Supported by NIH R01 ES009411 Donna Spiegelman, Sc.D. Professor of - - PowerPoint PPT Presentation

Donna Spiegelman, Sc.D.

Professor of Epidemiologic Methods

Departments of Epidemiology, Biostatistics, Nutrition and Global Health stdls@hsph.harvard.edu www.hsph.harvard.edu/donna-spiegelman/

Public Health and Statistics In India IISA-Harvard-SAMSI May 2016 Supported by NIH R01 ES009411

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Over the past 10 years, our group has developed methods that adjust for exposure measurement error in point and interval estimates of relative risk and other measures of association:

• Regression calibration for main study/external validation study designs
• Regression calibration for multiple surrogates for the same exposure
• Regression calibration with heteroscedastic error
• Regression calibration for main study/internal validation study designs
• Regression calibration for survival data analysis with baseline exposures, time-varying

point exposures, and exposure metrics that are functions of the exposure history Methods have been motivated by studies in environmental and occupational epidemiology conducted at the Harvard School of Public Health

Introduction

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Time permits a brief overview of a few of these:

• Regression calibration for main study/external validation study designs
• Regression calibration for main study/internal validation study designs
• Regression calibration for multiple surrogates for the same exposure
• Regression calibration with heteroscedastic error

In the future, I can discuss:

• Regression calibration for survival data analysis with baseline

exposures, time-varying point exposures, and exposure metrics that are functions of the exposure history

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𝑜1 : Number of participants in main study 𝑜2 : Number of participants in validation study 𝐸 : Binary health outcome 𝑌 : “True” exposure 𝑎 : Surrogate exposure 𝑉 : t perfectly measured covariates (e.g. age, race, smoking status) Measured on all participants in the main and validation studies 𝐸𝑗, 𝑎𝑗, 𝑽𝒋 , 𝑗 = 1, … , 𝑜1 Main study 𝑌𝑗, 𝑎𝑗, 𝑽𝑗 , 𝑗 = 𝑜1 + 1, … , 𝑜1 + 𝑜2 External validation study 𝐸𝑗, 𝑌𝑗, 𝑎𝑗, 𝑽𝑗 , 𝑗 = 𝑜1 + 1, … , 𝑜1 + 𝑜2 Internal validation study

Notation

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Assumptions

• True exposure (𝑌) and the t-vector of covariates (𝑉) are related to the

probability of binary outcome (𝐸) by the logistic function: 𝑚𝑝𝑕𝑗𝑢 Pr 𝐸 = 1 = 𝛾0 + 𝑌𝛾1 + 𝑽′𝜸𝟑 where 𝛾′2 = (𝛾21, 𝛾22, … , 𝛾2𝑢).

• Linear regression model is appropriate to relate the surrogates (𝑎) and

the t covariates (𝑉) to the true exposure: 𝑌 = 𝛿0 + 𝑎𝛿1 + 𝑽′𝜹𝟑 + 𝜁 where 𝐹 𝜁 = 0, 𝑊𝑏𝑠 𝜁 = 𝜏𝑌|𝑎,𝑉

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• 𝑎 is a surrogate if Pr 𝐸 𝑌, 𝑽, 𝑎 = Pr 𝐸 𝑌, 𝑽), that is, knowledge of the

surrogates provides no additional information if the true exposure is known.

• 𝜁~𝑂(0, 𝜏𝑌|𝑎,𝑉

2

) and Pr (𝐸) is small, or 𝛾1

2𝜏𝑌|𝑎,𝑉 2

small.

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Rosner et al. regression calibration method for MS/EVS

The (Rosner, Willettt, Spiegelman,1989; Rosner, Spiegelman, Willett, 1990; Rosner, Spiegelman, Willett, 1992) version of regression calibration for MS/EVS design:

3-step algorithm: 1. In the main study, regress 𝑍 on 𝒂 and 𝑽 to obtain

𝛾

∗, 𝜸

1

∗, 𝜸

2

∗ where now 𝒂 is a 𝑡 × 1 vector of mis-measured

continuous covariates and 𝑽 is a 𝑢 × 1 vector of perfectly measured covariates.

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Rosner et al. regression calibration method for MS/EVS

2. In the validation study, regress 𝒀 on 𝒂 and 𝑽 to obtain 𝛿

0, 𝛥

1, 𝛥 2 where 𝛿

0 is a 𝑡 𝑦 1 vector of regression intercepts, 𝛥

1 is a 𝑡 × 𝑡 matrix of slopes for the regression of 𝒀 on 𝒂, adjusted for 𝑽, and

𝛥

1 is a 𝑡 × 𝑢 matrix of slopes for the regression of 𝒀 on 𝑽, adjusted for 𝒂.

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Rosner et al. regression calibration method for MS/EVS

3. Correct estimates of effect for measurement error, by

𝛾 1 = 𝛾 1

∗

𝛿 1 , 𝛾 0= 𝛾

∗ − 𝛾

1𝛿 0, 𝛾 2= 𝛾 2

∗ − 𝛾

1𝛿 2

• r 𝛥

1 𝑈

𝛥

2 𝑈

1

−1 𝛾 1

∗𝑈

𝛾 2

∗𝑈

=

𝛾 1

𝑈

𝛾 2

𝑈

where 𝟏 is a 𝑡 × 𝑢 matrix of 0’s and 𝑱 is a 𝑢 × 𝑢 identity matrix,

𝑱 𝑢 × 𝑢 = 1 ⋯ 1 ⋮ ⋮ ⋱ ⋯ 1

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Rosner et al. regression calibration method for MS/EVS

4. Use multivariate delta method to derive variance, e.g.,

𝑊𝑏𝑠 𝛾 1 = 𝑊𝑏𝑠 𝛾 1

∗

𝛿

1 2

+ (𝛾 1

∗)2𝑊𝑏𝑠

𝛿 1 𝛿

1 4

See Appendices 2 and 3 of Rosner et al., 1990 for a derivation of the variance of 𝛾 1

𝑈

𝛾 2

𝑈

, again using the multivariate delta method.

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Regression calibration (Carroll et al.)

Given validation or reliability data, the Carroll et al. version of the regression calibration estimator follows (when 𝑜𝑠𝑗 = 𝑜𝑆𝐽 = 2): Sketch of Algorithm (univariate case) 1. Estimate 𝛿0 and 𝛿1 in the validation study from the regression of 𝑌𝑗 on

𝑎𝑗, 𝑗 = 1, … , 𝑜𝑊 or in the reliability study from the regression of 𝑎𝑗1

• n 𝑎𝑗2, 𝑗 = 1, , , 𝑜𝑆, where 𝑜𝑠𝑗 = 𝑜𝑆𝐽 = 2

2. Estimate 𝑌

𝑗 = 𝛿 0 + 𝛿 1𝑎𝑗 + 𝑓𝑗, 𝑗 = 1, … , 𝑜𝑁 in the main study.

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Regression calibration (Carroll et al.)

3. Run usual regression model for 𝑍 on 𝑌 in the main study to obtain estimates of effect adjusted for measurement error, i.e., fit model 𝑕 𝐹 𝑍 𝑗 𝑌𝑗

= 𝛾0 + 𝛾1𝑌 1

in the main study, where 𝑕[⋅] is a link function, e.g., identity for linear regression, log for Poisson and log-binomial regression, logit for logistic regression, probit for probit regression to obtain estimates of 𝛾1 and 𝛾0 that are corrected for measurement error, at least ‘approximately’. 4. Variance must be adjusted as well and cannot be obtained from the standard regression software.

RSW and Carroll et al. versions are identical in GLMs (Thurston SW,

Spiegelman D, Ruppert D. “Equivalence of regression calibration methods for main study/external validation study designs”. Journal of Statistical Planning and Inference, 2003; 113:527-539)

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An example

Home Endotoxin Exposure and Wheeze in Infants: Correction for Bias Due to Exposure Measurement Error

Nora Horick, Edie Weller, Donald K. Milton, Diane R. Gold, Ruifeng Li, and Donna Spiegelman

Department of Biostatistics and Department of Environmental Health, Harvard School of Public Health, Boston, Massachusetts, USA; Channing Laboratory, Harvard Medical School, Boston, Massachusetts, USA; Department of Epidemiology, Harvard School of Public Health, Boston, Massachusetts, USA

Environmental Health Perspectives Volume 114, Number 1, January 2006

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Download %blinplus SAS macro at

http://www.hsph.harvard.edu/donna-spiegelman/software/blinplus-macro/

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Regression calibration for logistic regression with multiple surrogates for one exposure

Edie A. Weller, Donna Spiegelman, Don Milton, Ellen Eisen Departments of Biostatistics, Epidemiology, and Environmental Health Harvard School of Public Health and Dana Farber Cancer Institute Journal of Statistical Planning and Inference, 2007; 137:449-461

• Occupational exposures often characterized by numerous factors of the workplace and

work duration in a particular area ==> multiple surrogates describe one exposure.

• Validation study: Personal exposure is commonly measured on a subset of the

subjects and these values are then used to estimate average exposure by job or exposure zone.

• No adjustment for bias or uncertainty in the exposure estimates.
• Standard methods typically assume that there is one surrogate for each exposure (for

example, Rosner et al, 1989, 1990).

• Propose adjustment method which allows for multiple surrogates for one exposure

using a regression calibration approach.

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Main Study

• To assess the relationship between exposure to metal working fluids

(MWF) and respiratory function (United Automobile Workers Union and General Motors Corporation sponsored study, Greaves et al, 1997).

• Outcome here is prevalence of wheeze
• Job characteristics include metal working fluid (MWF) type, plant and

machine operation (grinding or not).

• Assembly workers are considered the non-exposed group.
• Possible confounders include age, smoking status and race.

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Exposure Assessment Study (generically, the validation study)

• Exposure was measured in various job zones (Woskie et al., 1994).
• Intensity of exposure to MWF aerosol measured by the thoracic

aerosol fraction (i.e. the sum of the two smallest size fractions measured with the personal monitors).

• Full shift (8 hour) personal samples of aerosol exposure in breathing

zone of automobile workers were collected in various job zones.

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Assumptions

• True exposure (𝑌) and the 𝑢-vector of covariates (𝑽) are related to

the probability of binary outcome (𝐸) by the logistic function: 𝑚𝑝𝑕𝑗𝑢 Pr 𝐸 = 1 = 𝛾0 + 𝑌𝛾1 + 𝑽𝛾2 where 𝜸′𝟑 = (𝛾21, 𝛾22, … , 𝛾2𝑢).

• Linear regression model is appropriate to relate the 𝒔 surrogates (𝑿)

and the s covariates (𝒂) to the true exposure: 𝑌 = 𝛿0 + 𝑿′𝜹𝟐 + 𝑽′𝜹𝟑 + 𝜻 where 𝐹 𝜁 = 0, 𝑊𝑏𝑠(𝜁) = 𝜏𝑌|𝑽,𝑋

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• 𝑿 is a surrogate if Pr 𝐸 𝑌, 𝑿, 𝑽 = Pr 𝐸 𝑌, 𝑽 , that is, knowledge of

the surrogates provides no additional information if the true exposure is known.

• 𝜁~𝑂(0, 𝜏𝑌|𝑿,𝑽

2

) and Pr (𝐸) small, or 𝛾1

2𝜏𝑌|𝑿,𝑽 2

small

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Goal: to obtain point and interval estimates of 𝛾 and 𝑓𝛾 relating exposure (𝑌) to outcome (𝐸) adjusting for the covariates (𝑽)

Problem

• Quantitative measure of exposure (𝑌) is not measured on all subjects

– 𝑿 is measured on all 𝑜1 of the subjects – 𝑌 and 𝑿 measured on 𝑜2 subjects

• Multiple surrogates, 𝑿, describe exposure

Solution: An extension to two closely related approaches

• Rosner, Spiegelman and Willett (RSW, 1989, 1990)
• Carroll, Ruppert and Stefanski (CRS, 1995)

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Procedure

Propose the following approach which follows RSW and assumes normality of 𝜁 and rare disease, or that 𝛾1

2𝜏𝑌|𝑿,𝑽 2

is small 1. Estimate 𝜷 from a logistic regression model of 𝐸 on 𝑿 and in 𝑜1 subjects in main study

𝑚𝑝𝑕𝑗𝑢 Pr 𝐸 = 1 = 𝛽 0 + 𝑿′𝜷 𝟐 +𝑽′𝛽 𝟑

2. Estimate 𝛿 from a measurement error model among the 𝑜2 validation study subjects using ordinary least squares regression.

𝑌 = 𝛿 0 + 𝑿′𝛿 𝟐 +𝑽′𝛿 𝟑

SAS PROC GENMOD or PROC LOGISTIC for step 1, PROC REG for step 2

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• 3. Optimally combine the adjusted estimates for each

surrogate 𝛾 𝑿, where 𝛾 𝑿 = 𝛥

1 −1𝛽

1, 𝛥

1 = 𝑒𝑗𝑏𝑕(𝛿

1) 𝜐′ = (1′𝛵 𝛾𝑋

−1 1)−11′𝛵

𝛾𝑋

−1 . 1 = 1,1, … , 1 ′

𝛵 𝛾𝑋 is the estimated variance-covariance matrix of 𝛾 𝑿 𝛵 𝛾𝑋 = 𝜖𝜸𝑿 𝜖 𝛽1, 𝛿1

′ 𝛽 1,𝛿 1

𝛵 𝛽1 𝛵 𝛿1 𝜖𝜸𝑿 𝜖 𝛽1, 𝛿1

𝛽 1,𝛿 1

SAS macro downloadable from my website to accomplish step 3; input to the macro is the output from PROC LOGISTIC and PROC REG

http://www.hsph.harvard.edu/donna-spiegelman/software/multsurr-method/

ˆ ˆ ˆ   

W

τ β

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Results from logistic regression model for wheeze. GM/UAW main study (n1 = 1040). “True” Exposure (X) is thoracic aerosol fraction (mg/m3 ) measures on n2 = 83 workers Variable

Uncorrected

P-value

Corrected

P-value Exposure1 (mg/m3 ) 2.875 (1.353, 6.108) 0.006 Surrogates (W) Plant 2 Grinding Straight Synthetic 2.109 (1.391, 3.198) 0.706 (0.374, 1.332) 1.641 (1.119, 2.407) 1.851 (1.200, 2.854) < 0.001 0.282 0.011 0.005 Covariates (Z) Age 30-39 Age 40-49 Age 50+ Race Current Smoker 0.897 (0.615, 1.307) 0.834 (0.512, 1.358) 0.912 (0.544, 1.528) 1.173 (0.796, 1.728) 3.042 (2.210, 4.188) 0.571 0.465 0.726 0.420 < 0.001 0.965 (0.648, 1.437) 0.853 (0.513, 1.418) 0.914 (0.535, 1.561) 1.166 (0.782, 1.740) 2.978 (2.144, 4.137) 0.861 0.540 0.741 0.451 < 0.001

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Estimated GLS weights are 0.857 for straight, 0.127 for synthetic, 0.15 for grinding, and 0.0001 for plant

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Regression Calibration With Heteroscedastic Variance

Donna Spiegelman, Roger Logan, Douglas Grove International Journal of Biostatistics: 2011 Vol. 7, Issue 1, Article 4. PMCID: PMC3404553

Conclusion: For all practical purposes, no need to worry about heteroscedasticity, that is, if 𝑊𝑏𝑠(𝑌𝑗|𝑎𝑗, 𝑽𝑗) varies with 𝑗, little impact on bias or efficiency of RC method

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A comparison of regression calibration methods for designs with internal validation data

Sally W. Thurston , Paige L. Williams, Russ Hauser, Howard Hu, Mauricio Hernandez-Avila, and Donna Spiegelman

Department of Biostatistics and Computational Biology, University of Rochester Medical Center, 601 Elmwood Avenue, P.O. Box 630, Rochester, NY 14642, USA Department of Biostatistics, Harvard School of Public Health, USA Department of Environmental Health, Harvard School of Public Health, USA Centro de Investigaciones en Salud Poblacional, Instituto Nacional de Salud Publica, Cuernavaca, Morelos, Mexico Department of Epidemiology, Harvard School of Public Health, USA Channing Laboratory, Department of Medicine, Brigham and Women's Hospital, Harvard Medical School, US Journal of Statistical Planning and Inference, 2005; 131:175-190.

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ARE of optimal method compared to Carroll method

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Conclusions I

We can accommodate the following situations:

• multiple surrogates for a single mis-measured exposure
• heteroscedastic measurement error
• internal and hybrid validation study designs
• cumulative exposure variables and other functions of the exposure history in cohort

studies User-friendly SAS macros are available to implement many of these procedures

• http://www.hsph.harvard.edu/donna-spiegelman/software/blinplus-macro/
• http://www.hsph.harvard.edu/donna-spiegelman/software/multsurr-

method/

• http://www.hsph.harvard.edu/donna-spiegelman/software/rrc-macro/
• http://www.mep.ki.se/%7Emarrei/software/

(for optimal main study / validation study design)

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Conclusions II

• Bias due to exposure measurement error is a major limitation to the

validity of occupational and environmental studies

• Methods have been developed which accommodate the features of

study design and data distributions found in such studies

• These methods implement explicit adjustments for this source of

bias, using the exposure validation study to characterize the magnitude and other features of the measurement error

• Point and interval estimates of effect are adjusted
• Papers have been published applying these methods to the analysis
• f occupational and environmental studies: you won’t be the first!
• Just as we routinely adjust for confounding, we can routinely adjust

for measurement error

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Acknowledgements

• NIEHS
• Edie Weller, Ruifeng Li, Don Milton, Ellen Eisen, Barbara Valanis, Sally

Thurston, Jon Samet, Paige Williams, Russ Hauser, Roger Logan, Jon Samet, Doug Grove, Doug Dockery, Lucas Neas, Nora Horrick, Diane Gold, Mauricio Hernandez, Howard Hu, Aparna Keshaviah

• Xiaomei Liao, Molin Wang, Biling Hong
• Francine Laden, Helen Suh, Jaime E. Hart, Joel Kaufman, Adam Szpiro,

Lianne Sheppard, Ronald Williams, Robin C. Puett, Marianthi-Anna Kioumourtzoglou

• Alan Berkeley, Emily Long

Thank you!