Vol. 26, No. 2, (2005) Rule of Three Statistical Inference for the Occurrence Probability of Rare Events Rule of Three and Related Topics Manabu

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1 Vol. 26, No. 2, (25) Rule of Three Statistical Inference for the Occurrence Probability of Rare Events Rule of Three and Related Topics Manabu Iwasaki and Kiyotaka Yoshida Department of Computer and Information Science, Faculty of Science and Technology, Seikei University For the occurrence of a rare event A such as a severe adverse drug reaction, there exists the Rule of Three to remind practitioners that absence of evidence is not evidence of absence. The Rule of Three actually says that even if the event A was not observed among n patients it would be quite possible to observe three events among other n patients. The present paper examines this useful rule in detail and also extends it to a testing problem for occurrence probability of A. First, the Rule of Three is extended to the case that the number of the event observed among the first n patients is more than zero. We give rules that when k (> ) events were observed among n patients, n k events would be possibly observed among other n patients. Next, a testing procedure is introduced to examine whether the occurrence probabilities of A for two populations are the same under the condition that k events were observed among n patients for one population. It will be shown that the relevant probability distribution is a negative binomial, and then critical regions for small k s are given. For a possible application of the procedure, we mention the signal detection for spontaneous reporting system of adverse drug reaction. Key words: detection. adverse drug reaction, negative binomial distribution, rare events, signal. Rule of Three Uchiyama (25) absence of evidence is not evidence of absence (rule of three for zero events) R (Rule of Three) n n 3 Received August 25. Revised September 25. Accepted September 25.

2 54 n,, 3 3 absence of evidence is not evidence of absence Senn (997, Section 5.2) Rule of Three 2 A n p p 95% p U n 3 Rule of Three Rule of Three (R ) R n n 5 R 2 n 2 n 7 R 3 n 3 n 8 R k n k n n k k 2 2 n cn R k n k cn k 3 R k n n A p k n k n n k Rule of Three R n n 4 R, R 2 R 3 R n n 6

3 Rule of Three 55 R 2 n 2 n 8 R 3 n 3 n R k n k n m k 3 n n m k k =,,2 m k = n k + k = 3 n 3 = 8 m 3 = k 3 m k > n k + (24) 4 5 Rule of Three 2. Rule of Three Rule of Three A p n A X X = p p U α Pr(X = p = p U ) = ( p U ) n = α (2.) p U = α /n (2.) 2 nlog( p U ) = log α p U log( p U ) p U p U (log α)/n p U n E[X p = p U ] = np U = n( α /n ) (2.2) E[X p = p U ] = np U log α (2.3) n α =.5 log(.5) np U A 95% p U n 3 Rule of Three (2.2) (2.3) n p λ = np Rule of Three X λ

4 ( α)% n =,, (2.2) Poisson (2.3) X = λ ( α)% λ U Pr(X = λ = λ U ) = exp( λ U ) = α λ U = log α λ U (2.3) n α (2.2) (2.3) α =.5 n = 2.59 n = n Rule of Three (2.2) A k X = k ( α)% p U Pr(X 5 k p = p U ) = kx x= nc xp x U ( p U ) n x = α p U Clopper-Pearson p U F p U = k + k + + (n k)/f 2(k+),2(n k) (α) (98) F 2(k+),2(n k) (α) (2(k + ), 2(n k)) F α% p = p U n np U Pr(X k λ = λ U ) = kx x= (2.4) λ x U x! e λ U = α (2.5) λ U λ U k = (24) 2.2 α =.5 np U λ U k = Rule of Three k =, k = 5 k = 2 7 k = 3 8 R, R 2 R 3

5 Rule of Three k 95% 3. Ω A p n A X k Ω A p p c p = cp H : c = vs. H : c > (3.) Ω p Ω p c p p Ω n A Y Ω Ω n n n n n n Y 2 p p p Ω X = k p p Pr(X = k) = nc k p k ( p) n k f(p;k) = 8 >< >: B(k +,n k + ) pk ( p) n k ( < p < ) B(α,β) α, β B(α,β) = Z ( ) t α ( t) β dt p (3.) Y

6 58 p(y;n,k) p p(y;n,k) = Pr(Y = y X = k) = Z nc yp y ( p) n y B(k +, n k + ) pk ( p) n k dp = nc y B(k +, n k + ) Z = nc y B(y + k +, 2n y k + ) B(k +, n k + ) p y+k ( p) 2n y k dp = nc y (k + ) y(n k + ) n y (n + 2) n (3.2) (m) y (m) y = m(m + ) (m + y ) (3.2) (n, k +, n k + ) beta-binomial distribution n p(y;k) = lim n p(y;n,k) = y+kc k 2 y+k+ (3.3) (3.3) k + /2 negative binomial distribution NB(k +, /2) k = Ω A (3.3) p(y : ) = (/2) y+ /2 Ω A X λ X = k λ k + Pr(X = k) = λk k! e λ = Γ(k + ) λ(k+) e λ λ Y Pr(Y = y X = k) = = Z k! λk e λ y! λy e λ dλ = k!y! Z (y + k)! Γ(y + k + ) = 2 y+k+ k!y! k!y! (3.3) λ (y+k) e 2λ dλ y+k+ 2 = y+kc y+k+ x (3.4) 2 (3.) Y 3. Ω X k,, 2 3 Ω n A y P - P = Pr(y Y ) mid-p mid-p =.5Pr(Y = y ) + Pr(y < Y ).5 mid-p Armitage and Berry (994) (993) (24) Y P - α P - α

7 Rule of Three Ω P - mid-p 5% k = y = 4 P y = 5 P -.33 Y 5 5% 3.3% mid-p y = 4 mid-p.469 5% 5% k = Y 4 k = Y 6 k = 2 Y 8 k = 3 Y R, R, R 2 R 3 mid-p P - Y 5 (k = ) Y 7 (k = ) Y 9 (k = 2) Y (k = 3) Ω, Ω A (3.) c > Y Ω cλ Z Z Pr(Y = y X = k) = k! λk e λ y! (cλ)y e cλ dλ = k!y! cy λ (y+k) e (c+)λ dλ c y (y + k)! c y = Γ(y + k + ) = (c + ) y+k+ k!y! k!y! (c + ) y+k+ k+ c y = y+k C y. (3.5) c + c + k + /(c + ) NB(k +, /(c + )) c = (3.3) (3.4) (3.5) Ω Ω p n cn Y (3.5) Y n bn b p cp c (3.) Y NB(k +, /(b + )) NB(k +, /(bc + )) Y (3.5) (Yoshida and Iwasaki)

8 Senn (997) Chapter 23 signal detection (24) (25) p n n 2 n p n + 2 n 2 n 22 n 2p n 2+ m n m n m2 n mp n m+ n + n +2 n +p n ++ = / Proportional Reporting Ratios (PRR) (Evans, Waller and Davis, 2) n n 2 n + n 2 n 22 n 2+ n + n +2 n n (a) n (b) n PRR P RR = (n /n +)/(n 2/n 2+) 3

9 Rule of Three 6 () P RR 2 (2) n ++( n n 22 n 2n 2 n ++/2) 2 4 n +n 2+n +n +2 (3) n 3 PRR Van Puijenbroek, et al. (22) (3) n 4 3 k = 4 (24) (25) R k 5. Rule of Three Rule of Three log (24) p n.95 n 3/p a ( p) n a n n log( a)/log( p) p log( p) p n log( a)/p a =.95 log(.5) 3 n 3/p n p = 3/n p = Rule of Three 4

10 62 Rule of Three 2 Senn (997) (A) No Armitage, P. and Berry, G. (994) Statistical Methods in Medical Research, Third Edition. Oxford: Blackwell. Evans, S. W., Waller, P. C. and Davis, S. (2). Use of proportional reporting ratios (PRRs) for signal generation from spontaneous adverse reaction reports. Pharmacoepidemiology and Drug Safety,, (24). 23 (25). 24 (993). mid-p value (24). (25). No (24) (98) 2 Uchiyama, A. (25). Management of safety information from clinical trials. DIA Tutorial CIOMS Initiative for Safety Information (February 8, 25). Senn, S. (997). Statistical Issues in Drug Development. Chichester: John Wiley & Sons. Van Puijenbroek, E. P., Bate, A, Leufkens, H. G. M., Lindquist, M., Orre, R. and Egberts, A. C. G. (22). A comparison of measures of disproportionality for signal detection in spontaneous reporting systems for adverse drug reaction. Pharmacoepidemiology and Drug Safety,, 3-.

11 Rule of Three 63 (24) Yoshida, K. and Iwasaki, M. A new testing procedure for the probability of rare events (submitted for publication).

Vol. 29, No. 2, (2008) FDR Introduction of FDR and Comparisons of Multiple Testing Procedures that Control It Shin-ichi Matsuda Department of

Vol. 29, No. 2, (2008) FDR Introduction of FDR and Comparisons of Multiple Testing Procedures that Control It Shin-ichi Matsuda Department of Vol. 29, No. 2, 125 139 (2008) FDR Introduction of FDR and Comparisons of Multiple Testing Procedures that Control It Shin-ichi Matsuda Department of Information Systems and Mathematical Sciences, Faculty