Estimation of doubly labeled water energy expenditure with
confidence intervals
Richard H.
Jones,
Bakary J.
Sonko,
Leland V.
Miller,
Patti J.
Thureen, and
Paul V.
Fennessey
Departments of Preventive Medicine and Biometrics and of Pediatrics,
School of Medicine, University of Colorado Health Sciences Center,
Denver, Colorado 80262
 |
ABSTRACT |
Bivariate regression is used to estimate
energy expenditure from doubly labeled water data. Two straight lines
are fitted to the logarithms of the enrichments of oxygen-18 and
deuterium simultaneously as a bivariate regression, so that the
correlations between the oxygen and deuterium regression coefficients
can be estimated. Maximum likelihood methods are used to extend
bivariate regression to unbalanced situations caused by missing
observations and to include replicate laboratory determination from the
same urine samples, even if one of the replicates is missing. Use of maximum likelihood allows the determination of a confidence interval for the energy expenditure based on the log likelihood surface rather
than use of the propagation of variance methods for nonlinear transformations. The model is extended to include the subject's deviations from the two lines as a bivariate continuous-time
first-order autoregression to allow for serial correlation in the
observations. The analysis of data from two subjects, one without
apparent serial correlation and one with serial correlation, is presented.
deuterium; oxygen-18; maximum likelihood
 |
INTRODUCTION |
THE DOUBLY LABELED WATER TECHNIQUE, pioneered by Lifson
(7, 8) for energy expenditure measurement in animals, is now frequently
used for precise estimation of free living energy expenditure in humans
(2, 13). The technique uses two stable isotopes, oxygen-18
(18O) and deuterium (2H), which can exist in
water form as H218O and
2H2O, respectively. The mixed isotopes are
given to the subject to drink, and after the isotopes are allowed to
equilibrate with total body water, samples of body fluid, e.g., urine
or saliva, are taken to determine isotopic enrichment over the
measurement period, 14-21 days for adults and 5-8 days for
infants (10). The elimination of the isotopes from the body describes a
monoexponential function in which the 2H is eliminated only
as water, and the 18O is eliminated both as water and
carbon dioxide (CO2) (10). The difference in the
elimination rates of the two isotopes gives an estimate of the rate of
CO2 production in the body. The CO2 production
can be converted to energy expenditure by using either Weir's equation
(14) or Elia's equation (5), if the respiratory quotient (RQ) of the
subject over the study period is known. In an "ideal" experiment,
the isotope elimination curve would consist of as many points (samples)
as could be conveniently collected during the course of the experiment.
Under more practical conditions, the number of points in each curve is
limited by both the sample collection and sample processing
constraints, as well as the cost of 18O. Current practice,
as determined from the various methods sections of publications,
records as few as two (two-point method) data points to as many as
14-22 (multipoint method) data points (10) used to determine the
rate of the isotope elimination.
Cole and Coward (1) have reported a statistics technique to determine
confidence intervals for these approaches. Their model, in their
notation, follows the recommendation of Prentice (10)
|
(1)
|
|
(2)
|
for
i = 1...n, where the subscript D denotes deuterium,
and O denotes oxygen. Ei is the isotope enrichment
after subtraction of background expressed as a fraction of dose per
mole of body water, ti is the time (in
units of days), I is the enrichment at time 0, and
k is the flux rate. The error terms are assumed to have zero
means and constant variances. The multipoint method estimates log
I and k in each equation by linear regression, which also produces estimated standard errors of the estimates. This model is
appropriate when the original (untransformed) data have a constant
coefficient of variation, i.e., the standard deviation of the
observations,
EDi and
EOi are
proportional to the means, ID
exp(
kDti) and
IO
exp(
kOti). In
this case, the log transformation is a variance-stabilizing transformation. This assumption can be checked by plotting the residuals after fitting the lines to the transformed data (the differences between the observed log enrichment and the predicted value
based on the estimated regression coefficients). These residuals should
show a fairly random pattern. It is known that the error terms between
the two models are correlated; therefore, a bivariate regression is
appropriate where the observations are treated as pairs.
The CO2 production rate, with fractionation ignored, is
|
(3)
|
where
NO and ND are the body water
pool sizes obtained as
The
main statistical problem in obtaining the estimate of the
CO2 production rate in Eq. 3 is that this is a
highly nonlinear function of the estimated parameters in the two linear
regressions, and the covariance between the estimates of
kONO and
kDND is unknown. Cole and
Coward (1) argue that this covariance problem is much reduced by
working with two different linear regression equations, the difference
between Eqs. 1 and 2, and the sum of Eqs. 1 and 2
where
Ir = IO/ID (r denotes
ratio), Ip = IOID (p denotes
product), kr = kO
kD, and kp = kO + kD. The error terms in
these transformed equations will be uncorrelated if the variances in Eqs. 1 and 2 are equal, even if the errors in the
untransformed equations are correlated. The CO2 production
rate is now expressed as a function of the four parameters in these
last two linear regressions. The approximate variance of the
CO2 production rate is calculated using the delta method,
where the expression for the CO2 production rate is
expanded in a Taylor series (keeping only the first-order terms) about
the estimated values of the four parameters. The expression for the
approximate variance of the CO2 production rate is given as
Eq. 13 in Cole and Coward (1).
In this article, we use maximum likelihood estimation based on the
multipoint method. Confidence limits for energy expenditure are
obtained from the log likelihood surface so that methods of propagation
of error variances through nonlinear functions are not necessary. Also,
instead of treating the logs of the isotope enrichments as univariate
regressions, the pair of observations 2H and
18O are treated as a bivariate regression, with correlation
between the two observations in the pair. In the analysis of doubly
labeled water data, the problem can be somewhat more complicated if one of the measurements at a given time is missing. The other observation can be discarded, but this is a waste of information. If each urine
sample is split and analyzed twice, it is possible to estimate the
measurement error (analytical variance of the laboratory) as well as
the physiological variance of the subject. Prentice (Appendix 4.2, p.
290-293 of Ref. 10) discusses the effect of autocorrelated errors
on linear regression and states that "the effect of auto-correlation
is to reduce the advantage of the multi-point method." It is also
stated that
Because autocorrelation between errors may arise naturally or may
arise from attempting to fit an incorrect model, we recommend that if
there are ten or more equally-spaced data points, the autocorrelation
coefficient for the residuals is obtained. A high absolute value may
help to identify wrongly specified models.
Autocorrelation is also referred to as serial correlation. Serial
correlation can be observed in residual plots when the residuals seem
to have systematic excursions rather than a random appearance. Serial
correlation does not affect the position of the fitted line very much,
but it does affect confidence intervals of estimates in a
nonconservative way. If serial correlation exists and is ignored in the
model, the confidence intervals will be too narrow. In this paper we
take the approach that serial correlation in the errors can be modeled,
even for unequally spaced data, and tested using likelihood ratio tests
to determine whether the serial correlation is statistically
significant (6). If serial correlation is present in the data, the
error model gives more appropriate statistical tests and confidence intervals.
 |
METHODS |
The methods presented here are based on bivariate regression (a special
case of multivariate regression) in which there are two correlated
response variables. Classically, these methods require balanced data,
so there can be no missing observations, and, if there are replicate
analytical determinations at each time, there must be the same number
at every time, and the means of these replications are used as data.
Models of this type can be fitted using SAS (11) PROC GLM by using the
MANOVA statement. However, replicates must be averaged, and, if an
observation is missing at some time point, that time point is
discarded. This is due to the assumption of equal covariance matrices
at different times in the bivariate regression. Because there are two
components of variance, the analytical variance of the laboratory and
the physiological variation of the subject, it is not possible to weight the regression by the number of replications. A proper weighting
would be a function of these two components of variance.
Physiological variation causes both log enrichments to vary in a
similar way about the fitted lines. This is where serial correlation
may come into the model. This variation about the lines may not be
independent from observation time to observation time, as is required
by standard regression analysis. If an enrichment is above the line at
one time point, it is likely to be above the line at the next time
point. This variation about the line in a dependent fashion is serial
correlation, and the physiological variation is a random process. A
physiological model that fits the doubly labeled water data well is
where
yD(t) = logED(t),
yO(t) = logEO(t),
D = logID,
O = log
IO,
D =
kD,
O =
kO, and
(t) represents the
physiological variation about the lines as a function of time and is
assumed to have zero mean and unit variance. This random physiological variation enters both equations with different weights, gD
and gO, that are estimated from the subject's data;
gD and gO are the physiological standard
deviations of the two equations. This single random input entering both
equations produces correlation between the two equations. The model for
a single observational pair is
|
(4)
|
In
this model, ti denotes the time of the
observations, the subscript j denotes replications, and
Dj(ti) and
Oj(ti)
are the laboratory errors for replications and are assumed to have
means of zero, two different variances,
2D and
2O, and to be uncorrelated. The errors
are also assumed to be independent at different observation times and
independent of the physiological variation. In the Cole and Coward (1)
model, the error term in Eq. 1 would be
gD
(ti) +
D(ti) if there are no
replicates, and has variance g2D +
2D. If two replicates are averaged,
the error term would have variance g2D +
2D/2. This produces different error
variances if some of the replicates are missing.
Estimating the variance of the laboratory precision.
The variance of the laboratory precision can be estimated directly from
the replication data. If there are mi
replications at observation time i, these replications can
be averaged and the sum of squares of the deviations from the estimated
mean can be calculated. This estimate has
mi
1 degrees of freedom. The sums
of squares and degrees of freedom are summed over
observation times, and the variance is estimated as the final sum of
squares divided by the total number of degrees of freedom. This must be carried out separately for D and O, because the laboratory precision may be different for deuterium and oxygen-18 determination. Because these two variances can be estimated first, it eliminates two parameters from the maximum likelihood estimation. The standard deviations of the laboratory precision are estimated as the square roots of these variances.
Modeling the physiological variation.
The simplest model for physiological variation is that
(ti) is statistically independent at
different observation times and has a Gaussian distribution with zero
mean and unit variance. The variances of the two physiological inputs
are then g2D and
g2O, and the correlation is 1. This assumption means that the random input due to physiological variation is uncorrelated from time point to time point, but at each time, the
random deviation from the deuterium line is proportional to the random
deviation from the oxygen line. Because of the laboratory error, the
actual observations are not perfectly correlated.
This model has four parameters in the linear regression, two standard
deviation parameters for the laboratory precision,
D and
O, and two nonlinear parameters, gD and
gO, that are the standard deviations of the physiological
variation. Details of the maximum likelihood estimation, the method of
determining confidence intervals on the energy expenditure, and the
extension to models with physiological serial correlation are shown in
the APPENDIX.
 |
EXAMPLES |
Table 1 shows the enrichment data for a
subject without significant serial correlation (subject 1).
Missing values are indicated by periods. The details for the
calculation of the scale factors are reported by Prentice (p. 217, Ref.
10). The enrichments are represented as a fraction of the initial dose
giving a normalized enrichment by use of the formula
where
s is the enrichment of the sample,
p is
the predose enrichment of the subject,
a is the
enrichment of the dose,
t is the enrichment in the tap
water, a is the amount of the dose diluted for analysis in grams, W is
the amount of water used to dilute the dose (g), A is the amount of
dose administered (g), and 18.02 converts grams of water into moles. To
begin, the average of the two baseline O values is subtracted from each
O in the raw data, and the average of the two baseline D values is
subtracted from each D. This is the
s
p part of the conversion. The other parameters for
subject 1 are
a(O) = 219.51,
a(D) = 847.97,
t(O) =
12.20,
t(D) =
77.416, a = 0.0121, W = 2.3845, and A = 173.5. The RQ for this subject, needed to convert CO2
production to energy expenditure, is estimated from food intake data to
be RQ = 0.86.
The natural logarithms (base e) of the scaled data are
calculated, and the standard deviations of the laboratory precision are
estimated from the replications of the log data. For this data set, the
standard deviations of the laboratory precisions are estimated as
O = 0.00073 and
D = 0.01166. It should be
noted that the standard deviation of the laboratory precision based on
the logs of the data is the coefficient of variation (standard deviation divided by the mean) of the original data.
After the lines are fitted to the data, the predicted values of the
lines at each time point are calculated as well as the residuals. The
normalized residuals are calculated by dividing each residual by its
standard deviation. Plots of the fitted lines and the normalized
residuals are shown in Fig. 1
(left). A normalized residual that is >2.5 in absolute value
should be investigated as a possible error. Various investigators use
values of normalized residuals in the range of 2.0-3.0 for
detecting errors. The second D normalized residual at 7.06 days is
2.93, whose absolute value is >2.5. Going back to the raw
data, we see that the value 521.742 appears to be too low relative to
the other data. Setting this value to missing produces normalized
residuals, where the largest in absolute value is the second D
normalized residual of
2.06 at 12.06 days. This is considered to
be acceptable. The estimated coefficient of variation of the
measurement precision is decreased slightly to
D = 0.00969.

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Fig. 1.
Results for 2 subjects. Subject 1 (top left) has no apparent
serial correlation. Plot shows log data points and fitted straight
lines for subject 1. Plot (bottom left) shows
normalized residuals for subject 1. It was determined that the
one normalized residual at 7.06 days with a value of 3.03 was
probably an error, so this data point was declared missing in the final
analysis. Results for subject 2, with significant serial
correlation, are on right. Normalized residuals show systematic
trends. Recursive residuals use serial correlation to predict a
residual from the previous residuals and should show a more random
pattern. The low set of recursive residuals at 5 days is possibly a
timing error, but these data were included in the analysis.
|
|
From the regression analysis, 95% confidence limits on the slopes and
intercepts can be directly calculated and inverted to get confidence
limits on NO and ND
The
ND-to-NO ratio is estimated as
ND/NO = 1.033 (1.023, 1.042). This is in the range of physiological values. The
uncorrected CO2 production is estimated as
(NOkO
NDkD)/2 = 21.33 (20.81, 21.90)
mol/day. The confidence intervals on CO2 and energy expenditure are obtained by using the log likelihood surface, as
explained in APPENDIX. A fractionation correction is then
applied to correct for fractionation loss of both isotopes in breath
and transcutaneous water, and 18O in CO2. Let
the fractionation factors for 2H and 18O in
breath and transcutaneous water be f1 and
f2, respectively, and the fractionation factor for
18O in CO2 be f3. These
have values: f1 = 0.941, f2 = 0.992, and f3 = 1.040. The corrected
CO2 production is calculated (see p. 101 in Ref. 10) as
giving
a corrected CO2 estimate of 19.32 (18.84, 19.86) mol/day.
The estimated CO2 production is multiplied by 22.4 l/mol to
convert to liters/day of CO2. To convert from liters of
CO2 per day to total energy expenditure (kilojoules per
day), liters per day is multiplied by (5.55 + 15.48/RQ) (see p. 198 of
Ref. 10), where RQ is the subject's respiratory quotient. The
estimated total energy expenditure is 10,194 (9,938, 10,478) kJ/day.
This subject has a nonsignificant test for serial correlation by use of
a likelihood ratio test (see APPENDIX), and the normalized
residuals shown at the bottom left of Fig. 1 do not appear to have any
systematic trends.
The data for a second subject, who had significant serial correlation,
are shown in Table 2. The parameters for
subject 2 are
a(O) = 173.88,
a(D) = 649.969,
t(O) =
16.75,
t(D) =
117.745, a = 0.0121, W = 4.9257, and A = 211.52. The RQ for this subject is estimated to be 0.857.
The results of the regression for subject 2 are
The
ND-to-NO ratio is estimated as
ND/NO = 1.022 (1.010, 1.033). This is also in the range of physiological values. The
uncorrected CO2 production is estimated as
(NOkO
NDkD)/2 = 30.78 (29.79, 31.85)
mol/day. The corrected CO2 production is calculated as 28.17 (27.25, 29.17) mol/day. The estimate of energy expenditure is
14,902 (14,413, 15,431) kJ/day.
The normalized residuals for this subject are shown in Fig. 1
(upper right). These residuals show a more systematic behavior than those from the first subject. This second subject has significant serial correlation based on a likelihood ratio test. The change in
2 ln likelihood when the serial correlation parameter is
included is 4.93 and is tested as
2 with one degree of
freedom (P = 0.014). When serial correlation is modeled, the
physiological deviations from the lines are related at different times.
The estimated deviation at one time is used to predict the deviation at
the next time. The difference between the predicted deviation and
observed deviation can be called a recursive residual. These recursive
residuals should be nearly uncorrelated if the error structure is
properly modeled. Figure 1 (bottom right) shows the normalized
recursive residuals for the second subject. Modeling the serial
correlation appears to have removed the systematic behavior. The
residual at 5 days looks suspicious, because the data points are all
slightly below
2.0. This could be a timing error, possibly
caused by an improper urine collection. These data points were not
removed from the analysis.
Table 3 shows a comparison of the estimated
standard errors of the estimated energy expenditure obtained from the
2 ln likelihood surface and by the method of Cole and Coward (1)
for 10 subjects. The standard error of the likelihood method is
obtained by dividing the 95% confidence interval by 4 (±2 SE). A
computer program in FORTRAN is available from the first author. The
data input part of the program is specific for our data format from the
spreadsheet that we use. This part of the program would need to be
modified by a user with a different data format. The program is fairly large because it includes nonlinear optimization subroutines used to
search for maximum likelihood estimates of the nonlinear parameters.
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Table 3.
Estimates of energy expenditure with estimated standard errors for 10 subjects by the likelihood method and the method of Cole and Coward
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|
 |
DISCUSSION |
We have presented a new multipoint method for estimating energy
expenditure using doubly labeled water. The method is based on
bivariate regression with two components of variation, physiological variation and the variances of the laboratory precision. Unbalanced data caused by missing observations are handled using a maximum likelihood approach. Confidence intervals for the estimated energy expenditure are obtained directly from the log likelihood surface.
Table 3 shows that the estimated energy expenditures are very close to
the estimates obtained by the Cole and Coward (1) method. The standard
errors for the Cole and Coward method tend to be somewhat larger than
those of the likelihood method. This can be partly explained by the
fact that the likelihood method uses all available data when a
replicate or a replicate pair are missing. It is possible that the
approximations used when propagating variances through nonlinear
functions may tend to increase the estimated standard errors.
The model is extended to handle serial correlation in the physiological
variation. This model is an improvement over the model in the book by
Jones (see p. 181 in Ref. 6). The new error model based on a
continuous-time first-order autoregression fits the data better.
The variances of the logs of the laboratory precision are estimated
from the replicate data, where the urine samples are divided into two
samples and analyzed separately. Each replicate pair produces one
degree of freedom for the variance estimate. If one of the replicates
is missing, there is no information about the variance from that pair.
Two variances are estimated, one for deuterium and one for oxygen-18.
These unbiased estimates of the two variances are then held fixed in
the maximum likelihood estimates of the coefficients of the two lines
and the parameters of the physiological variation.
Normalized residuals from both the estimation of the variances of the
laboratory precisions and the fitting of the lines are used for error
detection. Based on the standard normal distribution, there is some
question about how to choose a cutoff value for detecting an outlier.
In our data we usually have ~14 observations on both deuterium and
oxygen-18. If there is an outlier in the data, it tends to pull the
fitted line toward itself, making the residual smaller. We recommend
that an outlier with an absolute value >2.5 be considered as a
possible error. A case can be made for a cutoff of anything in the
range of 2.5-3, because the number of observations increases the
probability of an observation in the range of 2-3 in absolute
value. In engineering, the value of 3 is often used, but there may be
thousands of data points.
If there is serial correlation in the physiological variation, it is
important that it be modeled in the error structure. Modeling the
serial correlation error structure affects the estimation of the lines
very little, but it does affect the width of the confidence intervals.
In general, confidence intervals are shorter if the data have serial
correlation and it is not modeled, and these shorter intervals are not
correct. It is possible that our model can underestimate the width of
the confidence intervals. Some of the parameters, such as RQ, used in
the model are assumed to be known, when they are actually estimates.
 |
APPENDIX |
Maximum Likelihood Estimation
To handle laboratory replications, physiological variation, and missing
observations, it is necessary to use maximum likelihood estimation.
These estimates are usually obtained by finding the maximum of the log
likelihood or the minimum of
2 log likelihood with respect to
the unknown parameters. The reason
2 log likelihood is often
used is that
2 likelihood ratio tests for significance
of parameters can be calculated from the change in
2 log
likelihood when parameters are added or eliminated from the model. In
textbook problems, maximum likelihood estimates are usually obtained by
differentiating the log likelihood with respect to the unknown
parameters, setting the derivatives to zero, and solving a system of
equations. In the real world there is often no closed-form solution,
and numerical searches need to be used.
In matrix notation, the regression model at a single time point,
ti, with a replicate for both the
2H and 18O, is
|
(5)
|
If
the replication error variances are
2D
and
2O, the total covariance matrix of
the errors in this model is
|
(6)
|
If we let
and
if we assume that the errors are Gaussian, the contribution to
2
log likelihood from this time point (dropping the constant terms
involving
), is (6)
where
|Vi| denotes the determinant, and ' denotes the transposed vector. Assuming that the physiological
variation does not have serial correlation, i.e., it is statistically
independent at different observation times, the
2 log likelihood
from all observation times is
|
(7)
|
To
obtain maximum likelihood estimates, it is necessary to find the
minimum of
with respect to the unknown parameters. These unknown
parameters are the four intercepts and slopes in the vector
,
gD and gO.
The search for a minimum of
2 log likelihood can be simplified,
because for given values of gD, gO,
2D and
2O,
has a closed form solution
that minimizes
2 log likelihood
Substituting
this into Eq. 7
which
is now a function of only gD and gO. The
nonlinear search need only be carried out with respect to these two parameters.
In the unbalanced case, consider the modification when one of the
replications is missing. In Eq. 5, suppose the second oxygen replication is missing. The resulting equation is
The
covariance matrix in Eq. 6 becomes
The
estimation then continues with these smaller arrays. This method can be
used for any pattern of missing observations, including that pattern
when all replications of deuterium or oxygen are missing. The pattern
of missing observations can be different at different observation
times. This methodology assumes that the missing observations are
missing at random (9). This means that the reason an observation is
missed is not related to the value of the missed observation.
Confidence Intervals From the Log Likelihood Surface
Maximum likelihood estimation also allows confidence intervals for the
estimated parameters to be estimated directly from the
2 log
likelihood surface. This method is sometimes referred to as
"likelihood ratio-based confidence intervals" (see p. 289 of Ref.
12) or "profile likelihood confidence intervals" (see p. 43 of
Ref. 4).
Model Eq. 4 has four linear parameters,
D,
kD,
O, and kO.
The main quantity of interest is
which
is a nonlinear function of the
-values, and the two terms are
correlated. To obtain a confidence interval on
2r'CO2, the maximum
likelihood estimates of the parameters are obtained first by minimizing
2 log likelihood. The value of
2r'CO2 is varied to find
the point at which
2 log likelihood increases by the
significance level for
2 with one degree of freedom. For
a 5% level of significance, this is 3.84. At each value of
2r'CO2 that is tried in
the search,
2 log likelihood is minimized with respect to
kO,
O, kD, and
D, under the constraint that
Serial Correlation
The physiological variation can be modeled as a continuous-time
first-order autoregression (see p. 92 of Ref. 6). Let
i = ti
ti
1 be the time
interval between observation time ti
1 and
observation time ti. The model is
In
this model the parameter must be positive and represents the rate of
decay of the physiological components toward the lines. There is one
random input to the model,
(ti) that
represents the unpredictability of the physiological component between
the two times. The random input has mean zero and variance, 1
exp(
2a
i), that depend on the time
interval. The variance of the autoregressive process is 1. Notice that
the variance of the random input approaches the variance of the process
2 as
i becomes large, and
approaches zero as
i approaches zero.
Given this model for the physiological variation, the parameters of the
model can be estimated using the state space methodology, as in Jones
(6). To obtain the likelihood ratio test for whether the inclusion of
the parameter for serial correlation is significant, the models with
and without serial correlation are fitted to the data, and the change
in
2 ln likelihood is tested as
2 with one degree
of freedom. The degrees of freedom are the number of additional
parameters included in the model, in this case the single parameter
"a."
 |
ACKNOWLEDGEMENTS |
We thank the reviewers for their comments.
 |
FOOTNOTES |
The first author is partially supported by National Institute of
General Medical Studies Grant GM-38519. The mass spectrometry core is
supported in part by the following National Institutes of Health
grants: HD-04024, HD-20716, DK-48520, and DK-49181. The Center for
Human Nutrition has supported one author (B. J. Sonko) for a pilot study.
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Address for reprint requests and other correspondence: R. H. Jones,
Dept. of Preventive Medicine and Biometrics, School of Medicine, Box
B-119, Univ. of Colorado Health Sciences Center, 4200 East 9th Ave.,
Denver, CO 80262 (E-mail: rhj{at}times.uchsc.edu).
Received 28 May 1999; accepted in final form 1 October 1999.
 |
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Coward, W. A.,
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207-212,
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Cox, D. R.,
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Elia, M.
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