1. An Application of Generalized Multiple Indicators, Multiple Causes Measurement Error Models to Adjust for Dose Error in RERF Data Carmen D. Tekwe Department of Biostatistics University at Buffalo Buffalo, NY This research is part of a collaborative between RERF and the University at Buffalo, Department of Biostatistics

2. Participants in the UB/RERF Collaboration Radiation Effects Research Foundation Harry Cullings, Kazuo Neriishi, Yoshiaki Kodama, Yochiro Kusunoki, Nori Nakamura, Yukiko Shimizu, Misa Imaizumi, Eiji Nakashima, John Cologne, Sachiyo Funamoto, Thomas Seed, Phillip Ross UB Department of Biostatistics Randy Carter, Carmen D. Tekwe, Austin Miller USC Department of Preventive Medicine Daniel Stram

3. Outline Background Classical Linear MIMIC Models G-MIMIC Models Conclusion

4. Background DS02 – current dosimetry system Based on physical dosimeter estimates Based on survivor recall of location and shielding at the time of explosion Self-reported measures are often plagued with classical measurement error, u. ln(DS02) = ln(True dose) + u, Or, in more convenient notation, X = x + u, where u is independent of x

5. Classical Measurement Error in Simple Linear Models Y = β 0 + β 1 x + ε X = x + u where x is independent of u, u is classical measurement error OLS estimates from regression of Y on X are biased. Model is not identified without additional information. Identifying information: Repeated observations Assume a known parameter Instrumental variables

6. Berkson Error in Simple Linear Models Y = β 0 + β 1 x + ε x = X + v where v is independent of X, v is Berkson error OLS estimates from regression of Y on X are unbiased. Model is identified. Variance is increased. Parametric inferences are robust.

7. Classical Linear MIMIC Model Multiple outcomes, an underlying latent variable, observations on causes of the latent variable are available Structural equations & factor analyses econometric settings/psychometrics Generalizations to nonlinear relationships have not been worked out.

8. Classical Linear MIMIC Model Y 1 = β 0 + β 1 x + ε 1 Y 2 = β 0 + β 2 x + ε 2 Y 3 = β 0 + β 3 x + ε 3 Y p = β 0 + β p x + ε p x = α 0 +α 1 Z 1 + α 2 Z 2 + + α k Z k + v x = unobservable latent variable Y 1,, Y 2, Y 3,, Y p p multiple indicators linearly related to x Z 1,, Z 2, Y 3,, Z k k multiple causes linearly related to x v = Berkson error If k=1 and α 0 = 0 and α 1 = 1, then this is a multivariate Berkson model..

9. Summary of Models Classical Measurement Error log(DS02) = log(true dose) + u corr(true dose, u) = 0 o Biased OLS estimates o Attenuation to null o Model not identified without additional information Berkson Measurement Error log(true dose) = log(DS02) + v corr(DS02, v) = 0 o Unbiased OLS estimates o Increased variance o Parametric inference is robust Classical Linear MIMIC Model Causal model: log(true dose) = α 0 + Zα + v Z contains distance and shielding indicators corr(Z, v) = 0 o Indeterminancy o Joreskog and Goldberger (1975): assume known parameter (e.g., σ v known) o Model is not identified without additional information

10. Illustration of Identifiability in the Classical Linear MIMIC Model indeterminancy is removed by transforming the structural causal model, let x * = x ÷ sd(x) Need kp + ½p(p+1) ≥ k+2p for model identification Parameterskp+½p(p+1)k+2pIdentifiability K=p=123Not identified K=1,p=255Just identified K=2, p=134Not identified K=3, p=145Not identified K=3, p=297Over-identified

11. Scientific Objectives Improve current physical dosimetry systems by including biological indicators of true dose (bio-dosimeters). Estimate dose response relationships between health outcomes and true dose after obtaining improved dose estimates based on regression calibration methods. Estimate dose response relationships between health outcomes and true dose after obtaining improved dose estimates based on MC-EM methods.

12. Available Biodosimeters in the RERF data set Stable chromosome aberrations in lymphocyte cells (CA) Erythrocyte glycophorin A gene mutant fraction (GPA) Electron spin resonance spectroscopy of tooth enamel (ESR) Epilation or other acute effects

13. Statistical Objectives Define the G-MIMIC model extend the classical linear MIMIC model to allow nonlinear relationships in the presence of Berkson error alone. Develop likelihood based parameters for the G- MIMIC model in the presence of both Berkson errors and classical measurement error in the structural causal equations (G-MIMIC ME models). Apply the newly developed methods to obtain unbiased estimates of A-bomb radiation dose on a variety of disease outcomes or risk indices.

14. Generalized Multiple Indicators and Multiple Causes Measurement Error Models Extends linear MIMIC model to allow non-linear relationships. Causal equation includes both Berkson and classical measurement errors. Observations of “causal” variables known to cause the latent variable exist in the data. Identifiability Instrumental variables Indeterminancy “Super” identifiability Assume a known parameter

15. G-MIMIC Models Y 1 = g(η 1 ) + ε 1 Y 2 = g(η 2 ) + ε 2 Y 3 = g(η 3 ) + ε 3 Y p = g(η p ) + ε p x = h(ξ) + v g(η i ), h(ξ) are monotone twice continuously differentiable functions with linear predictors η i = xβ i and ξ = α ’ Z respectively Note: if Y 1,, Y 2, Y 3,, Y p ̃ exponential family then this becomes the exponential G-MIMIC model If x = h(ξ) + v – u then we have the G-MIMIC measurement error model (G-MIMIC ME model)

16. Exponential G-MIMIC Models Y 1 = g(η 1 ) + ε 1 Y 2 = g(η 2 ) + ε 2 Y 3 = g(η 3 ) + ε 3 Y p = g(η p ) + ε p x = h(ξ) + v g(η i ), h(ξ) are monotone twice continuously differentiable functions with linear predictors η i = xβ i and ξ = α ’ Z respectively Y 1,, Y 2, Y 3,, Y p ̃ exponential family u = classical measurement error, v = Berkson error Model is not identified without additional information Indeterminancy

17. Applying the exponential G-MIMIC ME model to RERF data Biological indicators of true dose: chromosome aberrations (CA), epilation (EP), and glycophorin A (GPA). Causal variables: distance and shielding CA = g 1 (lp 1 ) + e 1 EP = g 2 (lp 2 ) + e 2 GPA = g 3 (lp 3 ) + e 3 true dose = h (lp d,s ) + v lp d,s = α 0 + α 1 shielding + α 2 distance + u Assuming distance and shielding where ascertained “imperfectly”.

18. Estimation of exponential G- MIMIC ME models Under the assumption that σ v 2 is known (e.g. can be estimated using external data) Construct the likelihood Use MC-EM methods to analyze data Obtain all parameter estimates including δ u 2 Obtain E(x|CA,GPA,X)

19. Application to RERF Data A biodosimeter can be obtained as the estimated value of E(x|CA,GPA,X) Estimated E(x|CA,GPA,X) = adjusted dose Use the estimated value of E(x|CA,GPA,X) as a substitute for x in disease outcome models in a regression calibration approach to risk assessment. Issue: regression calibration approaches are “exact” methods in linear settings but “approximate” methods in non linear settings

20. Future work Compare our G-MIMIC adjusted dose to the current adjusted doses in RERF data Use MC-EM methods rather than regression calibration methods for estimating dose response relationships i.e., add disease outcome of interest to G- MIMIC models Proceed with estimation Compare MC-EM approach to the regression calibration approach

21. Advantages of MC-EM approach Based on EM algorithm Allows modeling of dose-response curves in the presence of missing data Not an “approximate” method in non-linear settings

22. Conclusion Use biodosimeters as instrumental variables in the G-MIMIC models Obtain adjusted doses Use adjusted doses in dose response curves Use usual modeling techniques with disease outcome models