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JASA jasa v.2004/12/09 Prn:26/10/2006; 10:42 F:jasaap05194r.tex; (Diana) p. 1 Structured Measurement Error in Nutritional 1 Epidemiology: Applications in the Pregnancy, 60 2 61 3 Infection, and Nutrition (PIN) Study 62 4 63 5 Brent A. JOHNSON,AmyH.HERRING, Joseph G. IBRAHIM, and Anna Maria SIEGA-RIZ 64 6 65 7 66 8 Preterm birth, defined as delivery before 37 completed weeks’ gestation, is a leading cause of infant morbidity and mortality. Identifying 67 factors related to preterm delivery is an important goal of public health professionals who wish to identify etiologic pathways to target 9 68 for prevention. Validation studies are often conducted in nutritional epidemiology in order to study measurement error in instruments that 10 are generally less invasive or less expensive than “gold standard” instruments. Data from such studies are then used in adjusting estimates 69 11 based on the full study sample. However, measurement error in nutritional epidemiology has recently been shown to be complicated 70 12 by correlated error structures in the study-wide and validation instruments. Investigators of a study of preterm birth and dietary intake 71 13 designed a validation study to assess measurement error in a food frequency questionnaire (FFQ) administered during pregnancy and with 72 the secondary goal of assessing whether a single administration of the FFQ could be used to describe intake over the relatively short 14 73 pregnancy period, in which energy intake typically increases. Here, we describe a likelihood-based method via Markov chain Monte Carlo 15 to estimate the regression coefficients in a generalized linear model relating preterm birth to covariates, where one of the covariates is 74 16 measured with error and the multivariate measurement error model has correlated errors among contemporaneous instruments (i.e., FFQs, 75 17 24-hour recalls, and biomarkers). Because of constraints on the covariance parameters in our likelihood, identifiability for all the variance 76 18 and covariance parameters is not guaranteed, and, therefore, we derive the necessary and sufficient conditions to identify the variance and 77 covariance parameters under our measurement error model and assumptions. We investigate the sensitivity of our likelihood-based model to 19 78 distributional assumptions placed on the true folate intake by employing semiparametric Bayesian methods through the mixture of Dirichlet 20 process priors framework. We exemplify our methods in a recent prospective cohort study of risk factors for preterm birth. We use long-term 79 21 folate as our error-prone predictor of interest, the FFQ and 24-hour recall as two biased instruments, and the serum folate biomarker as the 80 22 unbiased instrument. We found that folate intake, as measured by the FFQ, led to a conservative estimate of the estimated odds ratio of 81 23 preterm birth (.76) when compared to the odds ratio estimate from our likelihood-based approach, which adjusts for the measurement error 82 (.63). We found that our parametric model led to similar conclusions to the semiparametric Bayesian model. 24 83 25 KEY WORDS: Adaptive rejection sampling; Dirichlet process prior; MCMC; Semiparametric Bayes. 84 26 85 27 1. INTRODUCTION often attenuate the regression coefficients toward 0 [although 86 28 the result is not true in general nonlinear models (Fuller 1987; 87 Measurement error is a common and well-known challenge 29 Carroll, Ruppert, and Stefanski 1995)]. Although several sta- 88 in nutritional epidemiology. One only has to glance at a re- 30 tistical methods have been proposed for the analysis of data 89 cent issue of any one of the leading epidemiological journals 31 where covariates are measured with error, regression calibration 90 to see this and to verify that there still are many unresolved 32 (Stefanski and Carroll 1985) seems to be the default method in 91 questions. One of the more intriguing recent developments in 33 nutrition (Willett 1998). The method is popular because it may 92 nutritional epidemiology concerns the fitness and applicability 34 be implemented using standard software assuming one has a re- 93 of traditional error models used to assess the validity and gen- 35 liable calibration model (Spiegelman, Carroll, and Kipnis 2001; 94 eralizability of estimated risks obtained from studies using the 36 Spiegelman, Zhao, and Kim 2004). In addition, much money 95 food frequency questionnaire (FFQ). 37 and energy have been spent on validation studies over the past 96 Despite many documented pitfalls (Block 2001; Byers 2001; 38 several decades; therefore, bias and variance parameters relat- 97 Willett 2001), including systematic biases and within- and 39 ing the FFQ to the true, long-term dietary intake can be esti- 98 between-subject variability, the FFQ is a common dietary in- 40 mated with some degree of precision. A related problem to the 99 strument because of its ease of administration and economy in 41 one considered here is the error in covariate misclassification 100 large nutritional studies. Naive regression methods that use the 42 (cf. Holcroft and Spiegelman 1999; Morrissey and Spiegelman 101 43 error-prone FFQ in place of the true long-term dietary intake 102 1999; Spiegelman et al. 2001; Zucker and Spiegelman 2004). 44 The traditional statistical analysis and inference proceeds by 103 45 104 Brent A. Johnson is Postdoctoral Fellow (E-mail: [email protected]), first regressing the FFQ on the outcome to obtain a naive esti- 46 105 Amy H. Herring is Associate Professor (E-mail: [email protected]), mate of the regression coefficient. Then we regress a reference 47 and Joseph G. Ibrahim is Alumni Distinguished Professor (E-mail: instrument—that is, an unbiased measure for the true dietary 106 48 [email protected]), Department of Biostatistics, University of North Car- 107 olina, Chapel Hill, NC 27599. Anna Maria Siega-Riz is Associate Professor, intake—on the FFQ to estimate the attenuation factor. It can be 49 108 Departments of Nutrition and Epidemiology, University of North Carolina, shown that dividing the naive estimated regression coefficient 50 109 Chapel Hill, NC 27599 (E-mail: [email protected]). We thank the asso- by the estimated attenuation factor leads to a corrected estimate 51 ciate editor and two anonymous referees for helpful comments and sugges- 110 tions that led to a much improved manuscript. The research of the first author of the desired regression coefficient, that is, one obtained if we 52 111 was supported in part by grants from the National Institute for the Environ- could have regressed the outcome on the true long-term dietary 53 112 mental Health Sciences (P30ES10126, T32ES007018). Dr. Herring’s research intake (Carroll et al. 1995; Kipnis et al. 2001). If the system- 54 was supported in part by grants from the National Institutes of Child Health 113 and Human Development (1R03HD045780, HD37584, HD39373) and NIEHS atic bias or the correlated errors in the FFQ or 24-hour recall is 55 114 (P30ES10126). The PIN study was funded by grants from NICHD (HD28684, 56 HD05798, DK55865, AG09525), UNC Clinical Nutrition Research Center 115 57 (DK56350), UNC General Clinical Research Resources (RR00046), and funds © 0 American Statistical Association 116 from the Wake Area Health Education Center in Raleigh, NC. Dr. Ibrahim’s Journal of the American Statistical Association 58 117 research was supported in part by NIH grants #CA 70101, #CA 69222, #GM ???? 0, Vol. 0, No. 00, Applications and Case Studies 59 070335, and #AI 060373. DOI 10.1198/00 118 1 JASA jasa v.2004/12/09 Prn:26/10/2006; 10:42 F:jasaap05194r.tex; (Diana) p. 2 2 Journal of the American Statistical Association, ???? 0 1 ignored, then the attenuation factor will be biased, and subse- measurement error in the FFQ. Women in the validation study 60 2 quently, the “corrected” regression coefficient estimate will no were enrolled in the first trimester and were asked to complete 61 3 longer be reliable. Although primary interest often lies in esti- three FFQs over the course of pregnancy, with each FFQ re- 62 4 mating this true regression coefficient, epidemiologists are also flecting intake over the past trimester. The purpose of the longi- 63 5 quite interested in the estimated attenuation factor. Because the tudinal component of the validation substudy was to determine 64 6 power of the study to detect a significant effect is a function of whether one FFQ measurement during the second trimester of 65 7 the attenuation factor, epidemiologists use this fact to make post pregnancy would be sufficient to characterize intake through- 66 8 hoc calculations to determine whether a null finding appears, in out pregnancy. In addition, three daily in-depth diet interviews 67 9 fact, be to the case or whether it seems to be a result of low (also called “24-hour recalls”) were collected proximal to each 68 10 power. FFQ, providing a maximum of 12 measurements over three 69 11 Our method uses models that allow for correlation in the er- time points. The replicate dietary records were collected in or- 70 12 rors for contemporaneous instruments as suggested in the lit- der to help quantify measurement errors in each FFQ. 71 13 erature (Kaaks, Riboli, Esteve, Van Kappel, and Vab Staveren Finally, we make two additional points regarding the PIN 72 14 1994; Kipnis et al. 2001, 2003). Our point and interval estima- study data. First, one serum folate biomarker was collected on 73 15 tion method is different from that considered in Kipnis et al. every woman in the study, that is, both in the main study and 74 16 (2001, 2003) in that we use a likelihood-based approach (also in the substudy. This feature of the PIN study is not common 75 17 called a structural measurement error model), whereas Kipnis among dietary studies, where a “typical” study collects bio- 76 18 et al.

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