 28

73 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: more results with shape restrictions? multivariate ordinal, like e.g. Yalonetzky (2013)? Bayesian inference with P(H 0 ) = 1/2? extend to settings like regression? more applications: health, happiness,...? Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 27 / 28

74 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: more results with shape restrictions? multivariate ordinal, like e.g. Yalonetzky (2013)? Bayesian inference with P(H 0 ) = 1/2? extend to settings like regression? more applications: health, happiness,...? Thank you! (And further questions or comments are welcome) Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 27 / 28

75 References I Abul Naga, R. H., Yalcin, T., Inequality measurement for ordered response health data. Journal of Health Economics 27 (6), URL 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 Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 28 / 28

76 References II Atkinson, A. B., On the measurement of poverty. Econometrica 55 (4), URL Bond, T. N., Lang, K., The sad truth about happiness scales. Journal of Political Economy, forthcoming. Deaton, A., Paxson, C., 1998a. Aging and inequality in income and health. American Economic Review (Papers and Proceedings) 88 (2), URL Deaton, A., Paxson, C., 1998b. Health, income, and inequality over the life cycle. In: Frontiers in the Economics of Aging. University of Chicago Press, pp URL Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 28 / 28

77 References III 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 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 Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 28 / 28

78 References IV 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 Lazar, A., Silber, J., On the cardinal measurement of health inequality when only ordinal information is available on individual health status. Health Economics 22 (1), 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 28 / 28

79 References V Lv, G., Wang, Y., Xu, Y., On a new class of measures for health inequality based on ordinal data. Journal of Economic Inequality 13 (3), URL 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. Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 28 / 28

80 References VI Reardon, S. F., Measures of ordinal segregation. In: Flückiger, Y., Reardon, S. F., Silber, J. (Eds.), Occupational and Residential Segregation. Vol. 17 of Research on Economic Inequality. Emerald Group Publishing Limited, pp URL 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 28 / 28

81 References VII Silber, J., Yalonetzky, G., Measuring inequality in life chances with ordinal variables. In: Rodríguez, J. G. (Ed.), Inequality of Opportunity: Theory and Measurement. Vol. 19 of Research on Economic Inequality. Emerald Group Publishing Limited, Ch. 4, pp URL Stoye, J., Partial identification of spread parameters. Quantitative Economics 1 (2), URL Yalonetzky, G., Stochastic dominance with ordinal variables: Conditions and a test. Econometric Reviews 32 (1), URL Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 28 / 28

82 References VIII Yalonetzky, G., Robust ordinal inequality comparisons with Kolm-independent measures. Working Paper 401, ECINEQ, Society for the Study of Economic Inequality. URL https: //ideas.repec.org/p/inq/inqwps/ecineq html Dave Kaplan (Missouri) and Longhao Zhuo Comparing inequality with ordinal data 28 / 28