BUCHER-CALCULATOR
to demonstrate the principles of the so called BUCHER method for indirect comparisons using a common comparator, how to extract the NNT (number needed to treat) out of Bucher's ratio
and how to construct a reliable NNT confidence for economic use by the Woolf modified Wilson Score (WmW).
12345678901234567890123456789012345678901234567890123456789012 123456789012345678901234567890123456789012345678 123456789012345678901234567890123456789012345678901234
1ststep: choose z-value to define the width of the confidence interval CI.  Choose the step of integration for the significancy-test.  Select the method to define the LR-CI (Bayes-Kernel).
%CI , z=
dx
Confidence interval of LR "by definition" or "by function"  
2ndstep: give inputs (absolute numbers n) to the fields a - d of the therapeutic 2x2 contingency table. The therapy B would be the so called "common reference comparator (f.i. warfarin)".
disease(+) disease(-)
bordersum
risk
odds
Therapy(+)
Therapy A
a
b
QS1
R1
H1
Placebo(-)
or Therapy B
c
d
QS2
R2
H2
bordersum
0-hypothesis
SS1
SS2
TOT
PVr
PVh
net benefit
lower lim
upper lim
risk difference R2-R1 = ARR
Number Needed to Treat = 1/ARR
hazard difference H2-H1 = AHR
reciprocal AHR = 1/AHR
123456789012345678901234567890 Relevancy of the comparison
Table 1) Therapy A : B   

  f.i. Dabigatran versus Warfarin
  in Patients with Atrial Fibrillation
  to prevent Hemorrhagic stroke
  Summary of RE-LY Trial, presented by
  Zack Dumont et al, Nov. 2010,
  www.RxFiles.ca
12345678901
log-transformation
risk ratios
RR=R1/R2
HR=H1/H2
LR=H1/PVh
value log(x=RR,HR,LR)
lower lim log(xll), lower lim
upper lim log(xul), upper lim
variance variance=SD²svSE²
do lnTF SD sv SE
gm=√(ll*ul) x = exp(log(x))
Significancy of the 2x2 contingency table 1), Therapy A : B
         Fisher's exact
Χ²-Test
123456789012345
e
Χ² value
bnk    *  10exp
p = α left of H0 p , ∫left 0→Χ²
[1-p]´= α´right of H0 [1-p],∫right Χ²→∞
gm = √(α*α´) α of Χ² = [1-p]/2
123456789012345678
log-transformation (lnTF)
123456789012345678901
x = log(x=RR,HR,LR)
log(xll), lower lim
log(xul), upper lim
variance = SD² sive SE²
SD sv SE
x = exp(log(x))
Significancy of the ratios in table 1)
Select the ratio to test by the onesided Gauss-test in log-transformation and to export down ↓ to table 3) :
RR HR LR
other ratio
Gauss-C
SD
p
[1-p]
Gauss α
1234567890123
disease(+) disease(-)
bordersum
risk
odds
Therapy(+)
Therapy C
a
b
QS1
R1
H1
Placebo(-)
or Therapy B
c
d
QS2
R2
H2
bordersum
0-hypothesis
SS1
SS2
TOT
PVr
PVh
net benefit
lower lim
upper lim
risk difference R2-R1 = ARR
Number Needed to Treat = 1/ARR
hazard difference H2-H1 = AHR
reciprocal AHR = 1/AHR
123456789012345678901234567890 Relevancy of the comparison
Table 2) Therapy C : B   

  f.i. Rivaroxaban versus Warfarin
  in Patients with Atrial Fibrillation
  to prevent Hemorrhagic stroke
  Summary of ROCKET Trial, presented by
  Margaret Jin, PHARMD, Sept. 2011,
  www.RxFiles.ca
12345678901
log-transformation
risk ratios
RR=R1/R2
HR=H1/H2
LR=H1/PVh
value log(x=RR,HR,LR)
lower lim log(xll), lower lim
upper lim log(xul), upper lim
variance variance=SD²svSE²
do lnTF SD sv SE
gm=√(ll*ul) x = exp(log(x))
Significancy of the 2x2 contingency table 2), Therapy C : B
clear for separate use of Χ²
Χ²-Test
123456789012345
Fisher's exact Test (binomial)
Χ² value
123456789012345678
p = α left of H0 p , ∫left 0→Χ²
[1-p]´= α´right of H0 [1-p],∫right Χ²→∞
gm = √(α*α´) α of Χ² = [1-p]/2
log-transformation (lnTF)
123456789012345678901
x = log(x=RR,HR,LR)
log(xll), lower lim
log(xul), upper lim
variance = SD² sive SE²
SD sv SE
x = exp(log(x))
Significancy of the ratios in table 2)
Select the ratio to test by the onesided Gauss-test in log-transformation and to export down ↓ to table 3) :
RR HR LR
other ratio
Gauss-C
SD
p
[1-p]
Gauss α
1234567890123
3rdstep: Compare Therapy A with Therapy C in the 2x2 contingency table 3) or by Bucher's method HRAC = HRAB / HRCB , using therapy B as common comparator.
disease(+) disease(-)
bordersum
risk
hazard
Therapy(+)
Therapy A
a
b
QS1
R1
H1
Placebo(-)
bs Therapy C
c
d
QS2
R2
H2
bordersum
0-hypothesis
SS1
SS2
TOT
PVr
PVh
net benefit
lower lim
upper lim
risk difference R2-R1 = ARR
Number Needed to Treat = 1/ARR
hazard difference H2-H1 = AHR
reciprocal AHR = 1/AHR
Table 3) Therapy A : C     direct comparison
select the method to define the ARR confidence interval
Standard Wald; Wilson Scores:BenderNewcombe; Katz;
traditional Wilson(1927), WmW new economic Standard
12345678901234567890123456 1234567890123456789012345
maths calculator ;  check CI-center (R or H ?)
12345678901
log-transformation
risk ratios
RR=R1/R2
HR=H1/H2
LR=H1/PVh
value log(x=RR,HR,LR)
lower lim log(xll), lower lim
upper lim log(xul), upper lim
variance variance=SD²svSE²
do lnTF SD sv SE
gm=√(ll*ul) x = exp(log(x))
Significancy of the 2x2 contingency table 3), Therapy A : C
clear for separate use of Χ²
Χ²-Test
123456789012345
Fisher's exact Test (binomial)
Χ² value
123456789012345678
p = α left of H0 p , ∫left 0→Χ²
[1-p]´= α´right of H0 [1-p],∫right Χ²→∞
gm = √(α*α´) α of Χ² = [1-p]/2
log-transformation (lnTF)
123456789012345678901
x = log(x=RR,HR,LR)
log(xll), lower lim
log(xul), upper lim
variance = SD² sive SE²
SD sv SE
x = exp(log(x))
Significancy of the ratios in table 3)
performed in the log-transformation (ln-TF)
1234567890123456789012345678901234567890
onesided Gauss-test: select the ratio of the
2x2 CT, which you want to test :
RR HR LR
other ratio
Gauss-C
SD
p
[1-p]
Gauss α
Indirect comparison of Therapy A with Therapy C by Bucher's method using a common reference comparator (Therapy B, f.i. warfarin). The formula is: HRAC = HRAB / HRCB.
Evaluation of the correct NNT out of BUCHER's ratioAC  
123456789012 disease + disease −
bordersum
ratio ID
CommonComp
TherapyB of A
TherapyB of C
c

d

QS2

R

H
Number Needed to Treat 1/ARR
LRAC ≠ LRAB / LRCB
1/AHR
BUCHER's method
ratio ID
Therapy A
Therapy C
  
ratioAC = xAB / xCB
  
Confidence → ratioAC-CI = exp(log(ratioAC) ± z*√VRAC)   
-
Difference (xCB − xAB) = 1 / (xCB − xAB) =   
for the NNT-CI we need number n of Therapy A & C (QS)
to get correct results for x=LR you have to put the vertical bordersums of PV
(SS1/SS2) into the fields of the common comparator (see left side) .
onesided Gauss-test of Bucher's ratio :
Gauss-C
SD
p
[1-p]
Gauss α
varianceAC

VRAC =

SD²AB +

SD²CB
= VRAC
(see table 1 and 2)
√VRAC
test gm of SD²AB*SD²CB
The so called Bucher method for "indirect comparisons using a common comparator" is recommended as a preferred statistical method to estimate the hazard ratio and its confidence. It is mostly used to compare the results of meta-analyses. In this calculator we present direct and indirect methods to compare the studyresults in the 2x2 contingency table and with Bucher's method. The significancy of the 2x2 contingency table is tested by Fisher's exact test and the Χ-square test and may be compared with the α-error of the onesided Gauss-test with the exponential variance of the efficiency-ratios in log-transformation. Furthermore we discuss several methods to evaluate the confidence of the absolute risk reduction (ARR) and present the so by us called "Woolf modified Wilson score" (WmW score) − scoring the reciprocal values of the four fields (Woolf variance) as exponential variance of the hazard and of the R-to-H-transformed risk (Wilson's principle) − as a new international "standard method for economic use" to evaluate the confidence of any proportion and espescially of the confidence of the absolute riskreduction (ARR = R2−R1). This calculator is thought to be a instrument easy to handle and to control studyresults, calculated by "blackbox" software programmes, which we often don't know exactly.

Die sogenannte Bucher Methode zum "indirekten Vergleich" von Studienresultaten mit "gemeinsamer Bezugsgrösse" wird allgemein als bevorzugte Methode zur Einschätzung von Hazard-Verhältnissen und ihrer Konfidenz in Metaanalysen empfohlen. In diesem Rechner stellen wir direkte und indirekte Methoden zum Vergleich von neuen Therapien in der 4-Feldertafel und anhand der Bucher-Methode vor. Die Signifikanz der 4-Feldertafel wird anhand des Fisher's exact und des Χ²-Testes ermittelt und derjenigen eines einseitigen Gauss-Testes der exponentiellen Varianz der hazard ratio (HR) oder riskratio (RR) in logarithmischer Transformation gegenübergestellt, so dass die Resultate der drei Signifikanz-Methoden miteinander verglichen werden können. Weiterhin erläutern wir einige Methoden zur Berechnung der Konfidenz der absoluten Risiko Reduktion (ARR) und präsentieren die von uns sogenannte "Woolf modifizierten Wilson Methode" (WmW Methode) − welche die Summe der reziproken Werte der 4-Felder-Tafel (Woolf-Varianz) als exponentielle Varianz der hazard-Proportion und der R-zu-H-transformierten risk-Proportion (Wilson-Prinzip) nutzt - als neue, für den ökonomischen Gebrauch taugliche, internationale Standard-Methode zur Konfidenzberechnung jedweder Proportion, speziell aber der Konfidenz der absoluten Risikoreduktion (ARR) und der Number Needed to treat (NNN, 1/ARR). Dieser Rechner ist gedacht als ein handliches Instrument im Praxisalltag zur Qualitätskontrolle und Überprüfung von Studienresultaten, die statistisch durch eine uns oft nicht genauer bekannte Software ("blackbox") aufgearbeitet sind.

Literature:
• 1. Lars Hvilsted Rasmussen, Torben Bjerregaard Larsen, Tina Graungaard, Flemming Skjøth, Gregory Y H Lip. Primary and secondary prevention with new oral anticoagulant drugs for stroke prevention in atrial fibrillation: indirect comparison analysis. BMJ 2012;345:e7097 doi: 10.1136/bmj.e7097 (Published 5 November 2012).
• 2. Bucher HC, Guyatt GH, Griffith LE, Walter SD. The results of direct and indirect treatment comparisons in meta-analysis of randomized controlled trials. J Clin Epidemiol 1997;50:683-91.
• 3. Diener HC, Connolly SJ, Ezekowitz MD, Wallentin L, Reilly PA, Yang S, et al, for the RE-LY Study Group. Dabigatran compared with warfarin in patients with atrial fibrillation and previous transient ischaemic attack or stroke: a subgroup analysis of the RE-LY trial. Lancet Neurol 2010;9:1157-63.
• 4. Margaret Jin, PHARMD. RXFILES TRIAL SUMMARY of RE-LY: Dabigatran versus Warfarin in Patients with Atrial Fibrillation (AF). RxFiles Academic Detailing Service, Saskatoon City Hospital, 701 Queen Street, Saskatoon, SK S7K 0M7 www.RxFiles.ca, Nov.2010
• 5. Margaret Jin, PHARMD. RXFILES TRIAL SUMMARY of ARISTOTLE: Apixaban vs Warfarin in patients with Atrial Fibrillation (AF). RxFiles Academic Detailing Service, Saskatoon City Hospital, 701 Queen Street, Saskatoon, SK S7K 0M7 www.RxFiles.ca, Sept.2011
• 6. Zack Dumont, Debbie Bunka. RXFILES TRIAL SUMMARY of ROCKET-AF: Rivaroxaban vs Warfarin in patients with Atrial Fibrillation (AF). RxFiles Academic Detailing Service, Saskatoon City Hospital, 701 Queen Street, Saskatoon, SK S7K 0M7 www.RxFiles.ca, Sept.2011
• 7. Ralf Bender, PhD Department of Epidemiology and Medical Statistics, School of Public Health, University of Bielefeld, Bielefeld, Germany (ralf.bender@uni-bielefeld.de). Controlled Clinical Trials 22:102–110 (2001), ©Elsevier Science Inc. 2001 0197-2456/01/, 655 Avenue of the Americas, New York, NY 10010
• 8. Bender R. Epidemiologie & Medizinische Statistik, Fakultät für Geisteswissenschaften Universität Bielefeld. Berechnung von Konfidenzintervallen für die Zahl "Number Needed to Treat" (NNT). Giessener Vorträge 2000.
• 9. Bender R. Interpretation von Effizienzmassen der Vierfeldertafel für Diagnostik und Behandlung. Medizinische Klinik 2001;96:116-21 (Nr.2).
• 10. Newcombe Robert G. Interval Estimation for the Differences between Indipendent Proportions: Comparison of eleven Methods. Statistics in Medicine, 17, 873-890 (1998).
• 11. Newcombe, Robert G. Two-Sided Confidence Intervals for the Single Proportion: Comparison of Seven Methods, Statistics in Medicine, 17, 857-872 (1998).
• 12. Katz D, Baptista J, Azen SP, Pike MC. Obtaining confidence intervals for the risk ratio in cohort studies. Biometrics 78:469-474 (www.epa.gov/ncea/ets/pdfs/appendd.pdf) .
• 13. Woolf B (1955). On estimating the relation between blood group and disease. Ann. Human Genetics 1955, 19, 251-253 .
• 14. Wilson E.B. Probable Inference, the Law of Succession and Statistical Inference. Journal of the American Statistical Association, 22,209-212(1927).
• 15. Bayes ET (1763). An assay toward solving a problem in the doctrin of chance. Philos Trans R Soc Lond (Biol) 53: 370 - 418
• 16. Fagan TJ. Nomogram for Bayes theorem. N Engl J Med 1975;293:257
• 17. Keiser O.M., Kirchgraber U. Testen von Hypothesen. ETH-Leitprogramm für das Gymnasium. Freies Gymnasium in Zürich und ETH Zürich, Departement Mathematik. Juni 1998.
• 18. Cavalli-Sforza L, Lorenz RJ. Biometrie. Grundzüge biologisch-medizinischer Statistik. Gustav Fischer Verlag, Stuttgart. 3.Aufl.1972.
• 19. Wissenschaftliche Tabellen Geigy, Vol 3 Statistik, 8.Auflage April 1980, Ciba-Geigy Limited, Basle, Switzerland.


Autor:
Dr. med. Franz Paul Ackermann-Ball
Spezialarzt FMH Innere Medizin
Postfach 105 , CH-4601 Olten.


7.02.2013
Letzte Revision 25.02.2013
BucherMethod.html, Version 2.3
Special Edition kardiolab Olten , VEMS (www.physicianprofiling)
Dr. med. M. Romanens , Facharzt Kardiologie FMH
Ziegelfeldstr.1, CH-4600 Olten / Switzerland


freeware