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

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

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Introduction 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 3 hsph.harvard.edu/donna-spiegelman/

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 4 hsph.harvard.edu/donna-spiegelman/

Notation 𝑜 1 : Number of participants in main study 𝑜 2 : 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 𝐸 𝑗 , 𝑎 𝑗 , 𝑽 𝒋 , 𝑗 = 1, … , 𝑜 1 Main study 𝑌 𝑗 , 𝑎 𝑗 , 𝑽 𝑗 , 𝑗 = 𝑜 1 + 1, … , 𝑜 1 + 𝑜 2 External validation study 𝐸 𝑗 , 𝑌 𝑗 , 𝑎 𝑗 , 𝑽 𝑗 , 𝑗 = 𝑜 1 + 1, … , 𝑜 1 + 𝑜 2 Internal validation study 5 hsph.harvard.edu/donna-spiegelman/

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 = 𝛾 0 + 𝑌𝛾 1 + 𝑽′𝜸 𝟑 where 𝛾′ 2 = (𝛾 21 , 𝛾 22 , … , 𝛾 2𝑢 ) . • Linear regression model is appropriate to relate the surrogates ( 𝑎 ) and the t covariates ( 𝑉 ) to the true exposure: 𝑌 = 𝛿 0 + 𝑎𝛿 1 + 𝑽 ′ 𝜹 𝟑 + 𝜁 where 𝐹 𝜁 = 0, 𝑊𝑏𝑠 𝜁 = 𝜏 𝑌|𝑎,𝑉 2 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. 2 2 2 𝜏 𝑌|𝑎,𝑉 𝜁~𝑂(0, 𝜏 𝑌|𝑎,𝑉 ) and Pr (𝐸) is small, or 𝛾 1 • small. 6 hsph.harvard.edu/donna-spiegelman/

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: In the main study, regress 𝑍 on 𝒂 and 𝑽 to obtain 1. 0 1 2 ∗ , 𝜸 ∗ , 𝜸 ∗ where now 𝒂 is a 𝑡 × 1 vector of mis-measured 𝛾 continuous covariates and 𝑽 is a 𝑢 × 1 vector of perfectly measured covariates. 7 hsph.harvard.edu/donna-spiegelman/

Rosner et al. regression calibration method for MS/EVS In the validation study, regress 𝒀 on 𝒂 and 𝑽 to obtain 𝛿 0 , 𝛥 1 , 𝛥 2. 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 . 8 8 hsph.harvard.edu/donna-spiegelman/

Rosner et al. regression calibration method for MS/EVS 3. Correct estimates of effect for measurement error, by 1 ∗ 1 = 𝛾 ∗ − 𝛾 ∗ − 𝛾 0 = 𝛾 0 1 𝛿 2 = 𝛾 2 1 𝛿 𝛾 , 𝛾 0 , 𝛾 2 𝛿 1 −1 𝑈 or 𝛥 0 1 1 ∗𝑈 𝑈 𝛾 𝛾 1 = 𝑈 2 ∗𝑈 2 𝑈 𝛥 1 𝛾 𝛾 2 where 𝟏 is a 𝑡 × 𝑢 matrix of 0’s and 𝑱 is a 𝑢 × 𝑢 identity matrix, 1 0 ⋯ 0 𝑱 0 1 ⋮ 𝑢 × 𝑢 = ⋮ ⋱ 0 0 ⋯ 0 1 9 9 hsph.harvard.edu/donna-spiegelman/

Rosner et al. regression calibration method for MS/EVS 4. Use multivariate delta method to derive variance, e.g., 1 1 𝛿 ∗ ∗ ) 2 𝑊𝑏𝑠 1 = 𝑊𝑏𝑠 𝛾 + (𝛾 1 𝑊𝑏𝑠 𝛾 2 4 𝛿 𝛿 1 1 See Appendices 2 and 3 of Rosner et al., 1990 for a derivation of 𝑈 1 𝛾 the variance of , again using the multivariate delta method. 𝑈 2 𝛾 10 hsph.harvard.edu/donna-spiegelman/

Regression calibration (Carroll et al.) Given validation or reliability data, the Carroll et al. version of the regression calibration estimator follows (when 𝑜 𝑠 𝑗 = 𝑜 𝑆 𝐽 = 2 ): Sketch of Algorithm (univariate case) Estimate 𝛿 0 and 𝛿 1 in the validation study from the regression of 𝑌 𝑗 on 1. 𝑎 𝑗 , 𝑗 = 1, … , 𝑜 𝑊 or in the reliability study from the regression of 𝑎 𝑗1 on 𝑎 𝑗2 , 𝑗 = 1, , , 𝑜 𝑆 , where 𝑜 𝑠 𝑗 = 𝑜 𝑆 𝐽 = 2 𝑗 = 𝛿 Estimate 𝑌 0 + 𝛿 1 𝑎 𝑗 + 𝑓 𝑗 , 𝑗 = 1, … , 𝑜 𝑁 in the main study. 2. 11 hsph.harvard.edu/donna-spiegelman/

Regression calibration (Carroll et al.) Run usual regression model for 𝑍 on 𝑌 in the main study to obtain estimates of 3. 1 effect adjusted for measurement error, i.e., fit model 𝑕 𝐹 𝑍 𝑗 𝑌 𝑗 = 𝛾 0 + 𝛾 1 𝑌 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 𝛾 1 and 𝛾 0 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) 12 hsph.harvard.edu/donna-spiegelman/

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 13 hsph.harvard.edu/donna-spiegelman/

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Download %blinplus SAS macro at http://www.hsph.harvard.edu/donna-spiegelman/software/blinplus-macro/ 16 hsph.harvard.edu/donna-spiegelman/

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. 17 hsph.harvard.edu/donna-spiegelman/

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. 18 hsph.harvard.edu/donna-spiegelman/