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Semiparametric Adjusted Exposure-Response Curves

Naimi, Ashley I.a; Moodie, Erica E.M.b; Auger, Nathaliec; Kaufman, Jay S.b

doi: 10.1097/EDE.0000000000000158

Exposure-response curves are central to understanding how continuous exposures relate to health outcomes. Common methods to plot such curves include the use of crude and adjusted regression, with the exposure specified using fractional polynomials or regression splines. These approaches are subject to important limitations. In this article, we illustrate the use of semiparametric marginally adjusted exposure-response curves obtained via inverse probability weighting. We explore the relation between interpregnancy interval and preterm birth in a cohort of over 720,000 live births in Quebec between 1989 and 2008. We include online supplementary material showing how mixed modeling routines in standard software packages can be used to implement the procedure, and how pointwise bootstrap confidence intervals can be obtained.

Supplemental Digital Content is available in the text.

From the aDepartment of Obstetrics and Gynecology, McGill University, Montreal, Quebec, Canada; bDepartment of Epidemiology, Biostatistics, and Occupational Health, McGill University, Montreal, Quebec, Canada; and cInstitut national de santé publique du Québec, Montreal, Quebec, Canada.

Submitted 9 May 2013; accepted 22 April 2014; posted 18 August 2014.

A.I.N. and N.A. received funding from Fonds de recherche du Québec-Santé; E.E.M.M. from Natural Sciences and Engineering Research Council; and J.S.K. from Canada Research Chairs Program.

The authors report no conflicts of interest.

Supplemental digital content is available through direct URL citations in the HTML and PDF versions of this article ( This content is not peer-reviewed or copy-edited; it is the sole responsibility of the author.

Correspondence: Ashley I. Naimi, Department of Obstetrics and Gynecology, McGill University, 687 Pine Ave West, Room F 432, Montreal, Quebec H3A 1A1, Canada. E-mail:

Exposure-response curves, central to understanding how continuous exposures relate to health outcomes, are often fit in 1 of 2 ways. A crude model relating the outcome to a function of the exposure may be used. Alternatively, a covariate-adjusted model may be used to obtain predicted outcome values for unique confounder values. For example, in a regression model adjusted for continuous age and an indicator of male sex, one can center the age variable at its mean and set the indicator for sex to zero to obtain an exposure-response function for average-aged females. In both approaches, the exposure is often specified using fractional polynomials1,2 or regression splines.3,4(p. 18−24)

Although almost universal, these approaches are subject to important limitations. First, a poorly specified functional exposure form can lead to bias, although the severity of this problem can be lessened by judicious selection of the polynomial or spline basis functions,1,2,5,6 or the implementation of a semi- or nonparametric approach.5–8 Second, while not adjusting for confounders may lead to bias, researchers and policy makers may wish to interpret the exposure-response function for the entire population rather than a specific subgroup defined by unique confounder combinations.

We present a simple approach to obtain semiparametric adjusted exposure-response curves for time-fixed exposures. We show how the generalized propensity score9 can be used to obtain exposure-response curves standardized to the total population,10 and we use a semiparametric approach to mitigate the problem of specifying an unknown functional form for the exposure-outcome relation. We estimate the exposure-response function for the relation between interpregnancy interval and the risk of preterm birth in a cohort of over 700,000 live births in Quebec, Canada, and provide supporting details in the accompanying eAppendix (

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The interpregnancy interval, or the elapsed duration between a woman’s previous pregnancy and the conception date of the next pregnancy, is related to a host of adverse perinatal outcomes, including preterm birth.11 As a potentially modifiable risk factor, the optimal timing of conception after a previous delivery is important.11,12 In the next section, we let Y be a binary indicator of preterm birth, X refer to the number of years between the index pregnancy and the previous pregnancy, and C = {C1, C2} represent 2 binary confounders for illustrative purposes.

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Standard Approach: Population-specific Curves

The simplest approach to characterize exposure-response functions is to estimate:

where s(•) is a user-specified function. To account for confounding, one could estimate:

where C is a vector of confounders. The parameter estimates from this model can be used to plot the risk of preterm birth against interpregnancy interval to obtain:

where expit{•} = 1/{1 + exp(−•)}. To plot the adjusted risk of Y as a function of X, one could choose the confounder combination that yields the lowest risk category or simply select an arbitrary subgroup (eg, with confounder values C1 = 0 and C2 = 1). Both cases will yield the same curve (on the log-odds scale) shifted along the y-axis.

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Weighted Approach: Population-standardized Curves

An alternative approach is to plot a marginally adjusted curve by standardizing it to a confounder distribution in the target population. If the target is the total population, one can compute a standardized curve as:

which can be re-written as:

where f (x | C = c) is the conditional density of the exposure.

For a binary exposure, the denominator in equation (2) reduces to the probability of the observed exposure, which is defined by the propensity score for the exposed, and the complement of the propensity score for the unexposed.13 When the exposure is continuous, the denominator is the generalized propensity score,9,14,15 which is defined as the exposure’s conditional density function evaluated at the observed exposure value. Using the generalized propensity score, equation (2) leads to an inverse probability–weighted estimator of the standardized exposure-response function defined in equation (1).16,17

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We extracted records on 726,294 live singleton multiparous births between 1989 and 2008 (inclusive) from Quebec birth file data to estimate exposure-response curves for the relation between interpregnancy interval (in years) and the risk of preterm birth (<37 weeks completed gestation). We used the quantile binning method to construct stabilized inverse probability weights.18

To obtain marginally adjusted exposure-response functions, we avoided conditioning the numerator of the weights on elements of C.19 Additional details are provided in eAppendix 1 (

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Semiparametric Smoothing Function

We used penalized smoothing splines20,21 to plot the exposure-response function for the relation between interpregnancy interval and preterm birth. With typical smoothing approaches (eg, fractional polynomials or restricted quadratic splines), one gains flexibility by increasing the number of polynomial terms or knot points at the expense of stability in the exposure-response function. Penalized smoothing splines use a large number of spline basis functions (yielding an unstable but highly flexible exposure-response function), with parameters estimated using penalized maximum likelihood.22 Using penalized maximum likelihood smooths the unstable exposure-response function, and thus balances the trade-off between stability and flexibility. The approach can be implemented using standard generalized linear mixed modeling routines, as explained in eAppendix 2 (

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Regression Models

We fit marginally adjusted logistic regression models to plot exposure-response functions for the overall relation between interpregnancy interval and the risk of preterm birth. We also fit marginally adjusted models for the relation between interpregnancy interval and the risk of preterm birth among mothers with 1 previous pregnancy (parity = 1) and 2 or more previous pregnancies (parity = 2+) separately. All models were marginally adjusted by weighting each individual’s contribution to the penalized likelihood with the stabilized inverse probability weights defined in eAppendix 1 (

We used R version 2.15.3 for all analyses. Additional technical details, including a description of our data and models, are provided in eAppendix 1 ( An illustration of the approach using SAS Software (SAS Institute, Cary, NC) is provided in eAppendix 2 (

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The Table shows descriptive statistics for continuous variables included in our models. The numbers of live births from mothers of parity 1, 2, 3, 4, and 5+ were 473,425, 175,696, 51,356, 15,030, and 10,787, respectively. The log-mean of the stabilized inverse probability weights was close to zero across the range of exposure values, with the largest log-weight no greater than 4.



The Figure shows the marginally adjusted exposure-response functions for the total relation between interpregnancy interval and the risk of preterm birth, and the corresponding exposure-response functions among women with 1 previous pregnancy (parity = 1) and 2 or more previous pregnancies (parity = 2+). The shape of the exposure-response functions was similar across models and coincided with expectations based on previous research.11 For all models, the lowest estimated risk of preterm birth was similar in magnitude (minimum [standard error] = 0.04 [0.005]) corresponding to an interpregnancy interval of 18.3 months.



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In this article, we illustrate the use of a simple method to obtain semiparametric marginally adjusted exposure-response curves. The approach can easily be implemented using generalized linear mixed model routines (eg, proc glimmix) and yields curves that can be interpreted as standardized to the total population.

Previous reports23–25 have outlined approaches to obtain marginally adjusted survival curves. Inverse probability weighting by the generalized propensity score has been used to plot survival curves for the relation between epoetin and mortality,26 epoetin and cardiovascular risk,27 and hormone replacement therapy on the risk of invasive breast cancer.28 Although Rothman et al29(p. 308, 444−8) suggest standardization as a solution to the problem of confounded exposure-response curves, few applications exist. Furthermore, only one prior study30 has combined inverse weighting with semiparametric approaches to simultaneously address issues of confounding and misspecification of the exposure function.

The propensity score for binary exposures has been generalized to the continuous exposure setting.9,14,15 Moodie and Stephens31 showed how exposure-response curves can be obtained by conditioning on the generalized propensity score in a study of the treatment of childhood amblyopia. Although conditioning on a scalar summary confounder score facilitates the process of plotting exposure-response curves, this approach still requires conditioning on specific generalized propensity score levels and assumes no heterogeneity across propensity score values.

Generalized propensity score weighting provides a simple method of obtaining standardized exposure-response curves, while use of a semiparametric approach renders curves more robust to misspecification of the exposure. Moreover, as illustrated in eAppendix 2 (, exploiting the well-established relations between penalized smoothing splines and mixed effects models32 makes fitting such models relatively straightforward. Pointwise confidence intervals for marginally adjusted curves can be obtained using the bootstrap,33 as we illustrate in eAppendix 2 (

Weighting by the inverse of the generalized propensity score may lead to well-known issues with highly variable weights.34 However, this problem may be improved by using normalized stabilized weights35 or a quantile binning approach.18 Although interpregnancy interval was highly skewed in our study, use of the quantile binning method yielded well-behaved inverse probability weights.

We combined generalized propensity score weights with semiparametric regression models to obtain exposure-response curves for the relation between interpregnancy interval and preterm birth. Our results illustrate the feasibility of obtaining marginally adjusted exposure-response curves while minimizing parametric assumptions.

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