Functional Clustering for Longitudinal Associations between Social Determinants of Health and Stroke Mortality in the US (2024)

We focus on the association between $Y_{i}(t)$ and $\bm{X}_{i\cdot(t)}$ for $t\in[0,1]$, referred to as longitudinal associations in what follows.To cluster the longitudinal associations of different counties $i$, we assume the samples $\{Y_{i}(\cdot),{\bm{X}_{i\cdot}(\cdot)};i=1,\ldots,n\}$ can be divided into $K$ clusters and follow a finite mixture of functional linear concurrent models

where $Z_{i}$ is the cluster membership for the $i$th subject,$\beta_{jk}(\cdot)$ is the functional coefficient for the $k$th group to capture the longitudinal associations between the response and the $j$th covariate,and $\varepsilon_{ik}(t)$ is a Gaussian white noise with variance $\sigma_{k}^{2}$ for each $k$.We assume that the white noise processes $\{\varepsilon_{ik}(\cdot);i=1,\ldots,n,k=1,\ldots,K\}$ are independent across different $i$s and $k$s, and $Z_{1},\ldots,Z_{n}$ are i.i.d. samples from a multinomial distribution on $\{1,\ldots,K\}$ with$\mathbb{P}(Z_{i}=k)=\pi_{k}$.

Since our dataset are only observed on a finite time grid, we assume that $\big{\{}\big{(}Y_{i}(t_{is}),\bm{X}_{i}(t_{is})\big{)};$ $i=1,\ldots,n,s=1,\ldots,S_{i}\big{\}}$ is the observed data for FMFLCM, where $\{t_{is};i=1,\ldots,n,s=1,\ldots,S_{i}\}\subset\mathcal{T}$.The log-likelihood of the parameters ${\bm{\Phi}}:=(\bm{\beta},\bm{\sigma}^{2},\bm{\pi})$ is then given as

where $f(\cdot;\mu,\sigma^{2})$ is a Gaussian density function with mean $\mu$ and variance $\sigma^{2}$.One may adopt the maximum likelihood estimator (MLE) of (2) to estimate $\bm{\Phi}$.However, given that the functional coefficient $\bm{\beta}$ is an infinite-dimensional parameter, the optimization of $\bm{\beta}$ in (2) is generally an ill-posed problem.Furthermore, the MLE of $\bm{\beta}$ may also suffer from the curse of dimensionality, i.e., $p$ is large. For this case, the MLE of (2) may be far from its true value, or even does not exist (Wangetal., 2008; Yi andCaramanis, 2015).

In this subsection, we propose a class of smoothly clipped absolute deviation (SCAD) type penalties (Fan andLi, 2001) to adjust the aforementioned ill-posed problems.The SCAD penalty takes the form

where $u$ is a scalar parameter to be penalized, $\lambda$ is a tuning parameter, and $\gamma$ is a hyperparameter chosen to be 3.7 as suggested in Fan andLi (2001).We extend the SCAD penalty for the functional objects $\bm{\beta}$, which is the functional SCAD (FSCAD) penalty

where $\|\beta_{jk}\|_{r}=\sqrt{\|\beta_{jk}\|^{2}+r\|\beta_{jk}^{{}^{\prime\prime}}\|}$ with$\|f\|=\sqrt{\int f^{2}(t)\ \mathrm{d}t}$ being an $L_{2}$ norm for a function $f$.Here, $\|\beta_{jk}\|_{r}$ measures the magnitude and smoothness of $\beta_{jk}$ simultaneously(Meieretal., 2009), in which $r$ leverages the function’s magnitude and smoothness in the norm.By penalizing $\bm{\beta}$ using (3),we can penalize the smoothness of each $\beta_{jk}$ to deal with its functional nature, and shrink the $\beta_{jk}$ with small norm $\|\beta_{jk}\|_{r}$ to zero (Huangetal., 2009) for the purpose of variable selections.

It’s worth noting that our SDOH functional covariates exhibit significant collinearity as illustrated in Figure 1, and the presence of the collinearity may enlarge variance and lead to misspecification in variable selections (Zou andHastie, 2005).To adjust for this issue, we adopt a similar strategy as the elastic net (Zou andHastie, 2005) to add an $L_{2}$ term to the FSCAD penalty (3).As such, we obtain the functional SCAD-Net (FSCAD-Net) penalty

where $\rho\in[0,1]$ is a tuning parameter controlling the proportions between the FSCAD and $L_{2}$ terms.The above penalty maintains the properties of the FSCAD penalty.In addition, the FSCAD-Net penalty can reduce variance and address misspecification in variable selections by penalizing the magnitude of $\sum_{j=1}^{p}\sum_{k=1}^{K}\|\beta_{jk}\|_{r}^{2}$.

It’s worth noting that the variable selections with FSCAD or FSCAD-Net do not guarantee that the selected covariates from different clusters are the same.This may not be reasonable for some applications, in which we need to identify the important variables for all clusters.In our case, the purpose of variable selections for SDOH is to facilitate the clustering on the selected SDOH covariates.To this end, we require the selected SDOH for all clusters to be the same within the estimation procedure.To facilitate such kind of investigation, we propose to shrink $\{\beta_{jk};k=1,\cdots,K\}$ to zero functions simultaneously within the variable selection, which is referred to as cluster-invariant sparsity.For this, we modify the FSCAD-Net penalty into the functional group SCAD-Net (FGSCAD-Net)penalty

where $\|\bm{\beta}_{j\cdot}\|_{r}=\sqrt{\sum_{k=1}^{K}\|\beta_{jk}\|_{r}^{2}}$.By grouping $\left\{\beta_{kj};k=1,\ldots,K\right\}$ for each $j$ together in the norm, the FGSCAD-Net not only yields cluster-invariant sparsity but also improves statistical efficiency for variable selection by borrowing strength from all clusters for estimating $\bm{\beta}$.

In this subsection, we apply an EM algorithm to conduct the parameter estimation of FMFLCM.We first introduce a latent variable $z_{ik}:=I(Z_{i}=k)$ to represent the cluster membership of the subject $i$, where $I(\cdot)$ is an indicator function.Denote $\bm{Z}$ as $\{z_{ik};i=1,\ldots,n,k=1,\ldots,K\}$.The complete log-likelihood of FMFLCM containing $\bm{Z}$ is then given by

In order to solve the ill-posed issue for estimating $\bm{\beta}$, we employ a regularized EM (REM) algorithm using the FGSCAD-Net penalty (5) as proposed in Section 2.3.In the following, we demonstrate the E-step and M-step of the REM algorithm.