DOE Review Release 08/25/2023

Page 1 of 48

Applications of Regression Models and 

Ratios

ORAUT-RPRT-0106 Rev. 00

Effective Date: 08/03/2023

Supersedes: None

Subject Expert(s): Nancy Chalmers, Daniel Sheanshang, and Thomas R. LaBone

Document Owner 

Approval: Signature on File

Nancy Chalmers, Document Owner

Approval Date: 08/01/2023

Concurrence: Signature on File

John M. Byrne, Objective 1 Manager

Concurrence Date: 08/02/2023

Concurrence: Vickie S. Short Signature on File for

Kate Kimpan, Project Director

Concurrence Date: 08/01/2023

Approval: Signature on File

Timothy D. Taulbee, Associate Director for Science

Approval Date: 08/03/2023

FOR DOCUMENTS MARKED AS A TOTAL REWRITE OR REVISION, 

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

New Total Rewrite Revision
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 2 of 48

 

PUBLICATION RECORD

EFFECTIVE

DATE

REVISION

NUMBER DESCRIPTION

08/03/2023 00 New report generated to illustrate the benefits and detriments of 

using regression models versus ratios. Incorporates formal internal 

and NIOSH review comments. Training is not required. Initiated by 

John M. Byrne and authored by Nancy Chalmers.

TRADEMARK INFORMATION

Minitab® is a registered trademark of Minitab in the United States and/or other countries.

All other trademarks are the property of their respective owners.
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 3 of 48

 

TABLE OF CONTENTS

SECTION TITLE PAGE

Acronyms and Abbreviations ............................................................................................................... 6

1.0 Introduction .............................................................................................................................. 7

2.0 Purpose ................................................................................................................................... 8

3.0 X and Y Have a Relationship and β = 1.................................................................................... 9

3.1 Linear Regression ...................................................................................................... 10

3.2 ROS with Ratios ......................................................................................................... 10

3.3 Comparison of Regression and Ratios ....................................................................... 11

4.0 X and Y Have a Relationship and 0 < β < 1............................................................................ 13

4.1 Linear Regression ...................................................................................................... 14

4.2 ROS with Ratios ......................................................................................................... 14

4.3 Comparison of Regression and Ratios ....................................................................... 15

5.0 X and Y Have a Relationship and β > 1.................................................................................. 17

5.1 Linear Regression ...................................................................................................... 17

5.2 ROS with Ratios ......................................................................................................... 18

5.3 Comparison of Regression and Ratios ....................................................................... 19

6.0 X and Y Have No Relationship, which Implies β = 0............................................................... 20

6.1 Linear Regression ...................................................................................................... 21

6.2 ROS with Ratios ......................................................................................................... 22

6.3 Comparison of Regression and Ratios ....................................................................... 24

7.0 Effect of β on the Ratio Method.............................................................................................. 25

8.0 Summary ............................................................................................................................... 25

References ........................................................................................................................................ 27

ATTACHMENT A FERNALD THORIUM DATA: X AND Y HAVE A RELATIONSHIP AND 

β = 1 ...............................................................................................................28

ATTACHMENT B PADUCAH TECHNETIUM DATA: X AND Y HAVE A RELATIONSHIP 

AND 0 < β < 1................................................................................................. 33

ATTACHMENT C MOUND NEUTRON SOURCE DATA: X AND Y HAVE A RELATIONSHIP 

AND β > 1 ....................................................................................................... 38

ATTACHMENT D ROCKY FLATS NEUTRON AND GAMMA DOSE DATA: X AND Y HAVE 

NO RELATIONSHIP, WHICH IMPLIES β = 0.................................................. 43

LIST OF TABLES

TABLE TITLE PAGE

3-1 Parameter estimates for regression using Equation 1-3 ......................................................... 10

3-2 Parameter estimates for lognormal ROS with ratios............................................................... 11
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4-1 Parameter estimates for regression using Equation 1-3 ......................................................... 14

4-2 Parameter estimates for lognormal ROS with ratios............................................................... 15

5-1 Parameter estimates for regression using Equation 1-3 ......................................................... 17

5-2 Parameter estimates for lognormal ROS with ratios............................................................... 18

6-1 Parameter estimates for regression using Equation 1-3 ......................................................... 21

6-2 Parameter estimates for lognormal ROS with ratios............................................................... 22

A-1 Parameter estimates for regression ....................................................................................... 29

A-2 Parameter estimates for lognormal ROS with ratios............................................................... 30

B-1 Parameter estimates for regression ....................................................................................... 34

B-2 Parameter estimates for lognormal ROS with ratios............................................................... 35

C-1 Parameter estimates for regression ....................................................................................... 39

C-2 Parameter estimates for lognormal ROS with ratios............................................................... 40

D-1 Parameter estimates for regression ....................................................................................... 44

D-2 Parameter estimates for lognormal ROS with ratios............................................................... 45

LIST OF FIGURES

FIGURE TITLE PAGE

3-1 Data simulated with β = 1, μ = 3, and σ = 0.2........................................................................... 9

3-2 Data simulated with β = 1, μ = 3, and σ = 0.2 and the associated linear regression fit............ 10

3-3 Data simulated with β = 1, μ = 3, and σ = 0.2 and the associated lognormal ROS with 

ratios fit ..................................................................................................................................11

3-4 Data simulated with β = 1, μ = 3, and σ = 0.2 and the associated linear regression and 

lognormal ROS with ratios fits ................................................................................................ 12

4-1 Data simulated with β = 0.5, μ = 3, and σ = 0.2...................................................................... 13

4-2 Data simulated with β = 0.5, μ = 3, and σ = 0.2 and the associated linear regression fit......... 14

4-3 Data simulated with β = 0.5, μ = 3, and σ = 0.2 and the associated lognormal ROS with 

ratios fit ..................................................................................................................................15

4-4 Data simulated with β = 0.5, μ = 3, and σ = 0.2 and the associated linear regression and 

lognormal ROS with ratios fits ................................................................................................ 16

5-1 Data simulated with β = 2, μ = 3, and σ = 0.2......................................................................... 17

5-2 Data simulated with β = 2, μ = 3, and σ = 0.2 and the associated linear regression fit............ 18

5-3 Data simulated with β = 2, μ = 3, and σ = 0.2 and the associated lognormal ROS with 

ratios fit ..................................................................................................................................19

5-4 Data simulated with β = 2, μ = 3, and σ = 0.2 and the associated linear regression and 

lognormal ROS with ratios fits ................................................................................................ 20

6-1 Data simulated from two independent lognormal distributions................................................ 21

6-2 Data simulated from two independent lognormal distributions and the associated linear 

regression fit ..........................................................................................................................22

6-3 Data simulated from two independent lognormal distributions and the associated 

lognormal ROS with ratios fit .................................................................................................. 23

6-4 Data simulated from two independent lognormal distributions and the associated linear 

regression and lognormal ROS with ratios fits........................................................................ 24

A-1 Fernald thorium data from 1994 to 1996 ................................................................................ 29

A-2 Fernald thorium data from 1994 to 1996 and the associated linear regression fit ................... 30

A-3 Fernald thorium data from 1994 to 1996 and the associated lognormal ROS with ratios fit .... 31

A-4 Fernald thorium data from 1994 to 1996 and the associated linear regression and 

lognormal ROS with ratios fits ................................................................................................ 32

B-1 Paducah annual air data from 1980 to 1999 (excluding 1996 and 1997)................................ 34

B-2 Paducah annual air data from 1980 to 1999 (excluding 1996 and 1997) and the 

associated linear regression fit............................................................................................... 35
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B-3 Paducah annual air data from 1980 to 1999 (excluding 1996 and 1997) and the 

associated lognormal ROS with ratios fit................................................................................ 36

B-4 Paducah annual air data from 1980 to 1999 (excluding 1996 and 1997) and the 

associated linear regression and lognormal ROS with ratios fits ............................................ 37

C-1 Mound neutron and photon data from NS in 1954 and 1957 .................................................. 39

C-2 Mound neutron and photon data from NS in 1954 and 1957 and the associated linear 

regression fit ..........................................................................................................................40

C-3 Mound neutron and photon data from NS in 1954 and 1957 and the associated 

lognormal ROS with ratios fit.................................................................................................. 41

C-4 Mound neutron and photon data from NS in 1954 and 1957 and the associated linear 

regression and lognormal ROS with ratios fits........................................................................ 42

D-1 Rocky Flats neutron and gamma data from 1969................................................................... 44

D-2 Rocky Flats neutron and gamma data from 1969 and the associated linear regression fit...... 45

D-3 Rocky Flats neutron and gamma data from 1969 and the associated lognormal ROS 

with ratios fit...........................................................................................................................46

D-4 Rocky Flats neutron and gamma data from 1969 and the associated linear regression 

and lognormal ROS with ratios fits ......................................................................................... 47
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ACRONYMS AND ABBREVIATIONS

ABRWH Advisory Board on Radiation and Worker Health

Bq becquerel

GM geometric mean

GSD geometric standard deviation

mrem millirem

NIOSH National Institute for Occupational Safety and Health

NS Neutron Source

ORAU Oak Ridge Associated Universities

ORAUT ORAU Team

ROS regression on order statistics

SRDB Ref ID Site Research Database Reference Identification (number)

yr year
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1.0 INTRODUCTION

Datasets often have a set of paired quantities Y and X, and it is sometimes helpful to determine a 

predictive relationship between them. For example, suppose a dataset contains the neutron dose Y

and the paired photon dose X for a number of dosimeters. A relationship can be established between 

the measured neutron doses and photon doses so that, when given a new measured photon dose, 

the model can be used to predict the associated unmeasured neutron dose.

If a proportional linear relationship exists between X and Y, a linear regression of Y on X through the 

origin is used to calculate the ratio of Y to X: 

Y X = + β ε (1-1)

where

Y = the response variable

β = the slope parameter

X = the predictor variable

ε = a random error term, assume ε is normally distributed with mean 0 and standard 

deviation σ, which can be denoted as ε ~ N(0,σ)

To estimate Y given X, simply multiply X by the slope β, which is the mean ratio Y/X. 

If a linear relationship exists between X and Y but is not proportional, a linear regression of Y on X

(not through the origin) is used to calculate the relationship of Y to X [Weisberg 2005, pp. 48–49]:

Y X =+ + μβ ε (1-2)

where

μ = the intercept parameter

To estimate Y given X, simply multiply X by the slope β and add the intercept μ.

When a linear relationship does not exist between X and Y, one option is to transform X and/or Y so 

that linear modeling techniques are appropriate. For data typically encountered in health physics, the 

log transform of X and Y is often a good choice:

log( ) ( ) Y X =+ + μ βlog ε (1-3)

where

log = natural log (base e) 

It is possible that the relationship between X and Y is nonlinear and cannot be transformed into a 

linear form. Those relationships would be handled with other statistical techniques that are outside of 

the scope of this report. The remainder of this report focuses on linear relationships like the one in 

Equation 1-3, which are referred to in this report as the “regression method.” For simplicity, the 

regression analyses for this report use linear regression but the task can be accomplished using 

quantile regression or other techniques [Oak Ridge Associated Universities (ORAU) Team (ORAUT) 

2018].
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In the dose reconstruction program, the standard method for estimating Y from X has historically been 

fitting a lognormal model to Y/X using the regression on order statistics (ROS) [Helsel 2011, pp. 52–

56], which is referred to in this report as the “ratio method.” This analysis gives the median of Y/X, 

which is the geometric mean (GM) of the fitted lognormal distribution. To estimate Y given X, simply 

multiply X by this GM. Lognormal ROS with ratios is widely used in the dose reconstruction program 

but does not appear to be used outside the program. 

2.0 PURPOSE

The remainder of this report demonstrates that using ROS with ratios can be problematic1 and should 

only be used with caution. The main problem with this method is that it collapses a bivariate 

relationship (Y versus X) to a univariate relationship (Y/X), which results in a loss of information. In 

practice, ROS with ratios causes the following problems:

1 Note that the issue is with the use of ratios, not ROS. Ratios are problematic even if the average (or some other 

summary statistic) of a group of observed ratios is used. The ROS technique is statistically sound and can be 

legitimately used in other applications. This report focuses on the use of ROS with ratios because that technique is so 

widely used in the dose reconstruction program.

1. Gives the correct answer only for one specific linear relationship (i.e., Equation 1-3 with β = 1). 

2. For β ≠ 1, will usually give a seemingly useful answer that is incorrect. This includes the case 

where there is no relationship between X and Y (i.e., when X and Y are independent). 

This problem can only be diagnosed by looking at the bivariate relationship between X and Y

(i.e., it cannot be diagnosed by examining the result of the ROS). 

Simulated data are used to illustrate the issues with ROS with ratios for various values of β (using 

Equation 1-3):

• X and Y have a relationship and β = 1 (Section 3.0), 

• X and Y have a relationship and 0 < β < 1 (Section 4.0), 

• X and Y have a relationship and β > 1 (Section 5.0), and

• X and Y have no relationship, which implies β = 0 (Section 6.0). 

Values of β < 0 are not addressed in this report. Mathematically, if β < 0, switching the roles of X and 

Y will result in positive β. In practice, if β < 0, there is no need to switch the roles of X and Y; model 

the bivariate relationship with negative β.

Simulated data are used in Sections 3.0 to 6.0 so that the true values of the parameters are known 

and can be compared with the estimated values from the models to show that the regression method 

is superior to the ratio method. Each section refers to an attachment that illustrates the same 

concepts with real data, where the true values of the parameters are not known but the regression 

estimates are taken to be the best estimates. No evaluation of the quality or source of the data is 

included in the attachments. The examples in the attachment are for illustrative purposes only. The 

supporting code and data for each example is in ORAUT [2023]. Section 7.0 describes how the value 

of β affects the performance of the ratio method as illustrated by the simulated examples in 

Sections 3.0 to 6.0. Section 8.0 summarizes the concepts in the report.

Note that the issues discussed in the following sections do not apply to modeling the distribution of Y

or the distribution of X separately – they apply only to modeling the two together. For example, there 
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are no problems associated with modeling the photon doses in a population with a lognormal 

distribution as long as the lognormal distribution provides an adequate fit to those data.

3.0 X AND Y HAVE A RELATIONSHIP AND β = 1

Assuming that there is a well-defined linear relationship between the logarithm of Y and the logarithm

of X, the linear model in Equation 1-3 can be rearranged to give:

(3-1) β log μ ε

Y

X

  = +    

which means that: 

( ) (3-2)       β log μ σ

Y ~N , X

In the special case where β = 1, the logarithm of the ratio Y/X is normally distributed, which means 

that the ratio Y/X is lognormally distributed. This means that for β = 1, the regression in Equation 1-3 

is equivalent to lognormal ROS of the ratio Y/X regardless of the value of μ or σ.

To illustrate this point, the 1,000 pairs of data shown in Figure 3-1 are simulated with β = 1, μ = 3, and 

σ = 0.2.

Figure 3-1. Data simulated with β = 1, μ = 3, and σ = 0.2. 
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3.1 LINEAR REGRESSION

Fitting the simulated data from Figure 3-1 using ordinary least squares and Equation 1-3 gives the 

parameter estimates in Table 3-1.

Table 3-1. Parameter estimates for regression 

using Equation 1-3. 

Parameter True Value Estimate

β 1 1.001

μ 3 3.014

σ 0.2 0.1980

As expected, the estimated parameters match the true parameters very well. Figure 3-2 shows the 

data and regression line on logarithmic scales, so the relationship appears linear.

Figure 3-2. Data simulated with β = 1, μ = 3, and σ = 0.2 and the associated linear regression fit. 

3.2 ROS WITH RATIOS

Taking the simulated data from Figure 3-1, calculating the ratio of Y/X for each of the 1,000 pairs, and 

modeling those ratios with lognormal ROS results in the estimates in Table 3-2, including the GM, the 

geometric standard deviation (GSD), and the lognormal probability plot in Figure 3-3.
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Table 3-2. Parameter estimates for lognormal ROS

with ratios. 

Parameter True Value Estimate

β 1 (a)

μ 3 3.015

σ 0.2 0.198

GM eμ = 20.1 20.39

GSD eσ = 1.2 1.219

a. β is assumed to be 1 when using ratios. The loss of 

information when collapsing bivariate data to univariate 

ratios will not allow estimation of β. 

As expected, the estimated parameters in Table 3-2 match the true parameters very well.

Figure 3-3. Data simulated with β = 1, μ = 3, and σ = 0.2 and the associated lognormal ROS with 

ratios fit. 

3.3 COMPARISON OF REGRESSION AND RATIOS

As mentioned in Section 3.0, for β = 1 the linear regression in Section 3.1 and the ROS with ratios in 

Section 3.2 are equivalent. The parameter estimates in Tables 3-1 and 3-2 match the true values very 

well. To further illustrate the equivalence of the two methods when β = 1, Figure 3-4 adds the line 

resulting from the ROS with the ratios to Figure 3-2. The two lines are essentially identical, which 

means the two methods are equivalent.
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Figure 3-4. Data simulated with β = 1, μ = 3, and σ = 0.2 and the associated linear regression and 

lognormal ROS with ratios fits. 

To illustrate how each of the models could be used, consider predicting Y for an X value of 10 (i.e., 

X0 = 10; dotted vertical line in Figure 3-4). The predicted value of reg Yˆ from the regression model in 

Equation 1-3 is: 

( ) (3-3) reg exp μ β log reg reg 0 ˆ ˆ Y X = +   ˆ

 

where

μreg ˆ = the estimate of μ from the regression in Table 3-1 

βreg

ˆ = the estimate of β from the regression in Table 3-1 

Therefore

( ) (3-4) Y .. ˆ

reg = +× exp 3 014 1 001 log 10    

(3-5) Y . ˆ

reg = 204 2

The predicted value of rat Yˆ from the ROS with ratios in Equation 3-1 is: 

( ) (3-6) rat exp μrat 0 ˆY X = × ˆ
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where

μrat ˆ = the estimate of μ from the ROS with ratios from Table 3-2 

Therefore

( ) (3-7) Y . ˆ

rat = exp 3 015 10 ×

(3-8) Y . ˆ

rat = 203 9

Because the two models agree very well when β = 1, there is little difference in the predicted values 

for X0 = 10. See Attachment A for an example that illustrates these concepts with Fernald thorium 

data.

When β ≠ 1, the linear regression and lognormal ROS with ratios models will not agree and will 

produce different predicted values. The degree and direction of the differences depends on the value 

of β. Sections 4.0, 5.0, and 6.0 present simulations to illustrate how changing β affects predictions. 

4.0 X AND Y HAVE A RELATIONSHIP AND 0 < β < 1

To illustrate the scenario when X and Y have a relationship and 0 < β < 1, the 1,000 pairs of data 

shown in Figure 4-1 are simulated with β = 0.5, μ = 3, and σ = 0.2.

Figure 4-1. Data simulated with β = 0.5, μ = 3, and σ = 0.2. 
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4.1 LINEAR REGRESSION

Fitting the simulated data from Figure 4-1 using ordinary least squares and Equation 1-3 gives the 

parameter estimates in Table 4-1.

Table 4-1. Parameter estimates for regression 

using Equation 1-3. 

Parameter True Value Estimate

β 0.5 0.4991

μ 3 3.007

σ 0.2 0.2048

As expected, the estimated parameters match the true parameters very well. Figure 4-2 shows the 

data and regression line on logarithmic scales, so the relationship appears linear.

Figure 4-2. Data simulated with β = 0.5, μ = 3, and σ = 0.2 and the associated linear regression fit. 

4.2 ROS WITH RATIOS

Taking the simulated data from Figure 4-1, calculating the ratio of Y/X for each of the 1,000 pairs, and 

modeling those ratios with lognormal ROS results in the estimates in Table 4-2 and the lognormal 

probability plot in Figure 4-3.
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Table 4-2. Parameter estimates for lognormal ROS

with ratios. 

Parameter True Value Estimate

β 0.5 (a)

μ 3 2.314

σ 0.2 0.405

GM eμ = 20.1 10.12

GSD eσ = 1.2 1.499

a. β is assumed to be 1 when using ratios. The loss of 

information when collapsing bivariate data to univariate 

ratios will not allow estimation of β. 

Because β ≠ 1, the estimated parameters in Table 4-2 do not match the true parameters very well. 

The fit in Figure 4-3 looks good, and there is no visible indication of the issue with the ROS with ratios 

giving the incorrect answers.

Figure 4-3. Data simulated with β = 0.5, μ = 3, and σ = 0.2 and the associated lognormal ROS with 

ratios fit. 

4.3 COMPARISON OF REGRESSION AND RATIOS

The parameter estimates in Table 4-1 match the true values very well, but the estimates in Table 4-2 

do not. To illustrate the performance of the two methods when β = 0.5, Figure 4-4 adds the line 

resulting from ROS with the ratios to Figure 4-2. The two lines intersect roughly in the middle of the 

horizontal axis. When β ≠ 1, the lines from the linear regression fit and the lognormal ROS with ratios 

will intersect at Xint [ORAUT 2023] where: 

(4-1) reg rat

int

reg

μ μ

exp 1 β

ˆ ˆ X ˆ

  − =     −  
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which for this simulated example is: 

(4-2) int

3 007 2 314

exp 1 0 4991

. . X .

  − =     −

(4-3) X . int = 3 985

This means that for any X value less than 3.985, the ratio method will underestimate the predicted 

value of Y by some amount. For any X value greater than 3.985, the ratio method will overestimate 

the predicted value of Y by some amount. The amount of under- or overestimation depends on how 

far the value of X is from 3.985.

Figure 4-4. Data simulated with β = 0.5, μ = 3, and σ = 0.2 and the associated linear regression and 

lognormal ROS with ratios fits. 

For example, suppose X0 = 10 (dotted vertical line in Figure 4-4). The predicted value of Y from the 

regression model in Equation 3-3 is: 

( ) (4-4) Y .. ˆ

reg = +× exp 3 007 0 4991 log 10    

(4-5) Y . ˆ

reg = 63 83

The predicted value of Y from the ROS with ratios in Equation 3-6 is: 

( ) (4-6) Y . ˆ

rat = exp 2 314 10 ×
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(4-7) Y . ˆ

rat = 101 2

See Attachment B for an example that illustrates these concepts with Paducah 99Tc data.

5.0 X AND Y HAVE A RELATIONSHIP AND β > 1

To illustrate the scenario when X and Y have a relationship and β > 1, the 1,000 pairs of data shown 

in Figure 5-1 are simulated with β = 2, μ = 3, and σ = 0.2.

Figure 5-1. Data simulated with β = 2, μ = 3, and σ = 0.2. 

5.1 LINEAR REGRESSION

Fitting the simulated data from Figure 5-1 using ordinary least squares and Equation 1-3 gives the 

parameter estimates in Table 5-1.

Table 5-1. Parameter estimates for regression 

using Equation 1-3. 

Parameter True Value Estimate

β 2 2.007

μ 3 2.995

σ 0.2 0.1976

As expected, the estimated parameters match the true parameters very well. Figure 5-2 shows the 

data and regression line on logarithmic scales, so the relationship appears linear.
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Figure 5-2. Data simulated with β = 2, μ = 3, and σ = 0.2 and the associated linear regression fit. 

5.2 ROS WITH RATIOS

Taking the simulated data from Figure 5-1, calculating the ratio of Y/X for each of the 1,000 pairs, and 

modeling those ratios with lognormal ROS results in the estimates in Table 5-2 and the lognormal 

probability plot in Figure 5-3.

Table 5-2. Parameter estimates for lognormal ROS

with ratios. 

Parameter True Value Estimate

β 2 (a)

μ 3 4.387

σ 0.2 0.7234

GM eμ = 20.1 80.43

GSD eσ = 1.2 2.061

a. β is assumed to be 1 when using ratios. The loss of 

information when collapsing bivariate data to univariate 

ratios will not allow estimation of β. 

Because β ≠ 1, the estimated parameters in Table 5-2 do not match the true parameters very well. 

The fit in Figure 5-3 looks good, and there is no visible indication of the issue with the ROS with ratios 

giving the incorrect answers.
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Figure 5-3. Data simulated with β = 2, μ = 3, and σ = 0.2 and the associated lognormal ROS with 

ratios fit. 

5.3 COMPARISON OF REGRESSION AND RATIOS

The parameter estimates in Table 5-1 match the true values very well, but the estimates in Table 5-2 

do not. To illustrate the performance of the two methods when β = 2, Figure 5-4 adds the line resulting 

from ROS with the ratios to Figure 5-2. The two lines intersect at: 

(5-1) int

2 995 4 387

exp 1 2 007

. . X .

  − =     −

(5-2) X . int = 3 986

which is essentially the same value in Equation 4-3 because the true value of β was changed from 0.5 

to 1/0.5 = 2 and the true values of μ and σ were not changed.

This means that for any X value less than 3.986, the ratio method will overestimate the correct 

predicted value of Y by some amount. For any X value greater than 3.986, the ratio method will 

underestimate the correct predicted value of Y by some amount.
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Figure 5-4. Data simulated with β = 2, μ = 3, and σ = 0.2 and the associated linear regression and 

lognormal ROS with ratios fits. 

For example, suppose X0 = 10 (dotted vertical line in Figure 5-4). The predicted value of Y from the 

regression model in Equation 3-3 is: 

( ) (5-3) Y .. ˆ

reg = +× exp 2 995 2 007 log 10    

(5-4) Y , ˆ

reg = 2 031

The predicted value of Y from the ROS with ratios in Equation 3-6 is: 

( ) (5-5) Y . ˆ

rat = exp 4 387 10 ×

(5-6) Y . ˆ

rat = 804 3

See Attachment C for an example that illustrates these concepts with Mound Neutron Source (NS) 

data.

6.0 X AND Y HAVE NO RELATIONSHIP, WHICH IMPLIES β = 0

To illustrate the scenario when X and Y have no relationship, which implies β = 0, 1,000 X values are 

randomly drawn from a lognormal distribution with GM = 4 and GSD = 2. One thousand Y values are 

randomly drawn from a lognormal distribution with GM = 3 and GSD = 3. Drawing randomly from 

these two distributions makes the pairs of data independent, meaning they have no relationship, and 

implies β is 0. The pairs of data are shown in Figure 6-1.
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Figure 6-1. Data simulated from two independent lognormal distributions. 

6.1 LINEAR REGRESSION

The data in Figure 6-1 clearly have no relationship. Essentially, the predicted value of Y will be the 

mean of Y despite the value of X, but for the sake of illustration the parameter estimates are included 

in Table 6-1. Note that the true values of μ and σ are simply the logarithm of the GM and logarithm of 

the GSD of Y, respectively, since β = 0.

Table 6-1. Parameter estimates for regression 

using Equation 1-3. 

Parameter True Value Estimate

β 0 0.069

μ log(3) = 1.099 0.9806

σ log(3) = 1.099 1.127

As expected, the estimated β parameter is fairly close to zero. Figure 6-2 shows the data and 

regression line on logarithmic scales, so the relationship appears linear and essentially flat.
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 22 of 48

 

Figure 6-2. Data simulated from two independent lognormal distributions and the associated linear 

regression fit. 

6.2 ROS WITH RATIOS

Taking the simulated data from Figure 6-1, calculating the ratio of Y/X for each of the 1,000 pairs, and 

modeling those ratios with lognormal ROS results in the estimates in Table 6-2 and the lognormal 

probability plot in Figure 6-3.

Table 6-2. Parameter estimates for lognormal ROS

with ratios. 

Parameter True Value Estimate

β 0 (a)

μ log(3) = 1.099 −0.3142

σ log(3) = 1.099 1.296

GM 3 0.7303

GSD 3 3.656

a. β is assumed to be 1 when using ratios. The loss of 

information when collapsing bivariate data to univariate 

ratios will not allow estimation of β.

The fit in Figure 6-3 looks good, and there is no visible indication of the issue with the ROS with ratios 

giving the incorrect answers. Nothing about the parameters in Table 6-2 or the fit in Figure 6-3 

indicates that X and Y have no relationship. By collapsing independent X and Y into a onedimensional ratio, two variables that have no relationship now appear to have a meaningful

relationship.
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 23 of 48

 

Figure 6-3. Data simulated from two independent lognormal distributions and the associated 

lognormal ROS with ratios fit. 

In fact, if X is lognormally distributed with parameters GMX and GSDX and Y is lognormally distributed 

with parameters GMY and GSDY, then: 

log( ) ( ) μ σ (6-1) X ~N , X X

log( ) ( ) μ σ (6-2) Y ~N , Y Y

If X and Y are independent, which is assumed throughout this section, it follows that [Casella and 

Berger 2002, p. 211]: 

(6-3) ( )     − +  

2 2 log μμ σσ YX Y X

Y ~N , X

This means that, mathematically, two independent lognormal distributions will always produce a ratio 

distribution that is also lognormal. 
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 24 of 48

 

6.3 COMPARISON OF REGRESSION AND RATIOS

To illustrate the performance of the two methods when X and Y have no relationship and β = 0, 

Figure 6-4 adds the line resulting from the ROS with the ratios to Figure 6-2. The two lines intersect 

at: 

(6-4) ( )

int

0 9806 0 3142

exp 1 0 069

. . X .

  − − =   −  

(6-5) X . int = 4 018

This means that for any X value less than 4.018, the ratio method will underestimate the correct 

predicted value of Y by some amount. For any X value greater than 4.018, the ratio method will 

overestimate the correct predicted value of Y by some amount.

Figure 6-4. Data simulated from two independent lognormal distributions and the associated linear 

regression and lognormal ROS with ratios fits. 

For example, suppose X0 = 10 (dotted vertical line in Figure 6-4). The predicted value of Y from the 

regression model in Equation 3-3 is: 

( ) (6-6) Y .. ˆ

reg = +× exp 0 9806 0 069 log 10    

(6-7) Y . ˆ

reg = 3 125
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 25 of 48

 

The predicted value of Y from the ROS with ratios in Equation 3-6 is: 

( ) (6-8) Y . ˆ

rat =− × exp 0 3142 10

(6-9) Y . ˆ

rat = 7 303

The essentially flat linear regression fit indicates the predicted value of Y does not depend on the 

observed value of X. However, the lognormal fit to the ratio indicates that the predicted value of Y

should change as a function of X, which is obviously inappropriate based on Figure 6-2 and the fact 

that X and Y have no relationship. In this case, the ratio creates the illusion of a meaningful 

relationship where none exists. See Attachment D for an example that illustrates these concepts with 

Rocky Flats neutron and gamma dose data.

In general, before statistical analysis of any kind is done, subject matter experts should consider 

whether there should be a relationship between two variables. Based on subject knowledge, if there is 

no reason to suspect that a relationship exists, there should be no attempt to do any modeling. If there 

is reason to suspect that a relationship should exist, and the data look similar to Figure 6-4, more 

exploratory analyses should be done to determine whether relationships exist amongst subgroups of 

the dataset.

7.0 EFFECT OF β ON THE RATIO METHOD

The value of β affects the degree and direction of the under- or overestimation of Y using lognormal 

ROS with ratios. For the remainder of this section, assume μ and σ are held fixed. The worst-case 

scenario, in terms of the amount of underestimation of Y from the ratio method for X values less than 

Xint and overestimation of Y for values larger than Xint, is when β = 0 (Section 6.0). For 0 < β < 1, the 

amount of under- and overestimation of Y will decrease as β increases (Section 4.0), with the ratio 

method being correct only for β = 1 (Section 3.0, no under- or overestimation). For β > 1, as β 

increases, the ratio will overestimate Y for X values less than Xint and underestimate Y for values 

greater than Xint (Section 5.0). These effects of changing β are also summarized using a simulation 

[ORAUT 2023].

8.0 SUMMARY

Overall, the key points of this discussion are:

• The standard method of fitting a lognormal distribution to the ratio of Y/X (i.e., ROS) is valid 

only for the model in Equation 1-3 when β = 1.

• For data generated by any statistical model other than that in Equation 1-3, ROS and other 

methods that might reasonably be used with ratios (like empirical quantiles) will give biased 

estimates of the GM, GSD, and other quantiles of interest (such as the 95th percentile). The 

lognormal probability plot of the ratio cannot indicate if the parameter estimates are biased.

• ROS with ratios will generate estimates of the GM and GSD even when there is no meaningful 

relationship between X and Y. The lognormal probability plot of the ratio cannot be used to 

diagnose this problem.

The main recommendations to draw from these conclusions are:

• Always examine the scatterplot of Y versus X to determine if a relationship exists between Y

and X. 
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 26 of 48

 

• If there is no meaningful relationship (linear or nonlinear) between Y and X, avoid creating the 

appearances of one by modeling the Y/X ratio.

• If a meaningful relationship exists, it is preferable to model the bivariate relationship between Y

and X using standard regression or quantile regression [ORAUT 2018] rather than model the 

relationship of Y/X as a univariate (e.g., lognormal) distribution.

• Reserve modeling the ratio Y/X for situations in which all other bivariate methods have been 

exhausted, and use it only with great care. 
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 27 of 48

 

REFERENCES

Casella G, Berger RL [2002]. Statistical inference. 2nd ed. Pacific Grove, CA: Duxbury. [SRDB Ref ID: 

195943]

Helsel DR [2012]. Statistics for censored environmental data using Minitab and R. 2nd ed. Hoboken, 

NJ: John Wiley & Sons. [SRDB Ref ID: 150353]

ORAUT [2018]. Applications of regression in external dose reconstruction. Oak Ridge, TN: Oak Ridge 

Associated Universities Team. ORAUT-RPRT-0087 Rev. 00, March 13. [SRDB Ref ID: 170100]

ORAUT [2023]. ORAUT-RPRT-0106 Rev. 00 support files. Zip file. Oak Ridge, TN: Oak Ridge 

Associated Universities Team. February 28. [SRDB Ref ID: 195944]

Weisberg S [2005]. Applied linear regression. 3rd ed. Hoboken, NJ: John Wiley & Sons. [SRDB Ref 

ID: 169440]
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 28 of 48

 

ATTACHMENT A 

FERNALD THORIUM DATA:

X AND Y HAVE A RELATIONSHIP AND β = 1 

TABLE OF CONTENTS

SECTION TITLE PAGE

A.1 Data ....................................................................................................................................... 29

A.2 Linear Regression.................................................................................................................. 29

A.3 ROS with Ratios..................................................................................................................... 30

A.4 Comparison of Regression and Ratios................................................................................... 31

LIST OF TABLES

TABLE TITLE PAGE

A-1 Parameter estimates for regression ....................................................................................... 29

A-2 Parameter estimates for lognormal ROS with ratios............................................................... 30

LIST OF FIGURES

FIGURE TITLE PAGE

A-1 Fernald thorium data from 1994 to 1996 ................................................................................ 29

A-2 Fernald thorium data from 1994 to 1996 and the associated linear regression fit ................... 30

A-3 Fernald thorium data from 1994 to 1996 and the associated lognormal ROS with ratios fit .... 31

A-4 Fernald thorium data from 1994 to 1996 and the associated linear regression and 

lognormal ROS with ratios fits ................................................................................................ 32
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 29 of 48

ATTACHMENT A

FERNALD THORIUM DATA:

X AND Y HAVE A RELATIONSHIP AND β = 1 (continued)

A.1 DATA

Figure A-1 shows a scatterplot of Fernald thorium data from 1994 to 1996.

Figure A-1. Fernald thorium data from 1994 to 1996. 

A.2 LINEAR REGRESSION

Fitting the data from Figure A-1 using ordinary least squares gives the parameter estimates in 

Table A-1.

Table A-1. Parameter 

estimates for regression. 

Parameter Estimate

β 0.9997

μ −0.1775

σ 0.3635

Figure A-2 shows the data and regression line on logarithmic scales, so the relationship appears 

linear.
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 30 of 48

ATTACHMENT A

FERNALD THORIUM DATA:

X AND Y HAVE A RELATIONSHIP AND β = 1 (continued)

Figure A-2. Fernald thorium data from 1994 to 1996 and the associated linear regression fit. 

A.3 ROS WITH RATIOS

Taking the data from Figure A-1, calculating the ratio of Y/X for each of the 39 pairs, and modeling 

those ratios with lognormal ROS results in the estimates in Table A-2 and the lognormal probability 

plot in Figure A-3.

Table A-2. Parameter 

estimates for lognormal ROS

with ratios. 

Parameter Estimate

β (a)

μ −0.1774

σ 0.3551

GM 0.8374

GSD 1.426

a. β is assumed to be 1 when 

using ratios. The loss of 

information when collapsing 

bivariate data to univariate 

ratios will not allow estimation 

of β.
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 31 of 48

ATTACHMENT A

FERNALD THORIUM DATA:

X AND Y HAVE A RELATIONSHIP AND β = 1 (continued)

Figure A-3. Fernald thorium data from 1994 to 1996 and the associated lognormal ROS with ratios fit. 

A.4 COMPARISON OF REGRESSION AND RATIOS

For β = 1, the linear regression in Section A.2 and the ROS with ratios in Section A.3 are equivalent. 

The parameter estimates of μ and σ in Tables A-1 and A-2 match each other very well. To further 

illustrate the equivalence of the two methods when β = 1, Figure A-4 adds the line resulting from ROS 

with the ratios to Figure A-2. The two lines are essentially identical, which means the two methods are 

equivalent.
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 32 of 48

ATTACHMENT A

FERNALD THORIUM DATA:

X AND Y HAVE A RELATIONSHIP AND β = 1 (continued)

Figure A-4. Fernald thorium data from 1994 to 1996 and the associated linear regression and 

lognormal ROS with ratios fits. 

Because the two models agree very well when β = 1, the models would produce little difference in the 

predicted values for any value of X, as seen in Section 3.3, so no example calculations are done here.
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 33 of 48

 

ATTACHMENT B 

PADUCAH TECHNETIUM DATA:

X AND Y HAVE A RELATIONSHIP AND 0 < β < 1

TABLE OF CONTENTS

SECTION TITLE PAGE

B.1 Data ....................................................................................................................................... 34

B.2 Linear Regression.................................................................................................................. 34

B.3 ROS with Ratios..................................................................................................................... 35

B.4 Comparison of Regression and Ratios................................................................................... 36

LIST OF TABLES

TABLE TITLE PAGE

B-1 Parameter estimates for regression ....................................................................................... 34

B-2 Parameter estimates for lognormal ROS with ratios............................................................... 35

LIST OF FIGURES

FIGURE TITLE PAGE

B-1 Paducah annual air data from 1980 to 1999 (excluding 1996 and 1997)................................ 34

B-2 Paducah annual air data from 1980 to 1999 (excluding 1996 and 1997) and the 

associated linear regression fit............................................................................................... 35

B-3 Paducah annual air data from 1980 to 1999 (excluding 1996 and 1997) and the 

associated lognormal ROS with ratios fit................................................................................ 36

B-4 Paducah annual air data from 1980 to 1999 (excluding 1996 and 1997) and the 

associated linear regression and lognormal ROS with ratios fits ............................................ 37
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 34 of 48

ATTACHMENT B 

PADUCAH TECHNETIUM DATA:

X AND Y HAVE A RELATIONSHIP AND 0 < β < 1 (continued)

B.1 DATA

Figure B-1 shows a scatterplot of Paducah 99Tc annual releases and intakes from 1980 to 1999, 

excluding 1996 and 1997.

Figure B-1. Paducah annual air data from 1980 to 1999 (excluding 1996 and 1997). 

B.2 LINEAR REGRESSION

Fitting the data from Figure B-1 using ordinary least squares gives the parameter estimates in 

Table B-1.

Table B-1. Parameter 

estimates for regression. 

Parameter Estimate

β 0.3787

μ −4.918

σ 0.8105

Figure B-2 shows the data and regression line on logarithmic scales, so the relationship appears 

linear.
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 35 of 48

ATTACHMENT B 

PADUCAH TECHNETIUM DATA:

X AND Y HAVE A RELATIONSHIP AND 0 < β < 1 (continued)

Figure B-2. Paducah annual air data from 1980 to 1999 (excluding 1996 and 1997) and the 

associated linear regression fit. 

B.3 ROS WITH RATIOS

Taking the data from Figure B-1, calculating the ratio of Y/X for each of the 18 pairs, and modeling 

those ratios with lognormal ROS results in the estimates in Table B-2 and the lognormal probability 

plot in Figure B-3.

Table B-2. Parameter 

estimates for lognormal ROS

with ratios. 

Parameter Estimate

β (a)

μ −16.96

σ 1.313

GM 4.321E−8

GSD 3.718

a. β is assumed to be 1 when 

using ratios. The loss of 

information when collapsing 

bivariate data to univariate 

ratios will not allow estimation 

of β.

The fit in Figure B-3 looks good, and there is no visible indication of the issue with the ROS with ratios 

giving the incorrect answers.
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 36 of 48

ATTACHMENT B 

PADUCAH TECHNETIUM DATA:

X AND Y HAVE A RELATIONSHIP AND 0 < β < 1 (continued)

Figure B-3. Paducah annual air data from 1980 to 1999 (excluding 1996 and 1997) and the 

associated lognormal ROS with ratios fit. 

B.4 COMPARISON OF REGRESSION AND RATIOS

To illustrate the performance of the two methods for this dataset, Figure B-4 adds the line resulting 

from ROS with the ratios to Figure B-2. The two lines intersect at: 

(B-1) ( )

int

4 918 16 96

exp 1 0 3787

. . X .

  − −− =   −  

(B-2) 8 X . int = × 2 603 10 Bq/yr

This means the ratio method will underestimate the best estimate of predicted intake for releases less 

than 2.603 × 108 Bq/yr and overestimate the best estimate of predicted intake for releases greater 

than 2.603 × 108 Bq/yr.
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 37 of 48

ATTACHMENT B 

PADUCAH TECHNETIUM DATA:

X AND Y HAVE A RELATIONSHIP AND 0 < β < 1 (continued)

Figure B-4. Paducah annual air data from 1980 to 1999 (excluding 1996 and 1997) and the 

associated linear regression and lognormal ROS with ratios fits. 

For example, for a release of 2 × 107 Bq/yr (dotted vertical line in Figure B-4), the predicted intake 

from the regression model is: 

( ) (B-3) 7 Y .. ˆ

reg =− + × × exp 4 918 0 3787 log 2 10    

(B-4) reg 4 256 Bq/yr Y . ˆ =

The predicted intake from the ROS with ratios is: 

( )( ) (B-5) 7 Y . ˆ

rat =− × exp 16 96 2 10

(B-6) rat 0 8642 Bq/yr Y . ˆ =

For a release of 2 × 107 Bq/yr, the ROS with ratio model underestimates by a factor of approximately

5. 
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 38 of 48

 

ATTACHMENT C 

MOUND NEUTRON SOURCE DATA: 

X AND Y HAVE A RELATIONSHIP AND β > 1 

TABLE OF CONTENTS

SECTION TITLE PAGE

C.1 Data ....................................................................................................................................... 39

C.2 Linear Regression.................................................................................................................. 39

C.3 ROS with Ratios..................................................................................................................... 40

C.4 Comparison of Regression and Ratios................................................................................... 41

LIST OF TABLES

TABLE TITLE PAGE

C-1 Parameter estimates for regression ....................................................................................... 39

C-2 Parameter estimates for lognormal ROS with ratios............................................................... 40

LIST OF FIGURES

FIGURE TITLE PAGE

C-1 Mound neutron and photon data from NS in 1954 and 1957 .................................................. 39

C-2 Mound neutron and photon data from NS in 1954 and 1957 and the associated linear 

regression fit ..........................................................................................................................40

C-3 Mound neutron and photon data from NS in 1954 and 1957 and the associated 

lognormal ROS with ratios fit.................................................................................................. 41

C-4 Mound neutron and photon data from NS in 1954 and 1957 and the associated linear 

regression and lognormal ROS with ratios fits........................................................................ 42
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 39 of 48

ATTACHMENT C 

MOUND NEUTRON SOURCE DATA:

X AND Y HAVE A RELATIONSHIP AND β > 1 (continued)

C.1 DATA

Figure C-1 shows a scatterplot of neutron and photon doses from the NS group at Mound in 1954 and 

1957.

Figure C-1. Mound neutron and photon data from NS in 1954 and 1957. 

C.2 LINEAR REGRESSION

Fitting the data from Figure C-1 using ordinary least squares gives the parameter estimates in 

Table C-1.

Table C-1. Parameter 

estimates for regression. 

Parameter Estimate

β 2.396

μ 3.353

σ 0.490

Figure C-2 shows the data and regression line on logarithmic scales, so the relationship appears 

linear.
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 40 of 48

ATTACHMENT C 

MOUND NEUTRON SOURCE DATA:

X AND Y HAVE A RELATIONSHIP AND β > 1 (continued)

Figure C-2. Mound neutron and photon data from NS in 1954 and 1957 and the associated linear 

regression fit. 

C.3 ROS WITH RATIOS

Taking the data from Figure C-1, calculating the ratio of Y/X for each of the 12 pairs, and modeling 

those ratios with lognormal ROS results in the estimates in Table C-2 and the lognormal probability 

plot in Figure C-3.

Table C-2. Parameter 

estimates for lognormal ROS

with ratios. 

Parameter Estimate

β (a)

μ 1.498

σ 1.008

GM 4.471

GSD 2.74

a. β is assumed to be 1 when 

using ratios. The loss of 

information when collapsing 

bivariate data to univariate 

ratios will not allow estimation 

of β.

The fit in Figure C-3 looks okay, and there is no visible indication of the issue with the ROS with ratios 

giving the incorrect answers.
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 41 of 48

ATTACHMENT C 

MOUND NEUTRON SOURCE DATA:

X AND Y HAVE A RELATIONSHIP AND β > 1 (continued)

Figure C-3. Mound neutron and photon data from NS in 1954 and 1957 and the associated lognormal 

ROS with ratios fit. 

C.4 COMPARISON OF REGRESSION AND RATIOS

To illustrate the performance of the two methods for this dataset, Figure C-4 adds the line resulting 

from ROS with the ratios to Figure C-2. The two lines intersect at: 

(C-1)   − =     − int

3 353 1 498

exp 1 2 396

. . X .

= (C-2) X . int 0 2647 rem

This means the ratio method will underestimate the best estimate of predicted neutron dose for 

photon doses less than 0.2647 rem and overestimate the best estimate of predicted neutron dose for 

photon doses greater than 0.2647 rem.
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 42 of 48

ATTACHMENT C 

MOUND NEUTRON SOURCE DATA:

X AND Y HAVE A RELATIONSHIP AND β > 1 (continued)

Figure C-4. Mound neutron and photon data from NS in 1954 and 1957 and the associated linear 

regression and lognormal ROS with ratios fits. 

For example, for a photon dose of 0.7 rem (dotted vertical line in Figure C-4), the predicted neutron 

dose from the regression model is: 

= +×   ( ) (C-3) reg   Y .. . ˆ exp 3 353 2 396 log 0 7

= (C-4) Y . ˆ

reg 12 16 rem

The predicted neutron dose from the ROS with ratios is: 

= ( ) × (C-5) Y .. ˆ

rat exp 4 471 0 7

= (C-6) Y . ˆ

rat 3 13 rem

For a photon dose of 0.7 rem, the ROS with ratio model underestimates by a factor of approximately 

4. 
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 43 of 48

 

ATTACHMENT D 

ROCKY FLATS NEUTRON AND GAMMA DOSE DATA: X AND Y HAVE NO RELATIONSHIP,

WHICH IMPLIES β = 0 

TABLE OF CONTENTS

SECTION TITLE PAGE

D.1 Data ....................................................................................................................................... 44

D.2 Linear Regression.................................................................................................................. 44

D.3 ROS with Ratios..................................................................................................................... 45

D.4 Comparison of Regression and Ratios................................................................................... 46

LIST OF TABLES

TABLE TITLE PAGE

D-1 Parameter estimates for regression ....................................................................................... 44

D-2 Parameter estimates for lognormal ROS with ratios............................................................... 45

LIST OF FIGURES

FIGURE TITLE PAGE

D-1 Rocky Flats neutron and gamma data from 1969................................................................... 44

D-2 Rocky Flats neutron and gamma data from 1969 and the associated linear regression fit...... 45

D-3 Rocky Flats neutron and gamma data from 1969 and the associated lognormal ROS 

with ratios fit...........................................................................................................................46

D-4 Rocky Flats neutron and gamma data from 1969 and the associated linear regression 

and lognormal ROS with ratios fits ......................................................................................... 47
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 44 of 48

ATTACHMENT D 

ROCKY FLATS NEUTRON AND GAMMA DOSE DATA: X AND Y HAVE NO RELATIONSHIP,

WHICH IMPLIES β = 0 (continued)

D.1 DATA

Figure D-1 shows a scatterplot of neutron and gamma doses from Rocky Flats in 1969.

Figure D-1. Rocky Flats neutron and gamma data from 1969. 

D.2 LINEAR REGRESSION

Clearly the data in Figure D-1 have no relationship. Essentially, the predicted neutron dose will be the 

mean of the neutron doses despite the value of X, but for the sake of illustration, the parameter 

estimates are included in Table D-1.

Table D-1. Parameter 

estimates for regression. 

Parameter Estimate

β 0.06333

μ 4.21

σ 0.5244

As expected, the estimated β parameter is fairly close to zero. Figure D-2 shows the data and 

regression line on logarithmic scales, so the regression line appears fairly flat.
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 45 of 48

ATTACHMENT D 

ROCKY FLATS NEUTRON AND GAMMA DOSE DATA: X AND Y HAVE NO RELATIONSHIP,

WHICH IMPLIES β = 0 (continued)

Figure D-2. Rocky Flats neutron and gamma data from 1969 and the associated linear regression fit. 

D.3 ROS WITH RATIOS

Taking the data from Figure D-1, calculating the ratio of Y/X for each of the 6,948 pairs, and modeling 

those ratios with lognormal ROS results in the estimates in Table D-2 and the lognormal probability 

plot in Figure D-3.

Table D-2. Parameter 

estimates for lognormal ROS

with ratios. 

Parameter Estimate

β (a)

μ 1.106

σ 1.556

GM 3.023

GSD 4.737

a. β is assumed to be 1 when 

using ratios. The loss of 

information when collapsing 

bivariate data to univariate 

ratios will not allow estimation 

of β.

The fit in Figure D-3 looks okay, and there is no visible indication of the issue with the ROS with ratios 

giving the incorrect answers. Nothing about the parameters in Table D-2 or the fit in Figure D-3 

indicates that X and Y have no relationship. By collapsing independent gamma and neutron doses 
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 46 of 48

ATTACHMENT D 

ROCKY FLATS NEUTRON AND GAMMA DOSE DATA: X AND Y HAVE NO RELATIONSHIP,

WHICH IMPLIES β = 0 (continued)

into a one-dimensional ratio, two variables that have no relationship now appear to have a meaningful 

relationship.

Figure D-3. Rocky Flats neutron and gamma data from 1969 and the associated lognormal ROS with 

ratios fit. 

D.4 COMPARISON OF REGRESSION AND RATIOS

To illustrate the performance of the two methods when gamma and neutron doses have no 

relationship, Figure D-4 adds the line resulting from ROS with the ratios to Figure D-2. The two lines 

intersect at: 

(D-1) int

4 21 1 106

exp 1 0 06333

. . X .

  − =     −

(D-2) X . int = 27 49 mrem

This means the ratio method will underestimate the best estimate of predicted neutron dose for 

gamma doses less than 27.488 mrem and overestimate the best estimate of predicted neutron dose 

for gamma doses greater than 27.488 mrem.
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 47 of 48

ATTACHMENT D 

ROCKY FLATS NEUTRON AND GAMMA DOSE DATA: X AND Y HAVE NO RELATIONSHIP,

WHICH IMPLIES β = 0 (continued)

Figure D-4. Rocky Flats neutron and gamma data from 1969 and the associated linear regression and 

lognormal ROS with ratios fits. 

For example, for a gamma dose of 5 mrem (dotted vertical line in Figure D-4), the predicted neutron 

dose from the regression model is: 

( ) (D-3) Y .. ˆ

reg = +× exp 4 21 0 06333 log 5    

(D-4) Y . ˆ

reg = 74 58 mrem

The predicted neutron dose from the ROS with ratios is: 

( ) (D-5) Y . ˆ

rat = exp 1 106 5×

(D-6) Y . ˆ

rat = 15 12 mrem

For a gamma dose of 5 mrem, the ROS with ratio model underestimates by a factor of approximately 

5 and is smaller than all of the observed neutron doses with associated gamma doses of 5 mrem. The 

essentially flat linear regression fit would indicate that the predicted neutron dose does not depend on 

the observed gamma dose. However, the lognormal fit to the ratio indicates that the predicted neutron 

dose should change as a function of gamma dose, which is obviously inappropriate based on the 

points in Figure D-4 and the fact that these Rocky Flats neutron and gamma doses have no 

relationship.
Document No. ORAUT-RPRT-0106 Revision No. 00 Effective Date: 08/03/2023 Page 48 of 48

ATTACHMENT D

ROCKY FLATS NEUTRON AND GAMMA DOSE DATA: X AND Y HAVE NO RELATIONSHIP,

WHICH IMPLIES β = 0 (continued)

As mentioned in Section 6.3, if there is reason to believe that neutron and gamma doses should have 

a relationship, exploratory analyses should be done to determine whether relationships exist amongst 

subgroups of the dataset.