Undirected graphical models are extensively used to study the conditional independence structure between random variables (Jordan, 1998; Liu & Page, 2013; Liu et al., 2014). Important applications include image processing (Mignotte et al., 2000), finance (Barber & Kolar, 2018) and neuroscience (Zhu & Cribben, 2017), among others. A major challenge in real world applications is that the underlying conditional independence structure can be distorted by confounders. Unfortunately, despite the large literature on graphical model estimation, there is limited work to date on estimation with observed confounding.
The observed confounding issue is ubiquitous. Consider the problem of estimating brain functional connectivity from functional magnetic resonance imaging (fMRI) (Biswal et al., 1995; Fox & Raichle, 2007; Shine et al., 2015, 2016) data. Here, the connectivity estimate is known to be susceptible to confounding from physiological noise such as subject motion (Van Dijk et al., 2012; Goto et al., 2016). We emphasize that although the amount of motion is observed, the resulting confounding can significantly distort the connectivity matrix when estimated using conventional means, leading to incorrect scientific inferences (Laumann et al., 2016). Another example is in social network analysis. The social contagion (the effects caused by the people close to each other in a social network) are shown to be confounded by the effect of an individual’s covariates on his or her behavior or other measurable responses (Shalizi & Thomas, 2011). As a result, a method which is able to recover the social contagion or the social network structure despite the individual confounding is clearly useful.
This manuscript is motivated by the question: is it possible to efficiently estimate sparse conditional independence structure between random variables with known confounders? We provide a positive answer for the important case of jointly Gaussian random variable models. Although prior works have studied the issue of hidden confounders in generative undirected graphical models (Jordan, 1998), to our knowledge, this manuscript is among the first to develop the methodology to deal with observed confounders. We propose a new class of graphical models: the partially linear additive Gaussian graphical models (PLA-GGMs), whose parameters capture the underlying relationships of random variables, and where these relationships take a partially linear additive form (Hastie, 2017). Further, we parametrize the model using two additive components: the target i.e. the non-confounded structure, and the nuisance structure induced by observed confounders.
Importantly, we do not impose a parametric form on the nuisance structure – only requiring smoothness to facilitate nonparametric estimation. This significantly improves on prior work which has required strong ad-hoc assumptions like the linear assumption (Van Dijk et al., 2012; Power et al., 2014) or the zero-expectation assumption (Lee & Liu, 2015; Geng et al., 2018a) on the nuisance parameter. PLA-GGMs are applicable as long as not all the observed samples are highly confounded, so that the proposed procedure can compare the confounded samples with the non-confounded ones in order to remove the confounding influence.
We propose a pseudo-profile likelihood (PPL) estimator for learning PLA-GMMs, which can be considered as a pseudo-likelihood version profile likelihood (Fan et al., 2005). By minimizing the -regularized negative log PPL, we derive a -sparsistent estimator of the target
structure under mild assumptions. The sparsistency of the estimator indicates that the proposed method recovers the true underlying structure with a high probability(Wainwright, 2009; Kolar et al., 2009; Kolar & Xing, 2011; Ravikumar et al., 2010). We also show that the convergence rate of the proposed estimator is faster than competing methods.
The proposed PLA-GGM can be considered as a generative-model counterpart of partially linear additive discriminative models (Fan & Zhang, 2008; Cheng et al., 2014; Chouldechova & Hastie, 2015; Lou et al., 2016). Compared with these discriminative models, PLA-GGM as a generative model focuses on estimating the relationships among random variables, thus can be used to recover the conditional independence structure, which discriminative models like Sohn & Kim 2012; Wytock & Kolter 2013 cannot. Recall that GGMs can be estimated as a collection of related regressions (Meinshausen & Bühlmann, 2006). Along similar linesm, our PLA-GGM approach requires studying multiple dependent discriminative models simultaneously.
Our main technical contributions are summarized as follows:
To the best of our knowledge, PLA-GMM is the first model to specifically deal with the observed confounders in generative undirected graphical models. Without assuming a parametric form for the confounding, PLA-GMM can accommodate a broad class of potential structure confounders.
We demonstrate that PLA-GGMs facilitate -sparsistent estimators by proposing the PPL method as a new objective for the parameter estimation. Further, since the corresponding minimization problem is shown to be equivalent to a regularized weighted least square, the optimization is shown to be efficient by leveraging the coordinate descent method (Friedman et al., 2010) and the corresponding strong screening rule (Tibshirani et al., 2012).
We demonstrate the utility of PLA-GGMs using both synthetic data and the 1000 Functional Connectomes Project Cobre dataset (COBRE, 2019), a brain imaging dataset from the Center for Biomedical Research Excellence. The proposed PLA-GGM demonstrates superior accuracy in terms of structure recovery and can effectively detect the abnormalities of the brain functional connectivity of subjects with schizophrenia.
We begin by formulating PLA-GGMs. For a continuous random vectorand a confounder variable , we assume that the conditional distribution follows a Gaussian graphical model (Yang et al., 2015a) with a parameter matrix, denoted by , that depends on . In particular, the conditional distribution of follows:
where we assume that the diagonal of the covariance matrix of is , without loss of generality (Yang et al., 2015a). Note that the parameter captures the conditional independence structure111We will refer to conditional independence structure as structure for the ease of presentation. of . Therefore, the structure of is allowed to vary based on the values of confounders. This characteristic makes the proposed PLA-GGM more general in scope and applicability compared to prior work where the structure of is assumed to be unrelated to the confounders (see discussion in Section 5).
Let represent the non-confounded structure. We assume that the parameter takes the partially linear additive form:
Our goal is to recover given independent observations
from the joint distribution of. The term is a nuisance component that arises due to confounding. Thus, while the structure of varies over observations, we are only interested in a specific one, , whose sparsity pattern encodes the target non-confounded structure.
It is clear that recovering is impossible without constraints on . For instance, estimates and result in the same likelihood, making it impossible to determine the true value of . To this end, we enforce a mild assumption – that the effect of confounders is trivial when the size of confounding itself is small. Specifically, for a known , we assume
for any satisfying .
The assumption states that the confounders with values smaller than do not have any effect on the structure of , and thus this serves as a constraint on . Then, as long as we can observe some samples with the confounders small enough (smaller than ), we should be able to distinguish from . Such an assumption is much weaker than those used in existing works including Van Dijk et al. (2012), Power et al. (2014), and Lee & Liu (2015), where or are often assumed. Note that when , (2) will degenerate to . Also, as is smooth, there should exist infinite ’s satisfying the definition. We do not require using the largest possible one. The selection of in practice is discussed in Section 6.2.
3 Pseudo-Profile Likelihood Method
PLA-GGMs facilitate fast-converging estimators. In this section, we propose an estimation procedure for in PLA-GGMs.
3.1 Pseudo Likelihood
For a PLA-GGM parameterized by with observations
, we first derive the log pseudo likelihood as a linear regression in Lemma1.
Define as the vector with the component replaced by , the column vector of , and the column vector of . The log pseudo likelihood of the PLA-GGM follows
It should be noticed that (3) has the same form as the objective function of linear regressions each with observations and covariates. Specifically, for the regression (), the covariate matrix is defined as
and the corresponding response is .
For graphical models without confounders it is known that minimizing -regularized negative log PL (Geng et al., 2017, 2018b, 2018c; Kuang et al., 2017) can lead to -sparsistent parameter estimators (Yang et al., 2015a). Unfortunately, this is no longer true for PLA-GGMs, since the number of unknown nuisance parameters, which are non-parametric, is far too large. Instead, we leverage kernel methods and propose an approximate PL.
3.2 Pseudo Profile Likelihood
We propose a new inductive principle to estimate . As mentioned in Section 3.1, the varying confounding ’s are an obstruction to estimating . We summarize the varying effects as , where denotes the row vector of . (3) is transformed to
There are two unknown parts in PL: and . Intuitively, if we can express ’s using , we will be able to omit and focus on estimating . This leads to the following Lemma on approximating ’s.
For the observation, we define an kernel weight matrix , , which is a diagonal matrix with . is a symmetric kernel density function, and is a user specified bandwidth. Then, we define an auxiliary matrix:
satisfying the smoothing assumptions in Section 4.1.
An estimator of can be derived as , where
Note that given the observations, is only dependent on . Therefore, by replacing with in (3) and some additional transformations, we can derive an approximate log pseudo likelihood whose only unknown parameter is . We define this as the log pseudo profile likelihood (PPL):
Definition 1 (Ppl).
Following the notations above, the log PPL is defined as
where is an vector, whose component is and others are ’s.
The proposed PPL shares a close relationship with the profile likelihood (Speckman, 1988; Fan et al., 2005): if the components of are independent of each other, the form of PPL is equivalent to the log profile likelihood. However, we do not make any assumptions on the independence here, which makes PPL a type of log pseudo likelihood. Such a rationale of intentionally overlooking the dependency is widely used in the derivation of various types of pseudo likelihoods including the one in Huang et al. 2012. However, Huang et al. 2012 focus on Cox regression for the longitudinal data analysis which is different from our setting. Also, the inductive principle in Huang et al. 2012 emphasizes the consistency, while we will show that a -sparsistent estimator can be achieved by using the PPL.
3.3 -Regularized MaPPLE
With the proposed PPL (4), we can now derive an estimator for . For the ease of presentation, we will use to denote . Then, the -regularized MaPPLE is derived as
where , and is the regularization parameter.
Note that (5) has the same form as a regularized weighted least square problem. Therefore, the optimization can be efficiently solved using the coordinate descent method (Friedman et al., 2010), combined with the strong screening rule (Tibshirani et al., 2012). We implement the optimization using the R package glmnet (Friedman et al., 2010).
4 Sparsistency of the -Regularized MaPPLE
The -regularized MaPPLE (5) is proved to be -sparsistent under some mild assumptions.
To start with, we discuss the assumptions for the estimator. Since the estimation of in Lemma 2 is based on kernel methods, we need some standard assumptions widely used in this literature (Mack & Silverman, 1982; Fan et al., 2005; Kolar et al., 2010b). The following assumptions are concerned with the order of , , and , and the smoothness.
Define with and . Then, we assume that there exists , so that .
For any , the following matrices are all element-wise Lipschitz continuous with respect to :
Also, since we do not pose parametric assumptions to and , we further need the following assumptions on both.
The random variable has a bounded support, and is Lipschitz continuous and bounded away from 0 on its support. has continuous second derivative.
Next, we introduce an assumption required for sparsistency. The following mutual incoherence condition is vital to the sparsistency (Ravikumar et al., 2010). Here, we define as the underlying parameter, and treat as a vector containing all the components without repeats.
Define as the index set of the non-diagonal and non-zero components of , as the index set of the diagonal components of , and as the index set of the non-diagonal and zero components of . Define the incoherence coefficient as . Then for , there exists , so that and , where we use the index sets as subscripts to represent the corresponding components of a vector or a matrix.
Define , where with , and for some . Then .
4.2 Main Theoretical Results
According to Theorem 1, the -regularized MaPPLE recovers the true structure of with a high probability. Also, the scale of the estimation error denoted by is less than , which converges to zero at a rate of . In other words, the smallest scale of the non-zero component that the PPL method can distinguish from zero in the true parameter converges to zero at a rate of . We refer to this result as -sparsistency.
Such a convergence rate is faster than ordinary nonparametric methods, which often have a convergence rate (Speckman, 1988; Kolar et al., 2010b). Also, the -sparsistency matches the results of semi-parametric methods (Fan et al., 2005; Fan & Zhang, 2008) for discriminative models, where the estimated parametric part is shown to be -consistent.
In Theorem 1, the value of does not affect the -sparsistency of the estimator. In practice, however, if is too small, the tends to be singular, since few samples are observed with . Accordingly, the PPL method will be not applicable. Therefore, we need to observe some non-confounded samples to implement the PPL method. The -sparsistency is not directly related to the selected either. Along the proof of Theorem 1 in the Supplements, we notice that (and thus ) can only affect some auxiliary constants. Since this relationship is neither significant, nor straightforward, we do not discuss it here.
5 Related Methods
After a thorough analysis on the proposed PLA-GGM and PPL method, we now study some related methods that fall into four categories: Gaussian graphical models incorporating confounders, denoted by CON-GGMs; the Gaussian graphical models using linear regression to deal with confounders (Van Dijk et al., 2012; Power et al., 2014) denoted by LR-GGMs; original Gaussian graphical models only using non-confounded samples, denoted by GGMs; and time-varying Gaussian graphical models (Kolar et al., 2010b; Yang et al., 2015b) denoted by TV-GGMs. Theoretically, the proposed PLA-GGM is more generalized and facilitates faster-converging estimators than the existing models.
5.1 CON-GGMs and LR-GGMs
Although not designed for this task, it is possible to apply more standard graphical modeling approaches to deal with some of the effects of observed confounders. For instance, a straightforward alternative to PLA-GGMs is to directly incorporate the confounder as a random variable into the GGM. Specifically, CON-GGMs assume that the confounder follows a GGM jointly with the random vector , which means
where the joint covariance matrix follows
Since the target structure is for , we can estimate by graphical Lasso (Friedman et al., 2008) first, and then derive the inverse conditional covariance matrix for .
LR-GGM is a model widely used in the neuroscience area (Van Dijk et al., 2012; Power et al., 2014), assuming that the confounders will cause a linear confounding to the observed samples. The model can be formulated as:
where follows a Gaussian graphical model with parameter , and satisfies Assumption 3. Since conditional on , is equivalent to , the target parameter for the non-confounded structure is just . LR-GGMs use linear regressions to recover , and further to regress out confoundings. Finally, LR-GGMs estimate by graphical Lasso.
By deriving the inverse covariance matrices of conditional on for both CON-GGMs and LR-GGMs, it should be noticed that the inverse conditional covariance matrices are irrelevant to the value of . In other words, the confounder does not affect the conditional independence structure of , which is often an unrealistic restriction. In contrast, PLA-GGM particularly deals with confounding of the structure by . Further following this direction, we can derive the following theorem which describes the the relationship among CON-GGMs, LR-GGMs, and PLA-GGMs.
follows a normal distribution,and ;
Thus, it is clear that CON-GGMs and LR-GGMs both assume a constant underlying structure irrelevant to , and are parametric special cases of the proposed PLA-GGMs. Also, since the two methods assume either exactly or asymptotically, they will treat the average of as the underlying and derive incorrect structures that are too dense.
5.2 GGMs and TV-GGMs
In PLA-GGMs, it is assumed that some non-confounded samples are observed. Therefore, we can directly apply GGM to the non-confounded samples and estimate the structure. However, by doing this, only the information from the non-confounded data are used. The estimators are obviously not -sparsistent, considering that most of the observed samples are confounded and not used by GGM. Thus, the method is less accurate than PLA-GGM.
Another class of relevant methods are time-varying graphical models (TV-GMs) (Song et al., 2009a, b; Kolar et al., 2010b; Kolar & Xing, 2012), used to estimate a different parameter at each time point or observation. Specifically, TV-GGMs assume that follows a varying Gaussian graphical model over . Methods like fused Lasso (Yang et al., 2015b; Zhu & Koyejo, 2018) and the kernel estimation (Kolar et al., 2010b) are applied to estimate the varying structure of . One may consider applying such a model, then perhaps averaging the time-varying graph to estimate the non-confounded component. However, since the target of such methods are multiple structures, the estimators can only be guaranteed to be -consistent. In contrast, PLA-GGMs use all the samples to recover the parameter representing the underlying non-confounded structure, and can achieve -sparsistency.
5.3 Graphical Models with Nonparametric Methods
PLA-GMM is not the first approach to incorporate nonparametric methods into graphical models. Prior works like Liu et al. 2009; Kolar et al. 2010a; Voorman et al. 2013; Wang & Kolar 2014; Suggala et al. 2017, and Lu et al. 2015, 2018 have tried to relax the parametric definition of graphical models to realize a more generalized model. However, these methods do not help much to deal with observed confounders, since the structure among the random variables is assumed to be independent of the values of the confounders. Partially linear additive models have also been combined with directed acyclic graphs in (Rothenhäusler et al., 2018), which was developed for causal inference and not the structure analysis.
To demonstrate the empirical performance of the proposed PLA-GGM and PPL method, we apply them to synthetic data for a structure recovery task in Section 6.1 and a real fMRI dataset for a brain functional connectivity estimation task in Section 6.2.
6.1 Structure Recovery
In this section, we use simulated data to compare PLA-GGM, TV-GGM and CON-GGM discussed in Section 5, with the proposed PLA-GGM for structure recovery. We simulate data from PLA-GGMs following the procedure provided in the Supplement. We consider the case of . For all these settings, we fix samples. Then, the four methods are applied to the generated datasets to recover the underlying conditional independence structure. The regularization parameter is selected by 10-fold cross validation from a series of auto generated ’s by glmnet. The bandwidth is determined according to Assumption 1. We use , where is selected according to the designated . We have also studied other forms for , which did not significantly affect the performance.
The achieved area under curve (AUC) of the receiver operating characteristic using the selected hyper parameters are reported in Figure 1. Consistent with the analysis in Section 5, the proposed PLA-GGM achieves higher AUCs on structure recovery than the competing methods. Also, as the number of variables increases, the advantage of PLA-GGM gets more significant. The phenomenon results from the -sparsistency of the -regularized MaPPLE, which is more accurate and requires less data. It should also be noticed that the AUC achieved by CON-GGM is always around 0.5. The reason is that, following the data simulation procedure, the true ’s are always dense, although is sparse. As suggested by the analysis in Section 5, CON-GGM treats as the , and thus tends to recover a wrongly dense .
6.2 Brain Functional Connectivity Estimation
We apply the PLA-GGM to the 1000 Functional Connectomes Project Cobre dataset (COBRE, 2019), from the Center for Biomedical Research Excellence. The dataset contains 147 subjects with 72 subjects with schizophrenia and 75 healthy controls. For each subject, resting state fMRI time series and the corresponding confounders are recorded. We use the 7 confounders provided in the dataset relate to motion for the analysis, and apply Harvard-Oxford Atlas to select the 48 atlas regions of interest (ROIs). Additional preprocessing details are deferred to the dataset authors (COBRE, 2019). The performance of PLA-GGM is compared to LR-GGM, which is the most widely-used method to deal with motion confounding in the fMRI literature (Van Dijk et al., 2012; Power et al., 2014).
We use for the following analysis, which is equivalent to . If the selected is less than the largest possible value, the estimation should still be accurate, since (2) is satisfied. However, a too small may induce a singular and thus the failure of the PPL method. We select the smallest where PPL can be successfully implemented, and use the corresponding . The results using other ’s and ’s are reported in the Supplements. Due to Theorem 1, the form of does not affect the sparsistency of the estimator, and thus has a limited effect on the performance.
We first generally analyze the brain functional connectivity by the PLA-GGM. Specifically, following the common practice in this area (Belilovsky et al., 2016), we assume that all the fMRIs from the subjects with schizophrenia follow a single PLA-GGM with the same brain functional connectivity, and thus combine the preprocessed fMRIs from the subjects into one dataset. Then, the PPL method is applied to the combined dataset to estimate an , which corresponds to the brain functional connectivity for all the subjects with schizophrenia. The same procedure is also implemented on the control subjects’ fMRI datsets.
For a comparison, we also apply the LR-GGM discussed in Section 5 by the aforementioned procedure. The estimated brain functional connectivity for subjects and the controls with the two methods are reported in Figure 2. The ROIs are denoted by nodes with different colors. Edges among nodes denote the estimated functional connectivity. Red edges denote the positive connections, while the blues ones denote negative connections. The darker the color, the stronger the connection. We only provide the figure from one angle here. The figures from other angles are provided in the Supplements.
Comparing the glass brain figure for controls with the one for subjects estimated by PLA-GGM, we find Occipital Pole and Central Opercular Cortex are the two areas differ the most. Interestingly, these two areas have been implicated in the literature as highly associated with schizophrenia (Sheffield et al., 2015). Also, by comparing the results of PLA-GGMs with those of LR-GGMs, the results of LR-GGMs are much denser and covered with lots of positive connections. This phenomenon is consistent with our analysis in Section 5: LR-GGMs will treat the average of ’s as the estimator to and derive incorrect over-dense estimates. This strongly suggests that the dense structure here is a result of the confounders which are not successfully accounted for by the regression.
Ideally, to demonstrate the accuracy of the proposed method for structure recovery, we should compare the estimated brain functional connectivity to the underlying ground truth, which, however, is not available in practice. Therefore, we consider a surrogate evaluation by using the estimated functional connectivity for schizophrenia diagnosis. Intuitively, if the recovered connectivity is more accurate due to effectively omitting the confounding, we should be able to improve schizophrenia diagnosis using the estimated connectivity as features. We apply PLA-GGMs and LR-GGMs to each subject respectively, and calculate ’s for every subject. Then, we use
’s as the input for classification methods to classify the subjects. For this two-class classification task, we use an
-regularized logistic regression as this is a common approach in the literature(Patel et al., 2016).
We consider using only the structure (signs without values) of ’s as the input for the classification. Although the classification is of course more challenging, we can more clearly see how helpful the structure itself is for the diagnosis. The AUCs for the diagnosis are reported in Figure 2(a). For a thorough comparison, we use the regularization parameters suggested by the R package glmnet for the penalized logistic regression and report all the AUCs. Clearly, PLA-GGMs resut in more accurate prediction. Therefore, the brain connectivity estimators derived by PLA-GGMs are more informative for schizophrenia diagnosis and more accurate than those of LR-GGMs.
We next include the values into the input, and evaluate the accuracy for schizophrenia diagnosis. Again, we report all the results using different regularization parameters in Figure 2(b). The green line indicates the best performance of using LR-GGM and -regularized logistic regression achieved in (Patel et al., 2016). Since we are using exactly the same dataset, we directly use their results for the LR-GGM combined with penalized logistic regression for a fair comparison. As a result, for most of the regularization parameters, PLA-GGMs derive more accurate diagnosis than LR-GGMs. Experiments using different are included in the Supplements. The results are similar. Therefore, we conclude that PLA-GGMs more accurately address confounding and derive more accurate estimators of brain functional connectivity.
We note, however, that some specialized alternative classifiers have been developed in the brain connectivity literature (Patel et al., 2016; Arroyo-Relión et al., 2017; Andersen et al., 2018), and expect that our approach will improve performance for those classifiers as well. We leave such further analysis to future work.
7 Conclusions and Future Works
We propose PLA-GGMS, to study the relationships among random variables with observed confounders. PLA-GGMs are especially generalized and facilitate -sparsisent estimators. The utility of PLA-GGMs is demonstrated using a real-world fMRI dataset for the brain connectivity estimation. While we have been taking GGMs as an example, the results can be generalized to other undirected graphical models, especially the univariate exponential family distributions (UEFDs) (Yang et al., 2015a). We leave the details to future work.
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Appendix A Proof of Theorem 1
Following the definition, we can derive the PL for the PLA-GGM as:
Then Lemma 1 can be proved by the definition of .
Appendix B Proof of Lemma 2
Appendix C Proof of Theorem 1
In this Section, we prove the -sparsistency of the -regularized MPPLE by following the widely-used primal-dual witness proof technique (Wainwright, 2009; Ravikumar et al., 2010; Yang & Ravikumar, 2011; Yang et al., 2015a). PDW is characterized by the following Lemma 3:
Let . For any , with probability of at least , there exists and satisfying the following two inequalities:
Proof of (8)
To begin with, we prove (8). We define
We use the to denote the PPL defined in Definition 1. Then, the derivative of is:
where denotes the component of the vector. For the ease of presentation, we define
Then, we consider
whose component is just the target value
We fist study : according ot Lemma 8, for any , there exists and satisfying
According to Assumption 1
Now, we study . According to Lemma 9, we have
uniformly for . Note that is a vector. Therefore, for the component, we have
It can be shown that and
are independent and follow chi-squared distribution with degree equal to.
By Lemma 1 in (Laurent & Massart, 2000), the linear combinition of chi-squared random variables satisfies:
for any . Combing the previous four probabilistic bounds, we can derive
Then, by the definition of , for any , there exists so that for :
Proof of (9)
To prove (9), we use the fixed point method by defining a map . If , by Taylor expansion of centered at ,
By Lemma 3, we can prove the sparsistency by building an optimal solution to (5) satisfying the strict dual feasibility (SDF) defined as , which is summarized. Therefore, we now build a solution by solving a restricted problem.
Solve a Restricted Problem
First of all, we derive the KKT condition of (5):
To construct an optimal primal-dual pair solution, we define as an optimal solution to the restricted problem: