Functional interaction-based nonlinear models with application to multi-platform genomics data

Abstract

Functional regression allows for a scalar response to be dependent on a functional predictor, however, not much work has been done when a scalar exposure that interacts with the functional covariate is introduced. In this paper, we present two functional regression models that account for this interaction and propose two novel estimation procedures for the parameters in these models. These estimation methods allow for a noisy and/or sparsely observed functional covariate and are easily extended to generalized exponential family responses. We compute standard errors of our estimators, which allows for further statistical inference and hypothesis testing. We compare the performance of the proposed estimators to each other and to one found in the literature via simulation and demonstrate our methods using a real data example.

1 INTRODUCTION

Functional covariates arise when multiple measurements are taken on a subject and these can be thought of as realizations of some unknown, smooth, underlying process. Functional data has been studied in a wide variety of fields such as environmental health,^1,2,3 entomology,⁴ genetics,⁵ biology and medicine,^6,7,8 geophysics,⁹ finance,^10,11 and more recently genomics.^12,13,14. There are many more applications dealing with functional data.¹⁵

A typical goal of functional regression models involves determining the relationship between a functional covariate and a scalar response of interest, and one of the most straightforward and widely used tools to achieve this goal is the functional linear model.^43,19,3,28 This model is analogous to the generalized linear model for vector valued covariates, only now, the effect of the functional covariate is quantified by the inner product between the covariate and an unknown functional parameter and the goal is to make inference on the functional parameter. Like its generalized linear model counterpart, functional linear models are simple and easy to interpret, but may be too restrictive in capturing the true effects, due to simplistic assumptions that aid in model fitting and computations.

In this paper, a scalar covariate is measured along with the functional predictor. The modeling challenges one faces is that the functional covariate may be noisy and sparsely observed, and that it may interact with the scalar covariate. Typical methods developed so far for functional linear models can account for the noise in the data, but can not capture the interaction effect, which can often arise in real-world applications and might be of interest. Thus, our objective is to accurately model the effects of the the scalar covariate and functional covariate of interest, while accounting for the interaction between the two, and adjusting for extraneous covariates that may have an effect the response. We do this by modeling the main effect of the scalar covariate nonparametrically using a smooth parameter function, and by modeling the main effect of the functional covariate jointly with the interaction effect using another bivariate parameter function. We model the extraneous covariates that do not interact with the predictors in a parametric way. The ultimate goal is to make inference on these model components.

There are a number of models and estimation procedures for fitting functional and scalar data. A flexible model has been presented in the literature²⁰ that we adopt in this paper, however the authors only present the case for Gaussian responses, they do not propose a way to obtain the standard errors of their estimators, and they assume that the entire functional covariate is observed without error, which may not hold in real world applications. A similar model for Gaussian responses has been fit using a different estimation scheme,⁵⁷ and the authors account for noisy or irregularly spaced observations in the functional covariate. However, they do not compute standard errors either, which is needed for statistical inference. Both^20,57 use local estimation procedures that are computationally intensive. Recently, a new estimation has been proposed that is similar to one in this paper,⁴⁹ but the focusus is on functional responses. We present a general model for scalar responses using two estimation procedures that can easily be extended to generalized responses and noisy or sparsely observed functional covariates. The generality of our proposed methodology allows us to efficiently find the standard errors of our estimators and thus, construct confidence bands and surfaces and develop testing procedures. Compared to local estimation methods, the global nature of our estimators makes them computationally inexpensive and have lower empirical variability.

The data that motivate this paper come from the Multiple Myeloma Research Consortium and consist of multi-platform data collected at genomics (DNA copy number) and transcriptomics (mRNA expression) levels on approximately 235 individuals with multiple myeloma (MM). The scientific question of interest is to leverage these correlated multi-platform data to find significant associations with patient-specific clinical outcomes. The specific goal is to identify genes associated with MM and to incorporate copy number variation (CNV) data to improve reliability. Because CNVs can change how genes are expressed and cooperate together with genes in known mutations,⁴¹ interest lies in investigating these two factors jointly. Analyzing CNV alone is best for precisely describing small focused events on a chromosome and may not give much insight when larger regions are of interest, which is common when studying human carcinomas.²¹ Accounting for gene expression alone is problematic because changes in expression resulting from passenger genes in CNV regions add noise to the data and the results from gene association studies alone are not consistent from study to study.^29,40 Recently, it has been shown^21,36,40 that analyzing CNV and gene expression together gives a more complete picture of tumorigenesis for patients with lung cancer and how incorporating CNV can help improve reproducibility of gene association studies.

In this paper, the outcome of interest is a biomarker used for MM staging (scalar) and the primary predictors of interest are gene expression (scalar) and copy number variation (functional). We regress the outcome on the two predictors using either a varying-coefficient functional linear model or a generalized model as published previously²⁰ (with an additional parametric term to fit extraneous covariates). We model CNV as functional covariate due to its high dimensional, serially correlated nature, and we smooth the noisy and possibly sparsely observed data using functional principal component analysis (FPCA).⁴⁷ We propose two global estimation methods using the eigenfunction basis from FPCA or the tensor product of B-spline bases and compute the standard errors of our estimators. We compare the linear and general models and the estimation methods in simulation and find that the tensor product basis is best suited for real-world applications when the true relationship between the predictors and response is unknown. We applied our estimation methods to the MM data and found them to yield similar results.

The rest of this paper is organized as follows: in Section 2 we present the varying coefficient model and propose two schemes to estimate the parameters of the model. We show how to extend the methodology in Section 3. We conduct simulation studies in Section 4, apply our methods to the MM data in Section 5, and give concluding remarks in Section 6.

2 GENERAL FUNCTIONAL MODEL FOR INTERACTING COVARIATES

Assume that for each of n subjects we have a scalar response Y, confounding variables V, a noisy functional covariate W, and another scalar covariate Z that can interact with W. For ease of presentation, we assume that the response, Y_i comes from a Gaussian distribution conditional on the covariates. Extension to the generalized case where Y_i comes from an exponential family is straightforward and is described in Section 3. The confounding variables, such as age and sex, are stored in a q × 1 vector V_i. The functional covariate is observed on a dense grid of points ${\left\{t_m\right\}}_{m=1}^M$ within a closed grid for each individual and these measurements can be thought of as discrete realizations of an unknown smooth curve, X_i(t). In practice, the functional covariate is measured with error, and thus what is actually observed is W_i(t) = X_i(t) + δ_i(t), where δ(t) is a white noise process with mean 0 and variance ${\sigma }_{\delta }^2$. The grid ${\left\{t_m\right\}}_{m=1}^M$ is assumed to be the same for each subject but there can be missing values.

Finally, there is a scalar covariate of interest Z_i that can possibly interact with the functional covariate X_i(t). The goal is to study the effects of Z and X(t) on the response Y while accounting for the possible interaction between these two covariates and adjusting for the effect of the other covariates measured.

Consider the following model:

$$Y_i={\mathbf{V}}_i^T\mathit{\gamma }+{\beta }_0(Z_i)+{\displaystyle \underset{t\in \tau }{\int }{X}_i(t)\beta (Z_i,t)dt+{\epsilon }_i,}$$ (1)

where we assume ε_i ~ N(0, σ²). The confounding variables are modeled parametrically with corresponding parameter vector γ, and the main effect of the interacting covariate Z is modeled nonparametrically by an unknown smooth function β₀(·). In (1), both the main effect of X and the interaction between Z and X are captured in β(z, t), a smooth, unknown parameter function. The goal is to estimate γ, β₀(z) and β (z, t). We present two ways of fitting this model below.

2.1 Estimation by the Spline and Empirical Bases

Our first approach is to use a univariate basis expansion for β (z, t). We first approximate X_i(t) using FPCA and obtain $\{{\widehat{\varphi }}_1(t),\dots ,{\widehat{\varphi }}_P(t)\}$, the eigenfunction basis. We then write the expansion of β as

$$\beta (z,t)=b_1(z){\widehat{\varphi }}_1(t)+b_2(z){\widehat{\varphi }}_2(t)+\cdots +b_P(z){\widehat{\varphi }}_P(t),$$ (2)

where ${\{b_p(z)\}}_{p=1}^P$ are coefficient functions of z. We are making the crucial assumption that X(t) and β(z, t) have the same degree of smoothness in the t direction, which can be restrictive, but the benefits of this estimation procedure are that it simplifies model fitting, there is software readily available for fitting VCMs using spline-based methods (see e.g. gam in the mgcv package), and as an alternative to spline fitting, one could estimate using local-polynomial kernel smoothing.^42,27

Because the set of eigenfunctions form an orthonormal basis, substituting (2) into the general model in (1) yields a standard varying coefficient model

$$Y_i={\mathbf{V}}_i^T\mathit{\gamma }+{\beta }_0(Z_i)+{\widehat{\xi }}_{i1}{b}_1(Z_i)+{\widehat{\xi }}_{i2}{b}_2(Z_i)+\cdots +{\widehat{\xi }}_{iP}{b}_P(Z_i)+{\epsilon }_i,$$

where the hats indicate that the scores, ξs, are estimated using FPCA (we found in simulation and in practice that this added variability is negligible). We fit the model by using B-spline basis approximations for ${\beta }_0(z)={\displaystyle {\sum }_{j=1}^J{\eta }_{0j}{\psi }_{0j}(z)}$ and $b_p(z)={\displaystyle {\sum }_{k=1}^{K_p}{\eta }_{pk}{\psi }_{pk}(z)}$ for p = 1, …, P (note that B-splines can be replaced with any choice of basis). When making these substitutions, after some manipulations, the mean model of the response variable has a linear form with (q + J + Σ_p K_p) × 1 parameter vector $\mathit{\eta }={({\mathit{\gamma }}^T,{\mathit{\eta }}_0^T,{\mathit{\eta }}_b^T)}^T$, where η₀ = {η₀₁, …, η_0J } and ${\mathit{\eta }}_b={({\mathit{\eta }}_1^T,\dots ,{\mathit{\eta }}_P^T)}^T$ is a Σ_p K_p × 1 vector where ${\mathit{\eta }}_p={({\eta }_{p1},\dots ,{\eta }_{p{K}_p})}^T$. Thus η_b is the parameter vector corresponding to the b_p functions.

Fitting standard linear model techniques will likely result in parameter function estimates that are undersmooth, thus we control this by adding a penalty to the objective. The integrated squared second derivative is a common way to penalize estimates that are too rough⁴⁷ and so we adopt it in our estimation. The penalized negative log-likelihood can be minimized using least squares and the smoothing parameters estimated by minimizing the REML criterion⁵⁶ to obtain the estimate of η and reconstruct β (z, t) as

$$\widehat{\beta }(z,t)={\displaystyle \sum _{k=1}^{K_1}{\widehat{\eta }}_{1k}{\psi }_{1k}(z){\widehat{\varphi }}_1(t)+}{\displaystyle \sum _{k=1}^{K_2}{\widehat{\eta }}_{2k}{\psi }_{2k}(z){\widehat{\varphi }}_2(t)+\cdots +{\displaystyle \sum _{k=1}^{K_P}{\widehat{\eta }}_{Pk}{\psi }_{Pk}(z){\widehat{\varphi }}_P(t).}}$$

We compute the Bayesian posterior covariance matrix $\mathrm{var}(\widehat{\mathit{\eta }})$ to estimate the standard errors.⁵⁵ We chose to use the Bayesian approach over the frequentist counterpart because the distributional results can be used directly. We then construct the pointwise standard errors for ${\widehat{\beta }}_0(·)$ and ${\widehat{b}}_p(·)$ as

$$\begin{gathered} \text{SE}\{{\widehat{\beta }}_0(z)\}={\left[{\{{\mathit{\psi }}_0(z)\}}^T\mathrm{var}({\widehat{\mathit{\eta }}}_0){\mathit{\psi }}_0(z)\right]}^{1/2} \\ \text{SE}\{{\widehat{b}}_p(z)\}={\left[{\{{\mathit{\psi }}_p(z)\}}^T\mathrm{var}({\widehat{\mathit{\eta }}}_p){\mathit{\psi }}_p(z)\right]}^{1/2} \end{gathered}$$

where ${\mathit{\psi }}_p(·)={\{{\psi }_{p1}(·),\dots ,{\psi }_{p{K}_p}(·)\}}^T$ for p = 1,…,P. Define the matrix

$${\mathbf{M}}^T={\{\mathit{\varphi }(t)\}}^T\text{diag}[{\{{\mathit{\psi }}_1(z)\}}^T,\dots ,{\{{\mathit{\psi }}_P(z)\}}^T],$$

where ϕ(t) = {ϕ₁(t), ϕ₂(t),…, ϕ_P (t)}^T. Then $\widehat{\beta }(z,t)={\mathbf{M}}^T{\widehat{\mathit{\eta }}}_b$ and the pointwise standard error is $\text{SE}\{\widehat{\beta }(z,t)\}={\mathbf{M}}^T\mathrm{var}({\widehat{\mathit{\eta }}}_b)\mathbf{M}$.

Note that the functional linear varying coefficient model is a special case of this framework by setting β(z, t) = β₁(t) + zβ₂(t) and following similar estimation procedures.

2.2 Estimation using Two Spline Bases

In practice, we do not know if the functional covariate X(t) and the parameter function β(z, t) have the same smoothness, so an alternative to the empirical basis is to use a tensor product basis to model β(z, t). Again, we start with FPCA to obtain the estimate ${\widehat{X}}_i(t)$ and then penalize the negative log-likelihood, but for the bivariate function, we smooth in both the z and t directions and use two penalty parameters to allow each direction to have a different amount of smoothness. We use the basis representation ${\beta }_0(z)={\displaystyle {\sum }_{j=1}^J{\eta }_{0j}{\psi }_{0j}(z)}$ as before, and for β (z, t), we use a tensor product between two B-splines,

$$\beta (z,t)={\displaystyle \sum _{k=1}^K{\displaystyle \sum _{l=1}^L{\psi }_{1k}}(z){\psi }_{2l}}(t){\eta }_{kl}.$$ (3)

Here we use two different sets of B-splines for z and t with cross products indexed by an unknown parameter η_kl. We can once again write (1) as a linear model with (q + J + KL) × 1 parameter vector, where η_p above is replaced by η_KL, a KL × 1 vector that is grouped along the K index. That is, ${\mathit{\eta }}_{\mathrm{KL}}={({\mathit{\eta }}_1^T,\dots ,{\mathit{\eta }}_K^T)}^T$ where η_k= (η_k1,…,η_kL)^T for k = 1,…, K.⁵⁵ We minimize the penalized negative log-likelihood and estimate the smoothing parameters using REML as before, then reconstruct β (z, t) as

$$\widehat{\beta }(z,t)={\displaystyle \sum _{k=1}^K{\displaystyle \sum _{l=1}^L{\psi }_{1k}}(z){\psi }_{2l}}(t){\widehat{\eta }}_{kl}.$$

The Bayesian posterior covariance matrix $\mathrm{var}(\widehat{\mathit{\eta }})$ is used to compute the standard errors and the pointwise standard error of $\widehat{\beta }(z,t)$ is

$$\text{SE}\{\widehat{\beta }(z,t)\}={\left[{\{{\mathit{\psi }}_1(z)\}}^T\otimes {\{{\mathit{\psi }}_2(t)\}}^T\right]}^T\mathrm{var}({\widehat{\mathit{\eta }}}_{\text{KL}})\left[{\{{\mathit{\psi }}_1(z)\}}^T\otimes {\{{\mathit{\psi }}_2(t)\}}^T\right],$$

where $\mathrm{var}({\widehat{\mathit{\eta }}}_{\text{KL}})$ is the appropriate sub-matrix of the covariance of η.

We denote the empirical basis formulation and the tensor product basis formulation as EB and TP, respectively, for the remainder of the paper.

3 EXPONENTIAL FAMILY RESPONSES AND SPARSE/NOISY OBSERVATIONS

For ease of presentation, we assumed that the response was Gaussian, but this does not have to be the case for our estimation methods to work. If the response’s conditional distribution belongs to a single-parameter exponential family, the general model in (1) can be written as

$$g({\mu }_i)={\mathbf{V}}_i^T\mathit{\gamma }+{\beta }_0(Z_i)+{\displaystyle \underset{t\in \mathcal{T}}{\int }{X}_i(t)\beta (Z_i,t)dt}$$ (4)

where μ_i = E(Y_i|V_i, Z_i, X_i) is the conditional mean of the response Y_i and g(·) is a link function from the generalized linear model framework.⁴⁴ The standard linear models with parameter η become generalized linear models with the left hand side replaced by g(μ_i) and the error terms omitted. The respective penalized likelihoods are minimized using penalized iterative reweighted least squares.⁵⁵ and the smoothing parameters are estimated by minimizing the REML as before.

If sparse or noisy realizations of the functional covariate are observed, we can still apply FPCA as long as the entire domain of t is dense across all subjects.^47,24 This is because our estimation methods are global, unlike local techniques where FPCA can potentially break down.

4 SIMULATION STUDY

We conducted simulation studies to evaluate our proposed methods. We generated three parametric covariates V_i = (V_i1; V_i2; V_i3)^T for each subject from a standard normal and let γ= (1; 0:5; −1)^T. We generated the interacting covariate Z_i uniformly over [−1; 1].We generated the functional covariates X_i(t) using four Fourier basis functions where the coefficients were normally distributed with mean 0 and variances (8; 4; 2; 1). The random error for the continuous response ε_i was generated from a standard normal distribution. We defined β₀(z) = (z − 1/2)² + 1 and the form of β(z; t) was either from the linear model (L), the empirical basis (EB) model, or the tensor product (TP) model. For the linear case, β₁(t) = 1, β₂(t) = 2 sin(πt), and β(z; t) = β₁(t)+z β₂(t). For the EB case in (2), b₁(z) = z+1, b₂(z) = z² and β(z; t) = b₁(z) sin(πt)+b₂(z) cos(πt). For the TP case in (3), $\beta (z,t)=e^{{(z-0.2)}^2}$ {1+sin(πt)}/4. For relativity purposes, the scaled norms were calculated, where ${\Vert {\beta }_0(z)\Vert }_2/2=2.93$, and ${\Vert \beta (z,t)\Vert }_2/2$ were 1.67, 0.77, and 0.52 for the linear, EB, and TP surfaces, respectively. Finally, the response Y_i was generated from either (1) or (4) using a high dimensional grid for t.

For our analysis, we used a noisy functional covariate W_i(t) = X_i(t) + δ_i(t), where δ_i(t) was a white noise process with mean 0 and variance 1/4 and W_i(t) was observed either on a dense grid of 41 equally spaced timepoints or on a sparse grid of 101 timepoints in [0,1]. For the sparse grid, we randomly selected a number m_i between 16 and 30 for each subject, and then randomly choose m_i of the 101 grid points to be the observed values, setting the rest to missing values. We fit the linear model, EB model, and TP model for all three choices of β(z, t) and chose 9 basis functions to keep the tensor product low and for consistency. We expected optimal performance when the true β(z, t) was the same as the fitted model.

We estimated the functions on an equally spaced grid for z ∈ [−1, 1] and t ∈ [0, 1]. We evaluate our estimates for each function based on the integrated squared bias, mean integrated squared error, empirical variance, estimated variance, and integrated coverage. We define ${\widehat{\beta }}_0(z)={\mathit{R}}^{-1}{\displaystyle {\sum }_r{\widehat{\beta }}_0^{(r)}(z)}$ and $\text{SB}=(1/2){\displaystyle \int {\{{\widehat{\beta }}_0(z)-{\beta }_0(z)\}}^{2}dz}$, $\text{MSE}={\mathit{R}}^{-1}{\displaystyle {\sum }_r\left[(1/2){\displaystyle \int {\{{\widehat{\beta }}_0^{(r)}(z)-{\beta }_0(z)\}}^{2}dz}\right]}$, $\text{Vem}=(1/2){\displaystyle \int {\mathit{R}}^{-1}{\displaystyle {\sum }_r{\{{\widehat{\beta }}_0^{(r)}(z)-{\widehat{\beta }}_0(z)\}}^{2}dz}}$, $\text{Ves}=(1/2){\displaystyle \int {\mathit{R}}^{-1}{\displaystyle {\sum }_r{[\text{SE}\{{\widehat{\beta }}_0^{(r)}(Z)\}]}^{2}dz}}$, and $\mathrm{C}=(1/2){\displaystyle \int {\mathit{R}}^{-1}{\displaystyle {\sum }_{r}I[{\widehat{\beta }}_0(z)\in \text{CI}\{{\widehat{\beta }}_0^{(r)}(z)\}}}]dz$, where R is the number of simulation replications and CI(x) = (x − 1.96 SE(x), x + 1.96 SE(x)) is the pointwise 95% confidence interval. All integrals are scaled by the the volume of Z. These definitions are similar for β (z, t) except that we integrate over both z and t and scale by the respective volumes.

We present the results for the continuous response here, and those for the binary response in the appendix. A comparison to the local spline method²⁰ is also given in the appendix.

4.1 Results for Continuous Response

The results for the continuous response are presented in Table 1. For our estimate of ${\widehat{\beta }}_0(z)$, we did not see much difference in bias across the three different fits for each surface and as sample size increased, the bias decreased. When fitting the linear model, the variability of our estimate for ${\widehat{\beta }}_0(z)$ increased as the structure of $\widehat{\beta }(z,t)$ moved from linear to EB to TP. For the other two fits there was no real change in variability across the three surface types, and both fits yielded similar estimated and empirical variance. Thus overall, the EB and TP fits had very similar biases and MISEs. In comparison, the linear fit had lower MISE when the linear surface was used, similar MISE when the EB surface was used, and higher MISE when the TP surface was used. As sample size increased, the integrated coverage for all three fits was similar and close to the nominal 95% value. We saw similar trends for ${\widehat{\beta }}_0(z)$ when the sparse grid of timepoints was used.

TABLE 1.

Gaussian response: Integrated squared bias (SB), mean integrated squared error (MSE), integrated monte carlo empirical variance (Vem), integrated estimated variance (Ves), and integrated coverage (C) for the two functions b₀(z) and β(z, t) when the functional covariate is densely observed and sparsely observed. We fit a linear model (L), an empirical basis model (EB), and a tensor product basis model (TP) to each of the three true surfaces for sample sizes 100, 200, and 400. We also estimated the residual error for the dense (D) and sparse (S) case, which should be close to 1. All values in the table are multiplied by 100. For relativity, the scaled norm for β₀(z) is 2.93, and the scaled norms for β(z, t) are 1.67, 0.77, and 0.52 for the linear, EB, and TP surfaces, respectively.

LINEAR SURFACE
		β0(z) Dense					β(z, t) Dense					β0(z) Sparse					β(z, t) Sparse					Residual
		SB	MSE	Vem	Ves	C	SB	MSE	Vem	Ves	C	SB	MSE	Vem	Ves	C	SB	MSE	Vem	Ves	C	D	S
	100	0.4	4.8	4.4	4.1	93.4	1.1	10.9	9.8	8.6	89.7	0.4	5.3	4.9	4.3	92.4	0.8	15.6	14.8	13.1	88.6	87.8	91.6
L	200	0.1	2.2	2.1	2.2	95.5	0.4	5.5	5.1	5.6	93.4	0.1	2.3	2.2	2.3	95.4	0.2	6.5	6.3	6.6	93.3	94.2	96.5
	400	0.1	1.2	1.1	1.2	96.1	0.1	2.6	2.5	4.0	97.3	0.1	1.2	1.1	1.2	96.1	0.1	3.2	3.1	4.2	96.6	97.4	99.3
	100	0.5	5.2	4.8	4.1	92.4	0.3	22.1	21.8	18.1	93.8	0.5	5.1	4.6	4.3	93.8	0.4	106.4	106.1	66.0	92.2	85.9	86.7
EB	200	0.2	2.4	2.2	2.2	95.1	0.2	9.4	9.2	8.3	94.8	0.1	2.4	2.3	2.3	94.9	0.2	38.7	38.6	26.4	93.0	92.8	95.0
	400	0.1	1.2	1.1	1.2	95.7	0.3	4.8	4.6	4.1	94.7	0.1	1.2	1.1	1.2	96.5	0.1	14.5	14.4	10.2	93.2	97.2	98.5
	100	0.4	5.1	4.7	4.0	92.5	2.7	12.0	9.2	8.1	90.2	0.4	5.1	4.7	4.2	93.1	2.1	15.4	13.3	13.3	88.6	86.8	90.6
TP	200	0.1	2.3	2.2	2.2	95.1	1.2	6.7	5.6	6.2	93.3	0.1	2.4	2.3	2.3	94.8	1.0	7.6	6.7	7.0	93.2	93.5	95.6
	400	0.1	1.2	1.1	1.2	96.2	0.3	3.3	3.0	4.9	97.7	0.1	1.2	1.1	1.2	95.7	0.3	3.9	3.6	5.1	97.3	97.3	98.9
EMPIRICAL BASIS SURFACE
	100	0.4	5.3	5.0	4.3	92.6	6.8	18.3	11.4	9.0	87.1	0.4	5.3	4.9	4.3	92.3	6.5	21.2	14.8	12.0	86.1	93.0	92.9
L	200	0.2	2.5	2.4	2.3	95.1	5.9	12.4	6.5	6.1	91.4	0.2	2.6	2.4	2.3	95.3	5.8	13.7	7.8	6.8	89.5	99.8	99.9
	400	0.1	1.2	1.2	1.3	96.1	5.4	8.8	3.4	4.3	95.8	0.1	1.2	1.1	1.3	96.3	5.4	9.8	4.4	4.6	94.3	102.6	103.4
	100	0.5	5.3	4.8	4.2	92.6	2.1	26.3	24.2	20.0	93.2	0.4	5.6	5.2	4.3	92.4	2.5	103.3	100.9	67.0	92.9	85.1	84.1
EB	200	0.2	2.4	2.2	2.3	95.1	1.0	12.4	11.4	10.4	94.0	0.1	2.4	2.2	2.3	95.4	1.0	40.7	39.8	29.8	93.5	91.5	91.9
	400	0.1	1.2	1.1	1.2	96.0	0.5	6.0	5.5	5.4	95.4	0.1	1.2	1.1	1.2	96.2	0.4	16.4	16.0	12.6	94.3	95.7	96.0
	100	0.5	5.4	4.9	4.1	92.2	4.5	15.6	11.1	9.0	87.7	0.5	5.2	4.7	4.1	92.8	3.8	18.0	14.3	11.2	87.1	86.9	87.8
TP	200	0.1	2.4	2.3	2.2	94.9	1.8	8.3	6.5	7.7	93.6	0.1	2.5	2.3	2.3	94.8	1.7	9.2	7.5	8.2	92.0	92.5	93.0
	400	0.0	1.2	1.1	1.2	96.1	0.8	4.2	3.4	6.2	97.9	0.1	1.2	1.1	1.2	96.2	0.7	4.6	3.9	6.4	97.5	95.8	96.0
TENSOR PRODUCT SURFACE
	100	0.4	8.9	8.4	5.7	91.3	6.8	19.7	13.0	8.2	89.7	0.5	9.5	9.0	5.7	90.4	6.9	27.4	20.5	16.1	90.3	129.6	130.3
L	200	0.3	4.4	4.1	3.1	92.6	6.8	13.2	6.4	4.3	89.0	0.2	4.3	4.1	3.1	92.8	6.8	14.0	7.2	4.6	88.8	140.6	140.9
	400	0.1	2.2	2.1	1.7	94.5	6.7	9.9	3.2	2.2	87.5	0.1	2.4	2.3	1.7	93.5	6.7	10.2	3.6	2.4	86.4	146.7	146.5
	100	0.5	5.4	4.9	4.3	92.6	0.3	22.9	22.6	23.7	96.0	0.4	5.3	4.9	4.4	92.9	0.4	101.7	101.4	70.8	95.4	81.3	81.2
EB	200	0.2	2.5	2.4	2.3	94.7	0.2	11.7	11.5	11.8	96.5	0.2	2.3	2.2	2.3	95.9	0.2	40.9	40.7	30.7	95.6	90.0	89.5
	400	0.1	1.2	1.1	1.2	95.7	0.1	6.1	6.0	6.5	96.7	0.1	1.2	1.1	1.2	96.4	0.1	16.6	16.5	13.9	96.4	94.6	95.0
	100	0.4	5.3	4.9	4.2	92.9	1.0	7.5	6.5	6.5	96.3	0.5	5.3	4.9	4.2	92.8	0.9	9.2	8.3	7.8	96.1	84.1	84.4
TP	200	0.2	2.3	2.2	2.2	95.5	0.8	4.5	3.7	3.9	95.8	0.2	2.4	2.2	2.2	95.5	0.8	4.6	3.8	3.9	95.6	91.4	91.6
	400	0.1	1.2	1.1	1.2	96.2	0.6	2.7	2.1	2.4	94.4	0.1	1.1	1.1	1.2	96.4	0.6	2.9	2.3	2.5	94.5	95.2	96.4

For the estimates of ${\widehat{\beta }}_0(z,t)$ we found that the integrated squared bias for the EB fit was uniformly lower than the other two fits for all surfaces and sample sizes, except for the dense case where the true surface was linear and the sample size was 400. All three fits had extremely low relative bias when the true surface was linear, the highest value being 0.016, corresponding to the TP fit when n = 100. When the true surface was nonlinear, the linear fit had uniformly higher relative bias and was especially high when the true surface was from the TP model (e.g. 0.067/0.52 = 12.8%). The other two fits performed well in terms of relative bias, the highest being 6% corresponding to the TP fit of the EB. surface when n = 100.

In terms of MISE, the linear fit perform the best when the true surface was linear, and the TP fit performed the best for the other two surfaces. The EB fit had high MISE for all three surfaces, due to the uniformly higher amount of variability of the fit. The linear and TP fits had comparable variability with the linear fit doing slightly better for the linear and EB surfaces. The linear fit generally had the worse coverage, even when the true surface was linear, while the EB and TP fits had comparable coverage. For the sparse grid of timepoints, we found a similar trend where the EB fit had uniformly lower bias, and the TP fit had lower MISE and variance when the true surface was not linear.

In summary, we found that the EB fit had uniformly lower bias than the other two fits, but also had uniformly higher variance, causing the MISE to be high. The linear fit performed well when the true surface was linear, but quickly fell apart for the nonlinear surfaces. The TP fit was stable and consistent for all three surfaces, had appropriate coverage, and naturally performed the best when the true surface was TP. Because of this, we feel the TP surface should be used in practice since the true structure of the bivariate surface is often unknown.

5 DATA ANALYSIS

To demonstrate the methods presented in this paper, we apply them to the Multiple Myeloma data described in the introduction. The data consist of over 17,000 gene expressions and CNV profiles for the entire genome. The goal was to study the effects of gene expression and CNV on β₂M, a biomarker for MM staging, while accounting for the interaction between gene expression and CNV and adjusting for the effects of two demographic covariates, age and sex. Thus, we denote the response β₂M as Y_i, the demographic covariates as the 2 × 1 vector V_i, the scalar valued gene expression as Z_i, and the CNV profiles as W_i(t). Of the 235 people in the study, 63 had missing response data, yielding a sample size of 172.

For each gene, we paired up CNV probes and gene expressions based on their locations along the genome by mapping the CNV probes that had a starting location that was within the starting and ending location of the gene. Because we wanted to model CNV as a functional covariate, any gene that had fewer than 20 associated CNV probes were dropped from the analysis, leaving 1467 candidate genes. We reduced this to 4 genes using two pre-selection methods. First, we identified genes that were historically known to be associated with MM³⁹ and of these, only two had corresponding CNV profiles with more that 20 probes, BRAF and TRAF3 (22 and 20 probes, respectively). Mutations in these genes, which regulate cellular response to growth signals, have been shown to induce growth factors that affect MM cells.^23,35 For the second selection method, we first smoothed all of the individuals CNV profiles for each gene using FPCA and obtained the scores. For each gene expression, we fit the additive model

$$Y_i=Z_i+{\widehat{\xi }}_i+Z_{i}f({\widehat{\xi }}_i)+{\epsilon }_i,$$

where, f is an unknown, smooth function. We tested for the effect of f and ranked the p-values from smallest to largest. After applying a Bonferroni correction, 116 genes had significant interaction effect and we chose the top two for further analysis, PRIM2 and GFRA1, which had 32 and 22 CNV measurements, respectively. There were no missing data in the CNV profiles corresponding to the 4 selected genes.

We fit the general model in (1) and used the linear, EB and TP estimation methods. The results for the BRAF gene are presented in Figure 1 and a random sample of observed CNV profiles with their corresponding smoothed estimates are given in the appendix. The estimates of β₀(z) were essentially the same for all estimation procedures. We also found the shape of β(z, t) to be nearly identical, suggesting that the true underlying surface is linear in both z and t. The color in the bivariate plots indicates the areas of the surface that were significantly different from zero, based on the corresponding 95% pointwise confidence surfaces. The green areas indicate that the surface was positive but not significantly different from zero and the yellow areas are negative but not significant. We emphasize that these interval surfaces are pointwise and temper any inference based on them. We found the estimate of $\widehat{\mathit{\gamma }}$ to be (0.003, 0.775) with pointwise standard error (0.02, 0.55) for both fits.

FIGURE 1.

Estimated β₀(z) (a) and estimated β(z, t) surface from the linear (b) EB (c), and TP (d) fits for the BRAF gene. The widest pointwise confidence bands were used in (a). For the surface plots, blue indicates the surface values are positive and significantly different than 0, green is positive but not significant, yellow is negative but not significant, and red is negative and significant.

The results for the PRIM2 gene are found in Figure 2. Here, we saw slightly different estimates of β₀(z). The shape of bivariate surface showed a strong quadratic trend in the t direction, and a linear trend in the z direction. Thus, the tail ends of the CNV profiles tended to have negative values for gene expression, and toward the middle of the CNV profile was when positive values of gene expression occurred. Again, the true surface seemed to be modeled well by all fits. The estimate of $\widehat{\mathit{\gamma }}$ was (0.023, 0.558) and (0.021, 0.576), for the EB and TP fits, respectively, with pointwise standard error (0.02, 0.52).

FIGURE 2.

Estimated β₀(z) (a) and estimated β(z, t) surface from the linear (b) EB (c), and TP (d) fits for the PRIM2 gene. The widest pointwise confidence bands were used in (a). For the surface plots, blue indicates the surface values are positive and significantly different than 0, green is positive but not significant, yellow is negative but not significant, and red is negative and significant.

The similarities between the estimation methods arose in the TRAF3 gene, found in Figure 3. The estimate of β₀(z) were virtually identical for both estimation methods. We saw similar shapes and significant regions for the estimated β(z, t) surfaces, with some slight curvature in the boundaries of the EB surface. This added curvature results from the curvature in PCs selected, but based on the overall shape it is probably safe to conclude that the true underlying surface is linear in z and t. The estimate of $\widehat{\mathit{\gamma }}$ was (0.003, 0.694) with pointwise standard error (0.02, 0.55).

FIGURE 3.

Estimated β₀(z) (a) and estimated β(z, t) surface from the linear (b) EB (c), and TP (d) fits for the TRAF3 gene. The widest pointwise confidence bands were used in (a). For the surface plots, blue indicates the surface values are positive and significantly different than 0, green is positive but not significant, yellow is negative but not significant, and red is negative and significant.

The results of the GFRA1 gene in our analysis are given in Figure 4. The TP estimate of β₀(z) had a slightly positive slope, while the EB fit had a flat slope, but the difference was not significant. For the bivariate surface, we saw more curvature in the EB fit compared to the TP fit. This could be due to the EB’s dependence on the shapes of PCs. In both surface estimates, we found a linear trend in z, and a nonlinear trend in t. Lastly, the estimate of $\widehat{\mathit{\gamma }}$ was (0.015, 0.692) and (0.014, 0.672), for EB and TP fits, respectively with pointwise standard error (0.02, 0.54).

FIGURE 4.

Estimated β₀(z) (a) and estimated β(z, t) surface from the linear (b) EB (c), and TP (d) fits for the GFRA1 gene. The widest pointwise confidence bands were used in (a). For the surface plots, blue indicates the surface values are positive and significantly different than 0, green is positive but not significant, yellow is negative but not significant, and red is negative and significant.

As a final analysis, we compared our interaction models presented in this paper to the standard functional linear model without interaction, $Y_i={\mathbf{V}}_i^T\mathit{\gamma }+Z_i{\beta }_0+{\displaystyle \int X_i(t)\beta (t)dt+{\epsilon }_i}$, by cross-validation of the prediction MSE. For each of the reported genes, the data were randomly partitioned in training (80%) and testing (20%) datasets and the prediction MSE was computed, then averaged over 100 repetitions of 80/20 splits. These results are in Table 2. Since estimated surfaces have a linear shape for these genes, there is little gain in using the general model over the linear varying coefficient model, however all models presented in this paper outperformed the basic functional linear model that ignores interaction.

TABLE 2.

Comparison of PMSE for the linear model (described in detail in the appendix), the general model with EB and TP estimation methods, and the standard functional linear model for each gene.

	LVCM	EB	TP	FLM
BRAF	11.87	11.87	11.90	14.20
TRAF3	12.89	12.85	12.91	17.51
PRIM2	11.05	11.71	11.39	12.47
GFRA1	12.46	12.61	12.68	27.57

6 DISCUSSION

In this paper, we established several methodologies for modeling the joint effect of a scalar covariate and a functional covariate and adjusting for any interaction between them. We proposed a varying coefficient model in the functional framework, useful for when the scalar covariate is categorical, and presented a more general model using a bivariate surface function for continuous covariates. Two estimation schemes were presented, each with their own benefits and drawbacks. We conducted a simulation study to compare the different estimation methods in this paper, and we also compared to other methods in the literature. Using a variety of simulation settings, we found that the TP estimation method was consistent and stable and thus is likely the best choice in real world applications. We then applied our methodology to the Multiple Myeloma data that motivated this work.

The methods presented in this paper can easily extend to case where several functional covariates, (X_i1, X_i2,…, X_iq) are observed for each subject. The interaction between the scalar covariate and each of these functional covariates can be modeled with a separate bivariate parameter function β_q(z, t), and this is just a matter of representing each function with a basis approximation and adding corresponding penalty terms. Methods for modeling the pairwise interaction between the functional covariates are currently being developed.^52,59). We do not address it in this paper because gene expressions do not typically interact with CNV regions that are not nearby and it is especially hard to justify interaction when the gene and CNV region are on different chromosomes. However, interaction can be argued for other types of functional predictors, such as environmental factors that can be modeled in a functional way.

This paper solely focuses on estimation of the unknown parameters and does not present a way to test for the effects of the covariates, an important aspect of any statistical analysis. The ultimate goal is to be able to reliably identify genes that have a significant effect on diseases. This paper is a stepping stone in that CNV was added to modeling, which has been shown to improve reliability in candidate gene and gene association studies.⁴⁰ In the future, we would like to be able to identify genes by testing for the main effects of the genes and the interaction between the genes and the CNV profile. Thus far, we found that testing in this framework is difficult, and encountered several problems when using a naive Wald statistic. Testing is a very open problem that we would like to consider going forward.

Finally, all the methods in this paper boil down to the estimation of several parameter functions with a single argument. We chose to use spline-based procedures but these can easily be replaced with any desired estimation scheme, such as local polynomial smoothing or kernel machine methods.

APPENDIX

A SIMULATION RESULTS OF BINARY RESPONSE

When the response was binary, the results from our simulation (Table C1) were more unstable due to sample size issues, thus we used relatively large sample sizes of 300 and 500. When the true surface was linear, we found that the linear and TP estimate of β₀(z) performed reasonably well for the dense and sparse cases and had similar results. The EB fit generally had the highest bias and variability in the estimate of β₀(z) and had the worst coverage. For the estimate of the linear β(z, t), we found that the TP fit generally had the highest bias and the lowest variability and MSE, but was comparable to the linear fit overall. The EB surface had the lowest bias for large sample sizes, but had three times as much variability in the dense case, and six times as much variability in the sparse case, compared to the other two fits.

When the true surface had the EB structure, we again saw very comparable results between the linear and TP estimation of β₀(z). The EB fit had uniformly higher variability and tended to have the worst coverage of all three fits. For the estimate of β(z, t), we saw slightly elevated bias in the linear fit compared to the other two fits, with overall relative bias around 10%. In comparison, the relative bias for the EB and TP fits were around 4% and 7%, respectively. In terms of variation, the linear and TP fits had similar results, with the TP fit doing slightly better. The EB fit had uniformly higher variability which resulted in better integrated coverage than the other two fits.

When the true surface had the TP type, we found a noticeable increase in the bias of β₀(z) when the linear estimation was used. This is because the excess nonlinear effects that are not captured by the linear fit is absorbed into the main effects term, which results in added bias. Also of note is that, as sample size increased, the bias of the linear estimate of β₀(z) also increased, showing that the bias is not a sample size issue. However, even though the linear fit had as much as 15 times the relative bias as the other two fits, the overall relative bias was only 3%. We found that the linear estimate of β₀(z) had uniformly lower variance than the other two fits while the TP fit had uniformly lower MISE. The EB estimate had the lowest relative bias (less that 1%) and the highest variability.

When we considered our estimate of β(z, t) for TP surface, we found that the linear fit naturally had high relative bias compared to the other two estimates with relative bias as high as 15%, corresponding to n = 500 using the dense grid. In comparison, the relative bias of the TP estimate was 4%, while the EB fit had the lowest relative bias at 2% of the scaled norm. Again, we found that the EB fit had the uniformly highest variability and MISE for both the dense and sparse designs which overshadowed any gains in bias when using this method. The TP fit had uniformly lower MISE and similar variance to the linear estimation, with the linear fit doing only slightly better. All three fits had very accurate 95% coverage. Because the TP fit was stable and consistent in the estimate of β(z, t) regardless of the true surface type, we again suggest using this fit in practice when the true structure of the surface is unknown.

B COMPARISON TO LOCAL SPLINES

We performed a small simulation to compare our TP estimation to the penalized local least squares splines estimation.²⁰ We generated n = 100 or 200 Gaussian responses from

$$Y_i={\beta }_0(Z_i)+{\displaystyle \int X_i(t)\beta (Z_i,t)dt+{\epsilon }_i,}$$

where β₀(z), X_i(t), and ε_i are the same as before. We used only the EB surface for β(z, t) and omit the parametric covariates. For estimation purposes, because²⁰ only consider smooth, densely observed functional covariates we fit their model and our TP model using the true X_i(t) observed on the dense grid of 41 points. We evaluated the estimates for each function using the integrated squared bias, MISE, and Monte Carlo variance. Since²⁰ do not present methodology for finding the variances of their estimators, we did not compute the estimated variance or integrated coverage.

The results of this small study are presented in Appendix C. For the estimates of β₀(z), we found that the local splines fit had three times the relative bias for n = 100 and five times the relative bias for n = 200. However the overall relative bias was very small (0.017/2.93 = 0.5%), thus both methods performed well in terms of integrated bias. We found that the local method had approximately nine times the variability as our TP fit for both sample sizes.

For the estimation of β(z, t), the bias of the two fits were comparable, with the local splines estimator doing slightly better for the smaller sample size. The relative bias for both fits was about 6% for the smaller sample size, and about 3% for the larger sample size. There was about half the amount of empirical variability in the TP estimate than in the local splines fit. Both methods performed reasonably well in terms of estimating the the residual variance.

C SAMPLE R CODE FOR TENSOR PRODUCT BASIS

Below is the R function used to fit the tensor product basis. We encourage the reader to contact the authors with questions about using the code to implement the methods in this paper.

FIGURE C1.

Observed CNV profiles along with their respective smoothed curves for 24 randomly selected individuals.

TABLE C1.

Binary response: Integrated squared bias (SB), mean integrated squared error (MSE), integrated monte carlo empirical variance (Vem), integrated estimated variance (Ves), and integrated coverage (C) for the two functions b₀(z) and β(z, t) when the functional covariate is densely observed and sparsely observed. We fit a linear model (L), an empirical basis model (EB), and a tensor product basis model (TP) to each of the three true surfaces for sample sizes 300, and 500. All values in the table are multiplied by 100. For relativity, the scaled norm for β₀(z) is 2.93, and the scaled norms for β(z, t) are 1.67, 0.77, and 0.52 for the linear, EB, and TP surfaces, respectively.

LINEAR SURFACE
		β0(z) Dense					β(z, t) Dense					β0(z) Sparse					β(z, t) Sparse
		SB	MSE	Vem	Ves	C	SB	MSE	Vem	Ves	C	SB	MSE	Vem	Ves	C	SB	MSE	Vem	Ves	C
	300	2.3	26.9	24.6	19.9	93.0	3.4	45.1	41.8	32.0	91.7	3.3	30.4	27.2	20.6	92.4	2.8	56.2	53.4	38.5	91.5
L	500	0.9	14.3	13.4	11.4	93.2	2.0	26.5	24.5	19.1	90.8	1.0	14.3	13.3	11.3	93.5	2.6	26.6	24.1	18.9	91.1
	300	4.1	33.6	29.5	21.5	92.5	4.0	122.7	118.9	96.5	94.6	5.2	37.5	32.3	22.5	91.4	5.3	474.6	469.8	265.3	93.9
EB	500	1.3	16.0	14.7	11.7	92.4	1.2	59.6	58.4	50.2	94.8	1.5	17.0	15.6	12.0	92.6	1.4	177.4	176.1	116.6	94.1
	300	2.8	30.0	27.2	19.4	91.9	6.2	44.5	38.3	29.6	93.2	2.8	30.1	27.3	19.4	92.3	5.5	49.9	44.4	31.9	92.9
TP	500	1.2	14.3	13.1	11.0	93.3	4.7	25.7	21.0	17.2	92.8	0.9	13.9	13.0	11.1	94.0	5.1	28.8	23.7	17.3	92.2
EMPIRICAL BASIS SURFACE
	300	1.8	21.2	19.4	15.8	92.8	7.3	42.5	35.2	25.5	90.0	1.8	21.4	19.7	15.7	92.6	8.1	45.8	37.7	26.5	90.6
L	500	0.7	11.0	10.4	9.1	93.3	7.4	25.5	18.1	14.3	89.7	0.8	11.0	10.2	8.9	93.8	7.1	27.3	20.2	15.3	89.9
	300	2.5	26.3	23.8	16.6	92.1	3.5	91.6	88.2	74.0	94.5	3.3	29.3	26.0	17.4	92.1	3.1	320.7	318.0	196.0	94.1
EB	500	0.7	12.7	12.0	9.6	92.6	2.5	46.8	44.4	40.2	95.2	0.9	12.3	11.4	9.7	93.3	2.4	140.6	138.3	94.8	94.4
	300	2.0	22.1	20.1	15.4	92.6	5.6	36.2	30.6	23.4	91.2	1.5	20.6	19.1	15.3	93.3	5.9	37.5	31.7	24.0	91.9
TP	500	0.8	11.2	10.4	9.0	93.6	5.3	22.6	17.2	14.4	90.9	0.7	12.0	11.3	9.1	92.3	5.2	24.6	19.4	15.2	90.9
TENSOR PRODUCT SURFACE
	300	10.0	21.5	11.5	10.8	94.4	7.7	26.3	18.6	14.3	94.6	10.3	22.3	12.0	11.1	94.0	7.5	26.8	19.3	15.1	94.2
L	500	11.4	18.3	6.9	6.2	94.2	7.9	18.3	10.4	8.1	94.1	10.9	17.8	6.9	6.2	93.6	7.8	19.6	11.8	8.9	94.5
	300	2.5	23.0	20.6	15.3	92.9	1.0	79.8	78.8	69.6	95.6	2.6	23.7	21.1	16.2	92.7	1.4	273.5	272.5	179.9	95.3
EB	500	1.8	12.5	10.7	8.9	93.5	1.0	39.2	38.3	37.5	96.1	1.6	12.6	11.0	9.1	92.7	1.0	114.1	113.2	85.7	95.6
	300	2.4	20.4	18.1	14.1	93.5	2.0	20.5	18.5	17.1	95.7	2.8	20.0	17.2	13.9	93.1	2.0	23.4	21.4	18.1	95.9
TP	500	1.9	12.2	10.2	8.6	93.1	1.6	13.0	11.4	10.7	95.9	2.0	11.8	9.8	8.6	93.7	1.7	14.4	12.7	10.7	96.1

TABLE C2.

Integrated squared bias (SB), mean integrated squared error (MSE), and integrated monte carlo empirical variance (Vem) for b₀(z) and β(z, t) when the functional covariate is densely observed. We fit the general model to the EB surface using local spline estimation method (Car) and our proposed tensor product estimation method (TP), and estimated the residual error. All values in the table are multiplied by 100.

		β0(z)			β0(z, t)			Residual
		SB	MSE	Vem	SB	MSE	Vem
n = 100	Car	1.7	42.7	41.1	4.2	26.0	21.8	116.0
	TP	0.5	5.3	4.8	4.5	14.5	10.1	89.5
n = 200	Car	1.1	22.5	21.4	2.3	13.0	10.7	106.5
	TP	0.2	2.3	2.1	1.9	7.9	6.0	93.3