Flexible regression models are useful tools to calculate and assess threshold values in the context of minimum provider volumes



      The aim was to review different approaches for the derivation of threshold values and to discuss their strengths and limitations in the context of minimum provider volumes.

      Study Design and Setting

      The following methods for the calculation of threshold values are compared and discussed: The value of acceptable risk limit, the value of acceptable risk gradient, the benchmark value proposed by Budtz-Jørgensen and Ulm's breakpoint model. The latter is extended to account for two different breakpoints. The methods are applied to German quality assurance data concerning total knee replacement.


      The discussed methods for calculating threshold values differ in the kind of information that has to be specified beforehand. For the value of acceptable risk limit approach an absolute number, the acceptable risk, has to be predetermined. The value of acceptable risk gradient approach and the method of Budtz-Jørgensen require the specification of a relative change expressed in gradient and in odds, respectively. On the other hand, the threshold value according to the method of Ulm is defined as a parameter of a statistical model and no a priori specification is required.


      Each of the proposed methods has benefits and drawbacks. The choice of the most appropriate approach depends on the specific problem and the available data.


      To read this article in full you will need to make a payment

      Purchase one-time access:

      Academic & Personal: 24 hour online accessCorporate R&D Professionals: 24 hour online access
      One-time access price info
      • For academic or personal research use, select 'Academic and Personal'
      • For corporate R&D use, select 'Corporate R&D Professionals'


      Subscribe to Journal of Clinical Epidemiology
      Already a print subscriber? Claim online access
      Already an online subscriber? Sign in
      Institutional Access: Sign in to ScienceDirect


        • Halm E.A.
        • Lee C.
        • Chassin M.R.
        Is volume related to outcome in health care? A systematic review and methodologic critique of the literature.
        Ann Intern Med. 2002; 137: 511-520
        • Gandjour A.
        • Bannenberg A.
        • Lauterbach K.W.
        Threshold volumes associated with higher survival in health care: a systematic review.
        Med Care. 2003; 41: 1129-1141
        • Davoli M.
        • Amato L.
        • Minozzi S.
        • Bargagli A.M.
        • Vecchi S.
        • Perucci C.A.
        Volume and health outcomes: an overview of systematic reviews.
        Epidemiol Prev. 2005; 29: 3-63
        • Altman D.G.
        • Lausen B.
        • Sauerbrei W.
        • Schumacher M.
        The dangers of using ‘optimal’ cutpoints in the evaluation of prognostic factors.
        J Natl Cancer Inst. 1994; 86: 829-835
        • Bender R.
        Quantitative risk assessment in epidemiological studies investigating threshold effects.
        Biom J. 1999; 41: 305-319
        • Wetzel H.
        Mindestmengen zur Qualitätssicherung: Konzeptionelle und methodische Überlegungen zur Festlegung und Evaluation von Fallzahlgrenzwerten für die klinische Versorgung.
        Z ärztl Fortbild Qual Gesundh wes. 2006; 100: 99-106
        • Bender R.
        • Grouven U.
        Möglichkeiten und Grenzen statistischer Regressionsmodelle zur Berechnung von Schwellenwerten für Mindestmengen.
        Z ärztl Fortbild Qual Gesundh wes. 2006; 100: 93-98
        • Schräder P.
        • Grouven U.
        • Bender R.
        Können Mindestmengen für Knieprothesen anhand von Routinedaten errechnet werden? Ergebnisse einer Schwellenwertanalyse mit Daten der externen stationären Qualitätssicherung.
        Orthopäde. 2007; 36: 570-576
        • Hosmer D.W.
        • Lemeshow S.
        Applied logistic regression.
        2nd Edition. Wiley, New York2000
        • Ulm K.
        A statistical method for assessing a threshold in epidemiological studies.
        Stat Med. 1991; 10: 341-349
        • Küchenhoff H.
        An exact algorithm for estimating breakpoints in segmented generalized linear models.
        Comput Stat. 1997; 12: 235-247
        • Budtz-Jørgensen E.
        • Keiding N.
        • Grandjean P.
        Benchmark dose calculation from epidemiological data.
        Biometrics. 2001; 57: 698-706
        • Crump K.S.
        A new method for determining allowable daily intake.
        Fundam Appl Toxicol. 1984; 4: 854-871
        • Jones R.H.
        • Molitoris B.A.
        A statistical method for determining the breakpoint of two lines.
        Anal Biochem. 1984; 141: 287-290
        • Crump K.S.
        Calculations of benchmark doses from continuous data.
        Risk Anal. 1995; 15: 79-89
        • Molinari N.
        • Daurès J.-P.
        • Durand J.-F.
        Regression splines for threshold selection in survival data analysis.
        Stat Med. 2001; 20: 237-247
        • Küchenhoff H.
        • Carroll R.J.
        Segmented regression with errors in predictors: semi-parametric and parametric methods.
        Stat Med. 1997; 16: 169-188
        • Heller R.F.
        • Dobson A.
        Disease impact number and population impact number: population perspectives to measures of risk and benefit.
        BMJ. 2000; 321: 950-952
        • Panageas K.S.
        • Schrag D.
        • Riedel E.
        • Bach P.B.
        • Begg C.B.
        The effect of clustering of outcomes on the association of procedure volume and surgical outcomes.
        Ann Intern Med. 2003; 139: 658-665
        • Royston P.
        • Ambler G.
        • Sauerbrei W.
        The use of fractional polynomials to model continuous risk variables in epidemiology.
        Int J Epidemiol. 1999; 28: 964-974
        • Sauerbrei W.
        • Meier-Hirmer C.
        • Benner A.
        • Royston P.
        Multivariable regression model building by using fractional polynomials: description of SAS, STATA and R programs.
        Comput Statist Data Anal. 2006; 50: 3464-3485
        • Royston P.
        • Sauerbrei W.
        Building multivariable regression models with continuous covariates in clinical epidemiology.
        Methods Inf Med. 2005; 44: 561-571
        • Urbach D.R.
        • Austin P.C.
        Conventional models overestimate the statistical significance of volume-outcome associations, compared with multilevel models.
        J Clin Epidemiol. 2005; 58: 391-400
        • Panageas K.S.
        • Schrag D.
        • Localio A.R.
        • Venkatraman E.S.
        • Begg C.B.
        Properties of analysis methods that account for clustering in volume-outcome studies when the primary predictor is cluster size.
        Stat Med. 2007; 26: 2017-2035
        • Shahian D.M.
        • Normand S.-L.T.
        The volume-outcome relationship: from Luft to Leapfrog.
        Ann Thorac Surg. 2003; 75: 1048-1058
        • Christian C.K.
        • Gustafson M.L.
        • Betensky R.A.
        • Daley J.
        • Zinner M.J.
        The Leapfrog volume criteria may fall short in identifying high-quality surgical centers.
        Ann Surg. 2003; 238: 447-457
        • Christian C.K.
        • Gustafson M.L.
        • Betensky R.A.
        • Daley J.
        • Zinner M.J.
        The volume-outcome relationship: don't believe everything you see.
        World J Surg. 2005; 29: 1241-1244