تست مقاوم در مدل رگرسیون لجستیک

We are interested in testing hypotheses that concern the parameter of a logistic regression model. A robust Wald-type test based on a weighted Bianco and Yohai [ Bianco, A.M., Yohai, V.J., 1996. Robust estimation in the logistic regression model. In: H. Rieder (Ed) Robust Statistics, Data Analysis, and Computer Intensive Methods In: Lecture Notes in Statistics, vol. 109, Springer Verlag, New York, pp. 17–34] estimator, as implemented by Croux and Haesbroeck [Croux, C., Haesbroeck, G., 2003. Implementing the Bianco and Yohai estimator for logistic regression. Computational Statististics and Data Analysis 44, 273–295], is proposed. The asymptotic distribution of the test statistic is derived. We carry out an empirical study to get a further insight into the stability of the pp-value. Finally, a Monte Carlo study is performed to investigate the stability of both the level and the power of the test, for different choices of the weight function.

In the binomial regression model we assume that the response variable YY has a Bernoulli distribution such that equation(1) View the MathML sourceP(Y=1|X=x)=F(x′β), Turn MathJax on where FF is a strictly increasing cumulative distribution function, View the MathML sourceX∈ℜp is the vector of explanatory variables and View the MathML sourceβ∈ℜp is the unknown regression parameter. When equation(2) View the MathML sourceF(t)=exp(t)1+exp(t) Turn MathJax on we have the logistic regression model, which is the model we will consider from now on. However, our results can be extended to other link functions. It is well known that the maximum likelihood estimator (MLE) of View the MathML sourceβ can be severely affected by outlying observations. Croux et al. (2002) discuss the breakdown behavior of the MLE in the logistic regression model and show that the MLE breaks down to zero when severe outliers are added to a data set. In the last few decades, a lot of work has been done in order to obtain robust estimates of the parameter in this model and also in the more general framework of generalized linear models. Among others, we can mention the proposals given by Pregibon (1982), Stefanski et al. (1986), Künsch et al. (1989), Morgenthaler (1992), Carroll and Pederson (1993), Christmann (1994) and Bianco and Yohai (1996) and more recently Croux and Haesbroeck (2003) and Bondell, 2005 and Bondell, 2008. We are interested in testing parametric hypotheses about the regression parameter of the logistic regression model. Robust testing in this setting has received much less attention than robust estimation. Testing procedures based on classical estimates inherit the sensitivity of these estimators to atypical data, in the sense that a small amount of outlying observations can affect the level or the power of the tests. Testing procedures that, under contamination, retain a stable level and also a good power under specified alternatives, are desirable. The works of Heritier and Ronchetti (1994) and Cantoni and Ronchetti (2001) go in this direction. Heritier and Ronchetti (1994) introduce robust tests for a general parametric model, which includes logistic regression. Cantoni and Ronchetti (2001) define robust deviances based on generalizations of quasi–likelihood functions and propose a family of test statistics for model selection in generalized linear models. They also investigate the stability of the asymptotic level under contamination. In this paper we propose a Wald–type statistic based on a weighted version of the Bianco and Yohai (1996) estimator introduced by Croux and Haesbroeck (2003). Our proposal is a natural robustification of the classical Wald–type test, in the sense that the statistic of the test is a quadratic form based on robust estimators of the regression parameter and its asymptotic covariance matrix. We show that the asymptotic behavior of the proposed test is the same as that of its classical counterpart, that is, central χ2χ2 under the null hypothesis and noncentral χ2χ2 under contiguous alternatives. This paper is organized as follows. In Section 2 we briefly review some estimators related to the weighted estimator introduced by Croux and Haesbroeck (2003) and in Section 3 we state its asymptotic properties. In Section 4 we define the test statistic and we state its asymptotic distribution. In Section 5 we analyze the behavior of the pp-values of the classical and proposed statistics when an outlying observation with increasing leverage is added to a data set. By means of a simulation study, we illustrate in Section 6 the performance of the proposed test in terms of both level and power. Finally, in Section 7 we provide some concluding remarks.

In this paper, a robust version of the Wald-type test statistic is introduced. This proposal is based on a weighted Bianco and Yohai (1996) estimator, as defined by Croux and Haesbroeck (2003). We investigate the asymptotic distributions for the proposed testing procedure and we show that the asymptotic behavior of the proposed test is the same as that for its classical counterpart. The simulation study shows that the proposed tests keep their asymptotic level for moderate sample sizes and that they are still reliable, even in the presence of outliers, especially when the weights are based on the bisquare or the hard rejection functions, suggesting that these two families may be a good choice for the practitioner.