Calculating the number needed to be exposed with adjustment for confounding variables in epidemiological studies

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6 530 R. Bender and M. Blettner / Journal of Clinical Epidemiology 55 (2002) of X,...,X k among the unexposed persons can be expressed as functions of the logistic regression coefficients, namely exp( ) and UER Thus, the absolute risk increase itself is a function of the logistic regression coefficients. Statistical software for logistic regression usually provides the covariance matrix for the estimated logistic regression coefficients, so it is possible to calculate the standard error (SE) of the estimated ARI by means of the multivariate delta method [6]. For these calculations matrix operations are required. Let B be the covariance matrix of the estimated logistic regression coefficients and be the row vector of the partial derivatives of ARI expressed as function of the logistic regression coefficients. The SE of the estimated ARI can be calculated by and an approximate 95% CI for ARI can be calculated by References n u = ---- p i n u i = SE( ARI) = B ' ARI ±.96 SE( ARI) [] Cook RJ, Sackett DL. The number needed to treat: a clinically useful measure of treatment effect. BMJ 995;30: [2] Laupacis A, Sackett DL, Roberts RS. An assessment of clinically useful measures of the consequences of treatment. N Engl J Med 988;38: [3] McQuay HJ, Moore A. Using numerical results from systematic reviews in clinical practice. Ann Intern Med 997;26: [4] Rembold CM. Number needed to screen: development of a statistic for disease screening. BMJ 998;37: [5] Bjerre LM, LeLorier L. Expressing the magnitude of adverse effects in case-control studies: The number of patients needed to be treated for one additional patient to be harmed. BMJ 2000;320: [6] Wu LA, Kottke TE. Number needed to treat: caveat emptor. J Clin Epidemiol 200;54: 6. [7] Smeeth L, Haines A, Ebrahim S. Numbers needed to treat derived from meta-analyses sometimes informative, usually misleading. BMJ 999; 38: [8] Feinstein AR. Indexes of contrast and quantitative significance for comparisons of two groups. Stat Med 999;8: [9] Altman DG. Confidence intervals for the number needed to treat. BMJ 998;37: [0] Rothman KJ, Greenland S. Modern epidemiology. Philadelphia: Lippincott-Raven Publishers; 998. p [] Altman DG, Bland JM. Treatment allocation in controlled trials: why randomise? BMJ 999;38:209. [2] Hosmer DW, Lemeshow S. Applied logistic regression. New York: Wiley; 989. [3] Bender R. Expressing the number needed to treat as a function of the odds ratio and the unexposed event rate (rapid response). ebmj (http:// [4] Lesaffre E, Pledger G. A note on the number needed to treat. Control Clin Trials 999;20: [5] Bender R. Calculating confidence intervals for the number needed to treat. Control Clin Trials 200;22:02 0. [6] Bishop YMM, Fienberg SE, Holland PW. Discrete multivariate analysis: theory and methods. Cambridge, MA: MIT Press; 975. [7] SAS. SAS/IML Software: Usage and Reference, Version 6, First Edition. Cary, NC: SAS Institute; 990. [8] Mühlhauser I, Bender R, Bott U, Jörgens V, Grüsser M, Wagener W, et al. Cigarette smoking and progression of retinopathy and nephropathy in type diabetes. Diabetic Med 996;3: [9] Bender R, Grouven U. Using binary logistic regression models for ordinal data with non-proportional odds. J Clin Epidemiol 998;5: [20] Sackett DL. On some clinically useful measures of the effects of treatment. Evidence-Based Med 996;:37 8. [2] Chatellier G, Zapletal E, Lemaitre D, Ménard J, Degoulet P. The number needed to treat: a clinically useful nomogram in its proper context. BMJ 996;32: [22] North D. Number needed to treat: absolute risk reduction may be easier for patients to understand. BMJ 995;30:269. [23] Newcombe RG. Confidence intervals for the number needed to treat: absolute risk reduction is less likely to be misunderstood. BMJ 999;38:765. [24] Hutton JL. Number needed to treat: properties and problems. J R Stat Soc A 2000;63: [25] Walter SD. Choice of effect measure for epidemiological data. J Clin Epidemiol 2000;53:93 9. [26] Altman DG. Statistics in medical journals: developments in the 980s. Stat Med 99;0: [27] Levy PS, Stolte K. Statistical methods in public health and epidemiology: a look at the recent past and projections for the next decade. Stat Meth Med Res 2000;9:4 55. [28] Altman DG, Deeks JJ. Comments on the paper by Hutton. J R Stat Soc A 2000;63:45 6. [29] McGettigan P, Sly K, O Connell D, Hill S, Henry D. The effects of information framing on the practices of physicians. J Gen Intern Med 999;4: [30] Altman DG, Andersen PK. Calculating the number needed to treat where the outcome is time to an event. BMJ 999;39: [3] Bender R, Grouven U. Logistic regression models used in medical research are poorly presented. BMJ 996;33:628. [32] Heller RF, Dobson A. Disease impact number and population impact number: population perspectives to measures of risk and benefit. BMJ 2000;32: [33] Davies HTO, Crombie IK, Tavakoli M. When can odds ratios mislead? BMJ 998;36: [34] Zhang J, Yu KF. What s the relative risk? A method of correcting the odds ratio in cohort studies of common outcomes. JAMA 998;280: 690.

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