Robust statistics is an extension of classical parametric statistics that specifically takes into account the fact that the assumed parametric models used by the researchers are only approximate. In this article, we review and outline how robust inferential procedures may routinely be applied in practice in the biomedical research. Numerical illustrations are given for the t-test, regression models, logistic regression, survival analysis and ROC curves, showing that robust methods are often more appropriate than standard procedures.
HuberPJ. Robust statistics. New York: Wiley, 1981.
2.
HampelFRRonchettiEMRousseeuwPJStahelWA. Robust statistics: the approach based on influence functions. New York: Wiley, 1986.
3.
MaronnaRAMartinRDYohaiVJ. Robust statistics: theory and methods. New York: Wiley, 2006.
4.
HeritierSCantoniECoptSVictoria-FeserM-P. Robust methods in biostatistics. Chichester: Wiley, 2009.
5.
RousseeuwPJLeroyAM. Robust regression and outlier detection. New York: Wiley-Interscience, 1987.
6.
MaronnaRA. Principal components and orthogonal regression based on robust scales. Technometrics2005; 47: 264–273.
7.
HubertMRousseeuwPJVanden BrandenK. ROBPCA: a new approach to robust principal components analysis. Technometrics2005; 47: 64–79.
8.
KaufmanLRousseeuwPJ. Finding groups in data. New York: Wiley, 1990.
9.
Cuesta-AlbertosJGordalizaAMatránC. Trimmed k-means: an attempt to robustify quantizers. Ann Stat1997; 25: 553–576.
10.
Garcia-EscuderoLAGordalizaA. Robustness properties of k means and trimmed k means. J Am Stat Assoc1999; 94: 956–969.
11.
GallegosMTRitterG. A robust method for cluster analysis. Ann Stat2005; 33: 347–380.
12.
FarcomeniA. Robust double clustering: a method based on alternating concentration steps. J Classif2009; 26: 77–101.
13.
HubertMVan DriessenK. Fast and robust discriminant analysis. Comput Stat Data Anal2004; 45: 301–320.
14.
CoptSVictoria-FeserMP. High-breakdown inference for mixed linear models. J Am Stat Assoc2006; 101: 292–300.
15.
HuberPJ. Robust estimation of a location parameter. Ann Math Statist1964; 35: 73–101.
16.
HeritierSRonchettiE. Robust bounded-influence tests in general parametric models. J Am Stat Assoc1994; 89: 897–904.
17.
VenablesWNRipleyBD. Modern applied statistics with S. New York: Springer, 2002.
18.
PironLTurollaAAgostiniMZucconiCSVenturaLToninPDamM. Motor learning principles for rehabilitation: A pilot randomised controlled study in poststroke patients. Neurorehabil Neural Repair2010; 24: 501–508.
19.
LiG. Robust regression. In: HoaglingDCTukeyJW (eds) Exploring data tables, trends, and shapes. Wiley: New York, 1985, pp.281–343.
20.
BrazzaleARDavisonACReidN. Applied asymptotics. Cambridge: Cambridge University Press, 2007.
21.
CookRDWeisbergS. Residuals and influence in regression. London: Chapman and Hall, 2007.
22.
AtkinsonACRianiM. Robust diagnostic regression analysis. New York: Springer-Verlag, 2000.
23.
WelshAHRonchettiE. A journey in single steps: robust onestep M-estimation in linear regression. J Stat Plan Infer1985; 103: 287–310.
24.
BiancoAMBoenteGdi RienzoJ. Some results for robust GM-based estimators in heteroscedastic regression models. J Stat Plan Infer2000; 89: 215–242.
25.
CarrollRJRuppertD. Robust estimation in heteroscedastic linear models. Ann Stat1982; 10: 429–441.
26.
GiltinanDMCarrollRJRuppertD. Some new estimation methods for weighted regression when there are possible outliers. Technometrics1986; 28: 219–230.
27.
WelshAHRichardsonA. Approaches to the robust estimation of mixed models. In: MaddalaGRaoC (eds) Handbook of statistics. Elsevier: Amsterdam, 1997, pp.343–385.
28.
CoptSHeritierS. Robust alternatives to the F-test in mixed linear models based on MM-estimates. Biometrics2007; 63: 1045–1052.
29.
RousseeuwPJ. Least median of squares regression. J Am Stat Assoc1984; 79: 871–880.
30.
HubertMRousseeuwPJVan AelstS. High-breakdown robust multivariate methods. Stat Sci2008; 23: 92–119.
31.
CarrollRJRuppertD. Transformation and weighting in regression. London: Chapman and Hall, 1988.
32.
BellioR. Algorithms for bounded-influence estimation. Comput Stat Data Anal2007; 51: 2531–2541.
33.
YohaiVJ. High breakdown-point and high efficiency robust estimates for regression. Ann Stat1987; 15: 642–656.
34.
MarazziA. Algorithms, routines, and S functions for robust statistics. Pacific Grove: Wadsworth and Brooks/Cole, 1993.
35.
MarkatouMBasuALindsayBG. Weighted likelihood equations with bootstrap root search. J Am Stat Assoc1998; 93: 740–750.
36.
AgostinelliCMarkatouM. A one-step robust estimator for regression based on the weighted likelihood reweighting scheme. Stat Prob Lett1998; 37: 341–350.
37.
McCullaghP. Quasi-likelihood and estimating functions. In: Hinkley DV, Reid N and Snell EJ (eds) Statistical theory and modelling. In Honour of Sir David Cox. London: Chapman and Hall, 1991, pp.265–286.
38.
AdimariGVenturaL. Quasi-profile loglikelihoods for unbiased estimating functions. Ann Inst Stat Math2002; 54: 235–244.
39.
BellioRGrecoLVenturaL. Adjusted quasi-profile likelihoods from estimating functions. J Stat Plan Infer2008; 138: 3059–3068.
40.
MarkatouMHettmanspergerTP. Robust bounded-influence tests in linear models. J Am Stat Assoc1990; 85: 187–190.
41.
MarkatouMHettmanspergerTP. Applications of the asymmetric eigenvalue problem techniques to robust testing. J Stat Plan Infer1990; 31: 51–65.
42.
HanfeltJJLiangKY. Approximate likelihood ratios for general estimating functions. Biometrika1995; 82: 461–477.
43.
OwenAB. Empirical likelihood. London: Chapman and Hall, 2001.
44.
AdimariGVenturaL. Quasi-likelihood from M-estimators: a numerical comparison with empirical likelihood. Stat Meth Appl2002; 11: 175–185.
45.
McKeanJWSheatherSJHettmanspergerTP. The use and interpretation of residuals based on robust estimation. J Am Stat Assoc1993; 88: 1254–1263.
46.
RuleADLarsonTSBergstralhEJSlezakJMJacobsenSJCosioFG. Using serum creatinine to estimate glomerular filtration rate: accuracy in good health and in chronic kidney disease. Ann Int Med2004; 141: 929–938.
47.
McCullaghPNelderJA. Generalized linear models, 2nd ed. London: Chapman and Hall, CRC, 1989.
48.
PregibonD. Resistant fits for some commonly used logistic models with medical applications. Biometrics1982; 38: 485–498.
49.
CopasJB. Binary regression models for contaminated data. J Roy Stat Soc, Ser B1988; 50: 225–265.
50.
MorgenthalerS. Least-absolute-deviations fits for generalised linear models. Biometrika1992; 79: 747–754.
51.
CarrollRJPedersonS. On robustness in the logistic regression model. J Roy Stat Soc, Ser B1993; 55: 693–706.
52.
BiancoAMYohaiVJ. Robust estimation in the logistic regression model. In: RiederH (ed.) Robust statistics, data analysis and computer intensive methods. New York: Springer, 1997, pp.17–34.
53.
MarkatouMBasuALindsayB. Weighted likelihood estimating equations: the discrete case with application to logistic regression. J Stat Plan Infer1997; 57: 215–232.
54.
Victoria-FeserM-P. Robust inference with binary data. Psychometrica2002; 67: 21–32.
55.
KünschHRStefanskiLACarrollRJ. Conditionally unbiased bounded influence estimation in general regression models, with applications to generalised linear models. J Am Stat Assoc1989; 84: 460–466.
56.
CantoniERonchettiE. Robust inference for generalised linear models. J Am Stat Assoc2001; 96: 1022–1030.
57.
WedderburnRWM. Quasi-likelihood function, generalised linear models, and the Gauss-Newton method. Biometrika1974; 61: 439–447.
58.
HeydeCC. Quasi-likelihood and its application. New York: Springer-Verlag, 1997.
59.
AdimariGVenturaL. Robust inference for generalized linear models with application to logistic regression. Stat Prob Lett2001; 55: 413–419.
60.
CoxDR. Regression models and life tables (with discussion). J Roy Stat Soc, Ser B1972; 34: 187–220.
61.
SamuelsS. Robustness for survival estimators. PhD thesis. Department of Biostatistics, University of Washington, 1978.
62.
BednarskiT. On sensitivity of Cox's estimator. Stat Decis1989; 7: 215–228.
63.
MinderCEBednarskiT. A robust method for proportional hazards regression. Stat Med1996; 15: 1033–1047.
64.
ReidNCrépeauH. Influence function for proportional hazards regression. Biometrika1985; 72: 1–9.
65.
ValsecchiMGSilvestriDSasieniP. Evaluation of long-term survival: use of diagnostics and robust estimators with Cox's proportional hazards models. Stat Med1996; 15: 2763–2780.
66.
CainKCLangeNT. Approximate case influence for the proportional hazards regression model with censored data. Biometrics1984; 40: 493–499.
67.
SchoenfeldD. Partial residuals for the proportional hazards regression model. Biometrika1982; 69: 239–241.
68.
GrambschPMTherneauTM. Proportional hazards tests and diagnostics based on weighted residuals. Biometrika1994; 81: 515–526.
69.
TherneauTMGrambschPMFlemingT. Martingale based residuals for survival models. Biometrika1990; 77: 147–160.
70.
NardiASchemperM. New residuals for Cox regression and their application to outlier screening. Biometrics1999; 55: 523–529.
71.
SasieniPD. Maximum weighted partial likelihood estimates for the Cox model. J Am Stat Assoc1993; 88: 144–152.
72.
SasieniPD. Some new estimators for Cox regression. Ann Stat1993; 21: 1721–1759.
73.
LinD. Goodness-of-fit analysis for the Cox regression model based on a class of parameter estimators. J Am Stat Assoc1991; 86: 725–728.
74.
SchemperMWakounigSHeinzeG. The estimation of average hazard ratios by weighted Cox regression. Stat Med2009; 28: 2473–2489.
BednarskiT. On a robust modification of Breslow's cumulated hazard estimator. Comput Stat Data Anal2007; 52: 234–238.
77.
BednarskiTNowakM. Robustness and efficiency of Sasieni-type estimators in the Cox model. J Stat Plan Infer2003; 115: 261–272.
78.
FarcomeniAVivianiS. Trimmed Cox regression for robust estimation in survival studies. Technical Report 7, Department of Statistics, Sapienza, University of Rome, 2010.
79.
ChakrabortyBChaudhuryP. On an optimization problem in robust statistics. J Comput Graph Stat2008; 17: 683–702.
80.
DavisonACHinkleyD. Bootstrap methods and their applications. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge: Cambridge University Press, 2006.
81.
AndrewsDFHerzbergAM. Data : a collection of problems from many fields for the student and research worker. New York: Springer-Verlag, 1985.
82.
BamberD. The area above the ordinal dominance graph and the area below the receiver operating characteristic graph. J Math Psy- col1975; 12: 387–415.
83.
KotzSLumelskiiYPenskyM. The stress-strength model and its generalizations: theory and applications. Singapore: World Scientific, 2003.
84.
PepeMS. The statistical evaluation of medical tests for classification and prediction. Oxford: Oxford University Press, 2003.
85.
AdimariGChiognaM. Partially parametric interval estimation of Pr(Y>X). Comput Stat Data Analysis2001; 51: 1875–1891.
86.
QinGSZhouXH. Empirical likelihood inference for the area under the roc curve. Biometrics2006; 62: 613–622.
87.
GrecoLVenturaL. Robust inference for the stress-strength reliability. Statistical Papers, 2010.
88.
MayerMPBukauB. Hsp70 chaperones: cellular functions and molecular mechanism. Cell Mol Life Sci2005; 62: 670–684.
89.
CorteseGVenturaL. Accurate likelihood on the area under the ROC curve for small samples. Technical Report 17, Department of Statistics, University of Padova, 2009.