1190 VOLUME 113 | NUMBER 9 | September 2005 • Environmental Health Perspectives

Research

Few researchers doubt that lead exposure has

significant health consequences at levels below

those considered medically acceptable just

decades ago, although there is still debate over

what levels of lead exposure, if any, can be considered harmless. Key to this debate is determining the form of the dose–response function

describing how the amount of exposure is

related to the magnitude of the health effect.

There are two basic forms of the dose–

response function for lead: a simple linear

model, in which the increase in health effect is

a linear function of increasing blood lead concentration (BPb), and a nonlinear model, in

which the amount of health effect change

attributable to lead changes according the

region of the dose–response curve studied. A

special case of the nonlinear dose–response

function is a threshold model in which the

response to lead decreases as a function of

decreasing dose until it reaches a lead dose

below which there is no further detectable

change in health. An alternative threshold

model is one in which the response to lead

changes as a function of increasing dose until

an upper lead bound is reached, at which point

the increase in health damage exceeds predictions, as in cases of high doses producing organ

damage.

Although epidemiologists have become

increasingly sophisticated in construction and

diagnosis of models describing their data, as a

whole, we generally pay much less attention to

systematically and rigorously addressing the

specification of the dose–response function. A

number of public health issues depend on

adequately specifying the form of the dose–

response function for lead, chief among them

regulatory action.

Cost–benefit analyses should form the

backbone of regulatory decisions regarding

permissible exposures or background concentrations of toxic substances in both population

and occupational settings. In such an ideal

world, the savings in health care, disability,

and productivity gain realized from reducing

exposure to a particular substance are compared to the cost required to achieve that

reduction in exposure. Policy analysts seek the

“sweet spot,” where the marginal costs of lead

reduction equal the marginal benefits (i.e.,

where the slopes of the cost function and benefits function are equal) (Pacala et al. 2003).

Even if in the real world less easily quantifiable

factors affect regulatory decisions, all parties to

regulation have some notion of costs and benefits in mind when presenting their cases to

regulatory agencies.

One recent publication (Grosse et al.

2002) presented data on the economic benefits

of nationwide lead reduction due to childhood

IQ (intelligence quotient) loss attributable to

lead over the last 25 years. These authors conservatively used a linear dose–response function of lead–IQ as part of their model, stating

that there was insufficient evidence to determine the shape of the dose–response function.

The economic savings predicted by their

model were in the range of hundreds of billions of dollars over the lifetime of a yearly

birth cohort.

The lead–health dose–response function

selected for the benefits model has clear implications for policy decisions based on it. A

threshold model suggests that once reductions

of population level of lead reach the threshold,

further lowering of lead would have no beneficial health or economic consequences. The

current Centers for Disease Control and

Prevention (CDC) action limit of > 10 μg/dL

for children (CDC 1991) would be justifiable

on health grounds alone if there were a threshold somewhere near that limit. A linear model

suggests that equal reduction in population

BPb is accompanied by equal reduction in

health consequence from any starting level of

lead. Under a linear dose–response model,

even though the health benefit would continue

to increase with further population lead reduction, the present CDC action limit might be

justifiable on economic grounds if the cost of

further population BPb reduction far exceeded

the recoverable economic benefits. A nonlinear

model, especially one in which health benefits

are greater for lead reduction nearer the population’s zero lead point than farther from it,

would argue for further reduction in population lead levels and CDC action limits if the

accelerated health benefit at lower lead levels

exceeded the increased costs of lead reduction

to those levels.

In this article, we present a critical examination of the dose–response function in a

widely studied area of epidemiologic research

with lead: childhood IQ. We present easily

accessible statistical techniques useful for

Address correspondence to S.J. Rothenberg, Center

for Research in Population Health, National Institute

of Public Health, Avenida Universidad 655, Sta.

María Ahuacatitlán, CP 62508 Cuernavaca, Morelos,

México. Telephone/Fax: 52-739-395-0662. E-mail:

drlead@prodigy.net.mx

We thank the pooled data study authors for permission to use the data. We especially thank R.

Canfield, who reviewed early versions of the manuscript and suggested residual confounding issues and

health benefit models. 

This work was supported in part by the Secretariat

of Health, Mexico; Consejo Nacional de Ciencia y

Tecnología (CONACyT) grant SALUD-2002-C01-

7159; and the U.S. Environmental Protection

Agency.

The authors declare they have no competing financial interests.

Received 22 October 2004; accepted 10 May 2005.

Testing the Dose–Response Specification in Epidemiology: Public Health and

Policy Consequences for Lead

Stephen J. Rothenberg1 and Jesse C. Rothenberg2

1National Institute of Public Health, Center for Research in Population Health, Cuernavaca, Morelos, Mexico; 2University of Sydney,

Faculty of Economics and Business, School of Economics and Political Science, Camperdown, New South Wales, Australia

Statistical evaluation of the dose–response function in lead epidemiology is rarely attempted.

Economic evaluation of health benefits of lead reduction usually assumes a linear dose–response

function, regardless of the outcome measure used. We reanalyzed a previously published study, an

international pooled data set combining data from seven prospective lead studies examining contemporaneous blood lead effect on IQ (intelligence quotient) of 7-year-old children (n = 1,333).

We constructed alternative linear multiple regression models with linear blood lead terms (linear–

linear dose response) and natural-log–transformed blood lead terms (log-linear dose response). We

tested the two lead specifications for nonlinearity in the models, compared the two lead specifications for significantly better fit to the data, and examined the effects of possible residual confounding on the functional form of the dose–response relationship. We found that a log-linear lead–IQ

relationship was a significantly better fit than was a linear–linear relationship for IQ (p = 0.009),

with little evidence of residual confounding of included model variables. We substituted the log-linear lead–IQ effect in a previously published health benefits model and found that the economic savings due to U.S. population lead decrease between 1976 and 1999 (from 17.1 μg/dL to 2.0 μg/dL)

was 2.2 times ($319 billion) that calculated using a linear–linear dose–response function ($149 billion). The Centers for Disease Control and Prevention action limit of 10 μg/dL for children fails to

protect against most damage and economic cost attributable to lead exposure. Key words: child IQ,

dose response, health benefit, health policy, lead. Environ Health Perspect 113:1190–1195 (2005).

doi:10.1289/ehp.7691 available via http://dx.doi.org/ [Online 10 May 2005]
deciding among alternative dose–response

functions and for testing whether residual confounding resulting from possible misspecification of model control variables affects the

dose–response function. We apply our dose–

response modeling results to the benefit model

noted above to calculate changes in economic

benefits realized from using a statistically adequate dose–response function. The results are

placed in the context of public health policy

and regulation.

Materials and Methods

Data sets. Eight prospective studies of lead

exposure have used child IQ or developmental index as the outcome measure, with

outcome measured at least to 5 years of age,

of which seven (Baghurst et al. 1992;

Bellinger et al. 1992; Canfield et al. 2003;

Dietrich et al. 1993; Ernhart et al. 1989;

Schnaas et al. 2000; Wasserman et al. 1997)

agreed to participate in a pooled analysis

study; combining the data sets produced a

study sample of 1,333 with a 0.1–71.7 μg/dL

range of lead exposure (Lanphear et al. 2005).

All studies producing data for the pooled

analysis were approved by an appropriate institutional review board. Child IQ as measured

by one of several versions of the Wechsler

Intelligence Scales for Children (Wechsler

1967, 1974, 1981, 1991) around 7 years of age

was regressed on different indices of BPb (child

BPb from 6 to 24 months, peak BPb during

the first 7 postnatal years, average BPb over the

same time, and contemporary BPb) in multiple

regression models controlling for maternal IQ

and education, quality of the home environment and child–caretaker interaction [Home

Observation for Measurement of the

Environment (HOME); Caldwell and Bradley

1984], birth weight, and study site. Other control and confounding variables, such as child’s

sex, tobacco exposure during pregnancy, alcohol use during pregnancy, maternal age at

delivery, marital status, and birth order, had

no significant effect in the models, did not significantly alter the IQ–lead relationship, and

were not included in the final models. All lead

variables in the models were natural-log transformed. All lead variables had highly significant effects on IQ (p < 0.0005) in the models.

We selected BPb measured contemporaneously with IQ for further analysis using the

pooled data set [adjusted estimate of naturallog lead (95% confidence interval; CI) on IQ =

–2.70 (–3.74 to –1.66)], because this was the

measure to which Lanphear et al. (2005)

devoted most attention, even though it had the

second smallest coefficient among the four

measures presented.

Statistical analyses. Multiple regression

modeling. The IQ data set was analyzed with

the original model specifications, including

log-transformed BPb, using multiple regression

analyses (STATA, version 8.2; Stata Corp.,

College Station, TX, USA). The IQ multiple

regression model was also respecified with a

linear lead term.

Specification tests for the functional form

of the lead variable (dose–response function).

The omitted variable test, or regression specification error test (Ramsey 1969), statistically

tests change in model fit when any polynomial

transformation of the variable in question is

used in place of the original functional form of

the variable. To test whether the polynomial

form is superior to the original form of the

variable, a chi-squared test is constructed by

using the difference in two models’ chi-squared

value (or the difference in two times the log

likelihood of the two models), with the number of degrees of freedom determined by the

number of additional variables added to the

polynomial model. This is a maximum likelihood evaluation of changes in model fit and is

a test of nested models, because the original

specification is nested within the polynomial

specification. Its principal disadvantage is that

it only tests whether a polynomial specification

is better than a simpler specification and does

not allow direct comparison of two non-nested

models each with a different specification, such

as linear and logarithmic. On the other hand,

it is easy to do even in the absence of “canned”

statistical routines and quickly indicates

whether the original variable specification can

be improved by adding polynomial terms.

An accessible approach for comparing variable specification between two non-nested

models is the J-test (Davidson and MacKinnon

1981). It can be realized by first obtaining predicted values for two models, each with a different specification of the same independent

variable, and then adding the prediction of the

first model to the specification of the second

model and vice versa (Appendix). A clear indication in favor of one or the other specification

would occur when one of these predictionadded models results in a significant value for

one specification of the variable and the other

prediction-added model results in a nonsignificant value. A disadvantage of this test is its low

power to detect a significant improvement in

variable specification. Hundreds or thousands

of observations might be needed if the difference between two alternative variable specifications is subtle or the variable is measured over a

limited range. Low power and limited range

are not limiting factors in the present study

(n = 1,333; BPb range, 0.1–71.7 μg/dL).

Testing for residual confounding. When

control or confounding variables either are

omitted or their functional form is misspecified, the resulting residuals in the model

could cause an alteration in the apparent functional form of the dose–response relationship

(Becher 1992). In the case of IQ, not accounting for variables such as the number of other

family members, family socioeconomic status,

birth or childhood trauma or serious illness,

IQ of the father, or other variables that might

control subject IQ could alter the measured

form of the dose–response relationship for

lead. If these variables are not accounted for in

the experimental design by becoming part of

the inclusion/exclusion criteria or they are not

tested for and, where appropriate, included in

the models, they may contribute to residual

confounding of the dose–response curve.

Another potential cause of residual confounding occurs when the functional form of

included control or confounding variables is

not correctly specified. Because much statistical

modeling in epidemiology is performed using

some variant of generalized linear models (we

used least-squares regression), modelers may

assume that the linear specification of these

other variables is correct. For instance, a truly

nonlinear relationship between maternal and

child IQ that is mistakenly modeled as a linear

relationship, significant or not, will alter the

residuals of child IQ over different parts of the

maternal–child IQ relationship. Because the

dose–response curve for lead–IQ is based on

those residuals, this confounding can modify

the modeled dose–response relationship.

When residual confounding is caused by a

variable omitted from the design, there is little

remedy available except to redesign the study

and collect the data anew. Fortunately, we can

account for residual confounding when it is

due to misspecification of included variables.

Generalized additive models (GAM) (Hastie

and Tibshirani 1990) can use smoothing

spline functions, among other smoothers, to

fit continuous and ordinal independent variables to the dependent variable instead of predetermined linear fits as with linear regression

models. Depending on the number of degrees

of freedom allotted to the splines, the technique can follow complex nonlinearity in the

relationship between independent variables

and the dependent variable, nonlinearity that

might be difficult to account for by parametric

functions. The penalty for increasing the complexity of the spline fit is the use of more

degrees of freedom in the model. GAM yields

no parameters readily summarizing the

relationship between independent variables

and the dependent variable. There is no disadvantage, however, if we want to use GAM

to characterize the possibly complex relationships among independent control variables

and the dependent variable to avoid having

incorrect residuals affect the parametric dose–

response relationship, as has been previously

shown with simulations (Benedetti and

Abrahamowicz 2004). 

GAM allows calculation of the gain from

the spline fit over a linear fit by assessing the

increase in deviance of the fit of the linear

characterization of the variable compared with

Dose–response curve for lead–health effects

Environmental Health Perspectives • VOLUME 113 | NUMBER 9 | September 2005 1191
the spline fit characterization. Under the null

hypothesis that nonlinearity of the smoothed

function is an artifact, the gain is approximately a chi-squared distribution. Thus,

approximate probability values can be calculated for improvement of fit using the spline

function. A significant gain indicates that the

original linear or any other specification of the

variable was a poorer fit to the data than is the

spline fit. The procedure also gives a total

model gain and model gain significance value.

We used cubic-spline GAM modeling of

IQ. We modeled all continuous and ordinal

variables with cubic splines with 2, 3, and 4

degrees of freedom. We constructed three

alternative models based on the basic model

above. In the first series of GAM models, we

used untransformed BPb (linear BPb) spline

modeled with the same number of degrees of

freedom as the control variables. A significant

gain in the spline-modeled untransformed

BPb term would indicate that the original linear BPb specification could be improved

upon, after correcting for any nonlinearity in

the control variables.

In the second series of GAM models, we

substituted the natural-log–transformed BPb

variable for the linear BPb variable of the first

model, allowing the number of degrees of freedom of the spline fit to vary as in the first

model. An insignificant gain of the naturallog–transformed lead variable would indicate

there was no improvement detected in the fit to

the dependent variable by spline modeling of

the log-transformed lead variable, correcting for

possible nonlinearity in the control variables.

Finally, the third series of GAM models

was constructed as above, except that the natural-log–transformed lead variable was held to

1 degree of freedom. This tested the original

natural-log specification of the lead variable in

a model where residual confounding from

possible misspecification of the control variables was corrected. Insignificant gains in the

other variables would suggest that their original specifications were adequate. The size of

the lead coefficient was compared between

the third series and the original multiple

regression model to determine how much

residual confounding of misspecified control

variables affected the estimated size of the

relationship between lead and the health outcome dependent variable.

All statistical procedures were carried out

using MATLAB (version 6.5.1; Mathworks,

Natick, MA, USA) and STATA 8.2.

The benefits model. We used a previously

published model (Grosse et al. 2002) of economic benefits showing expected dollar savings produced by population lead declines in

the United States from 1976 through 1999

solely through increased population cognitive

ability as measured by lead effects on child IQ.

The model posits that the dollar gain in the

affected cohort is a simple product of reduction in BPb over the period (micrograms per

deciliter), the IQ–BPb slope (IQ per micrograms per deciliter), the earnings–IQ slope

(%), the present value of lifetime earnings of a

2-year-old child (in year 2000 dollars), and the

size of the 2-year-old cohort. Grosse et al.

(2002) used linear IQ–lead slopes of

0.185–0.323 IQ points for each 1 μg/dL, calculated from published meta-analyses.

Instead of the linear IQ–lead slope, we

substituted the change in IQ expected over the

estimated 15.1 μg/dL decrease in population

lead in the United States, calculated by assuming both a linear–linear and a log-linear

lead–IQ dose–response function using the

results of the pooled analysis presented above

and then recalculated the cohort benefit.

Results

Lead and IQ. Table 1 shows the lead coefficients of the different IQ models. Both linear

and natural-log lead specifications were highly

significant (Table 1). The omitted variable test

using the linear lead variable showed a significant improvement in fit using the polynomial

lead specification (p = 0.020), whereas the

same test showed that a polynomial form of

the log lead variable offered no improvement

(p = 0.258).

The J-test showed that the log lead specification was still significant (p = 0.009) in a

model with the prediction from the linear

lead model added (Table 1). The alternative

model, the linear lead model with the prediction from the log lead model resulted in an

insignificant linear lead variable (data not

shown). The results indicate that the log lead

specification described the data significantly

better than did the linear lead specification.

Tables 2–4 show the results of the GAM

analyses. Presented results are limited to the

2 degrees of freedom spline fits because they

usually resulted in the largest gains and lowest

probability values, although results were similar

for the 3 and 4 degree of freedom spline fits.

Spline fit gains are shown only for ordinal and

continuous variables because dichotomous

variables cannot be fit by splines and they

remain in the model unmodified.

In Table 2 the linear lead model is

entirely fitted by splines. Note that the gains

of all control variables were nonsignificant,

suggesting adequate specification of these

variables as linear. The gain of the linear lead

variable was highly significant (p = 0.006),

and the total gain of the model was also significant (p = 0.0142). These findings indicate

that both the linear lead specification and the

model as a whole better fit the data when

splines were used than when the original variables were fit by linear regression. The results

from Table 1, that the linear lead term did

not adequately fit the data, was confirmed in

Rothenberg and Rothenberg

1192 VOLUME 113 | NUMBER 9 | September 2005 • Environmental Health Perspectives

Table 1. Lead coefficients for IQ as a function of model.a

Variable Coefficient 95% CI p-Value

Linear lead modelb –0.18 –0.26 to –0.10 < 0.0005

Natural-log lead modelc –2.70 –3.74 to –1.66 < 0.0005

Quadratic lead model 0.005 0.001 to 0.009 0.020

Quadratic-log lead model –0.25 –0.76 to 0.26 0.258

Ln(lead) with linear lead prediction –2.47 –4.30 to –0.63 0.009

(J-test) model

aControl variables for all models were HOME, birth weight, maternal IQ, maternal education, and site identification.

bModel with linear lead specification. cModel with natural-log lead specification.

Table 2. GAM results for IQ.

Probability

Variable dfa Gain of gain

HOME 2 2.621 0.106

Birth weight 2 2.587 0.108

Maternal IQ 2 0.596 0.440

Maternal education 2 0.961 0.327

Linear lead 2 7.467 0.006

df, degrees of freedom. Dichotomous variables (sites) not

shown: spline fit of linear lead specification and all independent variables with two degree of freedom splines.

Total gain (nonlinearity χ2

) = 14.232 (5.003 df), p = 0.0142. aApproximate.

Table 3. Spline fit of natural-log lead specification

and all independent variables with two degree of

freedom splines.

Probability

Variable dfa Gain of gain

HOME 2 2.646 0.104

Birth weight 2 2.515 0.113

Maternal IQ 2 0.603 0.438

Maternal education 2 0.690 0.406

Natural-log lead 2 1.438 0.230

df, degrees of freedom. Total gain (nonlinearity χ2) = 7.894

(5.005 df), p = 0.1626. aApproximate.

Table 4. Spline fit of all independent variables with

two degree of freedom splines with original naturallog lead variable modeled as is.

Linear Probability 

Variable dfa coefficient Gain of gain

HOME 2 4.51 2.740 0.098

Birth weight 2 1.48 2.523 0.112

Maternal IQ 2 4.91 0.609 0.436

Maternal 2 1.15 0.642 0.424

education

Natural-log lead 1 –2.62 — —

Abbreviations: —, not applicable (natural log lead was not

modeled as a spline function; thus, there is no gain or

probability of gain); df, degrees of freedom. Total gain

(nonlinearity χ2

) = 6.514 (4.006 df), p = 0.1644. aApproximate.
Table 2, and the nonlinear (spline) fit of the

lead variable was not due to residual confounding with included variables.

Table 3 shows the same spline-fit model as

Table 2, but the natural-log lead term is substituted for the linear lead term. Once again, no

control variable showed significant gain using

the spline fit, the log-transformed lead variable

gain was also nonsignificant (p = 0.230), and

the model itself was not significantly improved

by fitting the variables with splines (p = 0.163).

There was no significant improvement in the

log-linear lead–IQ fit by adjusting for possible

departures from that specification.

In Table 4 the natural-log–transformed

lead variable is allowed to maintain its original

specification while the remainder of the variables are fit with splines. Comparison of the

coefficient of the log lead variable in this

model (β = –2.62) with the coefficient of the

multiple regression model (β = –2.70; second

coefficient from Table 1) further supports the

result that there was no important misspecification of the control variables in the original

multiple regression model and the log-linear

form of the dose–response curve was not

affected by residual confounding of variables

included in the model.

These results strongly support the hypothesis that an adequate description of the

dose–response curve for the effect of lead on

child IQ is log-linear, not linear, and that

residual confounding of the dose–response

specification by possible misspecification of

included control variables played no role. The

log-linear dose–response relationship is compared with the linear dose–response relationship in Figure 1.

Economic benefits model for lead–IQ.

Grosse et al.’s (2002) benefit model of economic gains due to lead reduction effect on IQ

in the United States calculated the total year

2000 dollar savings as a result of the fall of BPb

over a 23-year period. Their model postulated

that the dollar benefit per cohort was benefit =

A × B × C × D × E, where A is the reduction in

BPb (micrograms per deciliter); B is the

IQ–BPb slope; C is the earnings–IQ slope (%);

D is the present value of earnings of a 2-yearold child (in 2000 US dollars); and E is the size

of the 2-year-old cohort. We used their “base

case” figures as follows: C, 2.0; D, $723,000;

E, 3,800,000. In place of Grosse et al.’s A of

15.1 μg/dL, we used the difference in the natural-log BPb values in 1976 and 1999: BPb

(1976), 17.1 μg/dL; natural-log BPb (1976),

2.84; BPb (1999), 2.0 μg/dL; natural-log BPb

(1999), 0.69; difference in BPb (1976 –

1999), 15.1 μg/dL; and difference in naturallog BPb (1976 – 1999), 2.15. In place of

Grosse et al.’s B of 0.257 IQ–BPb slope (every

decrease of 1 μg/dL BPb is associated with an

increase of 0.257 IQ points) used in their

“base case” analysis, we used the natural-log

lead coefficient calculated from the pooled

analysis study (Table 1, natural-log lead

model), 2.70 (every natural-log unit decrease

in BPb is associated with an increase of

2.70 IQ points). 

The original benefits model used uncertainty in the reduction of BPb over the period

studied (variable A) and the IQ–BPb slope

(term B) to calculate upper and lower bounds

on economic benefits. We used only the

uncertainty in the IQ–BPb slope, calculated

from the coefficients presented in Table 1 and

from the reported meta-analysis Grosse and

colleagues used (Grosse et al. 2002; Schwartz

1994) in their base case analysis. We present

Grosse et al.’s original calculations based on

their linear lead coefficient, the new calculations based on a log-linear dose–response relationship with 95% CIs of B (–3.74 to –1.66),

and, for comparison, the dollar savings per

cohort based on the demonstrably incorrect

linear lead specification calculated from the

pooled analysis study, 0.18 (95% CI, –0.26 to

–0.10) (Table 1, linear lead model). These

results are presented in Table 5.

Savings estimated using the correct loglinear dose–response relationship between

BPb and IQ are nearly 2.2 times those estimated using a poorly fitting linear dose–

response relationship for the same decrease in

population BPb.

Discussion

Model specification. Diagnosing model specification is an essential part of statistical modeling, particularly when ordinal and continuous

variables are part of the model. Compared

with the more commonly used diagnostic

tests for general linear models, such as testing

for distribution and homoskedasticity of

residuals, formal tests of the assumed functional form of any independent variables

against the dependent variable are scarcely

reported in the epidemiologic literature. We

often do not address functional form issues

except as a by-product of adjusting residual

diagnostics. For example, most lead–IQ studies in children, especially in the last 20 years,

have used a natural-log–transformed lead

variable to normalize the residual distribution

of the model and correct for heteroskedasticity

of residuals.

Although issues of the functional form of

the lead–health effect relationship have occasionally been raised in the literature, notably

by Schwartz (e.g., Schwartz 1994), and more

recently and extensively by Grosse et al.

(2002), it is common practice for the authors

of applications of these studies to economic

analysis to use linear approximations of the

lead effect over a limited range of BPb, as did

Grosse et al. (2002). Authors have not extrapolated health effects below the lower limits of

lead in their data sets in the past. Data sets

studying a wide range of BPb have only

recently become available. Because linear and

log lead specifications produce large differences in predictions only as BPb approaches

zero (Figure 1), data sets including substantial

numbers of very low BPb levels are required

to notice, appreciate, and test for adequacy of

alternative specifications. Implicit in the loglead specification is that change in health

effect with change in BPb at higher levels is

Dose–response curve for lead–health effects

Environmental Health Perspectives • VOLUME 113 | NUMBER 9 | September 2005 1193

Table 5. Economic savings (year 2000 dollars) per

cohort estimated from the Grosse et al. (2002) IQ

model according to dose–response specification.

Benefit/cohort

Study (billions $) 95% CIa

Grosse et al. (2002) 213.83 147.27–280.39

linear lead

Pooled analysis, 148.58 82.18–215.82

linear lead

Pooled analysis, 318.98 196.30–441.67

natural-log lead

aCIs cannot be used to compare linear and log lead specifications because the linear specification is incorrect and

the 95% CI calculated from it suffers from uncorrected

residual heteroskedasticity.

105

100

95

90

85

80

0 20 40 60 80

IQ

BPb (μg/dL)

Natural-log–transformed

Linear

Figure 1. Partial regression plot of adjusted IQ

(adjusted for natural-log lead model) and BPb (from

Lanphear et al. 2005). The two regression lines

(bold) with 95% CIs (narrow lines) represent the

best-fit estimates of the relationship between IQ

and BPb for natural-log–transformed BPb and linear BPb. Note that the linear BPb term overestimates the slope (change in IQ with change in BPb)

of the statistically superior natural-log lead function down to 15 μg/dL and underestimates the

slope < 15 μg/dL. The scatter plot does not show all

data points because the y-axis has been expanded

to show differences in regression functions. 
small, save when lead toxicity associated with

pathologic organ damage is reached.

Most studies using a log-linear dose–

response relationship also neglect to comment

on the public health implications of this functional form. As opposed to a linear dose–

response relationship, where equal changes in

health outcome are predicted for equal changes

in BPb across the entire range of BPb, the loglinear relationship has the steepest slope at the

lowest BPb. Health outcome changes are equal

for equal proportional changes in BPb across

the entire tested range of BPb. In the case of a

log-linear dose–response relationship, the

increase in population IQ predicted from a

decrease in population BPb from 2 to 1 μg/dL

is exactly the same as that predicted from a

decrease in BPb from 20 to 10 μg/dL or from

40 to 20 μg/dL, although populations exposed

to these different concentrations of lead will

likely have different mean IQs.

We calculated the BPb change in the U.S.

population between 1976 (17.1 μg/dL) and

1999 (2.0 μg/dL) used by Grosse et al. (2002)

at 2.15 natural-log units change. The pooled

analysis study has 38 subjects with BPb levels

< 2 μg/dL. If we project the 1999 population

BPb of 2.0 another 2.15 natural-log lead

units down to a population lead level of

0.24 μg/dL in the indeterminate future, we

can duplicate the health benefit of BPb reductions for the population already achieved by

the reductions between 1976 and 1999, at

least for IQ outcomes.

The log-linear dose–response function for

lead, especially if it generalizes across other

health outcomes, may also account for the failure of many older studies to find significant

lead-related effects. In occupational studies of

health effects of lead exposure, often the generally high mean BPb levels of the “exposed”

groups and even the “nonexposed” control

group place health comparisons among exposure levels on the flat end of the log-linear

dose–response curve, where the dose–response

curve approximates a nearly zero slope linear

trend. Under such conditions, a very large

sample size would be needed to detect significant differences among groups. If a linear

model were used to specify the dose–response

relationship at higher BPb, even significant

effects detected in large studies would have

small coefficients. The apparent “no-effect”

relationship predicted by the near zero slope of

a log-linear dose–response function at elevated

BPb is especially notable in the occupational

lead–blood pressure literature.

We do not propose that the log-linear

dose–response function for BPb effect on child

IQ is the “correct” dose–response function.

Our analysis indicates only that it is superior to

a linear–linear dose–response function. We

examined two other nonlinear dose–response

functions for this relationship, a third-order

polynomial and a logit dose–response function.

The logit function is attractive because it can

model a reduction of the dose–response slope

as BPb falls below currently modeled data, thus

providing for the possibility of an ultralow

threshold for lead effect on IQ. The polynomial function would permit modeling of a

new increase of slope of the dose–response

function beyond the upper limits of the data

set modeled here. This would allow accounting

for severe lasting effects of the pathologic

changes associated with lead-induced

encephalopathy. However, the alternative

nonlinear dose–response functions both modeled the present data set no better than did

the log-linear function, including the steeper

slope at low BPb. There was a difference of

< 0.2% of the variance in IQ accounted for

by the lead variable among the three specifications. Because the log-linear dose–response

relationship only required two parameters for

complete specification and the alternatives

required three parameters, we elected to use

the most parsimonious specification for

detailed analysis of the data set at hand.

Public health and policy implications.

There appears to be no support for a threshold

model for BPb effect on IQ. On the contrary,

instead of finding a no-effect lower limit, the

present study strongly suggests that most of

the damage attributable to BPb occurs within

the first few micrograms per deciliter of BPb

within the lead range studied. Any apparent

threshold will appear at the upper ranges of

BPb, where the dose–response curve flattens,

at least until BPb reaches the range producing

frank organ damage.

In prospective lead studies of child development, including the pooled study IQ effects

cited here, history of exposure is always available in the form of sequential BPb measurements of each child. We have good evidence

that the log-linear lead–IQ dose–response function is not an artifact of unmeasured history of

exposure and represents the best available functional description of BPb on IQ.

The drop in population lead exposure from

mean BPb of 17.1 μg/dL to 2.0 μg/dL over the

last quarter century produced the large health

benefits calculated in Table 5. Although lead in

paint and food has been specifically regulated

with the goal of reducing population lead

exposure, the reduction of lead in air and eventually in dust, one of the major contributors to

past urban population lead exposure (Mielke

et al. 1983, 1997, 1999), was by and large due

to the introduction of catalytic converters for

automobiles. Thus, a large part of the drop in

population lead was only coincidentally

achieved in response to the stated policy of

reducing gaseous automobile contaminants.

Fortunate though we may have been to have

benefited from this accidental process, it is

unlikely that further reduction in population

lead exposure will be achieved without

increased targeted effort.

Although many hailed the Occupational

Safety and Health Administration’s 1979 regulations seeking to limit occupational exposure

to 40 μg/dL (Occupational Safety and Health

Administration 1979) and the CDC’s promulgation in 1991 of action limits for childhood

lead exposure to 10 μg/dL (CDC 1991), it

appears very likely that these limits have prevented only a small percentage of the damage

associated with lead exposure. The total modeled increase in IQ from the pooled data study

over the BPb range of 0.1–71.7 μg/dL was

17.7 IQ points. The improvement in IQ predicted by the log-linear model down to the

CDC action limit for children was 5.3 IQ

points; the remainder of the IQ improvement

(12.4 IQ points) was found below the CDC

action limit. If we continue to permit children

and, by extension, pregnant women to maintain up to 10 μg/dL BPb without aggressive

intervention to lower exposure, we are still

allowing most of the preventable subclinical

damage to occur.

Economic benefits realized by lead exposure

reduction under a log-linear dose–response

function are more than twice that previously

estimated using a linear dose–response function. Updated cost studies of further population

and occupational exposure reduction are long

overdue. Using updated cost and benefit models, epidemiologists and health economists can

determine how much additional exposure

reduction is economically warranted.

In this article we only address the form of

the dose–response function for lead effect on

child IQ. Another well-studied area of health

effects of lead exposure is the effect of lead on

adult hypertension and blood pressure. A

recent meta-analysis of studies examining the

effect of contemporary BPb on adult blood

pressure (Nawrot et al. 2002) used a log-linear

dose–response relationship and found a significant effect. For every doubling of BPb, Nawrot

et al. (2002) calculated a 1.0 mm Hg increase

in systolic blood pressure and a 0.6 mm Hg

increase in diastolic blood pressure. The pattern of increased blood pressure with increase

in BPb was exactly the same as the decrease in

child IQ with increased BPb. The greatest

changes in predicted blood pressure occurred

in the first few micrograms per deciliter of

BPb. Because the authors performed no formal

testing of other forms of the dose–response

function, the log-linear dose–response relationship for BPb on adult blood pressure should be

examined in a pooled data study similar to that

used for the IQ study detailed here. If the original data in the studies contributing to the

meta-analysis for blood pressure were used in a

pooled analysis study, there would be > 50,000

subjects in the study. Such a large study sample

would allow testing other nonlinear forms of

Rothenberg and Rothenberg

1194 VOLUME 113 | NUMBER 9 | September 2005 • Environmental Health Perspectives
Dose–response curve for lead–health effects

Environmental Health Perspectives • VOLUME 113 | NUMBER 9 | September 2005 1195

the dose–response relationship for lead and

blood pressure against the log-linear form. Any

nonlinear dose–response relationship of the

same general form as the log-linear function for

blood pressure would have economic and public health repercussions similar to those discussed above for IQ. Over the U.S. population,

BPb change measured between 1976 and 1999,

Nawrot’s coefficient translates into a 4.1 mm

Hg decrease in population systolic blood pressure, a change with significant health and economic benefits. However, most of the health

and economic benefits would be realized only

by bringing population and occupational exposures well below currently permitted limits.

Conclusions

Correctly specifying the dose–response or

exposure–health relationship in all epidemiologic and toxicologic studies has important

scientific, economic, and policy implications.

Authors of such studies could take the initiative and apply statistical techniques similar to

those discussed in this article to test whether

the presented functional form of the dose–

response relationship cannot be ruled out by

definable statistical criteria. Journal editors and

their reviewers can also insist that authors provide such evidence regarding dose–response

curves in submitted manuscripts. Adopting

these practices will give toxicologists additional

clues about mechanisms of effect, will give

environmental economists more accurate data

for their models, and will give regulators the

needed information for evidence-based actions.

If the nonlinear form of the exposure–

health effect curve is more appropriate to the

data than a linear function, we still have most

of our work ahead of us to protect the population from the effects of lead.

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Appendix: 

Steps for Running J-test 

Selection between two non-nested models 

Linear Model 

Y = α • lead + [β • X ] + ε,

where [β • X ] is the vector of all other variables in the model. 

Generate the prediction of linear lead = Πlin.

Logarithmic Model 

Y = α • ln(lead) + [β • X ] + ε.

Generate the prediction of natural log lead

= Πlog. 

The J-test:

Y = γ • Πlog + α • lead + [β • X ] + ε,

Y = γ • Πlin + α • ln(lead) + [β • X ] + ε.

Then test the probability of the predicted

lead term and the original lead term in the

two artificial regressions above.