A new variant of the parallel regression model with variable selection in surveys with sensitive attribute

Highlights

• Broadening the application scope of the non-randomized response techniques.

• Examining the relationship between the sensitive attribute and other covariate(s).

• Overcoming the limitation of logistic parallel regression model (Liu et al., 2019).

• Providing methods for confounding variable identification and variable selection.

Abstract

In this paper, a new hidden logistic regression model, i.e., the variant of the parallel regression model, is developed to study the relationship between a sensitive binary response variable and a set of non-sensitive covariates, where the information about the sensitive attribute of interest is collected via the variant of the parallel model originally proposed by Liu and Tian (2013b). The EM–NR algorithm is provided to derive the maximum likelihood estimates of the regression coefficients. Furthermore, we also discuss the method of identifying confounding variable and the method of variable selection based on SCAD penalty for the variant of the parallel regression model. Finally, simulation studies and a real data example about premarital sexual behavior were conducted to illustrate the proposed approaches.

Introduction

In practice, some investigations deal with phenomena that are considered as violation of social morality or illegal activities (such as bribery, tax evasion, illegitimate child, illegal immigration, drug abuse and so on) will make interviewees feel embarrassed and refuse to provide truthful answer when they were asked directly. Thus, to better protect individuals’ privacy as well as encourage truthful answers, studies on helping collecting sensitive information have been developed rapidly during the last decades. Up to now, three main branches of techniques are introduced for collecting and analyzing data in sample surveys with sensitive questions: the randomized response techniques (Warner, 1965, Horvitz et al., 1967, Greenberg et al., 1969, Fox and Tracy, 1986, Chaudhuri and Mukerjee, 1988, Mangat, 1994, Chaudhuri, 2011), the item count techniques or the unmatched count techniques (Miller, 1984, Dalton et al., 1994, Kuklinski et al., 1997, Gilens et al., 1998, LaBrie and Earleywine, 2000, Tsuchiya, 2005, Janus, 2010, Imai, 2011, Petróczi et al., 2011, Tian et al., 2017, Liu et al., 2019) and the non-randomized response techniques (Tian et al., 2007, Tian et al., 2009, Tian G.L et al., 2011, Yu et al., 2008, Tan et al., 2009, Tang et al., 2009, Liu and Tian, 2013a, Liu and Tian, 2013b, Groenitz, 2014, Tian, 2015).

Only focusing on raising new survey models for sensitive information collection with well privacy protection is not enough, sometimes, researchers may also be interested in finding out which factors may have non-ignorable influence on such sensitive attributes of interest. Thus, to construct appropriate regression models is necessary and the logit model is often a good choice for binary response variable. For the randomized response approaches and the item count approaches, several works have been done on establishing a connection between the binary sensitive respond and other non-sensitive potential explanatory variables (see Maddala, 1983, Scheers and Dayton, 1988, Corstange, 2004, van den Hout et al., 2007, Hsieh et al., 2010). Note that the non-randomized response methods contain advantages such as reproducibility, low-cost, easy understanding for both interviewers and interviewees, strong operability, better privacy protection and so forth, it is worthy of our efforts to discuss regression analyzing ways under the non-randomized response models. However, in the field of the non-randomized response approaches, only Tian et al. (2019) proposed a so-called hidden logit parallel model that taking non-sensitive covariates into consideration while the collection of the sensitive information is aided by Tian’s non-randomized parallel model (Tian, 2015). Nevertheless, the parallel model given by Tian asked the successful proportions of the two auxiliary non-sensitive binary variables should be known before experiments, which may certainly restricted the application range of this model. Luckily, the variant of parallel model put forward by Liu and Tian (2013b) could better overcome such a limitation by letting one successful proportions of the two auxiliary non-sensitive variables to be known while the other one keeps unknown. Therefore, the first objective of this paper is to develop a new variant of parallel regression model to detect elements being responsible for the sensitive attribute of interest.

To our best knowledge, little work has been done on confounding variable identification and the variable selection for analysis in surveys with sensitive characteristics, especially for non-randomized response approaches. However, in regression analysis, other than providing effective regression estimators, these problems are two of the important issues that deserve our attention. Both contribute a lot to enhance the accuracy and precision of the regression model. For the former one, since the confounder has impact on both the explained variable and the explanatory variable(s), it may distort the observed relationship between the exposures and outcomes (Greenland et al., 1999, Pearl, 2009, VanderWeele and Shpitser, 2013). Thus, it is of great necessity to identify the confounding variable(s) in regression analysis such that the true associativity between dependent and independent variables can be revealed. On the other hand, for the latter one, it is helpful in reducing the complexity of the model to a great extend since a large number of covariates are usually chosen at the initial stage of modeling aiming at reducing possible modeling biases. However, too many redundancy variables go against the most basic modeling principle, i.e, the principle of parsimony; that is, the model used should require the smallest possible number of parameters that will adequately represent the data. Various kinds of variable selection techniques have been developed to help selecting significant factors by adding a penalty to the objective function (see Frank and Friedman, 1993, Tibshirani, 1996, Fan and Li, 2001, Zou and Hastie, 2005, Zou, 2006, Zhang, 2010, Wang and Wang, 2014, Wang and Wang, 2016). Fan and Li (2001) have pointed out that a good penalty function could help to obtain an estimator with the following three properties:

• Unbiasedness: the resulting estimator is unbiased when the unknown coefficient of the regression is large;

• Sparsity: the resulting estimator has an automatic thresholding rule that sets some estimated coefficients to zero in order to reduce the complexity of the model if the estimated values are small;

• Continuity: the resulting estimator is a continuous function of data to avoid instability in model prediction.

Consequently, the second objective of this paper is to provide a method of identifying the confounding variable(s) and the third objective is to derive the variable selection technique for the proposed variant of the parallel regression model while the Smoothly Clipped Absolute Deviation (SCAD) method introduced by Fan and Li (2001) is adopted.

The rest of the paper is organized as follows: In Section 2, we first propose the variant of the parallel regression model and then, a Expectation–Maximization embedded with Newton–Raphson (EM–NR) algorithm is derived for calculating the MLEs of the regression coefficients. In Section 3, we provide the approach for identifying confounding variable(s). The method for variable selection based on SCAD penalty was discussed in Section 4, and in the same section, we also present the asymptotic properties for the penalized estimators. Four simulation studies are performed in Section 5 and a real data example about premarital sexual behavior is analyzed in Section 6 to illustrate the proposed methods. Finally, a discussion is given in Section 7.