Hypothesis Testing in Nonlinear Function on Scalar Regression with Application to Child Growth Study

TL;DR

This paper introduces a kernel machine-based hypothesis testing method for nonlinear function-on-scalar regression, applied to child growth data to assess the impact of toxic metals and methylation regions.

Contribution

It develops a novel variance component test within a kernel machine framework for nonlinear function-on-scalar regression models.

Findings

Method effectively detects associations in simulation studies.

Application to NEST data reveals significant links between exposures and child growth.

Provides a flexible approach for nonlinear functional data analysis.

Abstract

We propose a kernel machine based hypothesis testing procedure in nonlinear function-on-scalar regression model. Our research is motivated by the Newborn Epigenetic Study (NEST) where the question of interest is whether a pre-specified group of toxic metals or methylation at any of 9 differentially methylated regions (DMRs) is associated with child growth. We take the child growth trajectory as the functional response, and model the toxic metal measurements jointly using a nonlinear function. We use a kernel machine approach to model the unknown function and transform the hypothesis of no effect to an appropriate variance component test. We demonstrate our proposed methodology using a simulation study and by applying it to analyze the NEST data.

Figures

Tables

Table 1. Table 1 : Percentage of the underlying variance-covariance structure of Σ Σ \Sigma explained by the first 2 2 2 principal component functions for the individuals with and without segregation by their gender for studying the association of growth curve with the metal content in cord blood as mentioned in the data analysis section.

	All	Female	Male
Number of children	75	38	37
Explained by first principal component function	95.4 %	98.1 %	99.9 %
Explained by second principal component function	3.2 %	1.9 %	0.1 %

Table 2. Table 2 : The p-values (rounded to fourth decimal place) for the test of association between growth curves and amount of metal in a child’s cord blood for either gender as mentioned in the discussion section.

	All Metals	Toxic Metals	Other Metals
Females	0.0163	0.2019	0.0174
Males	0.0282	0.2882	0.0303

Table 3. Table 3 : The p-values (rounded to fourth decimal place) for the test of association between growth curves and amount of metal in a child’s cord blood for both genders combined, and either gender separately for each of the 24 24 24 metals whose values are available to us, as mentioned in the discussion section.

	All	Female	Male
Number of children	75	38	37
Cadmium	0.4472	0.3489	0.4136
Lead	0.7191	0.4567	0.1715
Arsenic	0.7102	0.6798	0.3210
Iron	0.4734	0.0998	0.8637
Zinc	0.2612	0.6560	0.7337
Selenium	0.8298	0.0694	0.7281
Copper	0.9028	0.7353	0.0164
Calcium	0.2104	0.1559	0.3459
Strontium	0.6351	0.0676	0.7209
Barium	0.6884	0.9244	0.9779
Manganese	0.4178	0.8938	0.1069
Cobalt	0.5255	0.0657	0.9518
Molybdenum	0.9491	0.3438	0.9059
Scandium	0.6514	0.3574	0.6255
Titanium	0.4075	0.2335	0.5048
Chromium	0.2799	0.5231	0.5076
Aluminium	0.9435	0.8885	0.3141
Silicon	0.3530	0.0963	0.6416
Vanadium	0.4860	0.1063	0.1957
Rubidium	0.4089	0.8445	0.2812
Tin	0.4601	0.0135	0.2599
Mercury	0.4742	0.4186	0.4358
Nickel	0.5029	0.1059	0.3895
Magnesium	0.7958	0.2419	0.3440

Full text

**Supplementary Materials for

Hypothesis Testing in

Nonlinear Function on Scalar Regression with Application to Child Growth Study**

Mityl Biswas, Arnab Maity

Department of Statistics, North Carolina State University

1 $\mathbf{K}$ as a Kronecker product.

We have utilized the following property in order to make our code more computationally efficient while performing the simulation study.

Let’s consider the $(h,k)^{th}$ element of the $(i,j)^{th}$ block for some $h\epsilon\{1,$ $\ldots,$ $m_{i}\}$, $k\epsilon\{1,$ $\ldots,$ $m_{j}\}$, $i\epsilon\{1,$ $\ldots,$ $n\},$ $j\epsilon\{1,$ $\ldots,$ $n\}.$

$K\{(\mathbf{X}_{i},\mathbf{t_{ih}}),(\mathbf{X}_{j},\mathbf{t_{jk}})\}=L(\mathbf{X_{i}},\mathbf{X_{j}})e^{-|t_{ih}-t_{jk}|},$

$h\epsilon\{1,$ $\ldots,$ $m_{i}\}$, $k\epsilon\{1,$ $\ldots,$ $m_{j}\}$, $i=1,\ldots,n,$ $j=1,\ldots,n$. This is clearly a separable function in $(\mathbf{X}_{i},\mathbf{X}_{j})$, and $(t_{ih},t_{jk})$. Then, we can see that the $(i,j)^{th}$ block is $L(\mathbf{X}_{i},\mathbf{X}_{j})\otimes\mathbf{T_{ij}}$, where $\mathbf{T_{ij}}$ is the $m_{i}\times m_{j}$ matrix with the $(h,k)^{th}$ element as $e^{-|t_{ih}-t_{jk}|}$ for $h=1,\ldots,m_{i},k=1,\ldots,m_{j}$. Now, if we had $m_{i}=m$, with $t_{ih}=t_{h}$ for $h=1,\ldots,m,i=1,\ldots,n$, then, the $(i,j)^{th}$ block would have been $L(\mathbf{X}_{i},\mathbf{X}_{j})\otimes\mathbf{T}$, where $\mathbf{T}$ is the $m\times m$ matrix with the $(h,k)^{th}$ element as $e^{-|t_{h}-t_{k}|}$ for $h=1,\ldots,m,k=1,\ldots,m$. This results in the following corresponding kernel matrix: $\mathbf{K^{*}}=\mathbf{A}\otimes\mathbf{T}$, where $\mathbf{A}$ is the $n\times n$ matrix with the $(i,j)^{th}$ element as $L(\mathbf{X_{i}},\mathbf{X_{j}})$. Now, if we only consider those rows and columns of $K^{*}$ that correspond to the observed data, we would have the required kernel matrix $\mathbf{K}$.

2 Additional Tables and Figures

The first principal component function explained a large percentage of the underlying variance-covariance structure for either gender considered in testing the association of growth curve with metal content. This is evident in table 1. The first eigen vector for male children, and for female children was similar to that when all the children were considered together. This is illustrated in figure 1.

Table 1 here.

Figure 1 here.

The p-values obtained by performing the tests separately for the male and the female children show that they are similar for the two genders, although the values are slightly lower for the females. The statistics are listed in table 2.

Table 2 here.

The p-values obtained by performing the tests separately for each of the $24$ metals have been recorded in table 3. We obtained these numbers solely for the purpose of getting a vague idea of what the p-values would look like, and shall not be using them to draw conclusions as we have reused the dataset that we had used for our initial analyses.

Table 3 here.