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Multiple Imputation Applied to Bioassay 

Co-Exposure Models

ORAUT-RPRT-0096 Rev. 01

Effective Date: 06/11/2021

Supersedes: Revision 00

Subject Expert(s): Thomas R. LaBone

Document Owner 

Approval: Signature on File

Thomas R. LaBone, Document Owner

Approval Date: 05/17/2021

Concurrence: Signature on File

John M. Byrne, Objective 1 Manager

Concurrence Date: 05/17/2021

Concurrence: Vickie S. Short Signature on File for

Kate Kimpan, Project Director

Concurrence Date: 05/17/2021

Approval: Signature on File

Timothy D. Taulbee, Associate Director for Science

Approval Date: 06/11/2021

FOR DOCUMENTS MARKED AS A TOTAL REWRITE, REVISION, OR PAGE CHANGE, REPLACE THE PRIOR

REVISION AND DISCARD / DESTROY ALL COPIES OF THE PRIOR REVISION.

New Total Rewrite Revision Page Change
Document No. ORAUT-RPRT-0096 Revision No. 01 Effective Date: 06/11/2021 Page 2 of 20

 

PUBLICATION RECORD

EFFECTIVE 

DATE

REVISION 

NUMBER DESCRIPTION

01/24/2019 00 New report initiated to discuss applying the multiple imputation model 

to bioassay coworker models. Incorporates formal internal and 

NIOSH review comments. Training required: As determined by the 

Objective Manager. Initiated by Thomas R. LaBone.

06/11/2021 01 Revision initiated to incorporate experience gained from applying 

multiple imputation to recent co-exposure models. Only the new 

multiple imputation methods are discussed, whereas, Rev. 00 

included comparisons of the then current statistical methods for 

dealing with censored data with the new multiple imputation 

methods. In addition, Rev. 01 includes statistical methods (like the 

coworker R package) that were used for the first time in the INL coexposure statistical analysis. Incorporates formal internal and NIOSH 

review comments. Training required: As determined by the Objective 

Manager. Initiated by Thomas R. LaBone.
Document No. ORAUT-RPRT-0096 Revision No. 01 Effective Date: 06/11/2021 Page 3 of 20

 

TABLE OF CONTENTS

SECTION TITLE PAGE

Acronyms and Abbreviations ............................................................................................................... 4

1.0 Introduction .............................................................................................................................. 5

2.0 Purpose ................................................................................................................................... 6

3.0 Censored Data Using Informative Imputation........................................................................... 6

4.0 Censored Data Using Uninformative Imputation....................................................................... 9

5.0 Nonpositive Data Using Mixture Model Imputation ................................................................. 12

6.0 Nonpositive Data Using Lognormal Imputation....................................................................... 15

7.0 Summary ............................................................................................................................... 17

References ........................................................................................................................................ 20

LIST OF FIGURES

FIGURE TITLE PAGE

3-1 Normal q-q plot of 2,208 total uranium urine samples submitted by 427 workers in 1 year....... 7

3-2 Data from Figure 3-1 with CL of 25 dpm/d applied, generating censored data ......................... 7

3-3 Lognormal q-q plot of uncensored data with lognormal quantile line of lognormal fit to the 

uncensored data overlaid......................................................................................................... 8

3-4 Normal q-q plot of all data with lognormal quantile line of lognormal fit to the uncensored 

data overlaid ............................................................................................................................ 9

3-5 Uranium TWOPOS statistics derived from informative imputation for 427 workers 

generated from M = 10 iterations ............................................................................................. 9

4-1 Normal q-q plot of 61 241Am results from urine samples submitted by 60 workers .................. 10

4-2 Data from Figure 4-1 with CL of 0.04 dpm/L applied, generating censored data .................... 11

4-3 Figure 4-2 with loguniform quantile line overlaid..................................................................... 12

4-4 Americium TWOPOS statistics derived from uninformative imputation for 60 workers 

generated from M = 10 iterations ........................................................................................... 12

5-1 Figure 3-1 with the normal-lognormal quantile line overlaid.................................................... 13

5-2 PDF plots of the mixture distribution along with the normal and lognormal components of 

the mixture .............................................................................................................................14

5-3 Uranium TWOPOS statistics derived from mixture model imputation for 427 workers 

generated from M = 10 iterations ........................................................................................... 15

6-1 Lognormal q-q plot of the 49 positive 241Am results ................................................................ 16

6-2 Normal q-q plot of the 61 241Am results .................................................................................. 17

6-3 Americium TWOPOS statistics derived from log-normal imputation for 60 workers 

generated from M = 30 iterations ........................................................................................... 17
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ACRONYMS AND ABBREVIATIONS

CL censoring level

d day

DL decision level

dpm disintegrations per minute

GM geometric mean

GSD geometric standard deviation

IL imputation level

L liter

MDA minimum detectable amount

MI multiple imputation

NIOSH National Institute for Occupational Safety and Health

ORAU Oak Ridge Associated Universities

ORAUT ORAU Team

PDF probability density function

q-q quantile-quantile

SRDB Ref ID Site Research Database Reference Identification (number)

TWOPOS time-weighted one person–one statistic
Document No. ORAUT-RPRT-0096 Revision No. 01 Effective Date: 06/11/2021 Page 5 of 20

1.0 INTRODUCTION

Bioassay datasets that are used to calculate chronic intake summary statistics for a co-exposure 

model can contain nonpositive values (values less than or equal to zero) or censored values (values 

reported as less than a censoring level [CL]). The presence of nonpositive or censored values in a 

dataset complicates the statistical analysis and application of the data. Multiple imputation (MI) is a 

Monte Carlo method used to analyze datasets with missing data by replacing them with simulated 

data [Harel and Zhou 2006]. This versatile technique can also be used to replace nonpositive and 

censored data with positive (greater than zero) uncensored values below the CL. The positive 

uncensored datasets can then be analyzed with standard statistical methods for complete data. This 

report provides four illustrative examples of applying an MI approach based on the work of 

Krishnamoorthy et al. [2009] and Lubin et al. [2004] to uranium and americium urine bioassay 

datasets. The example cases are:

• Total uranium activity in urine bioassay data containing censored data, analyzed using an 

informative lognormal imputation model (Section 3.0);

• Americium-241 in urine bioassay data consisting almost entirely of censored data, analyzed 

using an uninformative loguniform imputation model (Section 4.0);

• Total uranium activity in urine bioassay data containing nonpositive data, analyzed using a 

normal-lognormal mixture imputation model (Section 5.0); and

• Americium-241 in urine bioassay data containing nonpositive data analyzed using a lognormal 

imputation model (Section 6.0).

An imputation model is a probability distribution that describes the distribution of the data (nonpositive 

or censored) that needs replacement with positive results. Random values are drawn from the 

imputation model, conditioned on the value being less than the CL for censored data or the imputation 

level (IL) for nonpositive data. The imputation model can be informative, based on observed positive 

or uncensored results, or uninformative, based on prior experience with the distribution of similar data.

Perhaps the most difficult part of using MI is developing the imputation model. For censored data, the 

probability distribution chosen for the imputation model should accurately describe the distribution of 

the data below the CL and, at the same time, return only positive values. There is no a priori preferred 

distribution to use for an imputation model, and the best model is suggested by what is known about 

the subject dataset. However, after experimenting with different EEOICPA datasets and different 

distributions, two default approaches were developed for use on the Project. The preferred imputation 

model is derived from fitting a lognormal distribution to the uncensored bioassay data in a given year 

to estimate the distribution of the censored data in that year. This is referred to as an “informative”

imputation model because it uses the uncensored data to estimate the distribution of the censored 

data. The imputed result is drawn from this lognormal distribution, conditioned on the result being less 

than the CL. The lognormal distribution was chosen because:

• It is arguable that this statistical distribution is the one most used in health physics to model 

right-skewed data, which means health physicists on the Project are comfortable working with 

it; and

• Less familiar, strictly nonnegative distributions, like the Weibull distribution, generate results 

similar to those obtained with the lognormal.
Document No. ORAUT-RPRT-0096 Revision No. 01 Effective Date: 06/11/2021 Page 6 of 20

 

The second type of imputation model is generated by assuming a distribution for the censored data 

that is not based on the uncensored data. This is referred to as an “uninformative” imputation model 

and is used for datasets that consist primarily of censored data.

The concept of bioassay results that are measured to be less than zero is problematic because some 

stakeholders misinterpret such results as implying that the true result is negative (which is not the 

case). An approach to solving this problem that is favorable to claimants is to impute positive results 

for nonpositive results in a way that is analogous to how positive results are imputed for censored 

results. In this application (where there are no censored data), the imputation model is created by 

fitting a lognormal model to the positive data and then using it to impute nonpositive results. 

Imputation models for nonpositive data are always informative because the values for all the data are 

known. 

2.0 PURPOSE

The purpose of this report is to present idealized examples that illustrate how MI can be used to 

convert datasets containing censored or nonpositive data into datasets that contain only positive 

uncensored data. This report is not intended to be a procedure that instructs how to perform MI with 

any given real-world dataset and all its inherent peculiarities. Therefore, specific parameters (i.e., CLs, 

distributions, number of iterations) used in the provided examples are for illustrative purposes only.

3.0 CENSORED DATA USING INFORMATIVE IMPUTATION

A quantile-quantile (q-q) plot of the daily excretion of total uranium activity in 2,208 urine samples 

submitted in 1 year by 427 workers is shown in Figure 3-1. This is a complete dataset that is not 

censored and contains nonpositive results. With a dataset consisting entirely of positive results with 

no censoring, the co-exposure modeling would proceed as follows:

1. Generate the time-weighted one person–one statistic (TWOPOS) results for each person for 

each year from the positive bioassay data.

2. Fit lognormal distributions to the TWOPOS data in each year to produce a geometric mean 

(GM) and geometric standard deviation (GSD) for each year.

The parametric 50th and 84th percentiles from the lognormal distributions in step 2 are the input 

parameters used to calculate the chronic intake rates. Calculating TWOPOS statistics with censored 

data is problematic, in the past requiring arbitrary substitutions that produce overly conservative 

results.1

1 Such as the maximum possible mean, for example.

In practice, complete datasets were often not reported by the site, which instead reported a 

censored dataset. The censored dataset resulted from applying one or more CLs to the complete 

dataset, with the censored results being reported simply as “<CL.”2 For this example, a simulated 

censored dataset was created by applying a CL of 25 dpm/d to the dataset in Figure 3-1, resulting in 

the censored dataset in Figure 3-2. 

 

2 Where CL could be a number or something like the decision level (DL) or minimum detectable amount (MDA), for 

example, <25, <DL, and <MDA.
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Figure 3-1. Normal q-q plot of 2,208 total uranium urine samples 

submitted by 427 workers in 1 year.

All the data below the dashed red line were reported as <25 dpm/d. That is, the actual values for the 

results represented by the grey dots were not known to the statistician analyzing the data.

Figure 3-2. Data from Figure 3-1 with CL of 25 dpm/d (red line) 

applied, generating censored data (grey points).

The key idea behind MI is to first determine the probability distribution that best approximates the 

distribution of the true values of the censored data, which is called the “imputation model.” Note that 

the imputation model can be any statistical distribution, but it must have its support on the positive real 
Document No. ORAUT-RPRT-0096 Revision No. 01 Effective Date: 06/11/2021 Page 8 of 20

 

line.

3

3 Note that this is not a constraint on the imputation model in general but rather one specific to the Project.

Next, the censored data were replaced with random draws from the imputation model 

conditioned on the draw being less than the CL. The dataset now consists of positive results that can 

be analyzed with methods for complete datasets. Note that MI does not pretend to create information 

through use of the simulated values but rather to present the data in a way that makes them 

amenable to analysis with those methods [Rubin 1996]. 

An example of an informative imputation model is shown in Figure 3-3, where a lognormal distribution 

is fit to the uncensored uranium data using regression on order statistics. The resulting lognormal 

imputation model has a GM of 13.403 and a GSD of 2.518. The statistician is most concerned with the 

imputation model’s goodness of fit in the region just above the CL because it is assumed that the 

distribution of censored results is best approximated by the fit in this region. Therefore, it is not 

necessary for the imputation model to have good agreement with the higher level values. In any 

event, this imputation model fits the entire uncensored dataset very well.

Figure 3-3. Lognormal q-q plot of uncensored data with lognormal 

quantile line (blue) of lognormal fit to the uncensored data overlaid.

Two results greater than 200 dpm/d were excluded from the fit.

The q-q plot in Figure 3-4 shows the lognormal quantile line from the imputation model overlaid on the 

complete dataset. The imputation model and data agree very well until the data approach zero and 

take on negative values. The imputation model asymptotically approaches zero in this region, which 

means that imputed values will overestimate the true values (i.e., be favorable to claimants).

After the censored values in the dataset are imputed, TWOPOS statistics are calculated and a 

lognormal distribution fit to them. For this example, this was repeated M = 10 times and the 10 GMs 

averaged to give the mean GM of 15.84 dpm/d and the 10 GSDs averaged to give the mean GSD of 

1.787.4

4 The values of the log(GM) are averaged and then exponentiated to give the mean GM. The same procedure is followed 

for the GSD.

The data from the 10 iterations and the final lognormal fit are shown in Figure 3-5.

 
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Figure 3-4. Normal q-q plot of all data with lognormal quantile line 

(blue) of lognormal fit to the uncensored data overlaid.

Figure 3-5. Uranium TWOPOS statistics derived from informative 

imputation for 427 workers generated from M = 10 iterations. The 

mean GM and mean GSD for the lognormal fit to the statistics are 

given along with the lognormal quantile line of the fit (blue line).

4.0 CENSORED DATA USING UNINFORMATIVE IMPUTATION

Bioassay datasets consisting primarily (or even entirely) of censored data are not uncommon. 

Although much is known about such datasets, they are difficult to use to assign a probability 

distribution to the data. For example, consider the uncensored 241Am data shown in Figure 4-1. 
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Figure 4-1. Normal q-q plot of 61 241Am results from urine samples 

submitted by 60 workers.

Applying a representative CL of 0.04 dpm/L to these data gives the plot in Figure 4-2. Note that the 

data below the CL (the dashed red line) are not observed (having been replaced by the CL). One way 

to approach this issue is to impute values for the censored results by drawing a random value from a 

probability distribution that is bounded at the upper end with a maximum equal to the CL. The 

loguniform LU distribution has this property and has proven to be very useful for this purpose. A 

random variable X is loguniformly distributed if its logarithm is uniformly U distributed:

log( ) (log( ),log( )) XU a b ∼ (4-1)

where

log(a) = the lower limit of U

log(b) = the upper limit of U

This is expressed in terms of X using the notation:

X LU a b U a b ∼ = ( , ) exp[ (log( ),log( ))] (4-2)
Document No. ORAUT-RPRT-0096 Revision No. 01 Effective Date: 06/11/2021 Page 11 of 20

 

Figure 4-2. Data from Figure 4-1 with CL of 0.04 dpm/L (red line) 

applied, generating censored data (grey points).

The value of the upper limit b should clearly be the CL, but there is no obvious way to select the lower 

limit a based on statistical theory. In practice, the lower limit is taken to be the fraction of the CL that 

results in a GSD of approximately 3 in the TWOPOS dataset. This GSD was selected to be consistent 

with the guidance given in Section 2.5.3 in ORAUT-OTIB-0060, Internal Dose Reconstruction [Oak 

Ridge Associated Universities (ORAU) Team (ORAUT) 2018]. Therefore:

= > , 0 CLa a

K (4-3)

where the choice of K gives a GSD of approximately 3 in the TWOPOS dataset. Therefore, K must be 

selected iteratively by choosing a value for K, examining how it affects the GSD of the TWOPOS data, 

and then selecting a new value of K. 

As an example, after several iterations, K = 50 was chosen to use in LU(0.04 ÷ 50,0.04) to generate 

the quantile function denoted by the blue line in Figure 4-3, which matches the positive portion of the 

data very well and gives a GSD of approximately 3. 

After imputing the censored data from the loguniform distribution, the TWOPOS statistics are 

calculated for each person and then a lognormal distribution is fit to the TWOPOS data. This is 

repeated M = 10 times, and the 10 GMs and 10 GSDs are averaged to give the final co-exposure 

model parameters for the year, as shown in Figure 4-4. 
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Figure 4-3. Figure 4-2 with loguniform quantile line (blue line) 

overlaid.

Figure 4-4. Americium TWOPOS statistics derived from 

uninformative imputation for 60 workers generated from M = 10 

iterations. The mean GM and mean GSD for the lognormal fit to the 

statistics are given along with the lognormal quantile line of the fit 

(blue line).

5.0 NONPOSITIVE DATA USING MIXTURE MODEL IMPUTATION

Figure 3-1 shows the normal q-q plot for a urine bioassay dataset for total uranium activity that was 

complete in the sense that none of the data were censored, with nonpositive and positive data being 
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reported. One way of obtaining the imputation model for the complete uranium dataset is to model it 

as a mixture of a normal distribution and a lognormal distribution (i.e., a normal-lognormal mixture 

model). The normal component of the mixture can be viewed as the analytical noise generated when 

samples containing approximately the same levels of uranium are subtracted from each other, while 

the lognormal component is viewed as being representative of the real exposures to uranium. The 

data plotted in Figure 3-1 were fit with a normal-lognormal mixture using an expectation-maximization

algorithm from the coworker R library [ORAUT 2020] with the following results: 

• A normal mixing fraction of fn = 0.712 (i.e., the probability of drawing a normally distributed 

random number from the mixture distribution is 0.712); 

• A mean of μ = 8.04 dpm/d for the normal distribution; 

• A standard deviation of σ = 9.99 for the normal distribution; 

• A lognormal mixing fraction of 1 − fn = 0.288; 

• A logarithmic mean for the lognormal distribution of μlog = 3.58, which equates to a GM of 35.9

dpm/d; and

• A logarithmic standard deviation for the lognormal distribution of σlog = 0.618, which equates to 

a GSD of 1.86. 

The quantile function of this mixture model is superimposed on the uranium data in Figure 5-1, 

showing excellent agreement with the data. 

Figure 5-1. Figure 3-1 with the normal-lognormal quantile line (in 

blue) overlaid.

It is useful to examine the probability density functions (PDFs) of the mixture model and its normal 

and lognormal components as shown in Figure 5-2. The PDF of the mixture model is simply the 

pointwise sum of the normal and lognormal PDFs. 
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Figure 5-2. PDF plots of the mixture distribution (black line) along 

with the normal (red line) and lognormal (blue line) components of 

the mixture.

Once the mixture model is obtained, the normal component is used to define the IL and the lognormal 

component is used to define the imputation model. The IL is the level below which nonpositive results 

will be imputed (note that positive results below the IL are not usually imputed). The imputation model 

is the distribution from which random draws are made, conditioned on the draw being below the IL, 

with a nonpositive result below the IL being replaced (imputed) with the positive random draw. For coexposure models, the IL is usually defined empirically as: 

IL = + 2σ μ (5-1)

where μ is the mean and σ is the standard deviation of the normal component. In this case: 

IL = += 2(9.99) 8.04 28.02 (5-2)

Therefore, all the nonpositive results less than 28.02 dpm/d are replaced with random draws from a 

lognormal distribution with a GM of 35.9 dpm/d and a GSD of 1.86, conditioned on being less than 

28.02 dpm/d.

After the nonpositive results are imputed, TWOPOS statistics are calculated and the process 

proceeds in exactly the same manner as for the censored data. The TWOPOS statistics for M = 10 

iterations and the resulting lognormal fit are shown in Figure 5-3. 
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Figure 5-3. Uranium TWOPOS statistics derived from mixture 

model imputation for 427 workers generated from M = 10 iterations. 

The mean GM and mean GSD for the lognormal fit to the statistics 

are given, along with the lognormal quantile line of the fit (blue 

line).

6.0 NONPOSITIVE DATA USING LOGNORMAL IMPUTATION

The best imputation model for some bioassay datasets with nonpositive values is based on a normal 

distribution rather than a normal-lognormal mixture (i.e., the data are all noise with no signal). In such 

cases, the upper tail of the normal distribution is fit with a lognormal distribution, giving a lognormal 

imputation model. For example, assume the complete 241Am dataset shown in Figure 4-1. As shown 

in Figure 6-1, there are 49 positive results of which 31 are greater than 0.01 dpm/L (the red line). 

These were fit with a lognormal distribution (the blue line). The lognormal imputation model has a GM

of 0.0113 dpm/L and a GSD of 2.02. 
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Figure 6-1. Lognormal q-q plot of the 49 positive 241Am results. The 

31 results above the red line at 0.01 dpm/L were fit with a 

lognormal distribution, which has the lognormal quantile line in 

blue. 

This lognormal imputation model is compared with the complete dataset plot in Figure 6-2. The IL is 

typically an empirically chosen value between zero and the lowest datum used in the lognormal 

imputation model. Note that the imputed dataset is favorable to claimants regardless of the IL chosen 

because the normally distributed data (with nonpositive values) have been replaced with lognormally 

distributed data (with all positive values). The resulting TWOPOS model for 1994 is shown in 

Figure 6-3.
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Figure 6-2. Normal q-q plot of the 61 241Am results. The lognormal 

quantile line of the fit to the 31 results above 0.01 dpm/L is shown 

in blue. 

Figure 6-3. Americium TWOPOS statistics derived from log-normal 

imputation for 60 workers generated from M = 30 iterations. The 

mean GM and mean GSD for the lognormal fit to the statistics are 

given, along with the lognormal quantile line of the fit (blue line).

7.0 SUMMARY

This report discusses the use of MI to analyze datasets containing censored or nonpositive data. The 

key step in MI is the imputation model, which is used to predict what the censored data should look 
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like or give a plausible overestimate of nonpositive data. The end result is a positive dataset that is 

analyzed using statistical methods for complete datasets. The typical MI procedure can be broken 

down into seven steps, the first two of which are different in each example case.

Censored Data Using Informative Imputation

1. Estimate how the bioassay results in each year are distributed based on the uncensored data, 

giving the imputation models for each year.

2. Replace censored bioassay results below the CLs in all years with random draws from the 

appropriate imputation models, conditioned on the value being less than the CL.

Censored Data Using Uninformative Imputation

1. Estimate how the censored bioassay results in each year are distributed using a loguniform

imputation model having an upper limit equal to the CL and a lower limit equal to the CL

divided by a constant K > 1, which results in a lognormal fit to the TWOPOS data with a GSD 

approximately equal to 3.

2. Replace censored bioassay results below the CL in all years with random draws from the 

appropriate imputation models, conditioned on the value being less than the CL.

Nonpositive Data Using Mixture Imputation

1. Estimate how the bioassay results in each year are distributed using a normal-lognormal 

mixture model. The imputation models are given by the lognormal components of the mixture 

and the ILs by the normal components.

2. Replace nonpositive bioassay results below the IL in all years with random draws from the 

appropriate imputation models, conditioned on the value being less than the IL.

Nonpositive Data Using Lognormal Imputation

1. Estimate how the positive bioassay results in each year are distributed by fitting a lognormal 

model to the upper tail of the data. The ILs are chosen empirically. 

2. Replace nonpositive bioassay results below the IL in all years with random draws from the 

appropriate imputation models, conditioned on the value being less than the IL.

The remaining steps are identical for all four cases. 

3. Generate the TWOPOS results for each person for each year from the now positive bioassay 

data.

4. Fit lognormal distributions to the TWOPOS data in each year to produce a GM and GSD for 

each year.

5. Repeat steps 2 to 4 M times, which generates M GMs and M GSDs in each year. 

6. The means of the M GMs and M GSDs in a given year are taken to be the final GM and GSD 

for that year. 

7. Repeat step 6 for each year to produce the summary statistics that are modeled to calculate 

the intake rate.

In conclusion, MI is not a panacea for nonpositive or censored datasets and, just like all methods, it 

can be difficult to implement with some datasets. Nevertheless, it can provide: 
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• Parameter estimates for highly censored and completely censored datasets that are more 

realistic than those given by previous methods (i.e., binomial fits); and

• Parameter estimates for nonpositive datasets, a problem for which there was previously no 

accepted solution.

In addition, MI does provide parameter estimates for censored datasets that are consistent with those 

obtained with previous methods.

Analysis of real-world bioassay datasets is likely to require modifications of the standard MI 

approaches illustrated in this report. Such modifications will require that the analysis be performed by 

someone with formal training in statistics.
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