# Predicting distributional profiles of physical activity in the NHANES
database using a Partially Linear Single-Index Fréchet Regression model

Aritra Ghosal<EMAIL_ADDRESS>Department Statistics and Applied
Probability, University of California, Santa Barbara.  Marcos Matabuena111 AG
and MM joint first co-authorship<EMAIL_ADDRESS>CiTIUS (Centro
Singular de Investigación en Tecnoloxías Intelixentes),Universidad de Santiago
de Compostela.  Wendy Meiring<EMAIL_ADDRESS>Department Statistics and
Applied Probability, University of California, Santa Barbara.  Alexander
Petersen<EMAIL_ADDRESS>Department of Statistics, Brigham Young
University.

###### Abstract

Object-oriented data analysis is a fascinating and developing field in modern
statistical science with the potential to make significant and valuable
contributions to biomedical applications. This statistical framework allows
for the formalization of new methods to analyze complex data objects that
capture more information than traditional clinical biomarkers. The paper
applies the object-oriented framework to analyzing and predicting physical
activity as measured by accelerometers. As opposed to traditional summary
metrics, we utilize a recently proposed representation of physical activity
data as a distributional object, providing a more sophisticated and complete
profile of individual energetic expenditure in all ranges of monitoring
intensity. For this purpose of predicting these distributional objects, we
propose a novel hybrid Fréchet regression model and apply it to US population
accelerometer data from NHANES 2011–2014. The semi-parametric character of the
new model allows us to introduce non-linear effects for essential variables,
such as age, that are known from a biological point of view to have nuanced
effects on physical activity. At the same time, the inclusion of a global for
linear term retains the advantage of interpretability for other variables,
particularly categorical covariates such as ethnicity and sex. The results
obtained in our analysis are helpful from a public health perspective and may
lead to new strategies for optimizing physical activity interventions in
specific American subpopulations.

##  1 Introduction

Medical science is living in a golden age with the expansion of the clinical
paradigms of digital and precision medicine [47, 25, 19]. In this new context,
it is increasingly common to record patient information that is most
faithfully represented by using complex statistical objects such as
probability distributions [14, 13, 32, 33] that contain enriched information
compared to traditional clinical biomarkers in predictive terms.
Distributional representations can be seen as natural digital functional
biomarkers to analyze wearable data information. In a series of papers, the
performance of the distributional representation was compared with that of
existing summary metrics, providing strong evidence of their advantages in
diabetes and physical activity domains [13, 32, 33, 31]. Distributional
representations are a direct functional extension of traditional compositional
metrics [3, 1] and allow the creation of synthetic profiles over a continuum
of intensities measured by wearable devices that provide an individualized
profile of the patient’s activity. Importantly, these representations overcome
the critical limitations of compositional metrics to define specific cut-off
points to categorize patient information that can introduce subjectivity and
be highly dependent on the population being analyzed.

This work is motivated by the desire to uncover factors that characterize the
physical activity patterns of the American population recorded with the new
distributional representation. As energetic expenditure behaves nonlinearly
with age [45], and other anthropometrical measures [34], more advanced and
flexible regression models are required to overcome the limitations of the
linear model. Here, in order to provide a good balance between the advantages
and disadvantages of linear models and non-linear models, we build on the
partially linear model for scalar responses [26] and propose the first
Partially Linear Feéchet Single Index model. As in the scalar response case,
this can be viewed as an extension of the recently proposed global Fréchet
regression and Fréchet single index models [40, 2, 12]. We incorporate the
survey weights from the complex survey design of the NHANES into the model
estimation to obtain reliable population-based results according to the
composition of the US population [29].

From a public health point of view, the proposed model is attractive because
it elucidates the impact certain variables exert on the American population’s
physical activity levels in all ranges of accelerometer intensities. Moreover,
these new findings can help to refine and plan specific health interventions
that reduce the gap in physical inactivity in different US sub-populations.
For example, one of the follow-up analyses conducted herein extracts clinical
phenotypes of individuals to characterize the patients who are more or less
active than predicted by the regression model.

The structure of the paper is as follows. Section 2 introduces the NHANES data
that will be analyzed, together with a background of the physical activity
distributional representations. Section 3 introduces the model and an
efficient, spline-based estimator. Section 4 reports the various analyses
performed. Finally, Section 5 discusses the results from a public health
perspective, this paper’s role in the broader statistical literature on
regression models in metric spaces, and its opportunities in the medical field
to analyze other complex statistical objects.

###  1.1 Contributions

We briefly summarize the methodological contributions of this paper as well as
the findings from analysis of the physical activity NHANES in regression
models whose responses are the distributional physical activity
representation.

  * •

To our knowledge, we propose the first Partially Linear Fréchet Single Index
regression model for responses that are probability distributions, viewed as
elements of the Wasserstein metric space. Moreover, for this particular
situation, we propose the use of splines for the first time in the Fréchet
regression modeling framework.

  * •

We propose an efficient modified optimization strategy to address the complex
survey sampling mechanism of the NHANES data that retains the estimator’s form
of a weighted least squares problem. The key idea of our approach is to
estimate the model’s non-linear component by means of regression splines after
projecting the variables in this term to a single covariate.

  * •

The primary findings of the NHANES analysis are:

    1. 1.

The proposed single-index model is shown to outperform the global Fréchet
model in terms of $R$-square.

    2. 2.

Interpretations are provided for the effects of ethnicity and other
interesting variables on distributional physical activity profiles. These
novel analyses can provide new insights into how inactivity varies among sub-
populations of the US country.

    3. 3.

We construct new physical activity phenotypes corresponding to individuals who
do more or less exercise than is predicted by the model using the
distributional representation. These analyses are new and help us examine how
well individuals adhere to the recommended physical activity guidelines.

###  1.2 Literature review

Statistical regression analysis of response data in metrics spaces is a novel
research direction in the statistical community [9, 4, 37, 49, 12, 39, 59, 7,
27, 20]. The first papers on hypothesis testing [6, 37], variable selection
[50], semi-parametric regression models [12, 2], and non-parametric regression
[15], have recently appeared. For classical regression models with univariate
response data, the partially linear Single Index model has been a topic with
particular popularity in the last twenty years in the statistical and
econometrics literature [55]. There are several works in this direction,
including recent extensions of the model to functional data [55, 53, 56, 60].
However, the authors are not aware of any existing extension to response data
in metric spaces, even for the special case of distributional response data,
or that incorporate the complex survey design into the analysis.

##  2 Motivation example: NHANES accelerometer data 2011-2014

We used data from the NHANES cohorts 2011 - 2014 [22]. The NHANES aims to
provide a broad range of descriptive health and nutrition statistics for the
non-institutionalized civilian U.S. population [22]. Data collection consists
of an interview and an examination; the interview gathers personal
demographic, health, and nutrition information; the examination includes
physical measurements such as blood pressure, a dental examination, and the
collection of blood and urine specimens for laboratory testing. Additionally,
participants were asked to wear a physical activity monitor, starting on the
day of their exam, and to keep wearing this device all day and night for seven
full days (midnight to midnight) and remove it on the morning of the 9th day.
The device used was the ActiGraph GT3X+ (ActiGraph of Pensacola, FL).

Physical activity signals were pre-processed by staff from the National Center
for Health Statistics (NCHS) to determine signal patterns that were unlikely
to be a result of human movement. Then, acceleration measurements were
summarized at the minute level using Monitor-Independent Summary (MIMS) units,
an open-source, device-independent universal summary metric [21]. In order to
further increase the reliability of the analysis, we use the following filter
criteria strategy extracted from [46] in order to remove participants with
poor quality in their accelerometry data. Those participants who i) had fewer
than three days of data with at least 10 hours of estimated wear time or were
deemed by NHANES to have poor quality data ii) had non-wear periods,
identified as intervals with at least 60 consecutive minutes of zero activity
counts and at most 2 minutes with counts between 0 and 5 were removed. These
protocol instructions were adapted from high-level accelerometer research
(see, for example, [48]).

###  2.1 Quantile distributional physical activity representations

We adopt a novel representation of the resulting data that extends previous
compositional metrics to a functional setting [32], aimed at overcoming their
dependency on certain physical activity intensity thresholds. This approach
also overcomes some previously known limitations of more traditional
approaches. Let $i\in\\{1,2,\dots,n\\}$ be the index for participants, where
$n$ is the total number of participants in the study. For the $i$-th
participant, let $M_{i}$ indicate the number of days (including partial days)
for which accelerometer records are available and $n_{i}$ be the number of
observations recorded in the form of pairs $\left(m_{ij},A_{ij}\right)$,
$j=1,\dots,n_{i}$. Here, the $m_{ij}$ are a sequence of time points in the
interval $\left[0,M_{i}\right]$ in which the accelerometer records activity
information, and $A_{ij}$ is the measurement of the accelerometer at time
$m_{ij}$.

(a)

(b)

(c)

Figure 1: (a) The plot of physical activity time series $A_{ij}$ of one
representative participant (one chosen $i$) in the NHANES 2011-2014 study
monitored during 8 days are plotted over the observed time intervals $m_{ij}$,
when the physical activity measurements are counted as described in the
section 2.1. (b) The empirical quantile representation $\hat{Q}_{i}$, of the
activity profile of the participant (chosen $i$) described in (a) above, also
described in the section 2.1. (c) The estimated empirical quantiles of
physical activity profiles are computed from the empirical distribution from
the raw time series for all 4616 participants in the study are plotted here.
This helps to understand the quantile representation of the participant $i$ in
comparison with the rest of the participants in our study.

In this paper, each individual’s accelerometer measurements
$\\{A_{ij}\\}_{j=1}^{n_{i}}$, $i=1,\dots,n$, are studied without regard for
their ordering. We consider the empirical quantile function,
$Y_{i}(t)=\hat{Q}_{i}(t)$, for $t\in[0,1]$, as the response in the regression
model. Here, $\hat{Q}_{i}=\hat{F}^{-1}_{i}$ is the generalized inverse of the
empirical cumulative distribution function
$\hat{F}_{i}\left(a\right)=\frac{1}{n_{i}}\sum_{j=1}^{n_{i}}1\\{A_{ij}\leq
a\\}$, $a\in\mathcal{R}$ for the physical activity values for the $i$-th
individual. In order to illustrate clearly the difficulty of analyzing raw
physical activity data in which different participants are monitored during
different periods and in different experimental conditions, Figure 1 shows the
plot of observed $A_{ij}$ against $m_{ij}$ for an arbitrary participant in our
study (top panel). The bottom panel shows the empirical quantile functions of
all participants after transforming the raw time series physical activity data
into distributional quantile physical activity representation. Quantile
physical activity representation overcomes the problem of summary physical
activity when the raw time series have different lengths. In addition, the new
representation uses all accelerometer intensities (over a continuum) to
construct the new physical activity functional profile, unlike traditional
representations of physical activity that summarize the information in the
form of a compositional vector.

###  2.2 Details of covariates

A total of 4616 individuals were chosen for our analyses with physical
activity monitoring available for at least 10 hours per day for four days. The
covariates used in the model include sociodemographic, physical activity,
dietary, and clinical variables such as age, Body Mass Index (BMI), Healthy
Eating Index (HEI), Total Activity Count (TAC), along with the categorical
variables Ethnicity and Sex. Age at the time of the analysis was in the range
20 to 80 years. The BMI ($\text{Kg/m}^{2}$) was restricted to the range
$18.5-40$ to study individuals ranging from healthy to highly
overweight/obese. The variable (HEI) was utilized indicating a Global score
about the diet quality. TAC was the average over all activity counts over all
the times activity was recorded for the device, in the notation of section
2.1, it is $\sum_{j=1}^{n_{i}}A_{ij}/n_{i}$ for each $i$. The ethnicity
variable reported racial origin of the participants divided in the following
categories: Mexican American, Other Hispanic, Non-Hispanic White, Non-Hispanic
Black, Non-Hispanic Asian, Other races including Multi-racial. To understand
the distribution of the covariates between the levels of the Sex variable we
constructed the Table 1.

Table 1: Summaries of the predictor variables Age, BMI (Body Mass Index), HEI (Healthy Eating Index), TAC (Total Activity Count), and Ethnicity used in the regression analysis, separated by the Sex. In the first column we distinguish the levels of the categorical variable ethnicity from the numerical covariates Age, BMI, HEI, TAC which are designated in the second column. In the third and fourth columns, the first four rows present the means and, in brackets, the standard deviations of the continuous variables (Age, BMI, HEI, and TAC) for Men and Women respectively. The rows 5 - 10 in the same columns represent the percentage breakdown of the sub-populations of Men and Women into their respective ethnicities. The description of the covariates are found in Section 2.2.  | Covariates | Men | Women
---|---|---|---
| Age | 47.45 (16.45) | 48.082 (16.50)
Numeric | Body Mass Index | 28.72 (5.73) | 29.18 (7.41)
Variables | Healthy Eating Index | 53.013 (14.13) | 56.63 (14.75)
| Total Activity Count | 14342023 (3985616) | 1382990 (3971762)
| Mexican American | 8.55 % | 6.42 %
| Other Hispanic | 5.42 % | 5.28 %
Ethnicities | Non-Hispanic White | 70.65 % | 72.3 %
| Non-Hispanic Black | 8.82 % | 10.29 %
| Non-Hispanic Asian | 3.77 % | 3.37 %
| Other Races Including Multi-racial | 2.79 % | 2.35 %

This paper aims to create a parsimonious and straightforward regression model
to interpret the several central aspects of energetic expenditure captured by
the Age and BMI variables that are expected to behave in a non-linear way with
the response. At the same time, we are interested in assessing the diet’s
effect on physical exercise. We have observed that sex and ethnicity
differences in the U.S population tend to interact in relation to physical
activity, e.g., Black, White and Asian women tend to be physically more active
than men while among the Mexican American and Other Hispanic ethnicities, men
tend to be more active. Hence, we considered an interaction between sex and
ethnicity to obtain reliable population-based conclusions about the
relationships between these predictors and physical activity. The sample
design of NHANES provides important advantages to obtain reliable population
measurements that we cannot guarantee due to selection bias with observational
cohorts such as the UK-biobank. In order to properly exploit this advantage,
however, we must incorporate survey design in the estimation procedure, as
described in Section 3.1 below.

##  3 The Partially Linear Fréchet Single-Index Regression model

Let $Y_{i}$ be the quantile function of daily activity levels corresponding to
the $i$-th participant. In what follows, we will build the regression by
directly modeling the pointwise mean function of $Y_{i}(t)$ on the covariates,
$t\in[0,1]$. The choice to use the quantile function as a characterization of
the physical activity distribution can be explained as follows. First, because
the distributions represented by the $Y_{i}$ are a mixture of a mass at 0 and
an absolutely continuous distribution for positive values, a density
representation that ignores inactivity time is not appropriate. Moreover, the
quantile function is practically less restrictive than, for example, the
cumulative distribution function, which must take values between 0 and 1.
Finally, and perhaps most importantly, the quantile function is known to be
intimately connected to the well-established Wasserstein geometry on the space
of distributions [52, 41, 36]. Briefly, if $\mu$ and $\nu$ are two suitable
measures on $\mathcal{R}$ with finite second moment, and if $Q_{\mu}$ and
$Q_{\nu}$ are their corresponding quantile functions, then
$d_{W_{2}}(\mu,\nu),$ the Wasserstein distance between $\mu$ and $\nu$, is
known to be equivalent to the $L^{2}$ distance between $Q_{\mu}$ and
$Q_{\nu}$, that is,

$d_{W_{2}}(\mu,\nu)=\left[\int_{0}^{1}(Q_{\mu}(t)-Q_{\nu}(t))^{2}\mathrm{d}t\right]^{1/2}.$
(1)

As a consequence, under this metric, the Fréchet mean [11] measure of a random
measure is characterized by the pointwise mean of the corresponding random
quantile process. Hence, by proposing a regression model for the random
quantile function $Y_{i},$ we are implicitly constructing a model for the
conditional (Wasserstein-)Fréchet mean of the underlying random physical
activity distribution measure [37].

Let $\boldsymbol{X_{i}}\in\mathcal{R}^{p}$ denote the $p$-dimensional
covariate vector that will appear in the single index part of the model, while
$\boldsymbol{Z_{i}}\in\mathcal{R}^{q}$ is the covariate vector considered for
the linear part. The Partially Linear Fréchet Single Index model is

$E(Y_{i}(t)|\boldsymbol{X}_{i},\boldsymbol{Z}_{i})=\alpha(t)+\boldsymbol{\beta}(t)^{T}\boldsymbol{Z}_{i}+g(\boldsymbol{\theta}_{0}^{T}\boldsymbol{X}_{i},t),\quad
t\in[0,1],$ (2)

where the vector $\boldsymbol{\theta_{0}}\in\mathcal{R}^{p}$, intercept
function $\alpha(t)$, coefficient function $\boldsymbol{\beta}(t)$ and link
function $g$ are the unknown parameters.

###  3.1 Model Estimation

For estimating the parameter $\boldsymbol{\theta}_{0}$, define parameter space

$\Theta_{p}=\\{\boldsymbol{\theta}\in\mathcal{R}^{p}:\left\lVert\boldsymbol{\theta}\right\rVert_{E}=1,\text{
first non-zero element being strictly positive}\\}$

where $\left\lVert\cdot\right\rVert_{E}$ is the Euclidean norm. To facilitate
estimation of the smooth bivariate function $g,$ we will use the expansion

$g(u,t)\approx\sum_{k=1}^{K+s}{\gamma}_{k}(t)\phi_{k}(u),$ (3)

where $\\{\phi_{k}\\}_{k=1}^{K+s}$ is a B-spline basis of order $s$ on a knot
sequence of length $K$, and ${\gamma}_{k}(t)$ are the coefficients of the
basis as a function of $t$. With this approximation, the approximation to (2)
that will motivate our estimator is

$E(Y_{i}(t)|\boldsymbol{X}_{i},\boldsymbol{Z}_{i})\approx\alpha(t)+\boldsymbol{\beta}(t)^{T}\boldsymbol{Z}_{i}+\boldsymbol{\gamma}(t)^{T}\boldsymbol{U}_{i}(\boldsymbol{\theta}_{0}),\quad
t\in[0,1],$ (4)

where $\boldsymbol{\gamma}(t)=(\gamma_{1}(t),\ldots,\gamma_{K+s}(t))^{T}$ and,
for any $\boldsymbol{\theta}\in\Theta_{p},$
$\boldsymbol{U}_{i}(\boldsymbol{\theta})=(\phi_{1}(\boldsymbol{\theta}^{T}\boldsymbol{X}_{i}),\ldots,\phi_{K+s}(\boldsymbol{\theta}^{T}\boldsymbol{X}_{i}))^{T}.$

The linear form of (4) suggests a semi-parametric least-squares approach for
estimation. However, one must remember that the individuals we analyze from
the NHANES database do not represent a simple random sample of the US
population. Instead, they are the result of a structured sample of a complex
survey design from a finite population of individuals. Therefore, in order to
perform inference correctly and obtain reliable results according to the
specific sample design of the NHANES dataset [29], it is necessary to adapt
the usual estimation approach.

Assume that a sample
$\mathcal{D}=\\{\left(Y_{i},\boldsymbol{X}_{i},\boldsymbol{Z}_{i}\right):i\in
S\\}$ is available, where $Y_{i}$ is a response variable, and
$\boldsymbol{X}_{i},\boldsymbol{Z}_{i}$ are vectors of covariates taking
values in a finite-dimensional space. The index set $S$ represents a sample of
$n$ units from a finite population. To account for this sampling, each
individual $i\in S$ will be associated with a positive weight $w_{i}$. In our
analyses, these weights were taken to be the inverse of the probability
$\pi_{i}>0$ of being selected into the sample, i.e. $w_{i}=1/\pi_{i}$ [24,
28]. In order to introduce the sampling mechanism into the estimation
procedure, we consider the Horvitz-Thompson estimator [17, 42], which is based
on weights $w_{i}$ that are proportional to the probability $\pi_{i}$ of
selecting the $i$-th individual in the experimental design, i.e.,
$w_{i}=1/\pi_{i}$.

Here, we adopt the Horvitz-Thompson approach in the estimation procedure
through a weighted least squares criterion. The full procedure can be broken
down into two steps. In the first step, for any unit-norm vector
$\boldsymbol{\theta}$ and any $t\in[0,1],$ we can readily compute

$\left(\hat{\alpha}_{\boldsymbol{\theta}}(t),\hat{\boldsymbol{\beta}}_{\boldsymbol{\theta}}(t),\hat{\boldsymbol{\gamma}}_{\boldsymbol{\theta}}(t)\right)=\underset{a\in\mathcal{R},\boldsymbol{b}\in\mathcal{R}^{q},\boldsymbol{c}\in\mathcal{R}^{K+s}}{\operatorname{argmin}}\,\sum_{i=1}^{n}w_{i}\left[Y_{i}(t)-a-\boldsymbol{b}^{T}z_{i}-\boldsymbol{c}^{T}\boldsymbol{U}_{i}(\boldsymbol{\theta})\right]^{2}.$
(5)

These estimates lead to initial fitted quantile functions

$Y_{i}^{*}(\boldsymbol{\theta},t)=\hat{\alpha}_{\boldsymbol{\theta}}(t)+\hat{\boldsymbol{\beta}}_{\boldsymbol{\theta}}^{T}(t)\boldsymbol{Z}_{i}+\hat{\boldsymbol{\gamma}}_{\boldsymbol{\theta}}^{T}(t)\boldsymbol{U}_{i}(\boldsymbol{\theta}),\quad
t\in[0,1].$ (6)

However, as a function of $t$, it may happen that
$Y_{i}^{*}(\boldsymbol{\theta},t)$ is not monotonically increasing, and hence
is not a valid quantile function. The typical solution for this is to project,
in the $L^{2}[0,1]$ sense, this fitted value onto the nearest monotonic
function [40, 37], yielding valid fitted quantile functions
$\hat{Y}_{i}(\boldsymbol{\theta},t).$

Once these initial quantities are formed for any $\boldsymbol{\theta}$ and
$t$, we can proceed to the estimation of $\boldsymbol{\theta}_{0}.$ As
justified in [12], one can use a generalized version of the residual sums of
squares to obtain the estimate. In the current context, we propose the survey-
weighted criterion

$W_{n}(\boldsymbol{\theta})=\sum_{i=1}^{n}w_{i}\int_{0}^{1}\left\\{Y_{i}(t)-\hat{Y}_{i}(\boldsymbol{\theta},t)\right\\}^{2}dt$
(7)

that constitutes a weighted average of the squared $L^{2}$ norms of the
quantile residuals (or, equivalently, of the squared Wasserstein distances
between observed and fitted physical activity distributions). Then the
estimated parameter is

$\hat{\boldsymbol{\theta}}=\underset{\boldsymbol{\theta}\in\Theta_{p}}{\operatorname{argmin}}\,W_{n}(\boldsymbol{\theta}).$
(8)

From this estimate of the index parameter, given any covariate pair
$(\boldsymbol{z},\boldsymbol{x}),$ we can estimate the conditional
Wasserstein-Fréchet mean quantile function as follows. First, the basis
functions are evaluated at the relevant input by computing
$\hat{\boldsymbol{u}}=(\phi_{1}(\hat{\boldsymbol{\theta}}^{T}\boldsymbol{x}),\ldots,\phi_{K+s}(\hat{\boldsymbol{\theta}}^{T}\boldsymbol{x}))^{T}.$
Then, as in (6), we construct the preliminary estimate

$Y^{*}(t;\boldsymbol{z},\boldsymbol{x})=\hat{\alpha}_{\hat{\boldsymbol{\theta}}}(t)+\hat{\boldsymbol{\beta}}^{T}_{\hat{\boldsymbol{\theta}}}(t)\boldsymbol{z}+\hat{\boldsymbol{\gamma}}_{\hat{\boldsymbol{\theta}}}^{T}(t)\hat{\boldsymbol{u}}.$
(9)

Finally, the estimated quantile function
$\hat{Y}(t;\boldsymbol{z},\boldsymbol{x})$ is obtained by projecting, in the
$L^{2}$ sense, $Y^{*}(t;\boldsymbol{z},\boldsymbol{x})$ onto the space of
quantile functions, meaning the nearest monotonically increasing function. In
particular, for any set of observed covariates
$(\boldsymbol{Z_{i}},\boldsymbol{X_{i}}),$ we obtain fitted values
$\hat{Y}_{i}(t)=\hat{Y}(t;\boldsymbol{Z_{i}},\boldsymbol{X_{i}}).$

###  3.2 Computational Details

We now provide details regarding our implementation of our estimator for the
NHANES data base. In the models implemented below in Section 4,the non-linear
covariate $\boldsymbol{X_{i}}$ for the $i$-th individual consists of their BMI
and age, so the dimension for this component of the model is $q=2,$ i.e.,
$\boldsymbol{\theta}_{0}\in\Theta_{2}$. For the spline basis in (3),
computations were internally performed using the dbs function in the package
splines2 [54]. In our experiments, we set $s=4$ and $K=5$, so the number of
spline regression parameters is $K+s=9$. Then, for a given
$\boldsymbol{\theta}$, we obtain, for each order of quantile t, we obtain the
sequence
$\mathcal{S}(\boldsymbol{\theta},t)=\\{\boldsymbol{\gamma}(t)^{T}\boldsymbol{u}_{i}:i=1,2,\ldots
n\\}$. After ordering the elements from smallest to the largest, the minimum
and the maximum values are considered the boundary knots, while the values
corresponding to the following percentiles are considered the interior knots
respectively: 16.667%, 33.333%, 50%, 66.667%, 83.333%. The covariates in the
linear component $\boldsymbol{Z}_{i}$ consist of HEI (continuous) and
indicator variables for sex and ethnicity, as well as the interaction between
these. The total number of covariates in this component is thus $p=12$.

The estimates of parameters in (5) can be efficiently computed as a weighted
least squares problem for any fixed $\boldsymbol{\theta}$ and $t\in[0,1]$, but
in practice this can only be done for a finite ordered grid of values $t\in
T_{m}=\\{t_{1},\ldots,t_{m}\\}\subset[0,1]$. These initial survey-weighted
least squares computations were done using R package survey [28, 29, 30], that
allows us to introduce splines into the regression model while simultaneously
computing and incorporating the weights $w_{i}$ that are necessitated by the
complex NHANES design. For any given $\boldsymbol{\theta}$ and grid point $t$,
computation of (6) is straightforward. To execute the projection step, observe
that monotonicity can only be achieved in the discrete sense in dependence on
the chosen grid $T_{m}$. We refer to [37] for a simple description of this
projection algorithm, which can be done using any basic quadratic program
solver. Consequently, for a given $\boldsymbol{\theta},$ (7) is approximated
by numerical integration. Finally, to perform the optimization in (8), we use
the function optim in R with the L-BFGS-B algorithm by repeatedly performing
the above steps to evaluate $W_{n}(\boldsymbol{\theta})$ for different values
of $\boldsymbol{\theta}$ across iterations. To deal with the possibility of
local minima, four different starting values (taken to be equally spaced in
their angular representation) in $\Theta_{2}$ were used for this optimization
step, yielding (potentially) four local minimizers. The final estimator was
taken as the one among these yielding the smallest value of $W_{n}.$

To understand the interaction of sex and ethnicity in physical activity
levels, we computed the model estimates of the intercept for the different
cases and plotted them in figure 3. The critical point in the interval
derivation is considered for each intercept and $t\in[0,1]$.

##  4 Experimental Results

We compare the performance between the new Partially Linear Fréchet Single-
Index model and the global Fréchet model [40] in order to examine the relevant
differences and advantages. The covariates used in the global Fréchet model
are same as those in the PL-FSI model, just that in the former, all covariates
are considered parts of the linear component. To begin, we evaluate the
capacity of the models to explain differences in physical activity
distributions across individuals using the survey-weighted $R^{2}$ metric

$R_{\oplus}^{2}=1-\frac{\sum_{i=1}^{n}w_{i}\int_{0}^{1}(Y_{i}(t)-\hat{Y}_{i}(t))^{2}\mathrm{d}t}{\sum_{i=1}^{n}w_{i}\int_{0}^{1}(Y_{i}(t)-\overline{Y}(t))^{2}\mathrm{d}t}$
(10)

where $w_{i}$ is the survey weight corresponding to $i$-th observation and
$\overline{Y}(t)=(\sum_{i=1}^{n}w_{i})^{-1}\sum_{i=1}^{n}w_{i}Y_{i}(t)$ is the
weighted sample Wasserstein-Fréchet mean of the observed physical activity
distributions. To compare models with different numbers of predictors, we
define the adjusted Fréchet $R^{2}$ as

$\bar{R}_{\oplus}^{2}={R}_{\oplus}^{2}-\left(1-{R}_{\oplus}^{2}\right)\frac{q}{n-q-1}$
(11)

where $n$ is the number of observations and $q$ is the number of unknown
parameters in the model [40]. In order to provide a measure of uncertainty for
the estimates of the functional parameter $\boldsymbol{\beta}$ corresponding
to the linear component of (4), we provide an ad hoc confidence interval, with
the following justification. For each fixed $j\in\\{1,\dots,q\\}$,
$\boldsymbol{\theta}\in\Theta_{p},$ and $t\in[0,1]$, if the order of splines
of the non-linear component is fixed and assuming no bias, the asymptotic
distribution of the estimator
$\hat{\boldsymbol{\beta}}_{\boldsymbol{\theta}}(t)$ has been shown to be
Gaussian in similar settings [5, 37]. We can then use the standard outputs of
the survey package, which adjusts for the survey weights, to construct
pointwise confidence bands for the functional coefficients
$\boldsymbol{\beta}(t)$. Importantly, standard re-sampling strategies like a
naive bootstrap do not work here with the two-step-sampling design of the
NHANES because the observational units are not exchangeable. We emphasize that
these confidence intervals are merely to guide our qualitative assessment of
uncertainty, although their asymptotic precision has not been theoretically
guaranteed due to sources of variability, including the effects of spline
parameters and the estimation of $\boldsymbol{\theta}_{0}$, for which this
procedure does not account.

###  4.1 Regression results

We began by fitting the Partially Linear Fréchet Single Index regression model
that includes sex, ethnicity, HEI, and the interaction of sex, ethnicity as
the variables in the linear component. Age and BMI were variables utilized in
the non-linear component. All numerical covariates were centered and
standardized prior to analysis. The estimated parameter for our model is
$\hat{\boldsymbol{\theta}}=(0.2661,0.9639)$ for the ordered variables BMI and
Age, reflecting a greater influence of age compared to BMI in the constructed
index. For the linear part of the model, the estimates
$\hat{\alpha}_{\hat{\boldsymbol{\theta}}}(t),\hat{\boldsymbol{\beta}}_{\hat{\boldsymbol{\theta}}}(t)$
from equation (5) are shown along with their 95% confidence bands in the
Figure 3 as functions of $t\in[0,0.97]$. We only consider in the results,
$t\in[0,0.97]$, due to the inherent noise of the quantile physical
representations in the far right tail. The pointwise confidence bands suggest
differences between the most active individuals of the American and the
Mexican individuals (reference group), where a significant decrease in
physical activity values is seen in all percentiles. Importantly, in some
ethnicities as Non-Hispanic Asian, women are more active than men. Regarding
the diet effect captured by the HEI variable, the results show that it has a
small potential effect, but only in the largest quantiles of physical
activity.

In terms of the adjusted $\overline{R}_{\oplus}^{2}$ metric from equation
(11), the Partially Linear Fréchet Single Index model attained a value of
$0.146$, an approximate increase of 24% relative to the value $0.118$ attained
by the global Fréchet model. Thus, while these results indicate that the
predictive capacity of the models are likely moderate, the additional
parameters introduced by the single index and spline representation lead to a
gain in the variance explained.

The estimates and their variance-covariance matrices are obtained from the
function vcov.svyglm in the survey package in R that takes into account the
complex survey design into the pointwise estimation and dependence structure
estimation. The pointwise 95% Confidence Intervals of the intercepts for
different cases of sex and ethnicity, as well as the parameter corresponding
to HEI are plotted for $t\in[0,0.97]$ in figure 3. Importantly, the findings
from these plots are interesting. As we can see, the physical activities vary
with respect to ethnicities and with respect to sex within the ethnicities in
different ways. Finally, For reproducibility of the results obtained, the code
with the methods proposal here are publicly available on GitHub
https://github.com/aghosal89/FSI_NHANES_Application.

###  4.2 Phenotypes of physical exercise performed

Following the model fitting, a clustering analysis was performed on the
quantile residuals $\hat{e}_{i}(t)=Y_{i}(t)-\hat{Y}_{i}(t)$ from the fitted
partially linear single-index Fréchet regression model, excluding the TAC
variable as a predictor. The goal of this analysis is to identify physical
activity clinical phenotypes [31] after adjusting for the effects of the
relevant clinical variables that are included as predictors. For this purpose,
we consider the clustering algorithm proposed in [10] that uses the energy
distance and is available in the R energy package utilizing the k-group
function. To select the number of clusters, we used the well-established elbow
rule. According to this criterion, we estimated the within cluster sum of
squares using the Gini mean difference for a different number of clusters, and
we plotted the results. The number of clusters was then selected where there
was a change in slope from steep to shallow (an elbow); in this case, $k=3$.

Figure 2 shows the functional residual profiles of each cluster. Cluster $1$
is composed of individuals who are more active than the model predicts
(quantile residuals are positive for nearly all percentiles), particularly in
the highest range of activity, and are therefore expected to be more healthy.
In cluster $2$ are the individuals of the American population that are less
active than the model predicted. Finally, cluster $3$ is a heterogeneous group
involving individuals whose physical activity patters are near those predicted
by the model. As this type of clustering is somewhat obvious (positive,
negative, and near zero), we further investigate the clusters relative to some
covariates that were not included as predictors, due to not having causal
biological effects on physical activity distributions. However, these
variables are still associated with the individuals’ health in terms of
metabolic profiles.

From a public health perspective, the most interesting clusters to describe
are the first two. Upon further analysis, the bottom row of Figure 2 indicates
that physical activity is related to the patient’s health condition in terms
of better and worse metabolic profiles. For example, the distribution of
glucose values and creatinine take values in healthier ranges in cluster $1$
compared to the others, as we observe in the boxplots. In addition, in order
to discriminate and interpret the differences between clusters, the cluster-
specific distributions of TAC are also helpful; the individuals belonging to
cluster $1$ are the most physically active in contrast to cluster $2$. From a
practical point of view, in terms of total energetic expenditure, we have
formal criteria to determine when individuals adhere to the physical
recommendations adjusting for potential confounders, i.e., the covariates used
in the model such as ethnicity, BMI, age, HEI, and sex, that can have a
significant effect in the comparison of the physical exercise performed. The
analysis of the residuals drawn by the regression model allows us to do a
standardized and harmonized of the deviation of physical exercise performed
and compare the individuals of different characteristics and then reclassify
the individuals by then expected physical activity performed.

Figure 2: In the top row, the plots represent the raw functional residuals of
the Partially Linear Fréchet Single Index model against the order of quantile
$t\in[0,0.97]$, belonging to the clusters 1, 2 and 3 respectively. The
clusters are described in the Section 4.2. In the bottom row, the boxplots of
distributions of the log of TAC variable, log of glucose (mg/ml), log of
creatinine (mg/ml), across the clusters is shown.

###  4.3 Association with the non-linear covariates

Figure 3: The estimates of the effects and their 95% Confidence Intervals
obtained from the subsection 3.2 from the regression coefficients in the model
(4) are shown here. The estimated effects are in orange color with a
horizontal grey line at zero for reference. The green and blue curves
represent the 95% pointwise upper and lower confidence bands, respectively as
a function of $t\in[0,0.97]$. We excluded the predicted quantiles for $t>0.97$
since near the boundary the estimates and their Confidence Intervals become
unreliable. From the plot, the pointwise 95% Confidence Intervals of the
regression parameters for the linear covariates do not include 0 for higher
quantiles, implying significance at level $0.05$.

Figure 4: Heatmap plot of $\hat{Y}(\hat{\boldsymbol{\theta}},t)$ across
different quantiles, $t=0.50$ (top left), $0.75$ (top right), $0.90$ (second
row left), $0.97$ (second row right) respectively. The plot in the third row
represnts the integral $\int_{0}^{1}\hat{Y}(\hat{\boldsymbol{\theta}},t)\,dt$.
A 2-dimensional grid was considered for the covariates BMI (in range
[18.5,40]) and age (in range [20,80]) for the single index component of the
Partially Linear Fréchet Single-Index regression model. The categorical
covariates in the linear component were fixed at their baseline levels (i.e.
sex male, ethnicity Mexican American) while the numerical covariate HEI was
fixed at median level. This plot describes the non-linear interaction in the
conditional mean function between age and BMI.

Figure 5: Left panel shows the plot of the derivative of the predicted
quantiles with respect to the single index,
$\boldsymbol{\theta}^{T}\boldsymbol{X}$ at the points on an equidistant grid
of values constructed on the range of
$\boldsymbol{\theta}^{T}\boldsymbol{X}_{i}$ for $i=1,2,...,n$. The selection
process of knots was described in the section 3.2. The red dotted vertical
lines represent these knots. This plot describes the non-linear interaction in
the conditional mean function between age and BMI. Due to the boundary effects
of the basis splines, the derivatives were not very reliable outside the
penaltymate knots. Hence, the derivatives between the $17.776$-th and
$83.333^{rd}$ quantiles of the Single Index were shown with the solid line and
outside the range were shown with dotted lines. The orders of the quantiles
$t=0.50,0.75,0.90,0.97$ being represented by colors black, red, green, and
blue respectively with the thickness also increasing with $t$ for better
visual understanding. Right panel shows the histogram of the density of the
Single Index value $\hat{\boldsymbol{\theta}}^{T}\boldsymbol{X}_{i}$ for
$i=1,2,\ldots,n$.

Since BMI and age are considered in the ranges $[18.5,40]$ and $[20,80]$
respectively and they are in the non-linear component of the model, we were
interested to interpret their effects on the mean of the physical activity
quantile function, how these variables would affect the model prediction and
performance. For the estimated index parameter $\hat{\boldsymbol{\theta}}$, we
obtained the single index $\hat{\boldsymbol{\theta}}^{T}\boldsymbol{X}$, when
$\boldsymbol{X}$ is considered over a 2-dimensional grid. To predict the
quantiles for each $\boldsymbol{X}$ in the grid, the covariates in the linear
component were fixed, as in, the categorical covariates sex and ethnicity were
fixed at their baseline level and the numeric covariate HEI was fixed at the
median level. Hence, we computed the basis functions evaluated at
$\boldsymbol{X}$,
$\hat{\boldsymbol{u}}=\left(\phi_{1}(\hat{\boldsymbol{\theta}}^{T}\boldsymbol{X}),\phi_{2}(\hat{\boldsymbol{\theta}}^{T}\boldsymbol{X}),\ldots,\phi_{9}(\hat{\boldsymbol{\theta}}^{T}\boldsymbol{X})\right)$.
Hence, $Y^{*}(t,\boldsymbol{z},\boldsymbol{X})$ are obtained from equation (9)
which then projected in $L^{2}$ sense to the element in Wasserstein space,
$\hat{Y}(t,\boldsymbol{z},\boldsymbol{X})$. Figure 4 presents a heatmap for
such predictions for selected values of the quantiles $t=0.50,0.75,0.90,0.97$.
The BMI and age being in the horizontal and vertical axes respectively. We did
plot for all quantiles on the grid, however, for $t<0.5$, there was very
little variation among the predicted quantiles and the predictions for
$t>0.97$ were not interesting due to boundary effect of the basis splines.

The four plots at the top of Figure 4 indicate that the non-linear
relationship between age and BMI is more substantial as the quantile increase,
and this relationship is more robust in middle age and intermediate BMI
values. As the value of the age increase and the BMI decrease, the individuals
are more inactive, especially in the extremes.

For a better understanding of effect of the covariates on the model, we
analyze the manner the prediction changes with respect to the projection
$\boldsymbol{\theta}^{T}\boldsymbol{X}$. For this purpose, we studied the
analytic derivative of $Y^{*}(t,\boldsymbol{z},\boldsymbol{X}_{l})$ with
respect to the single index $\hat{\boldsymbol{\theta}}^{T}\boldsymbol{X}_{l}$
in the equation (12) below:

$\frac{\partial}{\partial(\hat{{\boldsymbol{\theta}}}^{T}\boldsymbol{x})}Y^{*}(t,\boldsymbol{z},\boldsymbol{x})=\left[\sum_{j}\hat{\gamma}_{j}(t)\phi_{j}^{\prime}\left(\hat{\boldsymbol{\theta}}^{T}\boldsymbol{x}\right)\right]$
(12)

The derivative of $1^{st}$ order of the B-splines was computed in R using the
function dbs in the package splines2. This derivative was computed for each of
the points
{$\hat{\boldsymbol{\theta}}^{T}\boldsymbol{X}_{1},\hat{\boldsymbol{\theta}}^{T}\boldsymbol{X}_{2},\ldots,\hat{\boldsymbol{\theta}}^{T}\boldsymbol{X}_{n}$}
using the equation (12) and for the quantiles $t=0.50,0.75,0.90,0.97$. The
left panel of the figure 5 diplays the plot, which indicates that the index
involving BMI and age has little or no effect (ignoring the boundaries) until
the index value gets above 1, approximately, after which it has a negative
effect. This means that, given the other variables in the linear term, BMI and
age only start to have an effect for large BMI/age combinations, and the
effect gets stronger as this index increases. Also, the effect is stronger in
the tail (larger values of t) and not very strong for the median of the
Quantile function.

###  4.4 Variation in number of daily steps and non-linear covariates

A critical point in the interpretation of our results from a practical point
of view is to examine the clinical significance of the one-unit variation of
the Monitor Independent Movement Summary Unit (MIMS) in human health. To the
best of our knowledge, no paper has addressed this issue. However, the
following article [23] offers the equivalences between the various
accelerometer units, e.g., MIMS and EMMO. Then, according to the following
EMMO metric paper [44], which shows that a variation of $1$ mg is $500$ steps
per day and changes in the annual mortality of $2-3$ $\%$ and hazard ratio of
$0.95$; and extrapolate the overall change in steps and another clinical
outcome. In particular, in our setting, $1$ MIMS daily unit variations e.g
$\int_{0}^{1}Q(t)dt$ is equivalent in inactive people to $1000$ daily/steps
and $4$ percent in annual mortality.

The last graphic of Figure 4 shows the variation in the number of steps per
day of the non-linear variables that is equivalent approximately to the
integral of the quantile variable e.g $\int_{0}^{1}Q(t)dt$ (TAC variable). We
show that only for the effect of the age in terms of conditional mean
regression functions can be a variation of $2000$ steps between young and
elderly patients than corresponding changes in annual mortality around 8
percent. Regarding body mass, an increase of $20kg/m^{2}$ percent in the BMI
can induce causal variations of more than $1000$ steps and consequently,
increases the human mortality around $4$ percent.

##  5 Discussion

The core contribution of this paper is to propose a new Partially Linear
Fréchet Single Index regression model to analyze responses of distributional
functional nature. Importantly, the new methods have been implemented to
analyze the physical activity data from NHANES 2011-2014 that follow a complex
survey design. We incorporate survey weights within the new Partially Linear
Fréchet single index algorithm to handle the weighted squared loss function
according to the sampling mechanism induced by the estimator Horvitz-Thompson
[24].

The applications of the new methods in the NHANES dataset in individuals
between 20-80 years old drive new findings in the physical activity literature
that we summarized below:

  1. 1.

We examine the discrepancies between different ethnicities in the American
population’s physical activity levels across the sexes, and the continuous
variable HEI, BMI, and age, in all ranges of human physical activity
intensities thanks to the use of new physical activity quantile distributional
representations. For example, we show that diet is only important in the
physical activity levels in the hight-intensity range; a more positive diet
according to HEI score is related to more exercise. We also show that the
Mexican group is the most active individuals in the American population, and
we discover a non-linear interaction between age and BMI in the energetic
expenditure, specifically in higher quantiles probabilities.

  2. 2.

We characterize individuals that are more active and inactive than we expected
according to the results drawn by the regression model in cluster analysis.
These results can have significant implications from the public health
perspective because they can help program-specific interventions to combat the
inactivity in different subpopulations in U.S. populations.

  3. 3.

We show the modeling advantages of the new Fréchet Single Index algorithm in
predicting the distributional representations compared to the classical global
Fréchet regression model in terms of adjusted Fréchet R-squared and in terms
of interpretatability with the new tools introduced, e.g., the gradient of a
conditional mean function, in which we prove that the non-linearly effect is
in middle age individuals with low BMI.

From a methodological point of view, we propose the first Partially Linear
Fréchet Single Index regression model in the context of responses in metric
spaces to overcome some limitations of the lack of flexibility of the global
Fréchel model. At the same time, we preserve the interpretability advantages
of linear regression along with a subset of predictors.

To the best of our knowledge, this is also the first regression model to
incorporate survey data in the context of responses in metrics spaces and also
the first work that we consider the estimation of the gradient of the
conditional mean function in order to interpret the local contribution of each
predictor in each order of the Quantile functional response [16].

The finite-dimensional compositional metrics are the most popular approach to
analyzing accelerometer data. Here we consider using their functional
extension [32] to capture more information about physical activity from an
individual. By adopting the mathematical framework of statistical analysis in
metric spaces with the $L^{2}$-Wassertein or another Wasserstein metric to
handle the functional compositional representations, we overcome the problems
of zero with compositional data. In addition, the range of values measured by
the accelerometer varies widely between individuals and groups, which can
present difficulties when trying to apply the standard distributional data
analysis methods in our setting [32], for example, functional compositional
transformations [51, 38, 18] that can be an alternative strategy to create a
regression model about physical activity in a linear space using for such
purpose the vast literate of unconstrained functional response regression
models. From a methodological point of view, the distributional physical
activity representation arises from a mixed-stochastic process (see Figure 1
for more details) that prevents to use the of linear functional data methods
based on considering a basis of functions due to the discontinuity of the
quantile function in the transition of the inactivity to activity in the
physical exercise.

The analysis of complex statistical objects in biomedical science provides an
excellent opportunity to create new clinical biomarkers that enrich the
information more than existing variables that monitor the health and evolution
of diseases. Distributional representations are a significant advancement in
digital medicine [19] as a digital biomarker [58, 33]. However, the generality
of techniques introduced, provides users the opportunity to use the methods
developed here with other complex statistical objects such as connectivity
graphs, shape, and directional objects that can introduce new clinical
findings in a broad list of clinical situations for example in the brain and
phylogenetic tree analysis [43, 59, 35, 8, 8, 57]. Furthermore, with the
increasing analysis of large cohorts with richer designs such as complex
survey design, the methods provided here will gain more popularity among
practitioners, and the use of complex statistical objects will undoubtedly be
a daily statistical practice in biomedical applications.

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