Special Article - Biostatistics Theory and Methods

Austin Biom and Biostat. 2015;2(2): 1019.

# Multivariate Granger Causality Analysis of Obesity Related Variables

Nitai D Mukhopadhyay*, David Wheeler, Roy Sabo and Shumei S Sun

Department of Biostatistics, Virginia Commonwealth University, USA

***Corresponding author: ** Nitai D Mukhopadhyay,Department of Biostatistics, Virginia Commonwealth University, Virginia, USA.

**Received: **June 01, 2015; **Accepted: **June 11, 2015; **Published: **June 19, 2015

## Abstract

Obesity is a complex health outcome that is a combination of multiple health indicators. Here we attempt to explore the dependence network among multiple aspects of obesity. Two longitudinal cohort studies across multiple decades have been used. The concept of causality is defined similar to Granger causality among multiple time series, however, modified to accommodate multivariate time series as the nodes of the network. Our analysis reveals relatively central position of physical measurements and blood chemistry measures in the overall network across both genders. Also there are some patterns specific to only male or female population. The geometry of the causality network is expected to help in our strategy to control the increasing trend of obesity rate.

**Keywords:** Obesity; Granger causality; Network; Canonical correlation

## Introduction

In the US overweight or obesity affects two of three adults and one in three of their children. The authors of the recently published IOM Report on Obesity Prevention (2012) lamented that the epidemic is a “startling setback to major improvements in child health attained in the past century.”

Obesity impairs the metabolic and cardiovascular health of both adults and children and threatens to shorten the life span of the current generation of children [1]. The secular increases we have witnessed in the prevalence of childhood obesity presage an increase in the prevalence of T2DM as early as the second decade of life [2- 4]. The origins of obesity include individual genetic, neurohumoral, and physiological factors as well as familial, social, economic, environmental and policy decisions that influence children’s diet and physical activity.

Obesity is a complex phenotype that is captured through multiple surrogate measurements. Though focus on individual phenotypes of physical nature (such as BMI) can over-simplify this complexity, all of the body function measures that are seemingly inter-related with obesity are highly correlated among themselves. In this manuscript we intend to estimate the dependence pattern among many of the phenotypes and phenotypic groups in the context of obesity.

Aside from the presence or absence of any association, it is also important to understand the interplay between the various measures of body dysfunction and their downstream implications. Thus, among observed associations we will also focus on determining the direction of any causal relationships amongst the measures. As methods to infer causality are often labeled controversial, we instead intend to focus away from the methodological arguments and focus on the issue of emergence of obesity. We borrow the concept of Granger causality from econometrics [5] and adapt it to our context.

In the context of gene interaction, causality inference can be applied to decipher interactions within a network of hundreds of genes [6]. The size of the causality network for childhood obesity is smaller than gene networks, but remains large enough to derive causal inferences. Because the origins of childhood obesity have been widely studied, we have a reasonable understanding of the pathophysiology of childhood obesity that should provide our developed causality network with biological plausibility.

Studies of gene interactions use the correlation coefficient and its variations as a measure of interaction [7]. Used partial correlations, empirical Bayes methodology, and bootstrap methods to derive gene networks [8]. Used correlation as a primary tool for constructing networks and pathways among genes and for analyzing gene clusters. Correlation is an effective tool for computing direction-free linear dependence when a sample of independent data is available. In this proposal, we analyze longitudinal data as opposed to cross-sectional data. The time dependent autocorrelated measurements that characterize longitudinal data can be studied by time-lagged associations for the purposes of establishing causality. Graphical interaction models based on such analyses have been developed by [9] and applied to biological time series by [10-12]. Reported a detailed comparative study of techniques for directed interactions in multivariate time series. Based on these studies, we developed a longitudinal Granger causality network to establish causal relationships among genes [6]. The present paper uses the longitudinal Granger causality network method to analyze multivariate longitudinal data to study causal relations among factors associated with childhood obesity. In the context of childhood obesity, an inferred causality network that includes relevant biological variables could be used to identify those variables that would be most susceptible to interventions to prevent or delay the onset of childhood obesity.

## Materials & Methods

Our primary data for obesity inference is the National Heart, Lung and Blood Institute (NHLBI) Growth and Health Study (NGHS), which includes detailed growth profiles of 2380 girls (1,213 African-American and 1,166 Caucasian girls) between the ages of 9 and 21 years. Visits were scheduled annually during the 10-year enrollment, and at each visit measurements of Body Mass Index (BMI), waist circumference, skin fold thickness, blood pressure, blood chemistry, eating habits, and socioeconomic data were taken on each subject. The study population was 40% Caucasian and 51% African Americans. As the NGHS cohort is exclusively female, we also used data from the Fels Longitudinal Study (FLS), grouped into gender categories as our parallel study data for boys. The FLS started annual enrollment of 20-30 infants in 1929, and continues enrollment to follow the participants up to the present time. Like the NGHS, FLS participants provided measurements on body composition, blood pressure, blood chemistry, sexual maturity, cardio-vascular health, etc. over the life span. Visits are scheduled five times during the first year after birth, twice a year after that until age 18, and once every two years in adulthood. As we focus our analysis only on post-pubertal visits, our primary dataset excluded the visits in Tanner stages I and II of sexual maturity. Table 1 shows the mean and standard deviation of the baseline measurements of the demographic variables and the blood chemistry measures where baseline is the first visit after pubertal period.

**Table 1:**Baseline demographics and blood chemistry measures of the NGHS, and FLS boys and girls cohort after the end of pubertal period.

Table 1:Baseline demographics and blood chemistry measures of the NGHS, and FLS boys and girls cohort after the end of pubertal period.

Our approach to causal inference among the longitudinal
phenotypes captured in NGHS data is built upon the framework
of Granger causality inference. Based on its original formulation
[5], Granger causality is defined between pairs of time series data.
Suppose we have the following two kit order autoregressive [AR (k)]
time series model (*P _{1},P_{2t})*

$\begin{array}{l}{\text{P}}_{1\text{t}}={\alpha}_{11}{\text{P}}_{1\left(\text{t-1}\right)}+\cdot \cdot \cdot +{\alpha}_{\text{1k}}{\text{P}}_{\text{1(t-}k\text{)}}+{\beta}_{11}{\text{P}}_{\text{2(t-1)}}+\cdot \cdot \cdot +{\beta}_{\text{1k}}{\text{P}}_{\text{2(t-}k\text{)}}+{\epsilon}_{1}\\ {\text{P}}_{2\text{t}}={\alpha}_{21}{\text{P}}_{2\left(\text{t-1}\right)}+\cdot \cdot \cdot +{\alpha}_{\text{2k}}{\text{P}}_{\text{2(t-}k\text{)}}+{\beta}_{21}{\text{P}}_{\text{1(t-1)}}+\cdot \cdot \cdot +{\beta}_{\text{2k}}{\text{P}}_{\text{1(t-}k\text{)}}+{\epsilon}_{2}\end{array}$

where *P _{it}* is defined as the

*i*time series observed at time

^{th}*t, β*is defined as the linear dependence coefficient of series

_{ij}*P*on

_{i}*j*past observation of the other series, aij is the linear dependence coefficient of series

^{th}*P*on jth past observation of the same series, and Σ1 and Σ2 are error terms. If the hypothesis

_{i}*H*is rejected at a specified level of significance, we say the phenotype 2 (P

_{1}:β11=β12=…=β_{1k}=0_{2}) is Granger causing phenotype 1(P

_{1}). And if the hypothesis H

_{2}:β

_{21}=β

_{22}=…=β

_{2k}=0 is rejected at the specified level of significance, we say the phenotype 1 is Granger causing phenotype 2. If both hypotheses are rejected, we conclude that both series are Granger causing each other. These β coefficients objectively measure the influence of early values of one time series on the future values of the other time series. Rejecting the null amounts to accepting significant influence of one time series on the other in a time lagged manner, i.e. early values of the one series significantly influence the future of the other. Typically, the test can be performed for multiple values of k, but in our case, since the visits are one year apart, we perform it only for k = 1. In a previous application of Granger causality [6] in the context of gene networks, we selected only one direction of causality: accepting the direction with lower p-value. We also removed loops (circular paths) in the causality network by modeling the network as a weighted graph and reducing it to a minimal spanning tree, i.e., the sub network with minimal removal of edges necessary to make it free of loops. Such measures are ways to simplify the web of connections within the network so that only the most significant parts of the network remain. Pair wise causal inference has an additional disadvantage of multiple testing problems. All of the tests (involving repeated measurements) are likely to be inter-dependent, and modeling that dependence is challenging, with common corrective measures possibly being inadequate. Indeed, when applied to the NGHS data, application of this algorithm produces a dense network, which, even after multiple testing corrections, fails to show an interpretable or useful network.

A more acceptable route to make this joint inference would be to conduct multivariate Granger causality. The basic definition, aligned with the intuition leading to the bivariate causality, is as follows.

Suppose we have multiple time series given by ${Y}_{t}^{i}$
for *i=1,…, N*
and *t=1, … , T.* The series ${Y}_{t}^{i}$ is said to Granger cause the series ${Y}_{t}^{i}$ if

$MSE\left({Y}_{t}^{j}|{Y}_{t-h}^{(\cdot )}\right)<MSE\left({Y}_{t}^{j}|{Y}_{t-h}^{(\cdot )}\backslash {Y}_{s}^{i},s<t\right)$

For at least one *h=1,2,3, …* . We will consider the definition for

only *h=1*. Here *MSE(X|Y)* is the mean squared error for predicting
X based on a linear combination of Y,${Y}_{t}^{(\cdot )}$
is the multivariate time
series including${Y}_{t}^{i}\text{and}{Y}_{t}^{j}$
as components, and ${Y}_{t}^{(\cdot )}\backslash {Y}_{s}^{i}$
means all the
random variables in ${Y}_{t}^{(\cdot )}$except ${Y}_{t}^{i}\text{and}{Y}_{t}^{j}$ Note that this is not a statistical
hypothesis, as it is not stated in terms of fixed parameters. To
obtain a testable hypothesis, [13] restated this framework in terms
of the canonical correlation between the time series. The series ${Y}_{t}^{i}\text{and}{Y}_{t}^{j}$
is Granger non-causal for ${Y}_{t}^{j}$
if the canonical correlation denoted
$CCA\left({Y}_{t}^{i},{Y}_{t-1}^{j}|{Y}_{t}^{(\cdot )}\backslash {Y}_{t-1}^{j}\right)=\rho =0.$Both bootstrap- and likelihood-based
method can be implemented to test this hypothesis. Note that
${Y}_{t}^{i}\text{and}{Y}_{t}^{j}$
can be groups of time series, as canonical correlation is
well defined for groups of variables. For the current purpose such
group Granger correlation would render causal inference on groups
of variables, such as all the blood pressure variables on one group,
lipid profile defined by multiple blood chemistry measures, etc. We
have defined four such groups of homogeneous variables indicating
similar aspects of health (Table 2). Groups are conceived based on
the aspect of health measured by the available variables on the study.
BMI, skin folds thicknesses, waist circumference, all are physical
obesity phenotypes. Blood pressure is simply the direct measures of
hypertension. Blood chemistry is basically the lipid profile of subjects
restricted to the available measurements. Food intake measures the
eating habits of the subjects. We used a bootstrap-based approach
to simulate the distribution of the canonical correlation. A few
modifications of the straightforward application of Granger causality
were needed to further refine the outcome of the analysis. While
hypothesis testing can yield a significant p-value, thus indicating
canonical correlation different from 0, the actual value of the canonical
correlation is often very small. To understand the dependence among
these groups of health indicators, we rely on the final network derived
from the Granger causality tests and the canonical correlation values.
To bring the canonical correlation value in the process of building
the dependence network, we use values above the threshold of 0.44,
which represents the 3rd quartile of all the canonical correlations
computed here. When both directions in Granger causality tests
are significant, only the direction with higher canonical correlation
in retained. In analyzing FLS data, Granger causality testing rarely
achieved the desired significance level, leaving the network with no
edge. So we presented the network built solely from the canonical
correlations above 0.44, which again represents the 3rd quartile of all
the canonical correlations.

**Table 2:**Group definitions in NGHS and FLS cohort.

Groups

Variables in NGHS cohort

PhysicalBMI(kg/m

^{2}), Max below waist circ.(cm), Min waist circ.(cm) , Sum of skinfolds(mm), SF/(subscap ST + supraliac SF)

Blood Pressure

Blood Chemistry

Intake

Table 2:Group definitions in NGHS and FLS cohort.

In order to perform well, the causal model requires information about a large set of variables over an extended period of time. NGHS data are relatively well planned and collected in over a fixed duration of the age of the participants Table 3. The variables included are generally not missing; therefore the model is more powerful with relatively more subjects. The FLS cohort was populated over eight decades, so the subjects are spread out over a wide age range. The battery of measurements in the FLS population evolved by adding and replacing old technology with newer methods, so some variables were collected only during specific calendar years, leading to apparent missing values. Thus, not all the participants had all the information collected at different points of time. As a result the analysis of male of female subgroups, restricted to their pubertal stage often lacked the number of subjects needed to estimate the model parameters. In both cohorts, data during the pre-pubertal stage were sparse and the model was not estimable in either cohort. We defined the post-pubertal cohort to be beyond Tanner stage II. However, the female subgroup of FLS data, after all the variables of the model are included, is thin in terms of non-missing cases. As a result the network is sparse. We present the canonical correlation network in that case.

**Table 3:**Cannonical correlations of the groups among NGHS post-pubertal cohort with the response group in rows and lagged groups on columns. Values marked with * are significant at 5% level.

Physical

Blood.Pressure

Blood.chem

Intake

AGE

PhysicalNA

0.2745*

0.3014*

0.1500*

0.3222*

Blood.Pressure0.2886*

NA

0.0363

0.0524

0.1439*

Blood.chem0.2939*

0.1093

NA

0.1498*

0.0464

Intake0.1558*

0.0996*

0.1775*

NA

0.1005*

Table 3:Cannonical correlations of the groups among NGHS post-pubertal cohort with the response group in rows and lagged groups on columns. Values marked with * are significant at 5% level.

## Results

Table 4 shows the canonical correlations between the groups of variables among the post-pubertal subjects in NGHS cohort. These are all females within the age group 9 to 21 years, and only the observations after their Tanner stage II visit are used. The significant correlations are marked with *. Age is included in all models as a covariate; however, the causality network is drawn without the ‘Age’ node. Here we have a large cohort with complete observations, which may be the cause of the small correlations with significant test results. Consequently, the Granger network presented in Figure 1 is well connected, but most of the corresponding correlations are low. Regardless, most of the causalities depicted in Figure 1 are intuitive: change in intake is causing change in blood pressure and blood chemistry; change in physical parameters is causing change in intake and blood chemistry. Forward causality of blood pressure on physical variables is unexpected, as is the causality between intake and the physical variables. Another notable feature of the network is the lack of causality between blood chemistry and blood pressure, which suggests that the dependence is explained through other nodes.

**Table 4:**Cannonical correlations of the groups of variables in FLS postpubartal boys cohort with the response group in rows and causing groups on columns.’*’ indicates significant p value at 5% level of significance.

Physical

Blood. Pressure

Blood. chem

Intake

AGE

PhysicalNA

0.3251

0.6688*

0.1875

0.2404

Blood. Pressure0.5157

NA

0.5245

0.5864*

0.3936

Blood. chem0.5169

0.6070

NA

0.4766

0.5298

Intake0.1684

0.3248

0.5281

NA

0.0047

Table 4:Cannonical correlations of the groups of variables in FLS postpubartal boys cohort with the response group in rows and causing groups on columns.’*’ indicates significant p value at 5% level of significance.

**Figure 1:**Granger causality network among the NGHS participants during their post-pubertal period. Edge thickness is proportional to the canonical correlation between the two groups.

Figure 1:Granger causality network among the NGHS participants during their post-pubertal period. Edge thickness is proportional to the canonical correlation between the two groups.

Table 4 presents the canonical correlations among the groups defined in Table 1 in the male subgroup of FLS cohort. Here also, Tanner stage II was used to define the pubertal stage. Only the postpubertal data are used to define the subgroup, and Age is included in all models as a covariate. The final model included 115 male and 109 female subjects, with variable length of time of follow-up. Use of correlation requires multiple observations across time, but 12 of the female and 11 of the male subjects had 2 or fewer time points, thus reducing the effective sample size. Due to an insufficient number of subjects, few of the correlations are significant, although the actual values of the correlations are larger than those observed in the NGHS cohort. Both Figures 2 and 3 are generated by this analysis using the p values of the causality test and the value of the canonical correlations, respectively. The significant p-value network is naturally sparse with only two edges. Although the FLS cohort differs from the NGHS cohort, in gender composition and age range, it is interesting to note the similarities and differences between the networks. The causality in Figure 2 between physical variables and blood chemistry in participants in the FLS is also found in the NGHS network, but the causality between intake and blood pressure is reversed in direction. The canonical correlation network of the FLS participants presented in Figure 3 is similar to that in the NGHS network. Three of the causalities, namely blood pressure →physical parameters→ blood chemistry and food intake → blood chemistry, are also repeated in the NGHS network. Causality between physical parameters and intake is missing here, and the causality blood chemistry → blood pressure is the only new edge.

**Figure 2:**Granger causality network among FLS cohort boys using only the p values during their post-pubertal period. Edge thickness is proportional to the canonical correlation between the two groups.

Figure 2:Granger causality network among FLS cohort boys using only the p values during their post-pubertal period. Edge thickness is proportional to the canonical correlation between the two groups.

**Figure 3:**Granger causality network among FLS boys using only cannonical correlations above 0.44. Edge thickness is proportional to the canonical correlation between the two groups.

Figure 3:Granger causality network among FLS boys using only cannonical correlations above 0.44. Edge thickness is proportional to the canonical correlation between the two groups.

Table 5 shows the canonical correlations among the post-pubertal female population in the FLS cohort. Even though the canonical correlations are relatively larger compared to the NGHS cohort, none of the statistical tests reached significance, meaning the p-value induced network has no edges. The canonical correlation induced network, using the values above 0.44 is presented in Figure 4 and includes only one edge from physical parameters→ blood chemistry, which is common to all the networks.

**Table 5:**Cannonical correlations of the groups of variables in FLS postpubartal girls cohort with the response group in rows and causing groups on columns. ‘*’ indicates statistical significance at 5% level.

Physical

Blood. Pressure

Blood. chem

Intake

AGE

PhysicalNA

0.4129

0.4867

0.1334

0.4442

Blood. Pressure0.4229

NA

0.3111

0.2907

0.2894

Blood. chem0.3278

0.2717

NA

0.1632

0.3146

Intake0.1944

0.0732

0.2944

NA

0.0547

Table 5:Cannonical correlations of the groups of variables in FLS postpubartal girls cohort with the response group in rows and causing groups on columns. ‘*’ indicates statistical significance at 5% level.

**Figure 4:**Granger causality network among FLS cohort post-pubertal girls using their cannonical correlations above 0.44.

Figure 4:Granger causality network among FLS cohort post-pubertal girls using their cannonical correlations above 0.44.

## Discussion

Some of our findings align with biological intuition, notably the repeated patterns of food intake causing changes in blood chemistry, and physical parameters preceding change in blood chemistry. The repeated pattern of blood pressure causing physical parameters is a feature of our analysis that remains to be explored further through biological studies. Hypertension and BMI are other physical parameters whose has known association, though a causal direction has not been established.

Only in the subpopulation of post-pubertal boys did we find that blood chemistry affects blood pressure, and only in NGHS female subjects did we find that change in physical parameters precedes change in food intake. These directions can be explained as gender-specific effects. The interaction of blood pressure and food intake occurred in multiple networks with different directions. The biological basis of this interaction is not clear.

Regardless of the direction of interaction, the nodes ‘Physical parameters’ and ‘Blood chemistry’ remain the most connected nodes. Thus, these can be considered to be the central feature of this network. Implications of this connection can be used in strategies to manage overall health.

## Conclusion

Our analysis highlights the difficulty of performing a data based multivariate causality inference in cohort study. The main limitations are the dimension of the model and sparseness of some of the variables. We did not use any imputation for this analysis, which led to many partial observations not being used. FLS data has 2567 subjects in total, but this analysis uses only 115 male and 109 female subjects. NGHS data, were reduced from 2380 subjects to 2250 subjects, however, 112 of them had only 2 or fewer post-pubertal visits. Regardless, there is a trend in causality structure that reinforces our prior understanding, and there are some new features that are worth further exploration.

The bigger goal behind such causality inference is to identify a “source of causality” among all the correlated collective nodes. In a way, irregularity in all the variables considered in our analysis define the complex phenotype ‘obesity’ rather than rather than just being associated. Therefore the process of management of obesity should have an impact on all the nodes of our network. Interdependence among them is likely to better guide us in developing a strategy to manage obesity. The most connected nodes in our analysis have been consistently the ‘physical parameters’ and ‘blood chemistry’. Therefore any health management strategy should focus on these nodes. This does not downplay the need to manage the other nodes. We only emphasize that starting management of physical parameters or blood chemistry will be more likely to produce a cascade effect on the other nodes, than the reverse. A successful health management should have a positive effect on all the variables considered here.

## Acknowledgement

This study was supported by National Institutes of Health grant: U01HL101064, Bethesda, Maryland.

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