Mantel-Haenszel 2008 6 12 ) 2008 6 12 1 / 39
Mantel & Haenzel 1959) Mantel N, Haenszel W. Statistical aspects of the analysis of data from retrospective studies of disease. J. Nat. Cancer Inst. 1959; 224): 719 748. 1 ), ),,, ) 2 2 2 3 1 2 2 Mantel-Haenszel 1:1 2 j ) 4 2 2, Mantel-Haenszel R, R 1, R 2, R 3, R 4 1:1 ) 2008 6 12 2 / 39
1 ), ),,, ), pp.719 730 2 2 2 3 1 4 pp.730 746 2 2, i j ) 2008 6 12 3 / 39
,,,,, 対象集団 追跡 時間 発症無 発症有 ) 2008 6 12 4 / 39
,,,, 対象集団? 調査 時間 発症無 発症有 ) 2008 6 12 5 / 39
Mantel & Haenzel 1959) : John Snow,,, ) 2008 6 12 6 / 39
2 2 : q 1 1 q 1 1 q 2 1 q 2 1 Y 11 m 1+ Y 21 m 2+ m ++ Y 11 Bm 1+, q 1 ) Y 21 Bm 2+, q 2 ) 1) ) 2008 6 12 7 / 39
2 2 : p 1 1 p 1 1 p 2 1 p 2 1 X 11 n 1+ X 21 n 2+ n ++ X 11 Bn 1+, p 1 ) X 21 Bn 2+, p 2 ) 2) ) 2008 6 12 8 / 39
ϕ ris ratio) ϕ = q 1 q 2 3) ψ odds ratio), ψ = q 1/1 q 1 ) = p ) 1/1 p 1 ) q 2 /1 q 2 ) p 2 /1 p 2 ) 4) q 1 = 0.06, q 2 = 0.03 ϕ = 0.06 0.06/0.94 = 2, ψ = = 2.06 5) 0.03 0.03/0.97 ) 2008 6 12 9 / 39
[8] ψ D, Case-Control ψ E D, D, E, Ē 7) ψ D = PrD E)[1 PrD Ē)] PrD Ē)[1 PrD E)], ψ E = PrE D)[1 PrE D)] PrE D)[1 PrE D)] ψ D = Pr E D) Pr D) Pr Ē D) Pr D) Pr E D) Pr D)+Pr E D) Pr D) Pr Ē D) Pr D)+Pr E D) Pr D) Pr Ē D) Pr D) Pr E D) Pr D) Pr Ē D) Pr D)+Pr E D) Pr D) Pr Ē D) Pr D)+Pr E D) Pr D) = Pr E D)[1 Pr E D)] Pr E D)[1 Pr E D)] = ψ E 6) 7) ψ D ψ E ) 2008 6 12 10 / 39
, ψ p 2, X +1 = X 11 + X 21 p 2 /1 p 2 ) n1+ ) ψp ) x11 2 1 p ) n1+ x 2 11 PrX 11 = x 11, X 21 = x 21 ) = 1 p 2 1 ψ) 1 p 2 1 ψ) x 11 n2+ x 21 [ = 1 + exp ) p x 21 2 1 p 2) n 2+ x 21 { log ψ + log [ { 1 + exp log ) n1+ n2+ x 11 x 21 ) exp p 2 p 2 1 p 2 )}] n1+ )}] n2+ 1 p 2 { x 11 log ψ + x +1 log p )} 2 1 p 2 8) ) 2008 6 12 11 / 39
ˆψ ˆψ = X 11/n 1+ X 11 ) X 21 /n 2+ X 21 ) : c = 0, : c = 1/2 ) H 0 : ψ = 1 H 1 : ψ 1 9) 10) X 2 = X 11n 2+ X 21 ) X 21 n 1+ X 11 ) c) 2 n 1+ n 2+ X +1 n ++ X +1 ) χ 2 1) 11) ) 2008 6 12 12 / 39
) : X 11 n 1+ n 2+ X +1 = n +1 n ++ p 2 /1 p 2 ) X +1 = X 11 + X 21, ψ ) 2008 6 12 13 / 39
PrX 11 = x 11 X +1 = n +1 ) = PrX 11 = x 11, X +1 = n +1 ) PrX +1 = n +1 ) = n1+ ) x 11 n1+ u Ω X n 2+ n +1 x 11 ) u ) n 2+ n +1 u ψ x 11 ) ψ u 12) Ω X X +1 = n +1 Ω X = {u Z + max0, n +1 n 2+ ) u minn 1+, n +1 )} 13) Ω S Ω S = {u, v) Z 2 + 0 u n 1+, 0 v n 2+ } 14) ) 2008 6 12 14 / 39
ˆ ψ C 15) 15) =0 x 11 = E[X 11 X +1 = n +1, ψˆ C ] 15) ) ) n1+ n 2+ u u ψˆ C u n u Ω +1 u E[X 11 X +1 = n +1, ψˆ X C ] = ) ) 16) n1+ n 2+ u ψˆ C u n +1 u u Ω X ) 2008 6 12 15 / 39
Fisher s exact test), 10), X 11 12) P, ) ) PrX 11 = x 11 X +1 = n +1, ψ = 1) = n1+ x 11 n 2+ n +1 x 11 ) 17) n++ n +1 a) P Value = 2 PrX 11 x 11 ) b) P Value = PrZ F z F ), Z F = PrX 11 ), z F = Prx 11 ) c) P Value = Pr X 11 E 0 [X 11 ] x 11 E 0 [X 11 ] ) 18) ) 2008 6 12 16 / 39
Stratified analysis) 2 2? 2 2, Stratified analysis) Subgroup analysis) Multivariate analysis), ) 2008 6 12 17 / 39
2 2 2 2, : 2 2 p 1 1 p 1 1 p 2 1 p 2 1 X 11 n 1+ X 21 n 2+ n ++ ) 2008 6 12 18 / 39
, X +1 = X 11 + X 21 p 2 /1 p 2 ) PrX 11 = x 11, X 21 = x 21 ) = [ { 1 + exp log ψ + log [ { p 2 1 + exp log ) ) n1+ n2+ { exp x 11 x 21 p 2 1 p 2 )}] n1+ )}] n2+ 1 p 2 x 11 log ψ + x +1 log p )} 2 1 p 2 19) ) 2008 6 12 19 / 39
summary relative ris/common odds ratio) ψ 1 = ψ 2 = = ψ = ψ 20) PrX 11 = x 11, X 21 = x 21 ) = [ { p )}] 2 n1+ 1 + exp log ψ + log 1 p 2 [ { p 2 1 + exp log ) n1+ n2+ x 11 x 21 ) exp )}] n2+ 1 p 2 { x 11 ) log ψ + p )} 2 x +1 log 1 p 2 21) ) 2008 6 12 20 / 39
X 11 n 1+ n 2+ X +1 = n +1 n ++, p 2 /1 p 2 ) X +1 = X 11 + X 21 Ω X ) PrX 11 = x 11 X +1 = n +1 ) = n1+ u Ω X x 11 n1+ n 2+ n +1 x 11 ) x 11 n 2+ ) ψ x 11 n +1 x 11 ) ψ u 22) Ω X = {u Z + max0, n +1 n 2+ ) u minn 1+, n +1 )} 23) ) 2008 6 12 21 / 39
ˆ ψ C 24) 24) =0 x 11 = E[X 11 X +1 = n +1, E[X 11 X +1 = n +1, ˆ ψ C ] = u Ω X u u Ω X n1+ u n1+ u ) ˆ ψ C ] 24) n 2+ ) n +1 u ) ) n 2+ n +1 u u ψˆ C u ψˆ C 25) ) 2008 6 12 22 / 39
5 I Mantel-Haenszel 1959) R, Haenszel, et al. 1954) R 1, Wynder, et al. 1954) R 2, R 3 R 4 A = X 11 B = n 1+ X 11 26) C = n +1 X 11 D = n 2+ n +1 X 11 A D /n +1 R = B C /n +2 / A D R 1 = B R 2 = C D A n 1+ n 2+ ) C B n 1+ n 2+ ) E[A ] E[D ] E[B ] E[C ] 27) ) 2008 6 12 23 / 39
5 II A n 2+ n R 3 = 1+ ) D B n 2+ n 1+ ) C A n ++ n R 4 = 1+ ) D n ++ n 2+ ) B n ++ n 1+ ) C n ++ n 2+ ) 28) R, R 1 X 11 = E[X 11 n +1, ψ = 1] 1 R 1, 1 2 2., H 0 1. R 1 ψ = 1 R 4,. R 2, R 3,.? R 2 n 2+, R 3 n 1+, R 4 0. ) 2008 6 12 24 / 39
Mantel-Haenszel 1959) ψ C = 1 H 0 : ψ 1 = ψ 2 =... = ψ = ψ = 1 29) H 1 : ψ 1 = ψ 2 =... = ψ = ψ χ 2 MH = X 11 E[X 11 ] c) 2 V[X 11 ] E[X 11 ] = n 1+n +1 n ++ V[X 11 ] = n 1+n 2+ n +1 n +2 n 2 ++ n ++ 1) χ 2 1) 30) 31) ) 2008 6 12 25 / 39
Cochran-Mantel-Haenszel Cochran 1954; pp.443 446) [4]?, Mantel-Haenszel R np1 p) ) V C [X 11 ] = n 1+n 2+ n +1 n +2 n 3 ++ 32) Cochran 1954) χ 2,, ) Cochran-Armitage trend test ) 2008 6 12 26 / 39
1:1 Case Control 2 1, Mz ++, π 11, π 12, π 21 ) : 1:1 Control π Case 11 π 12 p 1 π 21 π 22 1 p 1 p 2 1 p 2 1 Control Z Case 11 Z 12 Z 21 Z 22 z ++ ) 2008 6 12 27 / 39
1:1 Mantel-Haenszel 2 2 = Z 11 + Z 12 + Z 21 + Z 22 ), ψ 2 [3, 7] Case 1 0 Z 11 Case 1 0 Z 12 Control 1 0 Control 0 1 Case 0 1 Z 21 Case 0 1 Z 22 Control 1 0 Control 0 1 ) 2008 6 12 28 / 39
1:1 Mantel-Haenszel Mantel-Haenszel R X11 n 2+ n +1 + X 11 )/n +1 R = = Z 12 33) n1+ X 11 )n +1 X 11 )/n +2 Z 21 34) 1:1 R = ψˆ C ) ) n1+ n 2+ u u u ψˆ C uψˆ C x 11 = u n u Ω +1 u X ) ) n1+ n = u Ω X u 2+ u ψˆ ψˆ C C u n +1 u u Ω X ψˆ C Z 11 + Z 12 + 0 + 0 = Z 11 + Z 12 + Z 1 + ψˆ 21 C u Ω X ˆ ψ C + 0 1 + ψˆ C 34) ψˆ C = Z 12 35) Z 21 ) 2008 6 12 29 / 39
1:1 Mantel-Haenszel Mantel-Haenszel χ 2 X 11 n 1+ n ) 2 +1 MH = n ++ n 1+ n 2+ n +1 n +2 n 2 ++ n ++ 1) = Z 11 + Z 12 + 0 + 0 2Z 11+Z 12+Z 21+0 2 ) 2 0+Z 12 +Z 21 +0 4 = Z 12 Z 21 ) 2 Z 12 + Z 21 36) ) 2008 6 12 30 / 39
McNemar [6] 2 2 marginal homogenity) 2 2, d H 0 : p 1 = p 2 d = π 12 π 21 = 0) 37) H 1 : p 1 p 2 d 0) X 2 Mc = ˆd 2 ˆV[ ˆd H 0 ] = Z 12 Z 21 c) 2 Z 12 + Z 21 38) ˆd = ˆπ 12 ˆπ 21 = Z 12 Z 21 z ++ V[ ˆd] = V[Z 12] + V[Z 21 ] 2Cov[Z 12, Z 21 ] z 2 ++ 39) ˆV[ ˆd H 0 ] = ˆπ 12 + ˆπ 21 z ++ = Z 12 + Z 21 z 2 ++ ) 2008 6 12 31 / 39
i j B 1 B 2 B j A 1 π 11 π 12 π 1 j 1 A 2 π 21 π 22 π 2 j 1........ A i π i1 π i2 π i j 1 B 1 B 2 B j c 1 c 2 c j A 1 r 1 X 11 X 12 X 1 j n 1+ A 2 r 2 X 21 X 22 X 2 j n 2+....... A i r i X i1 X i2 X i j n i+ n ++.. ) 2008 6 12 32 / 39
n i+! PrX i j = x i j ) = p x i j i j x i j! i j 40) j ψ i j = p i j p 11 41) p i1 p 1 j ) PrX i j = x i j ) = cψ, θ)hx) exp x i j log ψ i j + x + j θ j p1 ) j θ j = log, cψ, θ) = p 11 i=2 i j=2 1 + j=2 explog ψ i j ) expθ j ) j=2 n i+! hx) = j x i j! i ) ni+ 42) 43) ) 2008 6 12 33 / 39
[8] r 1 r 2 r i ) c 1 c 2 c j ), log ψ i j = c j c 1 )β i Mantel, log ψ i j = r i r 1 )c j c 1 )β 2, c j = c j Mantel [5] β i log ψ ij β 3 β 2 c 1 c 2 c j ) 2008 6 12 34 / 39
) PrX i j = x i j ) = cψ, θ)hx) exp x i j c j c 1 )β i + x + j θ j i i=2 j=2 44) ni+ cβ, θ) = 1 + exp[log{c j c 1 )β i }] expθ j )) 45) j β i W i = j x i j c j c 1 ) θ j X + j hx) exp i=2 W i β i ) PrX i j = x i j X + j = n + j ) = u Ω X j hu) exp 46) i=2 W i β i ) j=2 ) 2008 6 12 35 / 39
Mantel 2, c j = c j ), c 1 = 0, W = j X 2 j c j 2 1, H 0 : β = 0 47) H 1 : β 0 n+ ) j j x 2 j PrX i j = x i j X + j = n + j ; β = 0) = n++ ) 48) n 2+ E[W X + j = n + j ; β = 0] = n 2+ V[W X + j = n + j ; β = 0] = n 1+ n 2+ n ++ n ++ 1) χ 2 EMH = W E[W X + j = n + j ; β = 0]) 2 V[W X + j = n + j ; β = 0] j c j n + j n ++ { n++ j c 2 j n + j j c j n + j ) 2} 49) χ 2 1) 50) ) 2008 6 12 36 / 39
[1] Agresti A. A survey of exact inference for contingency tables. Statistical Science 1992; 71): 131 177. [2] Agresti A. Categorical data analysis. 2nd edition. New Yor: John Wiley & Sons 2002. [3] Breslow N. Odds ratio estimators when the data are sparse. Biometria 1981; 681): 73 84. [4] Cochran WG. Some methods for strengthening the common χ 2 tests. Biometrics 1954; 104): 417 451. [5] Mantel N. Chi-square tests with one degree of freedom; extensions of the Mantel-Haenszel procedure. Journal of the American Statistical Association 1963; 58303): 690 700. [6] McNemar Q. Note on the sampling error of the difference between correlated proportions or percentages. Psychometria 1947; 122): 153 157. [7],,,. Mantel-Haenszel 2 2. 1998; 461): 153 177. [8].. 1986. ) 2008 6 12 37 / 39
E[X 11 n +1, ψ = 1] E[X 11 ] ) ) E[X 11 ] = u Ω X u Pr[u] = n++ = 1 n +1 ) n1+ u u u Ω X n1+ u Ω X u n 2+ n +1 u ) ) n 2+ n +1 u n 1+n 1+ 1)! u 1)! n u Ω 1+ u)! X n 2+ n +1 u ) n++ 1 ) n 1+ n +1 1 = ) = n 1+n +1 n++ n +1 n ++ 51) t n ) m u=0 u t u ) = n+m ) t u+u ) 2008 6 12 38 / 39
V[X 11 ] = E[X11 2 ] {E[X 11]} 2 = uu 1) Pr[u] + u Pr[u] {E[X 11 ]} 2 u Ω X u Ω X = n 1+n 1+ 1)n +1 n +1 1) n ++ n ++ 1) = n 1+n ++ n 1+ )n +1 n ++ n +1 ) n 2 ++n ++ 1) + n 1+n +1 n ++ n2 1+ n2 +1 n 2 ++ 52), ) 2008 6 12 39 / 39