comparing methods for addressing limits of detection in environmental epidemiology roni kobrosly,...
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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
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A familiar diagram…
EnvironmentalExposure
InternalDose
BiologicallyEffective
Dose
AlteredStructure/Function
ClinicalDisease
Biomarker of Exposure
DeCaprio, 1997
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Biomarkers and Limits of Detection (LOD)
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It is difficult to quantify the concentration because it is so low
LOD
Higher concentration
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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
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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
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Citations in Google Scholar
Hornung & Reed, 1990
19901992
19941996
19982000
20022004
20062008
20102012
0102030405060708090100
Year
Nu
mb
er
of
Pu
blica
tio
ns
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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!
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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
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Statistical Bias
Nat’l Library of Med definition: “Any deviation of results or inferences from the truth”
Unbiased Biased
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Variable Definitions
• Four continuous variables:
• Y: Dependent variable (outcome)
• X: Independent variable (exposure, subject to LOD)
• C1, C2: Independent variables (covariates)
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6 “Experimental Conditions”
1) Dataset sample size: n = {100, 500}
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2) % of exposure variable with values in LOD region:
LOD% = {0.05, 0.25}
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3) Distribution of Exposure Variable:
Normal versus Skewed
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4) R2 of full model:
R2 = {0.10, 0.20}
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5) Strength & direction of exposure-outcome association:
Beta = {-10, 0, 10}
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6) Direction of confounding:
Strong Positive, versus Strong Negative, versus None
+-
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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)
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Method 1: Deletion
Y X C1 C2
167.7 25.8 13.5 12.9
-66.3 15.9 11.7 12.6
50.6 <LOD 10.4 10.8
-273.0 9.5 11.8 11.1
156.9 <LOD 12.6 9.0
Y X C1 C2
167.7 25.8 13.5 12.9
-66.3 15.9 11.7 12.6
50.6 <LOD 10.4 10.8
-273.0 9.5 11.8 11.1
156.9 <LOD 12.6 9.0
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Method 2: Sub with LOD/√(2)
Y X C1 C2
167.7 25.8 13.5 12.9
-66.3 15.9 11.7 12.6
50.6 <LOD 10.4 10.8
-273.0 9.5 11.8 11.1
156.9 <LOD 12.6 9.0
LODX = 9.0
Y X C1 C2
167.7 25.8 13.5 12.9
-66.3 15.9 11.7 12.6
50.6 6.4 10.4 10.8
-273.0 9.5 11.8 11.1
156.9 6.4 12.6 9.0
9.0/√2 = 6.4
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Method 3: Sub with LOD/(2)
Y X C1 C2
167.7 25.8 13.5 12.9
-66.3 15.9 11.7 12.6
50.6 <LOD 10.4 10.8
-273.0 9.5 11.8 11.1
156.9 <LOD 12.6 9.0
LODX = 9.0
Y X C1 C2
167.7 25.8 13.5 12.9
-66.3 15.9 11.7 12.6
50.6 4.5 10.4 10.8
-273.0 9.5 11.8 11.1
156.9 4.5 12.6 9.0
9.0/2 = 4.5
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Method 4: Sub with just LOD
Y X C1 C2
167.7 25.8 13.5 12.9
-66.3 15.9 11.7 12.6
50.6 <LOD 10.4 10.8
-273.0 9.5 11.8 11.1
156.9 <LOD 12.6 9.0
LODX = 9.0
Y X C1 C2
167.7 25.8 13.5 12.9
-66.3 15.9 11.7 12.6
50.6 9.0 10.4 10.8
-273.0 9.5 11.8 11.1
156.9 9.0 12.6 9.0
9.0
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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!)
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Method 5: Multiple Imputation
Y X C1 C2
167.7 25.8 13.5 12.9
-66.3 15.9 11.7 12.6
50.6 <LOD 10.4 10.8
-273.0 9.5 11.8 11.1
156.9 <LOD 12.6 9.0
Y X C1 C2
167.7 25.8 13.5 12.9
-66.3 15.9 11.7 12.6
50.6 3.0 10.4 10.8
-273.0 9.5 11.8 11.1
156.9 6.2 12.6 9.0
Y X C1 C2
167.7 25.8 13.5 12.9
-66.3 15.9 11.7 12.6
50.6 2.5 10.4 10.8
-273.0 9.5 11.8 11.1
156.9 6.8 12.6 9.0
Y X C1 C2
167.7 25.8 13.5 12.9
-66.3 15.9 11.7 12.6
50.6 3.3 10.4 10.8
-273.0 9.5 11.8 11.1
156.9 6.3 12.6 9.0
Y X C1 C2
167.7 25.8 13.5 12.9
-66.3 15.9 11.7 12.6
50.6 3.5 10.4 10.8
-273.0 9.5 11.8 11.1
156.9 6.0 12.6 9.0
Y X C1 C2
167.7 25.8 13.5 12.9
-66.3 15.9 11.7 12.6
50.6 2.8 10.4 10.8
-273.0 9.5 11.8 11.1
156.9 7.2 12.6 9.0
M = 5
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Method 5: Multiple ImputationY X C1 C2
167.7 25.8 13.5 12.9
-66.3 15.9 11.7 12.6
50.6 3.0 10.4 10.8
-273.0 9.5 11.8 11.1
156.9 6.2 12.6 9.0
Y X C1 C2
167.7 25.8 13.5 12.9
-66.3 15.9 11.7 12.6
50.6 2.5 10.4 10.8
-273.0 9.5 11.8 11.1
156.9 6.8 12.6 9.0
Y X C1 C2
167.7 25.8 13.5 12.9
-66.3 15.9 11.7 12.6
50.6 3.3 10.4 10.8
-273.0 9.5 11.8 11.1
156.9 6.3 12.6 9.0
Y X C1 C2
167.7 25.8 13.5 12.9
-66.3 15.9 11.7 12.6
50.6 3.5 10.4 10.8
-273.0 9.5 11.8 11.1
156.9 6.0 12.6 9.0
Y X C1 C2
167.7 25.8 13.5 12.9
-66.3 15.9 11.7 12.6
50.6 2.8 10.4 10.8
-273.0 9.5 11.8 11.1
156.9 7.2 12.6 9.0
= 10.01
β1 = 10.1
β2 = 9.5 β3 = 8.3 β4 = 12.1
β5 = 10.4
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Method 6: MLE-Imputation
Y X C1 C2
167.7 25.8 13.5 12.9
-66.3 15.9 11.7 12.6
50.6 <LOD 10.4 10.8
-273.0 9.5 11.8 11.1
156.9 <LOD 12.6 9.0
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Method 6: MLE-Imputation
Y X C1 C2
167.7 25.8 13.5 12.9
-66.3 15.9 11.7 12.6
50.6 <LOD 10.4 10.8
-273.0 9.5 11.8 11.1
156.9 <LOD 12.6 9.0
Assume normal distribution, estimate and Sx
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Method 6: MLE-Imputation
Y X C1 C2
167.7 25.8 13.5 12.9
-66.3 15.9 11.7 12.6
50.6 3.2 10.4 10.8
-273.0 9.5 11.8 11.1
156.9 5.8 12.6 9.0
Use estimated LOD value, , and Sx to randomly generate observations below LOD
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Two-step Data Generation Process
• 1st Step: Select “true” regression parameters for following two models:
o
o
• 2nd Step: Use “true” parameters to guide the drawing of random numbers
X 0 _ X C1_ X (C1) C 2 _ X (C2)
Y 0 _ Y X (X) C1_ Y (C1) C 2 _ Y (C2)
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“TRUTH”Y = 2.8 + 2(X) + 4.5(C1) + 6(C2)
Dataset1.1 Dataset1.2 Dataset1.3SIMULATED DATASETS
X = 1.3 - 6(C1) + 1.5(C2)
Obs # Y X C1 C2
1 24.67 5.44 -0.28 1.77
2 30.73 9.47 -1.55 -0.81
3 19.39 -0.98 0.96 0.92
4 -9.47 -8.20 1.72 0.49
i yi xi c1i c2i
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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
ˆ y 2.72 2.2(X) 4.2(C1) 5.98(C2)
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Help from Minerva
Minerva runtime ~ 5 minutes
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n = 100, 25% LOD, Skewed Dist, R2 = 0.20, Negative X-Y Association, Negative confounding
Mea
n B
ias
(wit
h 9
5% C
I)
3.0
4.0
5.0
6.0
Deletion
LOD/sqrt(2)
LOD/2
LOD
Multi Impu
2.0
0
1.0
-1.0
7.0
8.0
MLE Impu
-2.0
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Mea
n B
ias
(wit
h 9
5% C
I)
-3.0
-2.0
-1.0
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, R2 = 0.20, Positive X-Y Association, Negative confounding
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n = 100, 25% LOD, Skewed Dist, R2 = 0.20, Negative X-Y Association, No confounding
Mea
n B
ias
(wit
h 9
5% C
I)
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
-1.0
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n = 100, 25% LOD, Skewed Dist, R2 = 0.20, Positive X-Y Association, No confounding
Mea
n B
ias
(wit
h 9
5% C
I)
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
-1.0
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n = 100, 25% LOD, Skewed Dist, R2 = 0.20, Negative X-Y Association, Positive confounding
Mea
n B
ias
(wit
h 9
5% C
I)
3.0
4.0
5.0
6.0
Deletion
LOD/sqrt(2)
LOD/2
LOD
Multi Impu
2.0
0
1.0
-1.0
7.0
8.0
MLE Impu
-2.0
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n = 100, 25% LOD, Skewed Dist, R2 = 0.20, Positive X-Y Association, Positive confounding
Mea
n B
ias
(wit
h 9
5% C
I)
-3.0
-2.0
-1.0
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
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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
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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
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Deletion method with linear effects
Bottom 8% of X variable deleted
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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
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Multiple Imputation results
• Amelia II performs relatively well! Particularly when R2 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)
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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
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A Case Study…
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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
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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
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Anxiety/Dep
Withdrawn/Dep
Somatic
Social
Thought
Attention
Rule-Break
Aggressive
Internalizing
Externalizing
Total Problems
LOD/sqrt(2)
-0.2 0-0.4-0.6 0.2 0.4 0.6 0.8 1.0
Deletion