Positive breakdown point estimation in categorical regression mo dels Andreas Christmann University of Dortmund, HRZ and SFB 475, D-44221 Dortmund, GERMANY 1. Intro duction Positive breakdown point estimators for regresssion mo dels with discrete resp onse variables have not received the same attention as robust estimators for linear regression mo dels. It will be shown that there exist strongly consistent p ositive breakdown point estimators in many re- gression mo dels, if large strata assumptions are valid. Poisson regression and logistic regression are treated as imp ortant sp ecial cases. The idea is to transform the original data set by an appropriate transformation to an approximately linear regression mo del with approximately normal errors, and then use the least median of squares estimator LMS or the least trimmed squares estimator LTS, both prop osed by Rousseeuw घ1984ङ, for the transformed data set to estimate the unknown parameter vector. 2. Results T nࣾp Let n and m ;:::; m be p ositive integers, where n is xed. Let X = घx ;:::;x ङ 2 IR 1 n 1 n be a design matrix with full column rank p < n. Let Y , 1 ऊ i ऊ n, be random variables i;m i on the measurable space घ ; Aङ, where is either a subset of nonnegative integers and A the set of all subsets of , or = IR and A the Borel- algebra. De ne P = Y =m , i;m i;m i i i 1 ऊ i ऊ n, and h =[[ n=2]]+[[घp +1ङ=2]], where [[r ]] stands for the largest integer less than or 0 equal to r . Let h 2fh ;:::;ng be xed. Of sp ecial imp ortance is h = h or h ए [[0:75n]]. Let 0 0 g : IR ! A ई IR be a strictly increasing and continuously di erentiable function, such that ࣿ 1 g घuङ = घd=duङg घuङ 2 घ0; 1ङ, u 2 A. Let v : IR ! घ0; 1ङ be a continuous function. We consider parametric regression mo dels H with the following prop erties. The random variables 0 are indep endent with cumulative distribution function F such that ;:::;Y Y i;m n;m 1;m n 1 i T P g घx ई ङ i;m 1=2 i i T D wp1 =ङ Nघ0; 1ङ ; min घm ङ !1; घ1ङ P ! g घx ई ङ ; m i i;m i i i T 1ऊiऊn ई ङ v घx i p where ई 2 IR . De ne 1 g घP ङ 1 i;m 1=2 1=2 i ~ ~ Y = m ; X = m x 1 ऊ i ऊ n. घ2ङ i;m i;m i i i i i ࣿ 1 ࣿ 1 g घP ङv घg घP ङङ g घP ङv घg घP ङङ i;m i;m i;m i;m i i i i T ~ ~ ;:::;Y Under H the transformed resp onse vector घY ङ follows approximately a linear n;m 0 1;m n 1 T ~ ~ ~ regression mo del with sto chastic design matrix X = घX ;:::; X ङ , which dep ends on 1;m n;m n 1 1 ࣿ Y ,1ऊ i ऊ n, min घm ङ !1. In Table 1 the functions g , g , g , and v for two imp ortant i;m i i i parametric mo dels H are sp eci ed: logistic regression and Poisson regression. 0 Table 1: Examples of parametric mo dels H 0 1 ࣿ A g घuङ g घuङ g घuङ v घuङ P Y i;m i T 1 1=2 Biघm ; ࣿघx ई ङङ घ0; 1ङ ࣿघuङ logitघuङ [uघ1 uङ] [ࣿघuङघ1 ࣿघuङङ] i i T 1=2 Poiघm exp [x ई ]ङ घ0; 1ङ exp घuङ logघuङ 1=u [exp घuङ] i i The least median of weighted squares estimator घLMWSङ and the least trimmed weighted squares estimator घLTWSङ are de ned as the least median of squares estimator LMS and the least trimmed squares estimator LTS, b oth prop osed by Rousseeuw घ1984ङ, for the transformed data set to estimate the unknown parameter vector, c.f. Christmann घ1994,1998ङ. This ap- proach has several advantages. It can be shown that LMWS and LTWS inherit the p ositive breakdown point and exact t prop erty from LMS and LTS. Further, LMWS and LTWS are strongly consistent under a large sup ermo del of H which even allows some kind of over- or 0 underdisp ersion, the o ccurrence of multiple outliers and violations of the assumption that all resp onse variables are indep endent. There exist software to compute such estimates and cor- resp onding robust residual plots, e.g. S-PLUS. An S-PLUS function for LMWS and LTWS can be downloaded from StatLib घhttp://lib.stat.cmu.edu/S/ hbdp ङ. For group ed data the dimensions of the design matrix are often small to mo derate such that the least median of weighted squares estimator can sometimes be computed exactly even for very large groups. These estimators can b e useful as starting vectors for other robust estimators and can b e help- ful to nd outliers and leverage p oints. The b ehaviour of b oth estimators is compared with the b ehaviour of the ML-estimator and of M-estimators prop osed byKunsch, Stefanski and Carroll घ1989ङ in a simulation study under a logistic regression mo del and under a logistic regression mo del with approximately 25क mo derate or 25क extreme outliers in the sense of Davies n and Gather घ1993ङ. In the situations considered in the simulations LMWS and LTWS have a much smaller bias and mean squared error than the ML-estimator or M-estimators if there are extreme outliers. The application of b oth estimators is illustrated using a data set from a n multi-disciplinary pro ject, c.f. Jaeger et al. घ1997ङ. The aim is to detect risk factors for the capture rate of thrombi of inferior vena cava lters. REFERENCES Christmann, A. घ1994ङ. Least median of weighted squares in logistic regression with large strata. Biometrika, 81, 413-417. Christmann, A. घ1998ङ. On p ositive breakdown p oint estimators in regression mo dels with dis- crete resp onse variables. Habilitation thesis. University of Dortmund, Department of Statistics. Jaeger, H.J., Mair, T., Geller, M., Kinne, R.K., Christmann, A., Mathias, K.D. घ1997ङ. A Physiologic In Vitro Mo del of the Inferior Vena Cava with a Computer-Controlled Flow System for Testing of Inferior Vena Cava Filters. Investigative Radiology, 32, 511-522. Rousseeuw, P.J. घ1984ङ. Least Median of Squares Regression. J. Amer. Statist. Asso c., 79, 871-880. उ उ RESUME Deux estimateurs convergents avec un point de rupture positive ont etउe उetudiउe pour des modउeles de rउegression a variable dउependante catउegoriel le. Larउegression logistique et la rउegression de Poisson sont considउerउes plus en detail. Lecomportement pour des tail les nies est उetudiउepar une उetude de simulation. L'application des deux estimateurs est il lustउee dans deux examples du science mउedical..