1 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 (V b ) as a function of weight and height by the following equations

Men: V b= (20.4)( W) + (0.00683)(H 3 ) - 30 (1)
Women: V b= (14.6)( W) + (0.00678)(H 3 ) - 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+ (A 1 )(height) + (z)(S.E.) (3) where the values for A 0 , A 1 , and S.E. were obtained from Table 14 of Johnson et al.2

(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 = a 0+ (a 1 )[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 a 0, a 1, and s ewould 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)3

The R 2value was 0.1469. To estimate s e , the standard deviation of e, we used the equation

s e= [ S (O ij - E ij ) 2 (M ij )/(total males)] 0.5 (6) in which O ijwas the observed (listed) height associated with cell ij , E ij was the estimated height for cell ijbased on the regression equation, M ij was the number of males listed in Table 1 for cell ij , 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 R 2value 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 (O ij - E ij ) 2 (F ij )/(total females)] 0.5 (8) in which O ijwas the observed (listed) height associated with cell ij , E ij was the estimated height for cell ijbased on the regression equation, F ij was the number of females listed in Table 2 for cell ij , 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.4

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 M ij and F ij ) 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 eare larger than the estimates we would have obtained from an analysis in which the values of M ijand F ijwere determined for individual weights rather than for weight ranges. Consequently, the values of s epresented 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 willcontinue 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.5

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. W ijnberg. 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.6

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).7

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).8

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).9

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 VO 2max Ventilation rate Gender Age

SUBRATIO: VER at 65% of VO 2max Ventilation rate Gender Age MAXRATIO VO 2max Ventilation rate NVO 2max Weight (body mass) NVO 2max Ventilation rate Gender Age