A Novel Estimation Method in Generalized Single Index Models

The single index and generalized single index models have been demonstrated to be a powerful tool for studying nonlinear interaction effects of variables in the low-dimensional case. In this article, we propose a new estimation approach for generalized single index models E(Y | θ⊤X)=ψ(g(θ⊤X)) with ψ(·) known but g(·) unknown. Specifically, we first obtain a consistent estimator of the regression function by using a local linear smoother, and then estimate the parametric components by treating ψ(ĝ(θ⊤Xi)) as our continuous response. The resulting estimators of θ are asymptotically normal. The proposed procedure can substantially overcome convergence problems encountered in generalized linear models with discrete response variables when sparseness occurs and misspecification. We conduct simulation experiments to evaluate the numerical performance of the proposed methods and analyze a financial dataset from a peer-to-peer lending platform of China as an illustration.</p

The constructed gene regulatory networks for simulation studies with <i>N</i> = 360 and 100 iterations.

The legend is the same as in Fig 4.</p

Using single-index ODEs to study dynamic gene regulatory network

<div><p>With the development of biotechnology, high-throughput studies on protein-protein, protein-gene, and gene-gene interactions become possible and attract remarkable attention. To explore the interactions in dynamic gene regulatory networks, we propose a single-index ordinary differential equation (ODE) model and develop a variable selection procedure. We employ the smoothly clipped absolute deviation penalty (SCAD) penalized function for variable selection. We analyze a yeast cell cycle gene expression data set to illustrate the usefulness of the single-index ODE model. In real data analysis, we group genes into functional modules using the smoothing spline clustering approach. We estimate state functions and their first derivatives for functional modules using penalized spline-based nonparametric mixed-effects models and the spline method. We substitute the estimates into the single-index ODE models, and then use the penalized profile least-squares procedure to identify network structures among the models. The results indicate that our model fits the data better than linear ODE models and our variable selection procedure identifies the interactions that may be missed by linear ODE models but confirmed in biological studies. In addition, Monte Carlo simulation studies are used to evaluate and compare the methods.</p></div

Model Checking for Logistic Models When the Number of Parameters Tends to Infinity

We propose a projection-based test to check logistic regression models when the dimension of the covariate vector may be divergent. The proposed test achieves a reduction in dimension, and the proposed method behaves as if only a single covariate is present. The test is shown to be consistent and can detect root-n local alternatives. We derive the asymptotic distribution of the proposed test under the null hypothesis and establish the test’s asymptotic behavior under the local and global alternatives. The numerical performance is remarkably attractive comparing to the existing methods. Real examples are presented for illustration. Supplementary materials for this article are available online.</p

The constructed gene regulatory networks for simulation studies with <i>N</i> = 288 and 100 iterations.

<p>The legend is the same as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0192833#pone.0192833.g004" target="_blank">Fig 4</a>.</p

The value (standard deviation) of the monotonicity tests for the simulation study.

*<p>indicates that both -value and its associated standard deviation are less than .</p

The constructed gene regulatory networks for simulation studies with <i>N</i> = 180 and 100 iterations.

Solid lines: the true connections, numbers present: the times correctly identified using our procedure in 100 iteration, dots line: incorrectly identified connections.</p

The GRN identified by the linear ODE models for the time course yeast cell data set.

<p>Each node represents a module and the arrows presents the direction of influence.</p

The inward and outward regulations in the module-based regulatory network and RSS based on the linear ODE (L-ODE) and the single-index ODE (Si-ODE).

<p>The inward and outward regulations in the module-based regulatory network and RSS based on the linear ODE (L-ODE) and the single-index ODE (Si-ODE).</p

Dose response curves for various IC (A) or shape parameters, (B).

<p>Dose response curves for various IC (A) or shape parameters, (B).</p