An Application of Generalized Multiple Indicators, Multiple Causes

Measurement Error Models to Adjust for Dose Error in RERF Data

Carmen D. TekweDepartment of Biostatistics

University at BuffaloBuffalo, NY

This research is part of a collaborative between RERF and the University at Buffalo, Department of Biostatistics

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

Outline

Background

Classical Linear MIMIC Models

G-MIMIC Models

Conclusion

Background

DS02 – current dosimetry systemBased on physical dosimeter estimatesBased on survivor recall of location and

shielding at the time of explosionSelf-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

Classical Measurement Error in Simple Linear Models

Y = β0 + β1x + 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

Berkson Error in Simple Linear Models

Y = β0 + β1x + 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.

Classical Linear MIMIC Model

Multiple outcomes, an underlying latent variable, observations on causes of the latent variable are available

Structural equations & factor analyseseconometric settings/psychometrics

Generalizations to nonlinear relationships have not been worked out.

Classical Linear MIMIC Model

Y1 = β0 + β1 x + ε1

Y2 = β0 + β2 x + ε2

Y3 = β0 + β3 x + ε3

•

•

•

Yp = β0 + βp x + εp

x = α0+α1Z1 + α2Z2 + ••• + αkZk + v

• x = unobservable latent variable• Y1,, Y2, Y3 ,•••, Yp p multiple indicators linearly related to x• Z1,, Z2, Y3 ,•••, Zk k multiple causes linearly related to x• v = Berkson error

If k=1 and α0 = 0 and α1 = 1, then this is a multivariate Berkson model

..

Summary of Models

Classical Measurement Error

log(DS02) = log(true dose) + u

corr(true dose, u) = 0

oBiased OLS estimatesoAttenuation to nulloModel not identified without additional information

Berkson Measurement Error

log(true dose) = log(DS02) + v

corr(DS02, v) = 0

oUnbiased OLS estimatesoIncreased varianceoParametric inference is robust

Classical Linear MIMIC Model

Causal model:

log(true dose) = α0 + Zα + v

Z contains distance and shielding indicators

corr(Z, v) = 0

oIndeterminancyo Joreskog and Goldberger (1975): assume known parameter (e.g., σv known)

oModel is not identified without additional information

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

Parameters kp+½p(p+1) k+2p Identifiability

K=p=1 2 3 Not identified

K=1,p=2 5 5 Just identified

K=2, p=1 3 4 Not identified

K=3, p=1 4 5 Not identified

K=3, p=2 9 7 Over-identified

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.

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

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.

Generalized Multiple Indicators and Multiple Causes MeasurementError 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

G-MIMIC Models

Y1 = g(η1) + ε1

Y2 = g(η2) + ε2

Y3 = g(η3) + ε3

•

•

•

Yp = g(ηp) + εp

x = h(ξ) + v

• g(ηi), h(ξ) are monotone twice continuously differentiable functions with linear predictors ηi = xβi and ξ = α’Z respectively

• Note: if Y1,, Y2, Y3 ,•••, Yp S 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)

Exponential G-MIMIC ModelsY1 = g(η1) + ε1

Y2 = g(η2) + ε2

Y3 = g(η3) + ε3

•

•

•

Yp = g(ηp) + εp

x = h(ξ) + v

• g(ηi), h(ξ) are monotone twice continuously differentiable functions with linear predictors ηi = xβi and ξ = α’Z respectively

• Y1,, Y2, Y3 ,•••, Yp S exponential family • u = classical measurement error, v = Berkson error• Model is not identified without additional information• Indeterminancy

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 shieldingCA = g1 (lp1 ) + e1

EP = g2 (lp2 ) + e2

GPA = g3 (lp3 ) + e3

true dose = h (lpd,s ) + v

• lpd,s = α0 + α1 shielding + α2distance + u

• Assuming distance and shielding where ascertained “imperfectly”.

Estimation of exponential G-MIMIC ME models

Under the assumption that σv2 is known (e.g.

can be estimated using external data)Construct the likelihoodUse MC-EM methods to analyze dataObtain all parameter estimates including δu

2

Obtain E(x|CA,GPA,X)

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

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 relationshipsi.e., add disease outcome of interest to G-

MIMIC modelsProceed with estimation

Compare MC-EM approach to the regression calibration approach

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

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