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ABSTRACT |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
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The purpose of this analysis of previously published data was to identify a model that accurately predicts the mean ozone-induced FEV1 response of humans as a function of concentration (C), minute ventilation (
E), duration of exposure (T), and age. Healthy young adults (n = 485) were exposed for
2 h to one of six ozone concentrations while exercising at one of three levels. Candidate models were
fitted to portions of the data and evaluated on the basis of their ability to predict the mean response
of independent samples. A sigmoid-shaped model that is consistent with previous observations of
ozone exposure-response (E-R) characteristics was identified and found to accurately predict the
mean response with independent data. This model in a more general form may allow the prediction
of responses under conditions of changing C and E. We did not find that response was more sensitive to changes in C than in E, nor did we find convincing evidence of an effect of body size upon response. We did find that response to ozone decreases with age. In summary, we have identified a biologically plausible, predictive model that quantifies the relationship between the ozone-induced change
in FEV1, and C, E, T, and age. McDonnell WF, Stewart PW, Andreoni S, Seal E, Jr., Kehrl HR, Horstman DH, Folinsbee LJ, Smith MV. Prediction of ozone-induced FEV1 changes: effects of concentration, duration, and ventilation.
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INTRODUCTION |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
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The magnitude of the acute, reversible decrement in FEV1 induced in humans by short-term ozone exposure is known to be
a function of ozone concentration (C), minute ventilation
(E) during exposure, and duration of exposure (T). Although a number of attempts have been made to describe the
relationship between ozone exposure and FEV1 response,
these efforts have generally been limited by small ranges or
few data points for one or more of the exposure variables, and
no model has been identified that adequately describes response as a function of all three exposure variables simultaneously (1). In particular, the relationship between response and E has not been well characterized, and most
published models have ignored the increasing variability in response that is evident at higher levels of exposure (1, 2). On
the other hand, much information has been published that elucidates some characteristics of the relationships between
mean response and C and T.
Data from the literature suggest that ozone exposure- response (E-R) models for changes in lung function in humans should be consistent with the following observations.
- For exposures of less than 8 h duration, the response increases monotonically with increasing C, E, and T (1).
- The response is nonlinear in each of the three exposure variables, and the E-R curve is concave upward at low values of the three variables (1, 3).
- With increasing T, the response reaches a plateau, the magnitude of which is a function of the rate of exposure (8, 10, 17).
- With increasing C, the response appears to approach a plateau (1, 3).
- Individuals vary in their response to ozone, and this variability in response becomes more pronounced at higher levels of exposure (1, 2).
- Older adults tend to be less responsive than younger adults (11).
Two unanswered questions regarding the E-R relationship
are whether the response is more sensitive to changes in C
than to changes in E, as some have suggested (4, 5, 14), and whether the magnitude of response is independent of differences in lung size (15), or is a function of body surface area
(BSA) or lung size, as we and others have assumed.
The overall purposes of this study were: (1) to identify an E-R
model that is consistent with these prior observations and that
accurately predicts the mean change in FEV1 as a function of C,
E, T, and age; (2) to determine whether response can be described as a function of the dose rate (C × E), or whether the
sensitivities of the response to changes in C and E are unequal; and (3) to determine whether response is more directly related to absolute levels of E or to E normalized to some index of body or lung size.
Twelve similar, plausible models were identified a priori
that included all combinations of three model forms, two variance structures, and two methods of expressing E (with and
without adjustment for BSA). Dose rate was expressed as C × (E)z in the models, and the magnitude of the exponent z was
assessed to determine whether C and E affected response
differently. We fit all 12 models and assessed how well the
models predicted response, using a cross-validation technique
in which some of the data are used to fit the models and the
remaining data are used to assess the accuracy and precision
of the resulting model predictions.
Although all of the models did an adequate job of predicting mean response from the data available, we selected one
model that is biologically plausible and flexible in its applications and also consistent with previously published observations. A description of the other models is presented in APPENDIX. The results further indicated that response was not
significantly more sensitive to changes in C than to E, and
that correction of E for body or lung size did not substantially change the fit of the models.
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METHODS |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
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The exposure and response data that we modelled were generated in a series of eight experimental chamber exposure studies conducted at the U.S. Environmental Protection Agency Clinical Research Facility in Chapel Hill, North Carolina over the period 1980 through 1993 (1, 7, 12, 18). The subjects were 485 nonsmoking, healthy, white males, ages 18 to 36 yr. Potential subjects were excluded for any history of asthma, chronic disease, or symptoms of an acute respiratory infection within 4 wk prior to the beginning of the study, or for current medication use. All studies were approved by the Committee on the Protection of the Rights of Human Subjects of the University of North Carolina School of Medicine, all volunteers read and signed a statement of informed consent for the study in which they participated.
The details of the exposure protocol, which was nearly identical for all studies, are documented in the original manuscripts (1, 7, 12, 18). Briefly, for each study, each subject was exposed for 2 h on one occasion to one of six ozone concentrations (0.0, 0.12, 0.18, 0.24, 0.30, or 0.40 ppm) at one nominal level of exercise. In the event that a subject participated in more than one study, or had a clean air exposure in addition to an ozone exposure, only data from the first exposure in the facility were used for this manuscript, with one exception. In that study, in which each participant received both air and 0.40 ppm ozone exposures, the air data from every third subject were used and the ozone data from the other two-thirds were used.
FEV1 was measured in triplicate before exposure, at the end of the
first hour of exposure, and again at the end of the second hour of exposure. Using the largest FEV1 of the three trials in each session, the
percent decrement (100% × [pre 
post]/pre) in FEV1 (DELFEV1)
was calculated at the end of the first and second hours of exposures.
During the 2-h exposures, individuals alternated 15-min periods of
rest with 15-min periods of an activity chosen to produce a given E.
In one series of studies this activity was treadmill walking, and a level
of exercise was selected to produce a E of approximately 35 L/min/ m2 BSA; in another study the level of treadmill walking was selected to produce a E of 25 L/min/m2 BSA; and in yet two other studies the
activity was rest, which requires a E of approximately 5 L/min/m2
BSA. E was measured for 2 min during the latter part of each of the
four exercise periods, and the mean value for the first two periods was
taken to represent the exercise E for the first hour. The mean value
for all four periods was used for the 2-h exercise E. Because of the
difficulty in obtaining an accurate estimate of true resting E, each
subject whose prescribed activity level was rest was assigned an exercise E of 5 L/min × BSA. In order to account for the ventilation during the rest periods (50% of each exposure period), the 1- and 2-h exercise values were averaged with 5 L/min × BSA (the approximate
E during rest) to produce estimates of the overall E during exposure. These overall levels of E, reported here, will appear smaller
than those in the original reports, for which only the E measured during exercise was reported. The details of measurement of FEV1 and E, and of the production of ozone and maintenance of chamber conditions, are provided in detail in the original manuscripts.
We initially considered 12 similar E-R models for fitting the data. Because all of the models predicted independent data equally well, we describe one in the body of the manuscript and provide information on the other 11 in APPENDIX.
The form of the selected model (Equation 1) describes response as a function of the rate and duration, T, of exposure.
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(1) |
This model is based on the logistic function (12, 21) that describes a
sigmoid-shaped curve with a lower asymptote of 0 and an overall asymptote that is quantified by the numerator of Equation 1 and which
is a function of age. The values of the parameters 
3, 5, and 6 determine the position of the curve along the abscissa and define the steepness of the curve between the asymptotes. For fixed T, the response follows the logistic curve, reaching the overall upper asymptote defined
by the numerator at high levels of C and E. For fixed values of C and
E, with increasing T the response follows a sigmoid-shaped curve
with an asymptote that is a function of C × E, and which is less than
or equal to the overall asymptote in the numerator. (See Figure 2 for
examples in which the change in FEV1 is described as a logistic function of C at fixed values of E and T.)
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Because the present data violate some of the assumptions of ordinary least-squares regression (e.g., equal variance in response for all
treatment conditions, and independence of the observations), we explicitly modelled the variance structure with a "mixed models" approach (22). This method treats some coefficients as common to all
individuals (fixed coefficients) and the coefficient 1 (the overall asymptote of the sigmoid) as varying among individuals (random coefficient). The value of 1 presented for each model is the estimated mean
of the population of individual 1 values. The regression coefficients for
this variance structure were estimated with methods described by
Vonesh (23, 24), and statistical significance of these coefficients was
determined with the asymptotic covariance matrix of the estimated
parameter values (23, 24). Presentation of details of these statistical
methods is beyond the scope of this manuscript; further description
and appropriate references are, however, presented in APPENDIX.
The method that we chose for assessing the predictive ability of the model was that of a 4-fold cross-validation (25). The data were randomly divided into four samples (i.e., A,B,C,D), blocking on ozone concentration and three levels of exercise. Each candidate model was fitted to three-fourths of the data (e.g., A,B,C), and the ability of the resulting expression to predict the independent observations of the other one-fourth of the data (i.e., D) was evaluated. This was repeated for the other three possible data combinations (i.e., A,B,D used to predict C; A,C,D used to predict B; and B,C,D used to predict A). Thus, for each individual, we have an observed response and a predicted response that was generated from an independent set of data. We plotted and regressed the observed responses against these cross-validation prediction responses for all individuals. The slope and intercept of the resulting regression line were estimated, and the proportion of the variance explained (R2) was calculated. We also plotted and examined the cross-validation prediction errors (observed minus predicted responses) for evidence of any relationship between these prediction errors and each of the independent variables.
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RESULTS |
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ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
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A total of 485 individuals were exposed once to ozone, and
provided 1- and 2-h response data for analysis. Six studies contributed data for the group with the heaviest target exercise
level, one study contributed data for the group with the moderate target exercise level, and two studies contributed data
for the group at rest. There was considerable variability in the
absolute E and the E/BSA within each of the two groups
with exercise (Table 1), and there was considerable overlap
between these two groups. This variability was artificially
smaller in the resting group because individuals in this group
were assigned a E value of 5 L/min × BSA, and there was no
overlap of E between the resting and exercising individuals.
In Table 2, the mean decrements in FEV1 are shown for each
of three exercise groups, with subjects assigned to a group on
the basis of measured E rather than target level of exercise.
Very little response was noted in the resting group at any concentration of ozone. Otherwise, the magnitude and standard
deviation of the percent decrement in FEV1 generally increased with increasing C, T, and E.
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The estimates of the regression coefficients and standard
errors for the model fitted to all the data (A,B,C,D combined)
are listed in the first row of Table 3. Note that these coefficients are based on age having been centered (the mean subtracted from each individual value) at age 24 yr; and on E divided by 100. All model coefficients were significantly different
(p < 0.0001) from zero. The estimate of 1, the mean predicted
upper asymptote of the E-R function for age 24.0 yr, is an
18.6% decrement in FEV1. This maximal mean predicted decrement in FEV1 decreases by 1.1% (0.059 × 18.6%) for each
year of age up to 36 yr, and increases by the same amount
down to age 18 yr. This effect of age (2) on the upper asymptote of the model is the basis for the observed effects of age on
response at all levels of exposure. The estimated value of 4,
the exponent for E, was 0.91 ± 0.10, and was not significantly
different from 1 (95% confidence interval [CI]: 0.71 to 1.11),
indicating that response is not significantly more sensitive to
changes in C than in E. In a sensitivity analysis in which resting E was assumed to be 7 L/min × BSA rather than 5 L/
min × BSA, 4 = 1.08 (95% CI = 0.87 to 1.31). Division of E
by BSA did not substantially change the fit of the model.
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In Figure 1, the cross-validation predictions are plotted
against the observed individual responses. The observed and
predicted responses are generally in good agreement over the
entire range of response (R2 = 0.41, p < 0.0001), and the regression line of predicted versus observed response (slope = 0.96 ± 0.04, intercept = 0.13 ± 0.31) was not significantly different (p = 0.41) from the line of identity. Differences between the regression line and the line of identity represent differences in the percent decrement in FEV1 of less than 1%
over the range of exposures. The prediction errors (observed values minus cross-validation predictions) were also plotted
versus the independent variables (C, E, T, age, and BSA),
and no relationships between the prediction errors and the independent variables were observed (data not presented).
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In Figure 2, the observed mean responses given in Table 2
and the predicted functions (using all the data) at mean age
are plotted versus C for 2 h exposures for each of the exercise
levels. In general, the observed responses increase with increasing C and E, and the model appears to fit the observed
data well.
In Figure 3 the observed individual responses and the predicted (using all the data) responses at mean age are plotted
versus E for each concentration-duration combination. With
the exception of a slight overprediction of the response at C = 0.12 ppm for 1 h of exposure, the predicted response fits the
central tendency of the data very well. This demonstrates that
response increases in a sigmoid-shaped manner with increasing E, and that mean response can be accurately described as
a logistic function of Ez for any C and T within the data range
of this study.
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DISCUSSION |
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ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
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The model that we selected provided accurate predictions of
mean response as a function of the exposure conditions and
age over the range of the observed data (C = 0.0 to 0.40 ppm
ozone, E = 8 to 50 L/min, and T = 0 to 2 h, and age = 18 to
36 yr) for independent samples. We chose a model of the form
of Equation 1 because it is consistent with the previously observed characteristics of the E-R relationship listed at the beginning of this report; allows, in its general form, prediction of
the response when C and E are changing as a function of time;
and has characteristics that are common to a number of biologic systems. It should be noted that further rigorous evaluation of this model should include exposures of longer duration
than 2 h, and should include data at low levels of exercise,
which were lacking in this present study.
Equation 1 is the analytical solution of a dynamic model of
response for the special case in which C and E are held constant. This more general model describes the response of a
two-compartment system in which substance X enters Compartment 1 at a rate of C(t) × E(t), the rate at which ozone is
inhaled, and in which substance X is removed from Compartment 1 at a rate proportional to the concentration of X in
Compartment 1. Therefore, the rate of change in the concentration of X is the difference between the rates of input and
output, and is given by Equation 2:
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(2) |
Although E changes every 15 min, owing to alternating rest and
a fixed level of exercise, the average level of exercise over each
hour remains constant. At constant C and E, Equation 2 can be
solved for X as a function of time, as shown in Equation 3:
|
(3) |
where a is the inverse of the time constant of Compartment 1. In Compartment 2, an age-dependent, sigmoid-shaped FEV1 response is produced as a logistic function of the concentration of X in Compartment 1 as shown in Equation 4:
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(4) |
If one substitutes Equation 3 into Equation 4, this model reduces to Equation 1, with 3 = 
and 6 = a, the inverse
of the time constant of Compartment 1.
Although we are not suggesting that ozone behaves as simply as substance X in producing FEV1 decrements, or that the two functional compartments of the model correspond directly to anatomic compartments, it is plausible that both the rate-limiting steps that define the kinetics and the processes that shape the response curve could be approximated by such a model. The dynamic response of many biologic systems is consistent with that of Compartment 1. The logistic function of Compartment 2 describes a cumulative growth curve and is consistent with a description of an integrated neurologic response that depends upon activation of increasing numbers of neurons to produce an escalating response. A model such as this is amenable to biologically based modifications. For example, if antioxidants are found to modify the response to ozone exposure, or if the uptake of ozone is found to change as a function of time, this model provides several possible locations at which such effects could be modelled, and provides a framework for testing these options.
To model the variability in the data, we chose the random-coefficient (RC) variance structure, which is based on the notion that each individual has a unique E-R relationship, and
that some of the variability in response is due to individual differences in this relationship. Because decrements in FEV1
were measured only twice for each subject, only one model
parameter (1 in this case) could be allowed to vary among individuals. Although this model accurately described the mean
data, it is likely that individuals differ in more than one parameter, in which case full specification of the variability structure
would require more than a single random coefficient.
We found little evidence that response was related to measures of body or lung size. We could not distinguish between
models with or without adjustment of E for differences in
BSA. Furthermore, we could find no evidence that the cross-validation prediction errors for the model without adjustments for BSA were related to BSA over different ranges of
the E-R surface. In additional analyses, we substituted powers
of BSA, forced vital capacity (FVC), and height into the models. Any differences in model performance were very small,
and we concluded that any effect of BSA, height, or baseline
FVC on percent decrement in FEV1 in this population is small
if it exists at all. This is in agreement with the finding by Messineo and Adams (15) that among women exposed under the same conditions, there were no differences in response between those with larger and smaller measures of baseline
FVC. On the other hand, one might expect that the tissue dose
of ozone, and consequently the magnitude of response, would
be a function of the surface area of the conducting airways, the
most likely site of the neural receptors responsible for the
acute reductions in FEV1 observed with ozone exposure (26).
The absence of an observed relationship between response
and BSA, height, or FVC may have been due to the poor correlation between these variables and airway caliber (27, 28)
and to the relatively small range of heights and ages in the
sample of males subjects on whom our study was conducted. It
is possible that a relationship between response and body or
lung size would have been evident if our sample had included
individuals (such as children) who had increased the range of
body and airway sizes, or if we had made measurements of
dead-space volume. One implication of our findings in the
present study is that among adults who are exposed to ambient ozone while performing weight-bearing work, heavier individuals will require a larger E, and should consequently experience larger percent decrements in FEV1 than smaller
individuals.
We found that the estimated value of the exponent (4) for
the ventilation term was not significantly different from unity in any of the models we examined, indicating that response is not more sensitive to changes in concentration than in ventilation. When the overall data are fitted with the exponent of E
fixed at one, the other estimated parameters of Model 1 do
not change meaningfully (Line 2, Table 4). This finding contrasts with common perceptions (4, 5, 14). In two previous
studies in which C was reported to be a stronger predictor of
response than was E, alternating rest and exercise protocols
similar to those in the current study were utilized, and the E
measured during exercise, rather than that averaged over both
rest and exercise, was used to represent ventilation in regression models (4, 5). To explore the possibility that our use of
the mean of the exercising and resting E values had some effect on this issue, we refitted our data, utilizing the E measured only during the exercise portion of the exposure. This
resulted in mean E values of approximately 10, 48, and 65 L/min, rather than 10, 29, and 37 L/min, for the three nominal
exercise groups described in Table 1. The estimated exponent
for E calculated with these exercising E values was found to
be 0.70 ± 0.08 (95% CI: 0.53 to 0.86), which was significantly
less than unity (Line 3, Table 4). It thus appears that some of
the previous evidence for the response to ozone being more
sensitive to changes in C than in E is a result of ignoring the reduced E during the resting portions of exposure. This, however, does not address the results of a study in which continuous exercise was utilized (14). Partial explanations for the
findings of that study are that exposures were generally conducted over a wider range of C than of E, that there is more
error in the measurement of E than of C, and that there is
greater difficulty in controlling E than C. All of these factors
will contribute to less weight being placed on E than C in regression equations. In the current study, we utilized a wide
range of E values, and by using individual rather than mean
data for modelling, we took advantage of our inability to precisely control E to a single value for each subject in a group.
As we have previously observed in portions of these data
(12, 13), and others have observed in independent data (11), age was a strong predictor of response, with older individuals being less responsive than younger ones. We initially included age as a linear term in the model, in order to reduce the number of parameters to be estimated. Because the linear model
predicts increases in FEV1 beyond the age of 41 yr, we surmise
that the true relationship should be curvilinear. We refitted
the model, replacing the term (1 + 2 Age) with an exponential function (Exp[2 Age]), which fit the data well and resulted in minimal changes in other regression coefficients
(compare Line 4 with Line 2 in Table 4). Data for older individuals are necessary to further elucidate the age relationship.
Close examination of Table 2 and Figure 2 indicates that
lung function actually improved for resting individuals exposed to clean air and low levels of ozone, and that in clean
air, exercising individuals experienced slightly larger decrements than did resting individuals. This suggests that in our
studies, both small training effects and small direct effects of
exercise on lung function, perhaps due to fatigue, may have
been present. We refitted Model 1C (with the exponent of E
fixed at 1.0 and including the exponential age expression) with
an intercept and a linear term in E, and found that both the
intercept and main effect of E were significantly different
from zero, as shown in Equation 5 and Table 4, Line 5:
|
(5) |
The magnitudes of these non-ozone effects are quite small (a
2% increase in FEV1 over the course of the exposure, and a
1% decrease for E = 25 L/min), and are likely to be the result of specifics of testing in our laboratory. If desirable, one
may use Equation 5 to estimate ozone-induced FEV1 responses corrected for these other effects.
One of the goals of the present study was to test the validity
of a selected predictive model for the effects of ozone exposure on FEV1. We used a 4-fold cross-validation technique to
determine how well Model 1 could predict the responses of independent samples (25). Such model validation includes consideration of both the amount of bias in the predictions and
the precision of the predictions relative to independent observations. The plots of the observed versus the predicted responses (Figure 1), and plots of predictive errors versus C, E,
T, and age (not presented), indicate that the model is appropriate and provides generally unbiased predictions of the
mean response of FEV1 to ozone exposure. The unweighted proportion of the variability in individual responses explained by C, E, T, and age in the predictive model was 41% (Figure 1). The unexplained variance is largely due to individual differences in ozone responsiveness not explained by age, with
some additional variance due to measurement error, training
effects, and effects of exercise. It should be noted that mean
responses of groups of individuals can be predicted with
greater precision than can individual responses, as is evident
from comparing Figures 1 and 2.
We concluded that the mean FEV1 response to short-term
ozone exposure can be accurately predicted within the bounds
of the observed exposure conditions as a function of C, E, T,
and age through the use of any of the proposed model forms
and either of the variability structures. Of these, we found a
random-effects model of the form of Equation 1 to be most attractive. We found no convincing evidence that measures of
body or lung size were related to magnitude of response, although we acknowledge that the range in our data may not
have been adequate to permit the observation of such a relationship, and we did not have information on potentially relevant measures such as dead-space volume or airway surface
area. We found no evidence that the sensitivity of response to
changes in E was different than to changes in C, and concluded that mean response could be accurately predicted as a
function of the product of C and E. We found that the effects of ozone diminished with increasing age, and we concluded
than much individual variability in response to ozone remains
unexplained.
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Footnotes |
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Correspondence and requests for reprints should be addressed to William F. McDonnell, U.S. EPA (MD-58B), RTP, NC 27711.
(Received in original form November 12, 1996 and in revised form May 1, 1997).
Disclaimer: Although the research described in this article has been funded wholly by the U.S. Environmental Protection Agency, it has not been subjected to Agency review. Therefore, it does not necessarily reflect the views of the Agency.Acknowledgments: The authors would like to thank Melinda Eads for data management, Sa'id Abdul Salaam, Paulette DeWitt, and Martin Case for technical support, Ila Cote and John Vandenberg for overall support for this project, and the volunteers for their time and willingness to participate in these studies.
Supported by the U.S. Environmental Protection Agency.
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References |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
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1. McDonnell, W. F., D. H. Horstman, M. J. Hazucha, E. Seal, E. D. Haak, S. Salaam, and D. E. House. 1983. Pulmonary effects of ozone exposure during exercise: dose-response characteristics. J. Appl. Physiol. (Respir. Environ. Exerc. Physiol.) 54:1345-1352.
2. Kulle, T. J., L. R. Saunder, J. R. Hebel, and M. D. Chatham. 1985. Ozone response relationships in healthy nonsmokers. Am. Rev. Respir. Dis. 132: 36-41 [Medline].
3. Avol, E. L., W. S. Linn, T. G. Venet, D. A. Shamoo, and J. D. Hackney. 1984. Comparative respiratory effects of ozone and ambient oxidant pollution exposure during heavy exercise. J. Air Pollut. Control Assoc. 34: 804-809 [Medline].
4. Folinsbee, L. J., B. L. Drinkwater, J. F. Bedi, and S. M. Horvath. 1978. The influence of exercise on the pulmonary function changes due to low concentrations of ozone. In L. J. Folinsbee, J. A. Wagner. J. F. Borgia, B. L. Drinkwater, J. A. Gliner, and J. F. Bedi, editors. Environmental Stress. Academic Press, New York. 125-145.
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Hazucha, M. J..
1987.
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6. Horstman, D. H., L. J. Folinsbee, P. J. Ives, S. Abdul-Salaam, and W. F. McDonnell. 1990. Ozone concentration and pulmonary response relationships for 6.6-hour exposures with five hours of moderate exercise to 0.08, 0.10 and 0.12 ppm. Am. Rev. Respir. Dis. 142: 1158-1163 [Medline].
7. Seal, E. Jr., W. F. McDonnell, D. E. House, S. A. Salaam, P. J. DeWitt, S. O. Butler, J. Green, and L. Raggio. 1993. The pulmonary response of white and black adults to six concentrations of ozone. Am. Rev. Respir. Dis. 147: 804-810 [Medline].
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McDonnell, W. F., and
M. V. Smith.
1994.
Description of acute ozone response as a function of exposure rate and total inhaled dose.
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Silverman, F.,
L. J. Folinsbee,
J. Barnard, and
R. J. Shephard.
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10. Hazucha, M. J., L. J. Folinsbee, and E. Seal Jr.. 1992. Effects of steady-state and variable ozone concentration profiles on pulmonary function. Am. Rev. Respir. Dis 146: 1487-1493 [Medline].
11. Drechsler-Parks, D. M., J. F. Bedi, and S. M. Horvath. 1987. Pulmonary function responses of older men and women to ozone exposure. Exp. Gerontol. 22: 91-101 [Medline].
12. McDonnell, W. F., K. E. Muller, P. A. Bromberg, and C. M. Shy. 1993. Predictors of individual differences in acute response to ozone exposure. Am. Rev. Respir. Dis 147: 818-825 [Medline].
13. Seal, E. Jr., W. F. McDonnell, and D. E. House. 1996. Effects of age, socioeconomic status, and menstrual cycle on pulmonary response to ozone. Arch. Environ. Health 51: 132-137 [Medline].
14. Adams, W. C., W. M. Savin, and A. E. Christo. 1981. Detection of ozone toxicity during continuous exercise via the effective dose concept. J. Appl. Physiol. (Respir. Environ. Exerc. Physiol.) 51: 415-422 .
15.
Messineo, T. D., and
W. C. Adams.
1990.
Ozone inhalation effects in females varying widely in lung size: comparison with males.
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APPENDIX |
|---|
Twelve models were fitted to the data and included all combinations of three model forms and two variance structures, with and without division of E by BSA. All three models are
based on the logistic function and all allow an effect of age on
the level of the plateau. The first model form is as presented in the manuscript. The second (Equation 6) is a simple logistic
function of the total inhaled dose (C × Ez × T) of ozone:
|
(6) |
Models of the form of Equation 6 are nonlinear versions of previous "effective dose" models (4, 9, 14), but do not allow the plateau that occurs with long-duration exposure to be a function of rate of exposure (10). We have previously demonstrated that this simpler model form is able to provide reasonable approximations of response data resulting from exposures of short duration or a limited range of ozone concentrations (8). We believe that the adequate performance of this model form with the current data is the result of the limited exposure conditions.
The third model form (Equation 7) is an extension of a
model form that we have previously found to accurately describe mean response over a wide range of C and T at a fixed
E (8). Models of this form have very similar shape to those of
Equation 1 and are useful for prediction, but they have no
simple biologic interpretation and do not allow for C and E
to change with time.
|
(7) |
We explicitly modelled the variance structure in two ways for each of the three model forms. One model relies on a method of estimation known as "iteratively reweighted least squares" (IRLS), which is relatively easy to implement using nonlinear regression programs in widely available software packages (29). This model is simplistic in that it assumes a negligible correlation between the two measurements made on each subject. For this model we introduce the following notation. Let Yij be change from baseline on occasion j for individual i: j = 1 or 2 h, and i = 1,2,...,485. We write the expected value, E, of the response as:
![E[Y<SUB>ij</SUB>]=β<SUB>1</SUB>f(X<SUB>ij</SUB>)](/content/vol156/issue3/fulltext/715/img009.gif)
|
with function "1 f( )" defining a logistic curve that has 1 as
its upper asymptote. The arguments of the function (X ij) are
C, E, and T. Henceforth, fij will represent the numerical value of f( ):
|
Note that each of three models mentioned in the text is of this general form. For the IRLS method, we write the model as:
|
with error term eij being the difference between the observed
and expected value for individual i on occasion j. The model assumes that: (1) the population of mutually independent eij
values will seem to vary randomly according to a normal frequency distribution centered at zero; (2) these variations account for all of the variation in Yij; and (3) that variance, V, of
the eij values is a small constant, 
2, when fij = 0, but increases
with the square of fij:
![V[Y<SUB>ij</SUB>]=V[e<SUB>ij</SUB>]=(σ<SUP>2</SUP>+10 σ<SUP>2</SUP>f<SUB>ij</SUB><SUP>2</SUP>)](/content/vol156/issue3/fulltext/715/img012.gif)
|
The assumption of increasing variance accords with our observations of increasing variance in Yij, and is usual for data of this kind. Since the variance, V, depends on the parameters of fij that are to be estimated, this method requires a two-step iteration.
The random-coefficient (RC) variance structure was intended to more realistically address the data and to take advantage of the repeated measures on each individual. However, in order to fully realize this advantage, the analysis of the data required a statistical model for the variances and covariances of these correlated data (in addition to a nonlinear regression model for mean response). It was also necessary for the variability model to appropriately address the nonconstancy of variance that is apparent across increasing levels of the exposure variables, resulting in a need for more specialized software that required more computational effort.
Whereas the IRLS variability model assumed that the coefficient for the upper asymptote is a constant (1), the RC variability model assumes that the upper asymptote varies from
individual to individual; that is, the asymptote coefficient for
individual i is 1i. The "random coefficient" model should provide a more accurate fit to the data for three reasons: (1) our
observations in other studies have suggested that each individual has a characteristic value for his/her upper asymptote; (2)
the RC assumption implies that the pairs of measurements are
correlated; and (3) the RC model can fit the data more flexibly
and contains the IRLS fit as a special case. As in the IRLS
model, the expected value is
![E[Y<SUB>ij</SUB>]=β<SUB>1</SUB>f<SUB>ij</SUB>](/content/vol156/issue3/fulltext/715/img013.gif)
|
Here, 1 is the average of the population of 1i values. We
write the RC model as
|
The model assumes that: (1) the population of mutually independent eij values will seem to vary randomly according to a
normal frequency distribution centered at zero and having
V[eij] = 2; and (2) the population of independent 1i values
will vary according to a normal frequency distribution centered at 1 and V[1i] = 
2. These assumptions imply that the
variance of the observed values increases with the square of fij:
![V[Y<SUB>ij</SUB>]=σ<SUP>2</SUP>+φ<SUP>2</SUP>f<SUB>ij</SUB><SUP>2</SUP>](/content/vol156/issue3/fulltext/715/img015.gif)
|
These assumptions also imply that the correlation between the first-hour and second-hour outcomes is greater than zero:
![Corr[Y<SUB>i1</SUB>, Y<SUB>i2</SUB>]=φ<SUP>2</SUP>f<SUB>i1</SUB>f<SUB>i2</SUB>/(σ<SUB>1</SUB>σ<SUB>2</SUB>)](/content/vol156/issue3/fulltext/715/img016.gif)
|
in which j = ![<RAD><RCD>V[Y<SUB>ij</SUB>]</RCD></RAD>](/content/vol156/issue3/fulltext/715/img017.gif)
for j = 1,2. The RC model requires
more sophisticated random coefficient techniques (23, 24).
For each of these six models (three model forms with two
variance structures each), we fit the data utilizing E expressed for each subject in absolute terms (i.e., as L/min), and also normalized for differences among subjects in BSA (i.e., E/BSA in
L/min/m2). Note that for purposes of fitting the models and reporting the data, E in all models was divided by a factor of 100 and age was centered around the mean age (24.0 yr).
The mean squared error of prediction (MSEP), which is
defined as the average squared difference between independent observations (e.g., from D) and predictions from the
model (i.e., from A,B,C) for the corresponding values of the
independent variables (25), was used to evaluate the predictive ability of each of the 12 models that we examined for each
"fold" of the cross-validation. The mean of the four estimates
of MSEP for each model was used as an overall measure of
the performance of that model. The differences among models
in this mean measure of performance were negligible when
compared with the differences in MSEP across the four
"folds" of the cross-validation within each model. We concluded that for the data available, there were no substantial
differences in the predictive ability of the 12 models. The
model in which Equation 7 was coupled with the random coefficient variance structure, however, converged with difficulty
and provided unstable estimates of 1.
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