A goodness-of-fit test for variable-adjusted models

Abstract

This research provides a projection-based test to check parametric single-index regression structure in variable-adjusted models. An adaptive-to-model strategy is employed, which makes the proposed test work better on the significance level maintenance and more powerful than existing tests. With mild conditions, the proposed test asymptotically behaves like a test that is for classical regression setup without distortion errors in observations. Numerical studies with simulated and real data are conducted to examine the performance of the test in finite sample scenarios.

Introduction

Regression models are widely used to describe the relationship between response variable $Y$ and $p$-dimensional predictor $X={(X_1,\dots ,X_p)}^{\top }$. Generally, the response $Y$ and the predictor $X$ are assumed to be observable. But in some cases, obtaining the true values of $Y$ and $X$ are expensive or impossible, while the surrogates $\tilde{Y}$ and $\tilde{X}$ are available. This motivates and formulates a number of measurement error models. A classical structure is that the surrogate is the sum of the measurement error and the unobservable variable. Carroll et al. (2006), Cheng and Van Ness (1999) and Fuller (2009) are comprehensive references. Motivated by a real dataset, Şentürk and Müller (2005b) introduced another type of measurement error model, in which, the observed response and predictors are adjusted variables with distortion errors rather than in an additive manner. For the dataset, they investigated a linear relationship between the fibrinogen level and the serum transferrin level in haemodialysis patients. As the fibrinogen level and the serum transferrin level are both measured with a confounding effect from body mass index, they considered a way of model fitting in which the effect is multiplicative with an unknown function of body mass index. This measurement error model has several extensions in later developments. Cui et al. (2009) used a nonlinear regression model to determine the glomerular filtration rate by the serum creatinine level. With the body surface area as the distortion effect, a study of variable-adjusted nonlinear regression model was suggested by Cui et al. (2009).

Consider a general variable-adjusted regression model where the response $Y$, the predictor $X$ and their surrogates $\tilde{Y}$, $\tilde{X}$ are related to each other by the following relations:

$$Y=\mu \left(X\right)+\epsilon ,\tilde{Y}=\psi \left(U\right)Y,\tilde{X}=({\tilde{X}}_1,\dots ,{\tilde{X}}_d,X_{d+1},\dots ,X_p),$$

with ${\tilde{X}}_r={\varphi }_r\left(U\right)X_r$, $r=1,\dots ,d$. Here $d\le p$, $U$ is an observable confounder and $\psi \left(u\right)$ and ${\varphi }_r\left(u\right)$ are unknown distorting functions. The efforts are mainly devoted to estimation. A natural idea is to estimate the true values of $Y$ and $X$ by adjusting the observed surrogates $\tilde{Y}$, $\tilde{X}$ and then to further estimate $\mu \left(X\right)$ with $\hat{Y}$ and $\hat{X}$. The reference includes Nguyen and Şentürk (2008), Cui et al. (2009), (Zhang et al., 2012a), Zhang et al. (2013), Delaigle et al. (2016) and so on. However there is less attention on goodness-of-fit test for this model based on variable adjustment. Zhang et al. (2015) proposed a residual marked empirical process-based test that is a $\sqrt{n}$-consistent test. Their test requires a time-consuming bootstrap procedure and a user-specified weight function.  Zhao and Xie (2018) developed a local smoothing test for variable-adjusted models, which is very simple and easy to implement. The cost is that it can only detect local alternatives distinct from the null hypothesis at the rate of $n^{-1∕2}{h}^{-p∕4}$, which is slower than that of the test in Zhang et al. (2015) and greatly affected by the dimensionality $p$. This means when the dimensionality of the predictor is large or the sample size is moderate, this test may not work well.

In the case of no distortion error, the projection-based methods are broadly discussed to develop tests of dimension reduction type. These methods could be tracked back to Zhu and Li (1998), motivated from projection pursuit (see Huber, 1985). The later developments include Escanciano (2006), Stute et al. (2008), Lavergne and Patilea (2008), Lavergne and Patilea (2012), and Guo et al. (2016). A relevant reference is Delgado and Escanciano (2016).  Guo et al. (2016) proposed an adaptive-to-model test, which significantly improves the performances of local smoothing tests. Motivated by this work, we will take the advantages of the model adaptation strategy and combine the special construction for variable-adjusted models to develop a projection-pursuit test. To utilize dimension reduction structure in the hypothetical model, the problem of interest is to check whether the model is single-index and the hypotheses are formulated as

$${\mathcal{H}}_0:P\left(\mu \left(X\right)=m({\beta }_0^{\top }X,{\gamma }_0)\right)=1,\text{for some }{\theta }_0=({\beta }_0;{\gamma }_0)\in \Theta \subset {\mathbb{R}}^{p+p^{\prime }},$$$$\text{versus}$$$${\mathcal{H}}_1:P\left(\mu \left(X\right)=m({\beta }^{\top }X,\gamma )\right)<1,\text{for all }\theta =(\beta ;\gamma )\in \Theta \subset {\mathbb{R}}^{p+p^{\prime }}.$$

where $m$ is a given function, $\beta$ and $\gamma$ are the parameters of $p$-dimension and $p^{\prime }$-dimension respectively. From the viewpoint of sufficient dimension reduction (SDR, Cook, 2009), a general alternative regression model can be written as $\mu \left(X\right)=G\left(B^{\top }X\right)$ where $G(.)$ is an unknown function, different from $m({\beta }_0^{\top }x,{\gamma }_0)$ and $B$ is a $p\times q$ matrix(or vector) with unknown number $q$ of columns. Note that $1\le q\le p$ and usually $q$ is much smaller than $p$. If $q<p$, the alternative model has a dimension reduction structure. When $q=p$, it is just a general nonparametric model. To conveniently study the asymptotic properties of the proposed test, we consider the following sequence of models

$${\mathcal{H}}_{1n}:Y=m({\beta }_0^{\top }X,{\gamma }_0)+C_n\Delta \left(B^{\top }X\right)+\epsilon .$$

The case $C_n=0$ corresponds to the null hypothesis ${\mathcal{H}}_0$ and the alternative holds when $C_n\ne 0$. When $C_n$ is a fixed constant, the alternative is a global alternative under ${\mathcal{H}}_1$ and when $C_n\to 0$, ${\mathcal{H}}_{1n}$ specifies a sequence of local alternatives. The test in Zhao and Xie (2018) can only detect the local alternatives converge to the null model at the rate of $C_n$ such that $C_n{n}^{1∕2}{h}^{p∕4}$ is bounded above or goes to zero. Thus, $n^{-1∕2}{h}^{-p∕4}$ is the fastest rate to ensure that their test can detect the local alternatives. We will show that the proposed test can detect the local alternatives with the rate $C_n=n^{-1∕2}{h}^{-1∕4}$ and be consistent for any $C_n\gg n^{-1∕2}{h}^{-1∕4}$. On the other hand, according to the arguments in Guo et al. (2016), an estimate $\hat{B}$, which converges to $\pm {\beta }_0∕\Vert {\beta }_0\Vert$ under ${\mathcal{H}}_0$ and to $B$ when ${\mathcal{H}}_1$ is true, is the key to make the proposed test adaptive to the underlying model. In this paper, we use the sufficient dimension reduction technique proposed by Zhang et al. (2012b) to obtain $\hat{B}$, and systematically investigate its asymptotic properties under the local alternatives. We give more details in the following.

The paper is organized as follows. Section 2 describes the test problem for variable-adjusted model and proposes an adaptive-to-model test procedure. In Section 3, we present the large sample properties of the proposed test. Sections 4 Numerical studies, 5 A real data example report the simulation results and real data application to illustrate our method. The assumptions and proofs are postponed to Appendix A.