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Memorandum

To: Arlene Rosenbaum

Systems Applications International

From: Ted Johnson

TRJ Environmental, Inc.

Date: September 29, 1998 (Rev. 1.1)

Project: EPA Work Assignment 1-19

Memo No. 5: Equations for Converting Weight to Height Proposed for the

1998 Version of pNEM/CO 

Work Assignment 1-19 of EPA Contract No. 68-D6-0064 directs the ICF project team to 

propose and implement modifications to the Probabilistic NAAQS Exposure Model for

Carbon Monoxide (pNEM/CO). These modifications will include enhancements to the

algorithm which provides a “physiological profile” for each person simulated by

pNEM/CO. The enhanced algorithm will use probabilistic techniques to generate

values for all physiological parameters required by two other pNEM/CO algorithms: (1)

the algorithm that estimates ventilation (respiratory) rate and (2) the algorithm that

estimates carboxyhemoglobin (COHb) level. This memorandum provides equations

which can be incorporated into the physiological profile algorithm to estimate height

given a predetermined value for weight. 

Background

In the 1992 version of pNEM/CO (Johnson et al., 1992), the COHb algorithm estimates

blood volume (Vb) as a function of weight and height by the following equations

Men: Vb = (20.4)(W) + (0.00683)(H3) - 30 (1)

Women: Vb = (14.6)(W) + (0.00678)(H3) - 30. (2)

W is weight in pounds, and H is height in inches. We propose to use this same method

to determine blood volume in the 1998 version of pNEM/CO. Note that Equations 1

and 2 require estimates of weight and height specific to the simulated person. 

In the 1992 version of pNEM/CO, height was selected from one of six normal

distributions which varied according to age and gender (see Table 14 of Johnson et al.

1992). The value for weight was then determined by the equation

weight = A0 + (A1)(height) + (z)(S.E.) (3)

where the values for A0, A1, and S.E. were obtained from Table 14 of Johnson et al.
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(1992) according to the gender and age group of the simulated person. The value of z

was randomly selected from a unit normal distribution. 

In the 1998 version of pNEM/CO, weight will be used in both the ventilation and COHb

algorithms, whereas height will be used only in the COHb algorithm. As discussed in

Memorandum No. 3, we have developed lognormal distributions of weight for all

combinations of gender and age where age is given in one-year increments from zero

to 100. For these reasons, we would prefer that the physiological profile algorithm first

select a value of weight according to gender and age and then let that value determine

a corresponding value of height. The 1998 version of pNEM/CO will initially be applied

to adults in Denver, Colorado. Consequently, the physiological profile algorithm

currently requires gender-specific probabilistic equations which predict height as a

function of weight for people ages 18 years and older. 

Equations for Estimating Height from Weight

As noted in Memorandum No. 3, the lognormal distributions proposed for weight were

obtained from an article by Burmaster and Crouch (1997). In an article published five

years earlier, Brainard and Burmaster (1992) provide a method for sampling weight and

height simultaneously from a bivariate distribution based on data obtained from the

second National Health and Nutrition Examination Survey (NHANES II). Although

Brainard and Burmaster do not explicitly provide a method for estimating height given

weight, statistical results presented in the article indirectly suggest that height could be

determined from weight using a regression-derived equation of the form

height = a0 + (a1)[ln(weight)] + e (4)

in which e is a normally-distributed random term with mean = zero and standard

deviation = s e. Ideally, the values of a0, a1, and s e would be specific to age and gender. 

Table 1 is a facsimile of a table appearing in the article by Brainard and Burmaster

(1992). This table lists the number of males (18 to 74 years) associated with each of

210 combinations of height (inches) and weight (lbs). The authors suggest that an

analysis of weight vs. height based on these data be limited to the 132 cells that fall

within the indicated “box” (13 height categories times 12 weight categories). 

Consequently, we set up a data file containing 132 records, each corresponding to one

of the cells in the included in the box. Each record listed the indicated height value, the

midpoint of the indicated weight range (e.g., 115 = midpoint of 110 to 119 range), and

the number of males associated with the cell. 

As indicated in Equation 4, Brainard and Burmaster recommend relating height to

ln(weight). Consequently, we performed a weighted linear regression of the data file

using height as the dependent variable, ln(weight) as the independent variable, and

number of males as a weighting factor. The resulting regression equation for males (18

to 74 years) was

height, inches = 34.43 in + (6.67)[ln(weight,lbs)] + e. (5)
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The R2 value was 0.1469. 

To estimate s e, the standard deviation of e, we used the equation

s e = [S(Oij - Eij)

2

(Mij)/(total males)]0.5 (6)

in which Oij was the observed (listed) height associated with cellij, Eij was the estimated

height for cellij based on the regression equation, Mij was the number of males listed in

Table 1 for cellij, and total males was the total number of males in all cells. Equation 6

produced the estimate s e = 2.38 inches. 

To test the regression results, we performed a sample calculation using Equation 5 and

s e = 2.38 inches. For weight = 175 lb, we obtained height = 68.88 ± 2.38 inches. 

Based on this estimate, we would expect Table 1 to show a mode (most frequent value)

for 175 lbs near 68.88 inches, with approximately 68.3 percent of the values falling

between 68.88 - 2.38 = 66.5 inches and 68.88 + 2.38 = 71.26 inches. Consistent with

our estimate, Table 1 shows a mode of 69 inches in the height values listed for the

weight range (170 to 179 lb) centered on 175 lbs. Furthermore, 68.7 percent of the

males in the weight range have heights that fall between 67 and 71 inches. 

We repeated the analysis using the data for females provided by Table 2 of Brainard

and Burmaster (1992). Table 2 of this memo presents a facsimile of this table which

lists the number of females (18 to 74 years) associated with each of 270 combinations

of height and weight. Brainard and Burmaster suggest that an analysis of weight vs.

height based on these data be limited to the 208 cells that fall within the indicated “box”

(16 height categories times 13 weight categories). Consistent with the earlier analysis,

we set up a data file containing these 208 records in which each record listed the

indicated height value, the midpoint of the indicated weight range (e.g., 115 = midpoint

of 110 to 119 range), and the number of females associated with the cell. 

We performed a weighted linear regression of the resulting data file using height as the

dependent variable, ln(weight) as the independent variable, and number of females as

a weighting factor. The resulting regression equation for females (18 to 74 years) was

height, in = 48.07 in + (3.07)[ln(weight,lb)] + e. (7)

The R2 value for females was low (0.0477) relative to the value obtained above for

males (0.1469), but was still statistically significant. 

To estimate s e, the standard deviation of e, we used the equation

s e = [S(Oij - Eij)

2

(Fij)/(total females)]0.5 (8)

in which Oij was the observed (listed) height associated with cellij, Eij was the estimated

height for cellij based on the regression equation, Fij was the number of females listed in

Table 2 for cellij, and total females is the total number of females in all cells. Equation 8

yielded an estimate of s e = 2.48 inches for females, a value consistent with the value of

2.38 inches obtained previously for males. 
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We performed a sample calculation to test the regression results. For weight = 145 lb,

Equation 7 yielded an estimate of height = 63.35 in ± 2.48 in. Based on this estimate,

we would expect Figure 2 to show a mode for 145 lb near 63.35 inches, with

approximately 68.3 percent of the values falling between 63.35 - 2.48 = 60.87 inches

and 63.35 + 2.48 = 65.83 inches. Figure 2 shows a height mode for the corresponding

weight range (140 to 149 lb) at 63 in, with 75 percent of the females falling between 61

in and 66 in. 

It should be noted that the regression analyses described above were performed using

weighting factors (i.e., values of Mij and Fij) which pertained to ranges of weight values

(e.g., 140 to 149 lbs) rather than to individual weight values. Although each weight

range was represented by its midpoint in the regression analyses, the resulting

estimates of s e are larger than the estimates we would have obtained from an analysis

in which the values of Mij and Fij were determined for individual weights rather than for 

weight ranges. Consequently, the values of s e presented above should be considered

conservative in the sense that they indicate an overly wide distribution of possible

heights for a given weight. 

In addition, it should be noted that the data used in the regression analyses (Figures 1

and 2) represent subjects 18 to 74 years of age. We intend to apply the regression 

results to all adult cohorts in 1998 pNEM/CO analyses. As these cohorts include

simulated persons 18 to 100 years of age, we will occasionally apply the results to

simulated people who fall outside the range of data used in the regression analyses. 

Although we believe the proposed equations are adequate for use in pNEM/CO, we will

continue to look for alternative estimation equations in the literature. To facilitate ongoing model testing, Jim Capel has incorporated the proposed equations into the 1998

version of pNEM/CO. Table 3 presents a subroutine for performing the required

calculations. 

Parameters To Be Generated by the Physiological Profile Algorithm

At this stage in the developmental process, we need a “master list” of the components

which make up the physiological profile. Table 4 lists all adult physiological parameters

which are currently required by the COHb and ventilation rate algorithms in the 1998

version of pNEM/CO. This table may change as we revise these algorithms. 
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References

Brainard, J., and D. E. Burmaster. 1992. “Bivariate Distributions for Height and Weight

of Men and Women in the United States,” Risk Analysis, Vol. 12, No. 2, pp. 267 - 275.

Burmaster, D.E., and E.A.C. Crouch. 1997. “Lognormal Distributions for Body Weight

as a Function of Age for Males and Females in the United States,” Risk Analysis, Vol.

17, No. 4, pp. 499 - 508. 

Johnson, T., J. Capel, R. Paul, and L. Wijnberg. 1992. Estimation of Carbon

Monoxide Exposures and Associated Carboxyhemoglobin Levels in Denver

Residents Using a Probabilistic Version of NEM. Report prepared by International

Technology Air Quality Services under EPA Contract No. 68-D0-0062. U.S.

Environmental Protection Agency, Research Triangle Park, North Carolina. 
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Table 1. Number of Men 18-74 Years of Age by Weight and Height, 1976 - 1980

(Number of Persons in Thousands). Facsimile of Table I in Brainard and

Burmaster (1992). 
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Table 2. Number of Women 18-74 Years of Age by Weight and Height, 1976 -

1980 (Number of Persons in Thousands). Facsimile of Table II in

Brainard and Burmaster (1992). 
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Table 3. Subroutine for Determining Adult Height Values in the Physiological

Parameter Algorithm.

Estimation of Adult Height Given Weight (Version 1.0)

1. In previous steps, the “physiological profile” program has determined age and

gender of simulated person and selected a value for weight in kilograms

(wgt_kg) from the body mass distribution specified for the age and gender. 

2. Convert wgt_kg to wgt_lb by equation: wgt_lb = (2.2046)(wgt_kg). 

3. Select random value (z) from unit normal distribution with mean = 0 and

standard deviation = 1. 

4. If male: hgt_inches = 34.43 inches + (6.67)[ln(wgt_lb)] + (2.38 inches)(z). 

5. If female: hgt_inches = 48.07 inches + (3.07)[ln(wgt_lb)] + (2.48 inches)(z). 

6. If height in cm is required, convert hgt_inches to hgt_cm by equation: 

 hgt_cm = (2.54)(hgt_inches). 
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Table 4. Physiological Parameters for Adults To Be Included in the 1998 Version of

pNEM/CO.

Parameter Algorithm(s) Containing

Parameter

Other Parameters Required

for Calculating Parameter

Age COHb

Ventilation rate

Gender COHb

Ventilation rate

Weight (body mass) COHb

Ventilation rate

Gender

Age

Height COHb Weight

Gender

Menstrual phase COHb Gender

Age

Blood volume COHb Gender

Weight

Height

Total hemoglobin content of the blood COHb Gender

Age

Pulmonary CO diffusion rate COHb Gender

Height 

Age

Endogenous CO production rate COhb Gender

Menstrual phase

Resting metabolic rate (RMR) Ventilation rate Gender

Age

Weight (body mass)

Energy conversion factor (ECF) Ventilation rate Gender

MAXRATIO: VER at VO2max Ventilation rate Gender

Age

SUBRATIO: VER at 65% of VO2max Ventilation rate Gender

Age

MAXRATIO

VO2max Ventilation rate NVO2max

Weight (body mass)

NVO2max Ventilation rate Gender

Age