Wavelet-based functional linear mixed models: an application to measurement error–corrected distributed lag models

Abstract

Frequently, exposure data are measured over time on a grid of discrete values that collectively define a functional observation. In many applications, researchers are interested in using these measurements as covariates to predict a scalar response in a regression setting, with interest focusing on the most biologically relevant time window of exposure. One example is in panel studies of the health effects of particulate matter (PM), where particle levels are measured over time. In such studies, there are many more values of the functional data than observations in the data set so that regularization of the corresponding functional regression coefficient is necessary for estimation. Additional issues in this setting are the possibility of exposure measurement error and the need to incorporate additional potential confounders, such as meteorological or co-pollutant measures, that themselves may have effects that vary over time. To accommodate all these features, we develop wavelet-based linear mixed distributed lag models that incorporate repeated measures of functional data as covariates into a linear mixed model. A Bayesian approach to model fitting uses wavelet shrinkage to regularize functional coefficients. We show that, as long as the exposure error induces fine-scale variability in the functional exposure profile and the distributed lag function representing the exposure effect varies smoothly in time, the model corrects for the exposure measurement error without further adjustment. Both these conditions are likely to hold in the environmental applications we consider. We examine properties of the method using simulations and apply the method to data from a study examining the association between PM, measured as hourly averages for 1–7 days, and markers of acute systemic inflammation. We use the method to fully control for the effects of confounding by other time-varying predictors, such as temperature and co-pollutants.

1. INTRODUCTION

The study of the health effects of air pollution is an important public health problem. Many studies have linked air pollution exposures to mortality and morbidity (Dockery and others, 1993; Dominici and others, 2006). Moving forward, 2 scientifically pressing issues in this area are the identification of biologic mechanisms underlying these health effects and identification of particulate matter (PM) components, and ultimately pollution sources, responsible for these effects. With the US Environmental Protection Agency (US EPA) currently reviewing the national air quality standards (Kim and others, 2007), both questions are important for risk assessment and regulatory purposes.

Because the identification of biologic mechanisms underlying PM-induced health effects is a high priority, recent studies have examined subclinical health outcomes, such as blood pressure (Santos and others, 2005), heart rate variability (Gold and others, 2000), exhaled nitric oxide measurements (Adamkiewicz and others, 2004), and inflammatory markers in the blood (Pope and others, 2004). Reviews by Bernstein and others (2004) and Katsouyanni (2003) give excellent summaries of additional studies and also provide background on the different pollutants of interest. A popular study design for investigating subclinical effects of PM exposure is the panel design. Investigators repeatedly measure health outcomes on each subject over time, with pollution levels semicontinuously measured over some time period before examination. Because the time course of a health effect can relate to mechanism, the standard approach to analyzing such data uses models with time-varying (moving) averages, individual lag periods, or distributed lags of the pollution exposure in the mixed model framework. Moving average and individual lag models require the fitting of multiple longitudinal models sequentially, with each model including an exposure average (or lag) taken successively further back in time.

There are several statistical issues that arise in this sequential modeling strategy. First, use of sequential moving averages can result in an overly smoothed pattern of estimated health effects that make it difficult to determine the time of greatest influence on the outcome. Second, issues arise in drawing inferences from the multiple comparisons of the parameter estimates. Third, one must decide whether to fit models that use individual lags of exposure or moving averages, and this choice can sometimes impact the resulting overall conclusions of the analysis. Fourth, such models invariably control for potential confounding between pollution and other time-varying covariates, such as temperature and co-pollutants, but typically do so by including only a single scalar summary of the covariate recorded at a single time point. In fitting multiple health models, one usually decides whether to use the value of the confounding variable at the time of the health measurement or from the same time window used for the exposure, and the results may depend on this choice. There is no reason to expect that, if there is an extended, biologically relevant time window of exposure for a given health end point, there is not also a biologically relevant time window for potential confounders as well. Fifth, the potential for measurement error in the PM exposure measures can make the interpretation of the pattern of health effects unclear. Zeger and others (2000) noted that use of ambient pollution measurements typically involves a complex mixture of exposure errors, some having a classical structure that will bias the resulting health effect estimate and some having Berkson structure that does not. If sequential PM coefficients show no estimated association between health and exposure averaged over longer time periods, one cannot eliminate the possibility that larger classical measurement error associated with shorter averaging times attenuated the estimated associations at this timescale.

In this article, we propose the use of distributed lag models (DLMs) based on functional data analysis (FDA) methods to simultaneously address the above issues associated with sequential modeling of panel study data. Ramsay and Silverman (2005) coined the term “functional data” to describe samples that consist of curves. The pollution and temperature measurements recorded over time, as in Figure 1, are examples of functional data. The incorporation of such data in panel study settings is particularly challenging because the pollution measurements have much higher dimension than the number of subjects (typically 40–60) recruited into the study. Additionally, as demonstrated in Figure 1, the pollution curves often exhibit local features, such as spikes, and standard FDA techniques could potentially smooth these away. We develop a wavelet-based linear mixed DLM that addresses both these issues. The model incorporates multiple functional covariates measured at many lag times into a linear mixed model to examine the association between pollution exposure and a continuous health response while completely controlling for confounding by other time-varying covariates, such as temperature and co-pollutants. Moreover, as long as the measurement error contributes only fine-scale variability to the exposure profile and the time-varying effect of exposure is smooth, the wavelet-based decomposition of the exposure profiles effectively corrects for the measurement error. We take a Bayesian approach to model fitting, which allows us to implement automatic data-driven smoothing of the regression coefficient profiles for both the particle exposure and the potential confounders.

Fig. 1.

Ambient PM_2.5 and temperature measurements in St Louis, MO, for the 7 days prior to the morning of June 21, 2002.

Our work is related to existing work on models for functional predictors, DLMs, and Bayesian methods for wavelet-based nonparametric regression. Ogden and others (2002) looked at applications of linear regression models with functional predictors, and James (2002) examined generalized linear models with functional predictors. Zanobetti and others (2000) used DLMs to examine the association between daily pollution exposure and daily deaths while also addressing the issue of mortality displacement. Welty and Zeger (2005) used flexible DLMs to control for temperature confounding in mortality studies when examining a single lag of pollution exposure. Welty and others (2009) developed a DLM that includes prior knowledge on the lag function shape. All these distributed lag formulations focused on mortality time series data and did not address the issue of exposure measurement error. James (2002) and Zhang and others (2007) considered functional linear models for covariates with measurement error, but these authors took an approach of using spline-based methods to smooth the error out of the covariate profiles before estimating the time-varying effects of these covariates. Such smoothing strategies do not work particularly well in the environmental applications we consider as the splines would oversmooth the relatively spiky exposure profiles (Figure 1). We propose the alternative approach of isolating the error in the covariate function via a wavelet-based transformation and assuming a relatively smooth time-varying exposure effect, which serves to zero out the wavelet coefficients affected by the error and thereby correcting for this error. Our approach builds on that developed by Brown and others (2001), who used a Bayesian wavelet method to perform variable selection of wavelet coefficients but who also did not consider the possibility of measurement error.

In Section 2, we briefly describe data from a panel study of the association between air particles and markers of acute systemic inflammation, which motivated this work. Section 3 gives a brief review of wavelets and presents the model and methodology. We report the results of a simulation study in Section 4. In Section 5, we analyze the inflammation data. Further discussion of the methodology and the application is in Section 6.

2. DATA

We analyze data from an experiment designed to investigate the association between inhaled PM and acute systemic inflammation in older adults (Dubowsky and others, 2006). Data were collected on 44 senior citizens aged 60 years and older. Individuals participated in up to 4 trips aboard a diesel bus between March and June 2002. Indicators of inflammation, such as white blood cell (WBC) counts, were measured the morning after the bus trip for each individual on that trip. Ambient measurements of PM_2.5, PM with mass median aerodynamic diameter less than 2.5 μm, were collected as hourly averages and matched to each individual for up to 7 days prior to measurement of WBC counts. These were obtained from the US EPA–funded Supersite local to the participants in St Louis, MO. In addition, measurements of other time-varying covariates, such as ambient temperature, were also collected.

Figure 1 shows ambient PM_2.5 and temperature measurements for 1 event of the study. The right end point (time zero) corresponds to when WBC counts were measured, at approximately 9:00 AM on the morning following each bus trip. From this, we see some features typical of pollution and temperature data. The curves are cyclic with period of approximately 1 day. There are also many sharp peaks throughout the PM_2.5 curve and the occasional large spike, as demonstrated by the peak in the left plot at about − 4.25 days, corresponding to 4.25 days prior to measurement of the response.

Dubowsky and others (2006) analyzed these data using linear mixed models with time-varying (moving) averages for the pollution measurements. The moving averages that are predictors in these models were taken over successively larger intervals and used in a sequence of models. The authors used the 7 averaging periods of 1–7 days. The response variable was log transformed due to skewness. Models also controlled for gender, obesity status, diabetes status, smoking history (ever/never—no current smokers were included in the study), hour of WBC count measurement, trip number (as a proxy for season), vitamin intake, 24-h mean ambient and microenvironment apparent temperature, mold counts, and pollen counts. They found positive associations between ambient PM_2.5 and WBC counts that were significant at the 7-day mean. An open question with these data is the impact of controlling for confounding using the scalar 24-h mean rather than the complete time-varying profile of the confounders. That is, if PM_2.5 has a lagged effect on health, it is possible that this is true of temperature and PM co-pollutants as well. There is the further issue of the potential impact of measurement error. That is, in the presence of exposure measurement error, moving averages taken over short time intervals are likely to contain more measurement error than their longer term counterparts, which could lead to more severe attenuation of the resulting estimated health effects.

3. METHODS

3.1. Wavelets

Wavelets can be used as a basis expansion method for representing functions in L²(ℜ):

where φ_Sl(t) and ψ_sl(t) are the wavelet basis functions and c_Sl and d_sl are wavelet coefficients found by taking the inner production of x(t) with the basis functions. The coefficients describe features of x(t) at location l and scale s. The wavelet basis functions are derived from translations and dilations of the scaling function, φ(t), and the mother wavelet, ψ(t), so that φ_Sl(t) = 2^S/2φ(2^St − l) and ψ_sl(t) = 2^s/2ψ(2^st − l). These are fixed functions chosen from one of a number of families of wavelets. For example, the Daubechies and the Coiflets families are 2 well-known classes of wavelets (Daubechies, 1992).

Sampled curves typically are collected at discrete time points and not as continuously realized functions. The discrete wavelet transform (DWT) approximates the sample curve, x(t), using a finite basis expansion,

(3.1)

If x(t) is sampled on a grid of m values, then the discrete wavelet coefficients can be found by DWT in O(m) operations (Mallat, 1989). Given a row vector of discrete functional data values, x^′ = (x(t₁),x(t₂),…,x(t_m)) on some discrete time interval t₁,t₂,…,t_m, the DWT of x gives the discrete wavelet coefficients d^′ = (c_S1,…,c_SLS,d_S1,…,d_SLS,…,d₁₁,…,d_1L1) by multiplication of the orthogonal projection matrix W^′, that is, d^′ = x^′W^′. The c_Sl coefficients correspond to the coarsest scale and represent an approximation to the function. The d_sl coefficients correspond to the finer detail in the curve. The inverse discrete wavelet transformation (IDWT) can be used to reconstruct the original function and is found by multiplication of the IDWT projection matrix, W,x^′ = d^′W.

A wavelet analysis of functional data first requires transformation of the sample functional data into their corresponding sampled wavelet coefficients via the DWT. The analysis takes place in this wavelet space, and the IDWT can be used to transform the results back to the time domain. Wavelet representations of functional data provide numerous advantages over the time domain quantities. Wavelets are defined across scales and locations and therefore retain important local information about a function, such as spikes or discontinuities. Each function's wavelet coefficients are less correlated than the raw data, denoted by the whitening property of the DWT (Johnstone and Silverman, 1997).

3.2. Model

A functional linear mixed model with a single functional predictor measured over the interval 0 ≤ t ≤ T has the form

where y_ij is the jth response for participant i measured at time ω_ij, x_ij(ω_ij − t) is the jth functional predictor for participant i at lag t units from time ω_ij, T is the maximum lag time considered, and z_ij is a vector of nonfunctional covariates with corresponding parameter vector δ. There are i = 1,…,n participants with j = 1,…,J_i measurements per participant. All curves, x_ij(ω_ij − t), are assumed to be sampled on an equally spaced grid of lag values t₁ = 0,…,t_m = T. This gives the following discrete form of the model:

(3.2)

where implicit in β(t_k) is a scaling factor, Δt = T/m, which is the size of the increment of the discretized lag interval. That is, β(t_k) = λ(t_k)Δt in (3.2). If the unit of time measure is the lag number, then this gives m = T and a scaling factor of 1. As this is often the case for many pollution studies, we focus our method on estimation of β(t_k) but acknowledge that a rescaling of estimates by T/m will give the unscaled regression coefficient λ(t_k).

Standard mixed model assumptions on the error term and the random effect are ϵ_ij ∼ N(0,σ_ϵ²) and u_i ∼ N(0,σ_u²), with ϵ_ij independent of u_i. The elements of β are the values of the functional regression coefficient β(t) on the grid t₁,…,t_m. Note that this model is a special case of the linear mixed model for which we have a functional covariate with a nonparametric functional coefficient.

In this form, the model in (3.2) can be transformed to an equivalent wavelet-based model and the analysis carried out in the wavelet space. The DWT of the vectors x_ij and β requires multiplication of each by the DWT projection matrix, W:

(3.3)

(3.4)

where the equality in (3.3) follows from the orthogonality of W. Each vector d_ij consists of the wavelet coefficients for the jth measurement on individual i so that d_ij = (c_ij,S1,…,c_ij,SLS, d_ij,S1,…,d_ij,SLS,…,d_ij,11,…,d_ij,1L1)^′. The unknown wavelet coefficient vector for the functional regression coefficient has the analogous form β^* = (β_S1^c,…,β_SLS^c,β_S1^d,…,β_SLS^d,…,β₁₁^d,…,β_1L1^d)^′, where the superscripts c and d denote the course approximation and detail coefficients, respectively.

The final model in (3.4) can be fit using standard linear mixed model procedures (McCulloch and Searle, 2001) to obtain estimates for the wavelet coefficients β^*. The IDWT can then be applied to these coefficients to get the corresponding estimate of the regression function in the time domain. In many applications, the functional data are discretized over a fine grid of many values. Large data sets of this type cannot be fit using standard methods as there are often many more values of the functional data than observations in the data set. We propose fitting (3.4) using a Bayesian approach where the unknown regression function, β(t), is smoothed by placing a mean zero prior on the corresponding wavelet coefficients. Fitting the model in the wavelet space allows preservation of the information in the data curves, such as the peaks in the pollution curve of Figure 1. We outline the approach in the following section.

3.3. Bayesian smoothing

We use the DWT of the data curves x_ij(ω_ij − t) to obtain model (3.4). To preserve dominant features and to allow fitting of highly dimensional data curves, we adaptively regularize the regression coefficient by placing a mixture of a Normal random variable and a point mass at zero on the wavelet coefficients (Clyde and George, 1999; Brown and others, 2001; Morris and others, 2003; Morris and Carroll, 2006). For the element β_sl^d of β^*, we assume that

(3.5)

where ε₀ indicates a discrete measure concentrated at 0 (Ishwaran and Rao, 2005), γ_sl is an indicator of the importance of the wavelet coefficient and determines if it is nonzero, π_s gives the prior probability that the wavelet coefficient at scale s is nonzero, and τ_s is the prior variance at scale s. Thus, at a given scale all coefficients β_sl^d are assumed to have the same probability of being zero. This assumption may be relaxed if more flexibility is needed in fitting the model, so that each coefficient has its own prior probability, π_sl, of being nonzero. A similar prior can be placed on the approximation coefficients, β_Sl^c, although these are often not smoothed, in which case a Gaussian prior is used, β_Sl^c ∼ N(0,τ_c). To fully specify the Bayesian model, we place standard priors on the variance components and the coefficients for nonfunctional covariates,

(3.6)

The hyperparameters a₁,a₂,b₁,b₂,τ_c, and σ_δ are typically set at fixed values to give noninformative priors or estimated using empirical Bayes (EB) methods. We use EB methods to elicit values for the smoothing hyperparameters τ₁,…,τ_s,π₁,…,π_s, as discussed in Section 3.4, and Markov chain Monte Carlo methods to obtain posterior samples of the wavelet coefficients in the wavelet space. Derivations of the full conditional distributions for this model can be found in the supplementary material (available at Biostatistics online). The full conditional distributions are used in a Gibbs sampler to obtain posterior samples of the wavelet coefficients, β^*, covariate parameters, δ, random effects, u, and variance components, σ_u² and σ_ϵ². Posterior sample quantities of β^* are then transformed back to the time domain via the IDWT, yielding samples of the discretized regression coefficient, β(t). Estimation and inference on β(t) are performed using these time domain samples.

In many applications, multiple time-varying measurements are recorded and their collective association with an outcome is of interest. In addition, the interaction of the functional predictor with a time-invariant covariate, such as gender or health status (for short-term measurements), may be of interest. The Bayesian wavelet shrinkage method extends in a straightforward fashion to incorporate more than one functional predictor or interactions with functional predictors into the model. Due to the conditional independence of the wavelet coefficients, the full conditional distributions used in the Gibbs sampler are of the same form as for the single functional predictor model and implementation proceeds as before.

3.4. EB estimates of hyperparameters

The full Bayesian-specified wavelet space linear mixed model of Section 3.3 has 2S + 6 hyperparameters: a₁,a₂,b₁,b₂,τ₁,…τ_S,π₁,…,π_S,τ_c, and σ_δ. Prior specification of these may be difficult. An alternative would be to estimate these hyperparameters, or a subset of interest, using the observed data. Carlin and Louis (2000) described EB methods for hyperparameter estimation, and we borrow their notation and setup. The data y depends on a vector of parameters θ whose posterior estimates are of interest. In turn, these parameters depend on a vector of hyperparameters η,y ∼ f(y|θ) and θ ∼ g(θ|η). With η known, the posterior for θ is p(θ|y,η) = f(y|θ)g(θ|η)/f(y|η). The marginal distribution of y given η,f(y|η) = ∫f(y|θ)g(θ|η)dθ, is used to obtain an estimate . This is then used as a fixed quantity when deriving the posterior distribution of the parameters of interest, θ. Frequently, the method of moments or maximum likelihood is used to obtain an estimate for η.

As an example, we illustrate this method of finding EB estimates on the wavelet coefficient smoothing hyperparameters in the linear mixed model of Section 3.3. Thus, η = (π₁,…,π_S,τ₁,…,τ_S). Clyde and George (1999), Clyde and George (2000) discussed EB estimates for wavelet coefficients in the context of nonparametric regression. For simplicity, consider the wavelet space mixed model with a single functional predictor and no covariates. In matrix form, this is Y = Dβ^* + Z_uu + ϵ, where the design matrices are given by D, the wavelet coefficients and Z_u, the block diagonal design matrix corresponding to the random effects. Then the full Bayesian specification of the model is

Note that the prior on the detail coefficients β_sl^d can also be written as N(0,γ_slτ_s) and any nonfunctional covariates can be grouped with the approximation coefficients in β_Sl^c. As noninformative priors are typically used for β_Sl^c,σ_u², and σ_ϵ², in the following we consider σ_u²,σ_ϵ², and τ_c to be fixed quantities. This also allows for a closed-form expression for f(y|η) that often depends on a large number of parameters.

We find the maximum likelihood estimate of f(y|η) conditional on , and . A large value of τ_c is used for a noninformative prior for β_Sl^c. Reasonable values for and can be found by fitting a simpler model, such as the model with a moving average of the functional predictor, and using the resulting estimates for the variance components. Recall that β^* is the vector of L + L_S wavelet coefficients, β^* = (β_S1^c,…,β_SLS^c,β_S1^d,…,β_SLS^d,…,β₁₁^d,…,β_1L1^d)^′, where L = ∑_{s = 1}^SL_s. The density of interest is found by marginalizing over β^* and γ = (γ_S1,…,γ_SLS,…,γ₁₁,…,γ_1L1)^′, and using the mixture property of normal distributions, it can be shown that Y|γ,η ∼ N(0,DΣD^′ + V), where and Σ is an (L + L_S) × (L + L_S) diagonal matrix with entries on the first L_S diagonals, γ_S1τ_S,…,γ_SLSτ_S on the next L_S diagonals down to γ₁₁τ₁,…,γ_1L1τ₁ for the last L₁ diagonals. Thus,

(3.7)

where f(γ|η) = ∏_{s = 1}^S∏_{l = 1}^L_sπ_s^γ_sl(1 − π_s)^{1 − γ_sl} is the joint density of the L parameters in γ. The total number of wavelet coefficients, L + L_S, is approximately equal to the number of time points over which the functional predictor is discretized and this is typically large. Therefore, the number of terms in the sum in (3.7), 2^L, is quite large and maximizing this expression with respect to the 2S parameters in η becomes costly. We consider 2 approaches to addressing this computational issue. In the first, for the purpose of estimating η only, we assume that γ_sl = γ_s for all l = 1,…,L_s. This reduces the number of terms in (3.7) to 2^S, where S is typically much smaller than L. Maximization of this expression under this constraint can occur only at either , regardless of the values of τ_s. Furthermore, this results in either all coefficients at level s being smoothed to 0 (if ) or all coefficients being allowed to freely vary with variance (if ). Thus, this constraint effectively places the prior distribution on the γ_s parameters as corresponds to γ_s = 0 and corresponds to γ_s = 1. Although, for the latter case, smoothing can also occur when τ_s = 0. Note that optimization of (3.7) can be implemented using standard routines found in packages such as MATLAB.

This method of selecting π_s and τ_s will henceforth be referred to as the γ-EB method as it can be thought of as EB for the γ parameters. It does place a strong constraint on the coefficients within each level, effectively forcing them all in or all out of the model. As this may be too stringent a criteria, we also considered the approach of Morris and Carroll (2006) that does not depend upon this assumption. Morris and Carroll (2006) describe an EB method for estimation of the hyperparameters based on an iterative procedure which maximizes the joint likelihood f(y,γ|η) conditional on a current value of γ. With each update of the parameters in η, γ is also updated until a convergence criteria is met. It should be further noted that this method allows the variance hyperparameters, τ_s in (3.5), to differ across levels and coefficients, thus giving estimates for τ_sl. We refer to this method as the C-EB method as each step is conditional on a current estimate of the parameters in γ. Additional details of the implementation can be found in the supplementary material (available at Biostatistics online) or in the manuscript of Morris and Carroll (2006).

3.5. Impact of measurement error

As noted in Sections 1 and 2, a remaining issue is the potential impact of exposure measurement error on the estimates from the sequential model fits. Consider the wavelet coefficients d_ij = (c_ij,S1,…,c_ij,SLS,d_ij,S1,…,d_ij,SLS,…,d_ij,11,…,d_ij,1L1)^′ and corresponding regression coefficients β^* = (β_S1^c,…,β_SLS^c,β_S1^d,…,β_SLS^d,…,β₁₁^d,…,β_1L1^d)^′, introduced in Section 3.2. Given that the measurement error is likely to manifest itself as fine-scale variation in pollution profiles, in the wavelet space this error will be contained in the detail wavelet coefficients d_ij,S1,…,d_ij,SLS,…,d_ij,11,…,d_ij,1L1. However, in the event that β^* corresponds to an exposure effect that varies smoothly in the time domain, which is likely to be the case in practice, the nonlinear shrinkage prior (3.5) for these coefficients will yield an estimate of zero for the majority of the detail coefficients β_S1^d,…,β_SLS^d,…,β₁₁^d,…,β_1L1^d. Because these are the coefficients that correspond to the covariates (the d_ij) measured with error, this shrinkage effectively “zeros out” the covariates measured with error, thereby reducing the impact of the exposure error on the estimation of the functional effect of exposure.

To illustrate this point, Figure 2 displays a single PM_2.5 profile with and without measurement error and the corresponding wavelet coefficients. Details of the simulation setup is provided in Section 4. The heavy vertical line separates the approximation coefficients on the left and the detail coefficients on the right. One can note that the detail coefficients from the profile with measurement error are notably different from the wavelet coefficients from the true exposure profile, whereas the 2 sets of approximation coefficients closely match. Specifically, the average value of (d_ij,error − d_ij,true)² is 0.9809 for the approximation coefficients and 1.4661 for the detail coefficients, demonstrating the fact that the measurement error manifests itself in the fine-scale variability of the curve.

Fig. 2.

Example scaled PM_2.5 profile with and without measurement error used in simulations. Corresponding values of wavelet coefficients for the scaled PM_2.5 profile with and without measurement error. The coefficients to the left of the vertical bar are the approximation coefficients and to the right are the detail coefficients.

Another potential way to address possible measurement error in the exposure profiles is to use an initial de-noising of the functional covariate. Standard wavelet thresholding techniques accomplish this step with the resulting thresholded exposure profile “plugged-in” to the mixed model. The above argument suggests that, if the true functional regression coefficient is smooth, this step may be unnecessary as the wavelet decompositions place a large portion of the white noise in the detail coefficients which might then subsequently be smoothed away. In such settings, results based on a priori de-noising the exposure profiles will be virtual identical to those based on the raw profiles. We demonstrate this point in Sections 4 and 5.

4. SIMULATIONS

Data were simulated from a linear mixed model

(4.1)

with i = 1,…,25 subjects and j = 1,…,4 observations per subject. To examine the Bayesian wavelet shrinkage method under similar conditions as the pollution application of interest, we took x_ij(ω_ij − t_k) to be the ambient hourly PM_2.5 measurements described in Section 2 but scaled to have a variance of one at each hour. Note that time ω_ij is fixed for each individual and corresponds to the time that the response y_ij is measured. As only 16 subjects had completed all 4 trips, we replicated their profiles to give 25 subjects with 4 measurements each. We used m = 128 hourly time points from the pollution profile. The response was then generated based on the model in (4.1) using σ_u² = 0.05 and σ_ϵ² = 0.01. These were the estimated variance components from a linear mixed model using the data of Section 2 with the full 7-day moving average of PM_2.5 as the scalar covariate.

The DWT implemented in MATLAB (Misiti and others, 2004) uses boundary padding to conduct the transformation, in which the functional covariate is extended beyond the observed time domain. For this simulation, we selected zero padding for the scaled pollution profiles, where the covariate is set to zero outside of the observed time domain. This padding strategy in the wavelet-based DLMs can result in some edge effects on the resulting functional regression coefficient estimator. Although these edge effects occur on both the left and the right sides of the functional regression coefficient (i.e. for both short- and long-time lags), preliminary simulations suggested that these effects were nonnegligible primarily for longer lags (i.e. the right side of the functional regression coefficient) when one uses the nonsymmetric Daubechies wavelet of order 4 basis. In order to minimize edge effects incurred by the DWT of each exposure profile, we report estimates from only the first m = 108 time points.

One hundred data sets were simulated for each of 3 true regression coefficients in (4.1):

given as the solid curves in the left plots of Figure 4 and where t represents lagged time. In addition to using the scaled PM_2.5 profiles, we also examined the impact of having a functional covariate that was measured with error on the fitted regression coefficient by adding white noise to the scaled PM_2.5 curves and using these as the observed data (the response was still generated using the curves without error). Thus, we also examined models in which the observed functional covariate was , where e_ij(ω_ij − t_k) ∼ N(0,σ_ME²). We used 2 levels of measurement error, σ_ME² = 0.6 and σ_ME² = 2.0. Compared to a variance of 1.0 for the standardized exposure profiles, these values represent moderate and very large amounts of measurement error, respectively. As noted in Section 3.5, Figure 2 displays a single PM_2.5 profile with (σ_ME² = 2.0) and without measurement error and the corresponding wavelet coefficients.

Fig. 4.

Left column of plots: Estimated β(t) as a function of lag time and the middle 95th percentile of estimates from the models fit without measurement error. Right column of plots: Average bias in β(t) estimate as a function of lag time.

For each simulated data set, 3 methods were applied to estimate the time course of the exposure effect. First, the standard method that sequentially fits a series of mixed models was applied. This fits one model for each moving average of exposure but ignores the functional nature of the exposure. Second, the Bayesian wavelet shrinkage method of Section 3 using hyperparameters found by γ-EB was implemented for each simulated data set to obtain estimates for these β(t) coefficients. Third, the Bayesian wavelet shrinkage method was implemented with hyperparameters estimated using C-EB. We used the Daubechies wavelet of order 4 (db4) and S = 5 levels of decomposition. The hyperparameters for the variance component priors in (3.6) were fixed at a₁ = a₂ = 3 and b₁ = b₂ = 0.01. For the approximation coefficients, a Gaussian prior was used with a variance of 10¹⁰. After a burn-in of 10 000, a chain of an additional 10 000 iterations were sampled to compute estimates and perform inferences. Visual inspection of the chains was used to assess mixing.

Figure 3 shows the results of sequentially fitting a series of mixed models using moving averages of exposure over successively longer periods of time. This is one of the current methods frequently used for analyzing such data, and following standard practice, each regression coefficient estimate is scaled by the standard deviation of the moving average of exposure used in the model generating that estimate. As there is no closed-form expression for the functional form of the moving average coefficients as a function of average time, we compare the simulated form of these curves when there is measurement error to that when there is not. Large discrepancies between these 2 estimates suggest that the moving average approach is strongly affected by exposure error. As one would expect, in the presence of measurement error, the functional forms of lines connecting these moving average plots exhibit severe bias at short averaging times. This bias increases as the amount of error increases. It decreases as the amount of time over which the moving averages are constructed is increased to the extent that the amount of measurement error used in these simulations does not affect the estimates for moving averages longer than 40 h in the exponential scenario and past 20 and 60 h for the sigmoidal and normal scenarios, respectively.

Fig. 3.

Average estimated β coefficients, weighted by the standard deviation of the sequential moving averages of the covariate, versus lag time from models using sequential moving average as a scalar covariate.

Plots of the results from the wavelet-based functional linear mixed models using the γ-EB method are presented in Figures 4 and 5. Results using C-EB were virtually identical and are thus omitted. Figure 4 displays the true regression coefficient curves and the averages of the 100 estimates, taken as the median of the posterior samples, under each simulation scenario. The shaded regions in the left plots give the middle 95th percentile of the estimates based on the models fit using the covariates measured without error. The plots demonstrate that the percent bias in these estimates is much smaller than that incurred by the sequential modeling approach. Of special note, the estimates obtained under no and moderate measurement error appear to have approximately the same magnitude of bias, with only the case of very large measurement error exhibiting slightly more bias.

Fig. 5.

Left column of plots: Estimated sequential moving averages as a function of average time and the middle 95th percentile of estimates from the models fit without measurement error. Right column of plots: Average bias in moving average estimate as a function of lag time.

The corresponding mean moving average estimates of and pointwise bias are in Figure 5. These were found by successively averaging over longer lag periods and are analogous to the typical summary examined in panel studies (such as those in Figure 3). Again, some small bias is apparent at smaller averaging periods, over approximately the first 24 h, although as with the estimated β(t) curves, these curves fall within the middle 95th percentile of the estimates from models fit with covariates measured without error. Thus, the resulting estimates, although perhaps marginally more biased than using the error-free covariates, are still remarkably good.

Average EB estimates for the smoothing hyperparameters were such that the detail coefficients were uniformly smoothed, as was close to 0, on average, for all levels s. This was true for both the γ-EB and the C-EB methods. The largest average was 0.16, and it occurred with the γ-EB method for the s = 5 level of detail under the exponential scenario with no measurement error. That is, in 84% of the simulated data sets, the detail at level 5 was smoothed, while in 16% it was not, at least based on this average value. The corresponding average of of 0.0002 for this scenario and method suggests that this level 5 detail was uniformly smoothed across all simulated data sets. The C-EB method of estimating the hyperparameters not only had a consistently lower average for this level but also had higher for levels s = 1,2,3, and 4. It should be noted that the maximum average for C-EB was no more than 0.03 and the estimated probabilities decreased with level (i.e. ), thus, the detail here too was uniformly smoothed, on average. Measurement error had little obvious effect on the selection of these hyperparameters.

5. ANALYSIS OF INFLAMMATION DATA

We examined the association between WBC counts and ambient PM_2.5 as in Dubowsky and others (2006), but here we used the full pollution profile defined at hourly averages for 7 days. As typical of such data, the majority of subjects had some missing pollution values. Less than 4% of the PM_2.5 data values were missing when combined over all subjects and times measured. As no single individual had more than 18% of their PM_2.5 profile missing, we applied a simple linear interpolation technique for the missing values based on the nearest nonmissing pollution values and included all subjects in the analysis. When there is nonnegligible missingness, care must be taken in implementing a suitable imputation strategy, as discussed by Caffo and others (2010).

We applied the Bayesian wavelet-based functional data analytic approach of Section 3. Our model included the same scalar covariates as Dubowsky and others (2006), as described in Section 2. The Wavelet Toolbox in MATLAB (Misiti and others, 2004) was used to find the DWT of the pollution measurements. We used the db4 wavelet with S = 5 levels to determine the discrete wavelet coefficients of the wavelet basis expansion in (3.1). These appear to balance smoothness of the underlying wavelet with a support range sufficient enough to capture the local behavior of the pollution data. In addition to the model that included all covariates and the full 7-day PM_2.5 functional covariate, we considered 2 additional models, each of which included one addition functional covariate: ambient temperature and ozone. Periodic padding was used for the boundary conditions for the PM_2.5, ambient temperature, and ozone curves. This choice seems reasonable as there is periodicity displayed in the pollution and temperature profiles, such as in Figure 1. The ozone curves displayed similarly periodic behavior.

Noninformative priors were used for the hyperparameters in (3.6). The wavelet coefficient probabilities, π_s, and variances, τ_s, were estimated using the 2 EB method of Section 3.4. In all models considered, the γ-EB method estimated hyperparameters that uniformly smoothed the detail coefficients, while the C-EB method allowed for some detail in the level 5 coefficients. With these, posterior samples for the wavelet coefficients, nonfunctional fixed-effects parameters, and random-effects parameters were found using Gibbs sampling in WinBUGS (Lunn and others, 2000). After a burn-in of 10 000, a chain of an additional 10 000 iterations were sampled to compute estimates and perform inferences. The IDWT was performed on the coefficient chains for PM_2.5 to find time domain chains and then estimates of the discretized β(t) curve. Mixing was assessed by visual inspection of the chains.

The top left plot of Figure 6 shows the median (solid) of the posterior samples of β(t), along with pointwise 95% Bayesian credible intervals (dashed), for the model with only PM_2.5 treated as a functional covariate and hyperparameters estimated using γ-EB. In this model, the estimated association between WBC counts and ambient PM_2.5 is nonnegative for all times except during the second and third days prior to the measurement of WBC counts. A significant, positive association was found between approximately 3.2 and 4.6 days with a peak at 4 days, suggesting that elevated WBC counts are associated with elevated PM_2.5 measurements from 3.2 to 4.6 days prior. Regions of these left plots marked with * were found to be significantly greater than 0 using a 0.025 cutoff and controlling for false discovery rates (FDRs) using the method described in Morris and others (2008). The FDR method also suggests significant associations at the end points of the 7-day interval of measurement, but the edge effects displayed in the previous simulations suggest that these may be unreliable, particularly at the 7-day lag. While the plot for the C-EB selected hyperparameters is omitted, all levels of detail were smoothed except for the fifth level which has . Thus, the corresponding estimate is somewhat less smooth with significant positive associations found using the FDR method for the 0- to 0.3-day lag, 2.9- to 4.3-day lag, 5.2- to 5.8-day lag, and 6.3- to 7-day lag.

Fig. 6.

Left plots: Estimated median regression coefficient, , and pointwise (PW) 95% credible intervals (CI) for PM_2.5 from the models with (top) 24-h average of ambient temperature, (middle) complete ambient temperature profile, and (bottom) complete O₃ profile. Regions denoted by * indicate significance while controlling for multiple comparisons using an FDR control method. Hyperparameters were selected using the γ-EB method. Right plots are the corresponding estimated median moving average of PM_2.5β(t) and PW 95% CI versus the number of hours over which the average is computed.

To provide an alternative means of comparison to the standard moving average models used in panel studies, we also give the posterior median and 95% pointwise credible intervals for the moving averages of β(t) in the top right plot of Figure 6. Using the estimated coefficients from our Bayesian wavelet-based DLM, we compute estimates and pointwise credible intervals of ∫₀^Tβ(t)dt for T = 0,1,…,168 h. These show that after approximately 70 h there is an increasing trend in the estimated values. Along with the estimates of the full regression coefficient, β(t), these results are consistent with, though somewhat stronger, than those seen in Dubowsky and others (2006). Dubowsky and others (2006) saw that the association was only significant at the 7-day mean, at which point they estimated that an interquartile range increase in PM_2.5 was associated with a 5.5% increase in WBC counts (95% CI 0.10–11%).

The use of a 24-h mean for ambient temperature in the previous model assumes that only this single time period is important when adjusting for the confounding of temperature. Different averaging periods of temperature can be included in a model to examine the effect of confounding in an ad hoc fashion. To completely control for confounding by temperature, we used the multiple functional predictor setup mentioned at the end of Section 3.3 with the full 7-day temperature curve also included in the model. Both EB methods were implemented on each of the functional covariates separately to determine reasonable values for the smoothing hyperparameters. The middle left plot of Figure 6 gives the median estimate along with pointwise 95% Bayesian credible interval for the regression coefficients corresponding to PM_2.5 using the γ-EB method. The association between WBC counts and PM_2.5 remains elevated in a similar time interval as the previous analysis, although it is slightly smaller in magnitude and no longer significant at the 5% level. Examination of 90% credible intervals (output omitted) does suggest marginal significance at the 10% level, and based on the FDR method with a 0.05 cutoff, the region from 3.4 to 4.0 days prior is significant. The C-EB gives a similar region of significance using the same cutoff. The middle right plot gives the corresponding posterior median and 95% pointwise credible intervals for the moving averages of β(t), which shows a significant association at the 5% level beyond 140 h. The estimated temperature regression coefficient (output omitted) was not significant for any t.

Furthermore, we also considered both PM_2.5 and ozone (O₃) as the time-varying covariates. The PM_2.5 regression coefficient, bottom left plot in Figure 6, displays a similar association as the model without O₃ but compressed over early lags. A marginally significant positive association between PM_2.5 and WBC counts can be found near lag 0, in small intervals near 3 days lag (2.8–3.2), and just prior to 6 days lag. The effect on the latter interval is about half the size of the effect on the prior interval. The moving average of PM_2.5 is significant past 156 h at the 5% level. O₃ had a marginally significant positive association with WBC counts on 2 intervals, during the day immediately prior to measurement of WBC counts and a smaller positive association was also found just prior to the 6-day lag (output omitted). Dubowsky and others (2006) reported associations with PM_2.5 and O₃ that did not appear to be due to confounding as well. Thus, the overall conclusions from this earlier analysis and our analysis were generally similar. Interestingly, Dubowsky and others (2006) saw no effect for short lag times and an increasing effect at longer lag times. Our analysis estimates the magnitude of the effect to be largely constant across all lags, with the estimated effects at longer averaging times being slightly more precise. Based on the simulation results, this de-attenuation of the shorter term effects only is to be expected and suggests that health effects corresponding to shorter averaging times should not be ruled out for this outcome.

Although the 2 EB methods were fairly consistent in finding a significant positive association between an approximate 3- to 4-day lag of PM_2.5 exposure and WBC counts, the 2 estimates did display different levels of smoothness. As a sensitivity analysis, we also implemented a full Bayesian approach where we also placed a Uniform(0.0001, 0.9999) prior on each of the π_s hyperparameters in (3.5). With τ_s = 10⁴ and a burn-in of 100 000 iterations, we examined the same 3 models, PM_2.5 alone and with each of temperature and O₃. Results are given in Figure 7 and are consistent with, yet stronger than, the results in Figure 6 using γ-EB. The full Bayesian approach appears to be somewhat of a balance between the 2 EB methods examined, particularly in the PM_2.5-only model where the overall shape of the estimated coefficient is similar to that from the γ-EB method but the regions of significant associations are similar to the C-EB method. This is further reflected by the posterior estimates for π_s. In the model with PM_2.5 as the single functional covariate, level 5 had the highest median posterior probability with and these estimates decreased with level.

Fig. 7.

Left plots: Estimated median regression coefficient, , and pointwise (PW) 95% credible intervals (CI) for PM_2.5 from the models with (top) 24-h average of ambient temperature, (middle) complete ambient temperature profile, and (bottom) complete O₃ profile. Regions denoted by * indicate significance while controlling for multiple comparisons using an FDR control method. Models included a Uniform hyper-prior distribution on the wavelet coefficient probabilities, π_s. Right plots are the corresponding estimated median moving average of PM_2.5β(t) and PW 95% CI versus the number of hours over which the average is computed.

As an additional check on the potential effect of exposure measurement error on results, as discussed in Section 3.5, another approach for correcting for measurement error is to de-noise the PM_2.5 curves by thresholding their corresponding wavelet coefficients. We used a hard thresholding rule (Percival and Walden, 2006) where we examined a single level of thresholding on the magnitude of the coefficients, and we also considered an automatic thresholding using the method of Donoho and Johnstone (1995). The resulting de-noised PM_2.5 profiles were then used as plug-in estimators in the method of Section 3. Results from the 2 methods of thresholding were remarkably similar to the fit already presented here and are thus not shown. This result confirms that the wavelet decomposition can effectively control for measurement error that may be present in the exposure profiles.

6. DISCUSSION

Current technology allows for the collection of large data sets. It has become increasingly more common for exposure measurements to be taken over a fine grid of discrete values that collectively define a function. Analysis of such functional data has become an important area of statistical research. The Bayesian wavelet shrinkage methodology developed here incorporates repeated measures of fixed-effects functional predictors into a linear mixed model. Of particular interest is the situation where the functional covariates exhibit local features, such as spikes, and are measured at a relatively large number of time points, both of which are common in measurements of air pollution data. In addition, results may be sensitive to measurement error and confounding by other time-varying covariates, and fully adjusting for these is important in understanding these complex relationships. While our interest was in application to the health effects of pollution exposure, this methodology can be applied in any regression setting where a functional covariate has been measured.

Care must still be taken in employing wavelets when selecting the appropriate wavelet family and the number of levels over which to decompose the function and may be problem specific. These are similar in spirit to selecting the degree of the polynomial and the number of knots in a DLM. The simulation results suggest that the methodology is capable of determining the overall shape of the regression coefficient with high precision. Moreover, smooth regression coefficients were also estimated with small error even though the functional predictors were not smooth. These suggest that application of this methodology to highly dimensional functional data can lead to interesting and useful results, especially when multiple time-varying functions have been measured. This was also illustrated in the analysis of Section 5 where multi-pollutant effects were examined simultaneously and confounding of PM with temperature was examined more directly than in previous studies. Careful selection of the smoothing hyperparameters is one key component, and data-driven methods, such as EB, may be employed to aid in this.

SUPPLEMENTARY MATERIAL

Supplementary material is available at http://biostatistics.oxfordjournals.org.

FUNDING

National Institutes of Health (ESO7142 to E.J.M., ES012044 and ES000002 to B.A.C., CA107304 to J.S.M., and P01ES009825 to B.A.C., S.D.A., and H.S.); Environmental Protection Agency (R827353 to B.A.C., S.D.A., and H.S.).