Decomposing racial differences in outcomes from nonlinear models - PowerPoint PPT Presentation

Decomposing racial differences in outcomes from nonlinear models Paul L. Hebert, PhD Research Associate Professor University of Washington School of Public Health Department of Health Services AcademyHealth Annual Research Meeting 2017 Supported by the Eunice Kennedy Shriver National Institute of Child Health & Human Development of the National Institutes of Health under Award Number R01HD078565 and by the National Institute on Minority Health and Health Disparities of the National Institutes of Health under Award Number R01MD007651. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.

Motivation • From 2010-14 there were 2775 very Neonatal morbidity and mortality among very preterm (24-32 gestational weeks) preterm neonates of White and Black mothers neonates born to black mothers and in New York City, 2010-2014 1418 to white mothers in NYC. • The rate of morbidity and mortality White Black mothers mothers (“events”) was 9.7 percentage points higher for black mothers than for N 1418 2775 white mothers. Died<28 days or neonatal 319 893 • This implies there were 269 morbidity (“Events”), n (9.7%*2775=269) excess events for Event rate,% 22.5% 32.2% Black mothers. • Q: What are the most important Excess events among black 269 mothers modifiable factors that contribute to these 269 excess deaths/morbidity among black neonates?

There are many candidate causes of the excess events White (n=1418) Black (n=2775) • Black mothers differ Baby’s risk substantial from white Birthweight, mean 1274 (485) 1159 (433) mothers in terms of: (SD) A. Baby risk characteristics APGAR<7 39%(550) 48% (1344) (e.g., weight, gender, Mothers Risk APGAR, gestational age, Age<20, %(n) 1.6% (23) 5.7% (158) etc..) Overweight/obese, 32% (454) 61%(1682) B. Mother’s health risk %(n) (e.g, age, diabetes, Mother’s SES obesity,…) <HS education, %(n) 7% 20% C. Mother’s SES (e.g., insurance, education,…) Medicaid Insurance 26% 68%

Typical, unsatisfying solution • Estimate a series of logisitic Odds ratio on black race from regressions of the event as a sequence of logistic regressions function of race, (P<0.001) (P<0.001) • Sequentially add sets of variables 1.64 (P=0.03) 1.55 to the model each time (P=0.171) 1.36 0.21 • See how the odds ratio on race 1.15 changes with each set of new variables. UNADJUSTED MOM SES MOM SES+ MOM MOM SES + MOM RISK RISK+ BABY RISK

Why this is unsatisfying • The order matters Odds ratio on black race from • The results are expressed in terms asequence of logisitic regressions of odds ratios. • We need a more useful method for (P<0.001) 1.64 0.29 decomposing the number of excess (P=0.001) events into various factors that (P=0.033) (p=0.171) 1.35 contribute to those events. E.g., 1.23 1.15 • How many excess black neonate deaths/morbidity would be avoided if the birthweight of neonates of minority and white mothers did not differ? • How many would be avoided if insurance status of minority and UNADJUSTED BABY RISK BABY RISK +MOM BABY RISK +MOM white mothers did not differ? RISK RISK + MOM SES

Today’s presentation • Problems with decomposing outcomes from nonlinear models • Solution proposed by Fairlie (2005) • Solution proposed through application of the Shapley Value • Summary

Decomposing racial differences in linear models is relatively straight-forward Probability of event P P’ Birth weight Black White mean mean

Decomposing racial differences in linear models is relatively straight-forward • The effect of birth weight on the event is the same for Medicaid and non-Medicaid recipients, and P| Mcaid • The effect of Medicaid is the same at every level of birth weight Probability of event P’| Mcaid Medicaid recipients Private insurance Birth weight Black White mean mean

Oaxaca-Blinder decomposition for linear models (1973) • You can decompose racial differences estimated from linear models using only the means of the explanatory variables and the coefficients from the model. 𝑍 𝑋 − ത 𝑍 𝐶 = 𝑌 𝑋 − ത 𝑌 𝐶 ∗ መ 𝛾 𝑋 + 𝛾 𝑋 − መ 𝛾 𝐶 ∗ ത መ ത ത 𝑌 𝐶

Nonlinearity of logit model complicates things • The effect of birth weight on the probability of the event depends on the level of birth weight, and on the level of every other }- Effect of Medicaid at black variable in the model. Probability of event • Effect of Medicaid depends on birth weight mean birth weight • Effects sizes are path dependent } – Effect of Medicaid at white mean birth weight Birth weight Black White mean mean

A consequence of this nonlinearity is Jensen’s inequality 𝐹 𝑔 𝑦 ≡ 𝑔 𝐹 𝑦 iff 𝑔 . is linear • For example, event rate for black mothers in our sample was 0.322 • But, if we estimate a logistic regression and apply the coefficients to the mean of the variables in the model, we get 𝑚𝑝𝑕𝑗𝑢 −1 መ 𝛾 1 𝐶𝑏𝑐𝑧 𝐶 + መ 𝛾 2 𝑇𝐹𝑇 𝐶 + መ 𝛾 3 𝑁𝑝𝑛 𝐶 + መ 𝛾 0 + መ 𝛾 4 = 0.276 Where 𝐶𝑏𝑐𝑧 𝐶 is the mean — i.e., expected value — of baby risk factors for black mothers, and መ 𝛾 4 is the coefficient on black race • On the other hand, the mean of the individual predictions for black mothers is identical to the black event rate 𝑜 𝑐 1 𝑚𝑝𝑕𝑗𝑢 −1 መ 𝛾 0 + መ 𝛾 1 𝐶𝑏𝑐𝑧 𝑗 + መ 𝛾 2 𝑇𝐹𝑇 𝑗 + መ 𝛾 3 𝑁𝑝𝑛 𝑗 + መ ෍ 𝛾 4 = .322 𝑜 𝑐 𝑗=1 • This is the inspiration for the decomposition method proposed by Fairlie (2005)

Today’s presentation • Problems with decomposing outcomes from nonlinear models • Solution proposed by Fairlie (2005) • Solution proposed through application of the Shapley Value • Summary

Fairlie RW (2005), “An Extension of Blinder -Oaxaca Decomposition Technique to Logit and Probit Models” J of Econ and Soc Measure, 30(4), 305-316 • Don’t use the means of explanatory variables for black and white mothers, swap explanatory variables between a white and black mothers at the individual level. • To estimate the contribution of groups of variables A, B, and C on an event Y for a sample that contains N b black and N w white mothers: 1. Estimate a logistic regression of the probability of the event as a function of A, B, and C for each race group. Calculate the predicted probability the event for every mother. Call this P 0 . 2. Draw at random N b white mothers. Match black mothers 1:1 to sampled white mothers based on predicted probability of the event 3. Replace each black mother’s data for variables in A with data from her matched white counterpart, and re - calculate the predicted probability of the event for the black mothers. Call this P A . 4. Replace each black mother’s data for variables in B with data from her matched white counterpart , and re - calculate the predicted probability of the event for the black mothers. Call this P AB 5. Replace each black mother’s data for variables in C with data from her matched white counterpart , and re - calculate the predicted probability of the event for the black mothers. Call this P ABC 6. The contribution of A to the racial difference in the probability of the event is P A - P 0 , the contribution of B is P AB - P A , and C is P ABC -P AB .

1. Estimate a logistic regression of the event as a function of A, B, and C. Calculate the predicted probability the event for every mother. White Mothers Black Mothers Baby risk factors (A) Mother's SES (B) Mother's risk (C) Baby risk factors (A) Mother's SES (B) Mother's risk (C) Birth Estimated Birth Estimated id weight sex … Education Insurance … Age diabetes … prob(event) id weight sex … Education insurance … Age diabetes … prob(event) 1 1386 F … 12 Medicaid 31 Y … 0.85 XXX 554 F … 8 Medicaid 20 Y … 0.97 2 761 M … 12 Private 17 N … 0.85 12 815 F … 13 Private 16 N … 0.66 3 1487 F … 14 Private 34 N … 0.83 13 1122 M … 15 Medicaid 24 N … 0.54 4 1197 M … 10 Private 27 Y … 0.82 14 1044 F … 10 Medicaid 19 Y … 0.51 5 1287 F … 9 Medicaid 35 N … 0.78 15 556 M … 11 Medicaid 24 N … 0.42 6 613 F … 8 Private 21 Y … 0.75 … … … … … … … … … … YYY 606 F … 15 Private … 34 N … 0.72 7 1412 F … 11 Private 34 N … 0.51 8 1261 M … 10 Private 32 Y … 0.27 P 0 =Mean Prob(Event|A blk , B blk , C blk ) 0.322 9 1447 F … 10 Private 24 N … 0.21 P A =Mean Prob(Event|A wht , B blk , C blk ) 10 721 F … 14 Medicaid 24 N … 0.04 P AB =Mean Prob(Event|A wht , B wht , C blk ) … … … … … … … … … … P ABC =Mean Prob(Event|A wht , B wht , C wht ) XXX 1492 F … 16 Private 30 N … 0.12