Functional data analysis (FDA) has become increasingly important in modern statistics and has been successfully applied in a variety of scientific fields. Apart from books on general introductions to FDA (e.g., Bosq, 2000; Ramsay and Silverman, 2005; Ferraty and Vieu, 2006; Horváth and Kokoszka, 2012; Hsing and Eubank, 2015; Kokoszka and Reimherr, 2017), recent advances of FDA, including innovative methodologies, profound theories, efficient algorithms, and successful applications, have been illustrated by numerous survey papers (e.g., Guo, 2004; Müller, 2008; Delicado et al., 2010; Geenens, 2011; Hörmann and Kokoszka, 2012; Cuevas, 2014; Marron and Alonso, 2014; Shang, 2014; Wang et al., 2016; Chen et al., 2017; Nagy, 2017; Vieu, 2018; Kokoszka and Reimherr, 2019).
A majority of FDA methods can only reveal correlations primarily via either functional regression models (for reviews see e.g., Morris, 2015; Greven and Scheipl, 2017; Paganoni and Sangalli, 2017; Reiss et al., 2017) or correlation measures (e.g., Leurgans et al., 1993; Dubin and Müller, 2005; Cupidon et al., 2008; Eubank and Hsing, 2008; Lian, 2014; Shin and Lee, 2015; Zhou et al., 2018). However, FDA methods for causal inference is underdeveloped despite the importance of causation in many scientific studies. Among very few exceptions, almost all of them focus on randomized clinical trials (e.g., Lindquist, 2012; McKeague and Qian, 2014; Ciarleglio et al., 2015, 2018; Zhao et al., 2018; Zhao and Luo, 2019). In classical causal inference for observational studies where multivariate data are of primary interest, the propensity score (Rosenbaum and Rubin, 1983) plays an important role and has been widely applied in epidemiology and political science among others. Despite its popularity, its use in FDA to study causations in observational studies is nearly void.
The main contribution of this paper is to introduce and adapt various state-of-art propensity score methods to observational functional data. We consider the scenario where the treatment is binary and at least one covariate is functional. We generalize the definition of the propensity score to functional data, and study two types of propensity score estimations. The propensity score is estimated by either directly fitting a functional regression model or balancing appropriate functions of the covariates. This paper in particular focuses on propensity score weighting (e.g., Rosenbaum, 1987; Robins et al., 2000; Hirano et al., 2003), although the propensity score may be used to adjust for confounding through other means, e.g., matching (e.g., Rosenbaum and Rubin, 1985; Rosenbaum, 1989; Abadie and Imbens, 2006) and subclassification (e.g., Rosenbaum and Rubin, 1984; Rosenbaum, 1991; Hansen, 2004). A systematic comparison of two popular propensity-score-weighted average treatment effect estimators is provided in both a simulation study and a real data application.
The rest of the paper proceeds as follows. Section 2 provides the problem setup and generalizes the classical definition of the propensity score to functional data where the treatment is binary and one covariate is functional. Section 3 introduces two types of propensity score estimations via direct modeling and covariate balancing respectively and two widely used average treatment effect estimators via propensity score weighting. The two average treatment effect estimators based on a variety of estimated propensity score weights are comprehensively compared in a simulation study in Section 4. They are also applied in a real data analysis in Section 5 to study the causal effect of duloxitine on the pain relief of chronic knee osteoarthritis patients. Discussion in Section 6 concludes the paper.
Suppose that is a continuous outcome, is a binary treatment variable which equals either (control) or (treatment), is a multivariate covariate, and is a functional covariate defined on a compact domain . Suppose that and is smooth, e.g., continuous or twice-differentiable. Without loss of generality , and for all .
Let and represent the potential values of when and respectively. In practice is observable but and are not both observable. Based on , which are independently and identically distributed (i.i.d.) copies of , we aim to estimate the average treatment effect .
We assume that each is fully observed, but the methods below are also applicable for densely measured since its entire trajectory can be accurately recovered by smoothing (e.g., Zhang and Chen, 2007). In this paper we only consider low-dimensional . The handling of high-dimensional multivariate covariates is beyond the scope of this paper but is a promising topic for future research (e.g., Belloni et al., 2014; Farrell, 2015; Belloni et al., 2017; Chernozhukov et al., 2018; Ning et al., 2018).
In this paper we make the following two assumptions:
where “” represents independence.
The propensity score satisfies
for all vectors
for all vectorsand all functions defined on such that .
Assumptions 1 and 2 are straightforward generalizations of the commonly used strong ignorability and positivity assumptions in classical causal inference respectively. Assumption 1 implies that there is no unmeasured covariate, while Assumption 2
essentially requires that every sample has a positive probability of receiving the treatment or being in the control group.
In this section, we introduce various methods for propensity score estimation and two average treatment effect estimators via propensity score weighting.
3.1 Propensity Score Estimation
3.1.1 Direct Modeling
To estimate the propensity score
, one may assume a parametric model forand fit it with an appropriate estimation procedure.
The simplest model might be the generalized functional partial linear model (GFPLM):
To fit (2), functional principal component analysis (FPCA) may be performed to approximate the functional covariate by where
are eigenfunctions corresponding to the topeigenvalues of the covariance function , and are corresponding FPC scores. Thus
where . The maximum likelihood method can be used to find the parameter estimates and thus the propensity score estimate . The number of FPC scores can be determined by various means, including cross-validation, the fraction of variation explained (e.g., 95% or 99%), the Akaike information criterion (AIC), etc.
1. The terms above are all population quantities. In practice one can only obtain their sample versions.
2. The aforementioned FPCA-regularized maximum likelihood method is also applicable to fit a GPFLM if is a multidimensional functional covariate, i.e.,
where is a generic and multidimensional index for . With the FPC scores obtained by FPCA, this multidimensional GPFLM can also be approximated by (3) and fitted by the maximum likelihood method.
where the unknown parameters are , a scalar, , a vector, and , a bivariate function. To fit (4), one may apply the maximum likelihood method after approximating
by a set of tensor products of B-spline basis functions.
3.1.2 Covariate Balancing
In the classical literature on causal inference, it is well known that parametric methods for propensity score estimation may suffer from model misspecification substantially (e.g., Smith and Todd, 2005; Kang and Schafer, 2007). Recently covariate balancing methods, which aim to mimic randomization, have been proposed as important alternatives (e.g., Qin and Zhang, 2007; Hainmueller, 2012; Imai and Ratkovic, 2014; Zubizarreta, 2015; Li et al., 2018; Wong and Chan, 2018; Zhao, 2019). To the best of our knowledge, all existing covariate balancing methods so far are developed to handle multivariate covariates and cannot be directly used for functional covariates.
To balance functional covariates, we propose to substitute the functional covariate by a multivariate covariate. For example, by FPCA as in Section 3.1.1, the functional covariate can be approximated by with a proper integer such that possesses a majority of the information of . Thus may be used as a substitute of and we can define the substitute propensity score:
where . Obviously if is of finite rank in terms of its spectral decomposition, i.e., FPCA, the substitute propensity score is equivalent to the propensity score defined in (1) when is chosen to be the rank of .
With the substitute propensity score defined in (5), one may estimate it using any existing covariate balancing method. For instance, to apply the covariate balancing propensity score (CBPS) method by Imai and Ratkovic (2014), one may assume a logistic model for , e.g.,
where and are unknown parameters to be estimated. This model is equivalent to the approximate GFPLM in (3).
To estimate the unknown parameters in (6), one may solve the following covariate balancing equation:
where is a user-defined vector-valued function to reflect how the multivariate covariates are balanced. For example, one may define
to balance the first moment of. Alternatively to balance both the first and second moments of , one may use where contains the entry-wise square of .
3.2 Average Treatment Effect Estimation
To estimate the average treatment effect , a variety of estimators via propensity score weighting have been proposed, such as the Horvitz-Thompson estimator (Horvitz and Thompson, 1952), the inverse propensity score weighting estimator (Hirano et al., 2003), the weighted least squares regression estimator (Robins et al., 2000; Freedman and Berk, 2008), and the doubly robust estimator (Robins et al., 1994) among others.
In the simulation experiments and real data application below, we will consider two representative average treatment effect estimators, the Horvitz-Thompson (HT) estimator and Hájek estimator, to numerically evaluate and compare the propensity score estimation methods in Section 3.1.
Explicitly, for each propensity score estimate , the HT and Hájek estimators are respectively defined by
where if it is obtained by direct modeling or if any covariate balancing approach is used for propensity score estimation. Apparently the Hájek estimator, which normalizes the HT estimator, is a special inverse propensity score weighting estimator.
In this section we present a simulation study to evaluate and compare a few propensity score estimation methods in terms of the performances of their resulting average treatment effect estimations.
We had simulation runs where we generated independent subjects with sample size and respectively. For the th subject,
were i.i.d. sampled from the standard normal distribution. The multivariate covariatewas generated by , , . The functional covariate was generated by where and , , . Note that and .
We generated the treatment using the three propensity score models (PSMs) for as follows.
PSM 1: The treatment
follows a Bernoulli distribution with the probability
where and .
PSM 2: The treatment follows a Bernoulli distribution with the probability
where and .
PSM 3: The treatment follows a Bernoulli distribution with the probability
We generated the outcome based on the following two outcome models (OMs).
OM 1: where is generated from the standard normal distribution independently of . The true average treatment effect is
OM 2: where follows the standard normal distribution which is independent of . The true average treatment effect is
We compared the performances of five propensity score estimation methods in the simulation study, denoted by GFPLM, FGAM, CBPS1, CBPS2 and KBCB respectively. The first two methods are via direct modeling while the last three are via covariate balancing. By the FPCA approximation and maximum likelihood method as in Section 3.1.1, GFPLM fits (2) to estimate the propensity score. The number of FPC scores was selected as the smallest integer such that the fraction of variation explained by the top FPC scores is at least 95%. FGAM obtains the propensity score estimate by fitting (4) directly, where tensor products of seven cubic B-spline basis functions were used to approximate before the maximum likelihood method was applied. Apparently GFPLM is subject to model misspecification when data are generated from PSM 2, while both GFPLM and FGAM fit incorrect models when data are generated from PSM 3.
Both CBPS1 and CBPS2 are based on the CBPS method as introduced in Section 3.1.2. The multivariate substitute for the functional covariate was obtained by FPCA which explains at least 95% of the variation of , and (6) was assumed for the substitute propensity score . CBPS1 balanced the first moments of while CBPS2 balanced both first and second moments of , and they were performed using the CBPS R package. KBCB is another covariate functional balancing method recently proposed by Wong and Chan (2018), which controls the balance of over a reproducing kernel Hilbert space (RKHS). KBCB was implemented using the ATE.ncb R package downloaded from https://github.com/raymondkww/ATE.ncb where the RKHS was chosen as the second-order Sobolev space.
|n = 200|
|GFPLM (99.9%, -)||4.82||100.17||2.28||11.28|
|FGAM (99.9%, -)||8.44||17.12||9.58||10.47|
|CBPS2 (99.8%, -)||1.77||28.58||3.75||7.29|
|n = 500|
|CBPS2 (99.8%, -)||1.98||21.26||2.73||5.74|
With the propensity score estimate obtained by each of the five methods above, we achieved the HT and Hájek estimates for the average treatment effect, i.e., and as in Section 3.2. Note that KBCB does not give an estimate for the substitute propensity score . Instead it provides estimates for both and , but they suffice to obtain both HT and Hájek estimates.
|n = 200|
|GFPLM (99.9%, -)||1.26||57.49||-0.68||8.20|
|FGAM (99.8%, -)||-1.05||59.70||-0.93||8.22|
|CBPS2 (99.9%, -)||-4.19||22.22||-1.81||5.62|
|n = 500|
. For each average treatment effect estimate based on any propensity score estimation method, we removed the simulation runs of which average treatment effect estimates are ten standard deviations away from the mean, and used the remaining simulated data to calculate bias and RMSE values.
|n = 200|
|GFPLM (99.9%, -)||0.15||22.19||-0.15||5.66|
|FGAM (99.9%, -)||-4.22||9.16||-4.37||5.96|
|CBPS2 (99.8%, -)||-1.69||14.80||-1.11||4.83|
|n = 200|
|GFPLM (99.9%, 99.8%)||0.05||18.26||0.27||12.48|
|FGAM (99.9%, 99.9%)||3.28||11.83||3.29||11.75|
|CBPS1 (99.9%, 99.9%)||0.59||9.74||0.65||10.71|
|CBPS2 (99.8%, -)||0.99||7.92||1.04||8.26|
|n = 500|
|GFPLM (99.8%, 99.9%)||-0.07||11.59||-0.04||10.76|
|FGAM (99.8%, 99.8%)||4.01||10.40||4.02||10.43|
|CBPS1 (99.9%, 99.9%)||0.11||8.49||0.22||8.63|
|CBPS2 (99.8%, 99.9%)||0.51||6.74||0.64||6.58|
The six tables show that for any propensity score estimation methods but KBCB, a larger sample size generally improves the average treatment effect estimation accuracy measured by RMSE, but it unnecessarily improves the bias. With respect to RMSE, the three covariate balancing methods are generally better than the two directly modeling methods, although FGAM occasionally outperforms the two CBPS methods (see Tables 1 and 3). Between the two direct modeling methods, FGAM almost always performs better than GFPLM in terms of RMSE even when the latter correctly specifies PSM 1, but the former is often worse in terms of bias. The results for the two CBPS methods indicate that balancing additional covariate moments can typically improve average treatment effect estimation. Among all five propensity score estimation methods, KBCB performs overall the best in terms of both bias and RMSE with the only exceptions for PSM 1 and OM 2 (see Table 4) and for PSM 2 and OM 2 with (see Table 5).
|n = 200|
|GFPLM (99.8%, 99.8%)||-0.54||10.24||-0.52||9.29|
|FGAM (99.7%, 99.9%)||-0.21||10.17||-0.24||9.91|
|CBPS2 (99.9%, -)||-0.35||5.95||-0.34||6.70|
|n = 500|
|GFPLM (99.8%, 99.9%)||-0.53||8.46||-0.54||8.25|
|FGAM (99.8%, 99.9%)||-0.54||9.70||-0.55||8.96|
|CBPS2 (99.9%, 99.9%)||-0.55||6.24||-0.60||6.35|
|n = 200|
|GFPLM (99.9%, 99.9%)||-0.43||10.91||-0.39||10.49|
|FGAM (99.8%, 99.8%)||-0.29||10.19||-0.30||10.31|
|CBPS2 (99.9%, 99.9%)||-0.37||10.43||-0.31||8.94|
|n = 500|
|CBPS2 (99.9%, 99.9%)||-0.59||5.89||-0.60||6.04|
In terms of computational stability, all propensity score estimation methods perform satisfactorily, but KBCB is the most robust method since it never produces outlying average treatment effect estimates. CBPS1 is slightly less likely to produce extreme average treatment effect estimates than CBPS2. This is somewhat unsurprising since the latter additionally balances the second moments of covariates. Compare to the HT estimates, the Hájek estimates generally have fewer outlying values and smaller RMSE values for all propensity score estimation methods but KBCB. This observation demonstrates the benefit of inverse propensity score weighting.
5 Data Application
We applied three propensity score weighting methods introduced above to a pain relief dataset (Tétreault et al., 2016), which was downloaded from OpenNeuro (https://openneuro.org/datasets/ds000208/versions/1.0.0). The dataset consists of chronic knee osteoarthritis pain patients in two separate clinical trials. The first trial was single-blinded where all subjects took placebo pills, while the second trial was double-blinded where subjects were randomized to take either duloxetine (30/60mg QD) or placebo. With the observational data obtained by combining the two trials, we aimed to estimate the average treatment effect of duloxitine compared to placebos on chronic knee osteoarthritis pain relief. The pain relief was measured by the visual analog scale (VAS) score, and the Western Ontario and McMaster Universities Osteoarthritis Index (WOMAC) score, and we studied the average duloxitine effect on both measures separately.
|[2.5% , 97.5%]|
|GFPLM||-5.99||11.62||[-22.77 , 6.49]|
|CBPS||-5.26||4.12||[-12.21 , 3.17]|
|KBCB||0.39||3.43||[-6.29 , 7.21]|
|GFPLM||-0.52||4.27||[-9.04 , 7.58]|
|CBPS||-0.23||3.64||[-6.87 , 7.18]|
|KBCB||0.28||3.37||[-6.14 , 7.17]|
The HT and Hájek estimates for the average treatment effect of duloxitine on pain relief measured by the VAS score. Bootstrap standard errors (SE) and 95% bootstrap percentile confidence intervals were obtained by 1,000 bootstrap samples.
A subject is considered to receive the treatment if he/she took duloxitine; those who took placebo pills are regarded to be in the control group. The multivariate covariates are age and gender. Each subject also underwent pretreatment brain scans, via both anatomical magnetic resonance imaging (MRI) and resting state functional MRI (rsfMRI). Using the FMRIB Software Library v6.0 (https://fsl.fmrib.ox.ac.uk/fsl/fslwiki), we preprocessed the rsfMRI scans of each subject, registered them to template MNI152 through his/her anatomical MRI scan, and then downsampled each registered rsfMRI scan to the spatial resolution of voxel size . Finally, inspired by Tétreault et al. (2016), we obtained a connectivity network/matrix for each subject which contains the Pearson correlation of the brain signals from every pair of voxels, and we treated this network as a functional covariate . Since each voxel is indexed by a three-dimensional spatial coordinate, the functional covariate is six-dimensional.
|[2.5% , 97.5%]|
|GFPLM||-8.41||11.23||[-26.25 , 5.35]|
|CBPS||-7.63||4.79||[-16.22 , 2.08]|
|KBCB||1.16||4.18||[-7.54 , 8.45]|
|GFPLM||-3.18||4.72||[-11.66 , 6.50]|
|CBPS||-2.85||4.31||[-10.56 , 6.52]|
|KBCB||1.05||4.17||[-7.76 , 8.50]|
We considered three methods for propensity score estimation, GFPLM, CBPS and KBCB. GFPLM refers to the direct modeling method where the propensity score is estimated by fitting the model in Remark 1.2. The top FPC scores of , denoted by , were used in the approximate model (3) for GFPLM. They were also used as the multivariate substitute of to define the substitute propensity score as in (5) to perform CBPS and KBCB. To apply CBPS, (6) was assumed for the substitute propensity score, and only the first moments of were balanced due to a small sample size. We used in all three methods, which was selected as the smallest integer such that the corresponding AIC value no longer decreases when the top FPC scores are sequentially added to (3).
For each propensity score estimation and each pain relief measure, i.e., VAS or WOMAC score, we obtained its corresponding HT and Hájek estimates for the average treatment effect of duloxitine. We used bootstrap samples to provide uncertainty measures, including standard errors and confidence intervals.
Tables 7 and 8 provide the HT and Hájek estimates, bootstrap standard errors and 95% bootstrap percentile confidence intervals for the average treatment effect of duloxitine on VAS and WOMAC scores respectively. They indicate no significant treatment effect of duloxitine over placebo pills on pain relief. This is consistent with Tétreault et al. (2016), although their conclusion was made from a double-blinded clinical trial, i.e., the second trial, while we based our finding on an observational dataset.
The bootstrap HT and Hájek average treatment effect estimates are illustrated in Figures 1 and 2 for VAS and WOMAC scores respectively. Both figures show that the HT estimates based on GFPLM for propensity score estimation have a much larger variation than the two covariate balancing methods, but inverse propensity score weighting can substantially reduce their differences as revealed by the Hájek estimates. The median of the Hájek estimates is shifted towards zero compared to that of the HT estimates when the propensity score is estimated by either GFPLM or CBPS. The two average treatment effect estimates essentially show no difference for KBCB.
To the best of our knowledge, this paper has made the first attempt to study average treatment effect estimation via propensity score weighting for functional data in observational studies. The paper introduces both direct modeling and covariate balancing methods for propensity score estimation and systematically evaluates their performances via a simulation experiment and a real data application. The results confirm the benefits of both inverse propensity score weighting and covariate balancing methods as advocated for multivariate data.
The methods introduced in this paper for average treatment effect estimation only focus on the scenario where the outcome is a continuous scalar variable and there is only one functional covariate. However, with straightforward modifications, they may be generalized to handle multiple functional covariates and continuous functional outcomes.
The covariate balancing methods introduced above rely on a satisfactory multivariate substitute for the functional covariate, which requires the functional covariate to be either fully observed or densely measured (e.g., Dauxois et al., 1982; Hall and Hosseini-Nasab, 2006; Hall et al., 2006). A future research topic is to develop covariate balancing methods for sparsely measured functional covariates (e.g, James et al., 2000; Yao et al., 2005) or a unified approach for all types of functional covariates (e.g., Li and Hsing, 2010; Zhang and Wang, 2016; Liebl, 2019).
Xiaoke Zhang’s research is partly supported by USA National Science Foundation grant DMS-1832046.
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