1. Comparing methods for addressing limits of detection in environmental epidemiology Roni Kobrosly, PhD, MPH Department of Preventive Medicine Icahn School of Medicine at Mount Sinai

2. A familiar diagram… Environmental Exposure Internal Dose Biologically Effective Dose Altered Structure/ Function Clinical Disease Biomarker of Exposure DeCaprio, 1997

3. Biomarkers and Limits of Detection (LOD)

4. It is difficult to quantify the concentration because it is so low LOD Higher concentration

5. Handling LODs in analysis Easiest approach: simply delete these observations Problems with this: o However, values < LOD are informative: analyte may have a concentration between 0 and LOD o Studies are expensive and you lose covariate data! o Excluding observations from analyses *may* substantially bias results Chen et al. 2011

6. Handling LODs in analysis Hornung & Reed describe approach that involves substituting a single value for each observation <LOD Three suggested substitutions: LOD/2, LOD/√2, or just LOD Problem: Replacing a sizable portion of the data with a single value increases the likelihood of bias and reduces power! Helsel, 2005; Hughes 2000; Hornung & Reed, 1990

7. Citations in Google Scholar Hornung & Reed, 1990

8. Comparing LOD methods While there are many studies testing individual methods, relatively little work comparing performance of several methods Even fewer studies have compared methods in context of multivariable data Comparative studies that do exist provide contradictory recommendations. No consensus!

9. Simulation Study Objectives Compare performance of LOD methods when independent variable is subject to limit of detection in multiple regression Compare performance across a range of “experimental” conditions Create flowchart to aid researchers in their analysis decision making

10. Statistical Bias Nat’l Library of Med definition: “Any deviation of results or inferences from the truth” UnbiasedBiased

11. Variable Definitions Four continuous variables: Y: Dependent variable (outcome) X: Independent variable (exposure, subject to LOD) C1, C2: Independent variables (covariates)

12. 6 “Experimental Conditions” 1) Dataset sample size: n = {100, 500}

13. 2) % of exposure variable with values in LOD region: LOD % = {0.05, 0.25}

14. 3) Distribution of Exposure Variable: Normal versus Skewed

15. 4) R 2 of full model: R 2 = {0.10, 0.20}

16. 5) Strength & direction of exposure-outcome association: Beta = {-10, 0, 10}

17. 6) Direction of confounding: Strong Positive, versus Strong Negative, versus None + -

18. LOD methods considered 1. Deletion of subjects with LOD values 2. Substitution with LOD/√(2) 3. Substitution with LOD/2 4. Substitution with just LOD value 5. Multiple imputation ( King’s Amelia II ) 6. MLE-imputation method ( Helsel & Krishnamoorthy )

19. Method 1: Deletion YXC1C2 167.725.813.512.9 -66.315.911.712.6 50.6<LOD10.410.8 -273.09.511.811.1 156.9<LOD12.69.0 YXC1C2 167.725.813.512.9 -66.315.911.712.6 50.6<LOD10.410.8 -273.09.511.811.1 156.9<LOD12.69.0

20. Method 2: Sub with LOD/√(2) YXC1C2 167.725.813.512.9 -66.315.911.712.6 50.6<LOD10.410.8 -273.09.511.811.1 156.9<LOD12.69.0 LOD X = 9.0 YXC1C2 167.725.813.512.9 -66.315.911.712.6 50.66.410.410.8 -273.09.511.811.1 156.96.412.69.0 9.0/√2 = 6.4

21. Method 3: Sub with LOD/(2) YXC1C2 167.725.813.512.9 -66.315.911.712.6 50.6<LOD10.410.8 -273.09.511.811.1 156.9<LOD12.69.0 LOD X = 9.0 YXC1C2 167.725.813.512.9 -66.315.911.712.6 50.64.510.410.8 -273.09.511.811.1 156.94.512.69.0 9.0/2 = 4.5

22. Method 4: Sub with just LOD YXC1C2 167.725.813.512.9 -66.315.911.712.6 50.6<LOD10.410.8 -273.09.511.811.1 156.9<LOD12.69.0 LOD X = 9.0 YXC1C2 167.725.813.512.9 -66.315.911.712.6 50.69.010.410.8 -273.09.511.811.1 156.99.012.69.0

23. Method 5: Multiple Imputation “Amelia II” by Dr. Gary King Assumes pattern of observations below LOD only depends on observed data (not unobserved data) Lets you constrain imputed values (very helpful when working with LODs!)

24. Method 5: Multiple Imputation YXC1C2 167.725.813.512.9 -66.315.911.712.6 50.6<LOD10.410.8 -273.09.511.811.1 156.9<LOD12.69.0 YXC1C2 167.725.813.512.9 -66.315.911.712.6 50.63.010.410.8 -273.09.511.811.1 156.96.212.69.0 YXC1C2 167.725.813.512.9 -66.315.911.712.6 50.62.510.410.8 -273.09.511.811.1 156.96.812.69.0 YXC1C2 167.725.813.512.9 -66.315.911.712.6 50.63.310.410.8 -273.09.511.811.1 156.96.312.69.0 YXC1C2 167.725.813.512.9 -66.315.911.712.6 50.63.510.410.8 -273.09.511.811.1 156.96.012.69.0 YXC1C2 167.725.813.512.9 -66.315.911.712.6 50.62.810.410.8 -273.09.511.811.1 156.97.212.69.0 M = 5

25. Method 5: Multiple Imputation YXC1C2 167.725.813.512.9 -66.315.911.712.6 50.63.010.410.8 -273.09.511.811.1 156.96.212.69.0 YXC1C2 167.725.813.512.9 -66.315.911.712.6 50.62.510.410.8 -273.09.511.811.1 156.96.812.69.0 YXC1C2 167.725.813.512.9 -66.315.911.712.6 50.63.310.410.8 -273.09.511.811.1 156.96.312.69.0 YXC1C2 167.725.813.512.9 -66.315.911.712.6 50.63.510.410.8 -273.09.511.811.1 156.96.012.69.0 YXC1C2 167.725.813.512.9 -66.315.911.712.6 50.62.810.410.8 -273.09.511.811.1 156.97.212.69.0 β 1 = 10.1 β 2 = 9.5β 3 = 8.3β 4 = 12.1 β 5 = 10.4

26. Method 6: MLE-Imputation YXC1C2 167.725.813.512.9 -66.315.911.712.6 50.6<LOD10.410.8 -273.09.511.811.1 156.9<LOD12.69.0

27. Method 6: MLE-Imputation YXC1C2 167.725.813.512.9 -66.315.911.712.6 50.6<LOD10.410.8 -273.09.511.811.1 156.9<LOD12.69.0

28. Method 6: MLE-Imputation YXC1C2 167.725.813.512.9 -66.315.911.712.6 50.63.210.410.8 -273.09.511.811.1 156.95.812.69.0

29. Two-step Data Generation Process 1 st Step: Select “true” regression parameters for following two models: o 2 nd Step: Use “true” parameters to guide the drawing of random numbers

30. “TRUTH” Y = 2.8 + 2(X) + 4.5(C1) + 6(C2) Dataset1.1Dataset1.2Dataset1.3 SIMULATED DATASETS X = 1.3 - 6(C1) + 1.5(C2) Obs #YXC1C2 124.675.44-0.281.77 230.739.47-1.55-0.81 319.39-0.980.960.92 4-9.47-8.201.720.49 iyiyi xixi c1 i c2 i

31. Y = 2.8 + 2(X) + 4.5(C1) + 6(C2) Create a set of “true” parameters Dataset1.1 Dataset1.2 Dataset1.3 Dataset1.1000 Create 1500 simulated datasets for set of “true” parameters, using specific set of experimental conditions Apply a LOD correction method and run regression for each dataset Bias = 2.2 – 2 = 0.2 Take difference of estimated coefficient and “true” parameter. Produce 1000 bias estimates with 95% CI’s

32. Help from Minerva Minerva runtime ~ 5 minutes

33. n = 100, 25% LOD, Skewed Dist, R 2 = 0.20, Negative X-Y Association, Negative confounding Mean Bias (with 95% CI) 3.0 4.0 5.0 6.0 Deletion LOD/sqrt(2) LOD/2 LOD Multi Impu 2.0 0 1.0 7.0 8.0 MLE Impu -2.0

34. Mean Bias (with 95% CI) -3.0 -2.0 0 Deletion LOD/sqrt(2) LOD/2 LOD Multi Impu -4.0 -6.0 -5.0 -7.0 1.0 2.0 MLE Impu -8.0 n = 100, 25% LOD, Skewed Dist, R 2 = 0.20, Positive X-Y Association, Negative confounding

35. n = 100, 25% LOD, Skewed Dist, R 2 = 0.20, Negative X-Y Association, No confounding Mean Bias (with 95% CI) 0 0.2 0.4 0.6 Deletion LOD/sqrt(2) LOD/2 LOD Multi Impu -0.2 -0.6 -0.4 -0.8 0.8 1.0 MLE Impu

36. n = 100, 25% LOD, Skewed Dist, R 2 = 0.20, Positive X-Y Association, No confounding Mean Bias (with 95% CI) 0 0.2 0.4 0.6 Deletion LOD/sqrt(2) LOD/2 LOD Multi Impu -0.2 -0.6 -0.4 -0.8 0.8 1.0 MLE Impu

37. n = 100, 25% LOD, Skewed Dist, R 2 = 0.20, Negative X-Y Association, Positive confounding Mean Bias (with 95% CI) 3.0 4.0 5.0 6.0 Deletion LOD/sqrt(2) LOD/2 LOD Multi Impu 2.0 0 1.0 7.0 8.0 MLE Impu -2.0

38. n = 100, 25% LOD, Skewed Dist, R 2 = 0.20, Positive X-Y Association, Positive confounding Mean Bias (with 95% CI) -3.0 -2.0 0 Deletion LOD/sqrt(2) LOD/2 LOD Multi Impu -4.0 -6.0 -5.0 -7.0 1.0 2.0 MLE Impu -8.0

39. An overview of results Relative bias of methods is highly dependent on experimental conditions (i.e. no simple answers) Covariates and confounding matters! Simulations that only consider bivariate, X-Y relationships with LODs are limited

40. Deletion method results Surprisingly… provides unbiased estimates across all conditions! If sample size is large and LOD % is small, this may be a good option. As LOD % becomes larger, deletion is more costly Important caveat: deletion method works well if true associations are linear

41. Deletion method with linear effects Bottom 8% of X variable deleted

42. Substitution method results Not surprisingly… these methods are generally terrible! Just LOD substitution is worst type In most scenarios, these will bias associations towards the null … but, works reasonably well when distribution is highly skewed, no confounding, and LOD% is low

43. Multiple Imputation results Amelia II performs relatively well! Particularly when R 2 is higher Does well even when LOD% is high Problematic when there is no confounding (reason: this indicates there are no/weak associations between variables)

44. MLE Imputation results Associated with severe bias in most cases Highly reliant on parametric assumptions and the code is daunting: recommend avoiding this method However, performed reasonably well when exposure is normally distributed, no confounding, and LOD% is low

45. A Case Study…

46. Sarah’s SFF Analysis Study for Future Families (SFF): a multicenter pregnancy cohort study that recruited mothers from 1999-2005 Sarah Evans’ analysis: prenatal exposure to Bisphenol A (BPA) and neurobehavioral scores in 153 children at ages 6-10 28 (18%) children have BPA levels below the LOD

47. Sarah’s SFF Analysis Maternal urinary BPA collected during late pregnancy Neurobehavioral scores obtained through School- age Child Behavior Checklist (CBCL). Used multiple regression adjusting for child age at CBCL assessment, mother’s education level, family stress, urinary creatinine

48. Anxiety/Dep Withdrawn/Dep Somatic Social Thought Attention Rule-Break Aggressive Internalizing Externalizing Total Problems LOD/sqrt(2) -0.20-0.4-0.60.20.40.60.81.0 Deletion