Mantel-Haenszel 2008 6 12 1 / 39 1 (, (, (,,, pp719 730 2 2 2 3 1 4 pp730 746 2 2, i j 3 / 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; 22(4: 719 748 1 (, (, (,, 2 2 2 3 1 2 2 Mantel-Haenszel 1:1 2 j ( 4 2 2, Mantel-Haenszel R, R1, R2, R3, R4 1:1 2 / 39,,,,, 対象集団 追跡 発症無 発症有時間 4 / 39
7 / 39 8 / 39,,,, 対象集団? 調査 発症無 発症有時間 Mantel & Haenzel (1959 : John Snow,,, 5 / 39 2 2 : q1 1 q1 1 q2 1 q2 1 6 / 39 2 2 : p1 1 p1 1 p2 1 p2 1 Y11 m1+ Y21 m2+ m++ Y11 B(m1+, q1 Y21 B(m2+, q2 (1 X11 n1+ X21 n2+ n++ X11 B(n1+, p1 X21 B(n2+, p2 (2
11 / 39 12 / 39 ϕ (ris ratio ψ (odds ratio ϕ = q1, ( q1/(1 q1 p1/(1 p1 ψ = = q2/(1 q2 p2/(1 p2 q2 q1 = 006, q2 = 003 ϕ = 006 006/094 = 2, ψ = = 206 (5 003 003/097 (3 (4 [8] ψd, Case-Control ψe D, D, E, Ē (7 Pr(D E[1 Pr(D Ē] ψd = Pr(D Ē[1 Pr(D E], ψe = Pr(E D[1 Pr(E D] Pr(E D[1 Pr(E D] ψd = Pr (E D Pr (D Pr (E D Pr (D+Pr (E D Pr ( D Pr (Ē 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] = ψe Pr (Ē 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 ψd ψe (6 (7 9 / 39, ψ p2, X+1 = X11 + X21 p2/(1 p2 ( n1+ ( x11 ( ψp2 1 n1+ x11 p2 Pr(X11 = x11, X21 = x21 = x11 1 p2(1 ψ 1 p2(1 ψ ( n2+ p x21 2 (1 p2n2+ x21 x21 [ { ( }] n1+ p2 = 1 + exp log ψ + log 1 p2 [ { ( }] n2+ p2 1 + exp log 1 p2 ( ( { ( } p2 exp x11 log ψ + x+1 log x11 x21 1 p2 (8 10 / 39 ˆψ X11/(n1+ X11 ˆψ = X21/(n2+ X21 : c = 0, : c = 1/2 ( H0 : ψ = 1 (10 H1 : ψ 1 X 2 ( X11(n2+ X21 X21(n1+ X11 c2 = χ 2 (1 (11 n1+n2+x+1(n++ X+1 (9
15 / 39 16 / 39 ( : X11 n1+ n2+ X+1 = n+1 n++ p2/(1 p2 X+1 = X11 + X21, ψ ( ( ψ x11 Pr(X11 = x11, X+1 = n+1 x11 n+1 x11 Pr(X11 = x11 X+1 = n+1 = = Pr(X+1 = n+1 ( ( ψ u (12 ΩX X+1 = n+1 ΩX = {u Z+ max(0, n+1 n2+ u min(n1+, n+1} (13 ΩS ΩS = {(u, v Z 2 + 0 u n1+, 0 v n2+} (14 13 / 39 ψc ˆ (15 (15 =0 x11 = E[X11 X+1 = n+1, ψc] ˆ (15 ( ( u ψc ˆ u E[X11 X+1 = n+1, ψc] ˆ = ( ( (16 ψc ˆ u 14 / 39 (Fisher s exact test, (10, X11 (12 P, ( ( x11 n+1 x11 Pr(X11 = x11 X+1 = n+1, ψ = 1 = ( n++ (17 n+1 (a P Value = 2 Pr(X11 x11 (b P Value = Pr(ZF zf, ZF = Pr(X11, zf = Pr(x11 (18 (c P Value = Pr( X11 E0[X11] x11 E0[X11]
19 / 39 20 / 39 (Stratified analysis 2 2? 2 2, (Stratified analysis (Subgroup analysis (Multivariate analysis, 2 2 2 2, : 2 2 p1 1 p1 1 p2 1 p2 1 X11 n1+ X21 n2+ n++ 17 / 39, X+1 = X11 + X21 p2/(1 p2 Pr(X11 = x11, X21 = x21 = [ { ( 1 + exp log ψ + log [ { 1 + exp ( ( x11 x21 ( log }] n2+ p2 1 p2 p2 1 p2 { x11 log ψ + exp }] n1+ ( x+1 log p2 1 p2 } (19 18 / 39 (summary relative ris/common odds ratio ψ1 = ψ2 = = ψ = ψ (20 Pr(X11 = x11, X21 = x21 = [ { ( }] n1+ p2 1 + exp log ψ + log 1 p2 [ { ( }] n2+ p2 1 + exp log 1 p2 ( ( {( ( } p2 exp x11 log ψ + x+1 log x11 x21 1 p2 (21
23 / 39 24 / 39 X11 n1+ n2+ X+1 = n+1 n++, p2/(1 p2 X+1 = X11 + X21 ( ( ψ x11 x11 n+1 x11 Pr(X11 = x11 X+1 = n+1 = ( ( n1+ (22 n2+ ψ u x11 n+1 x11 ΩX ψc ˆ (24 (24 =0 x11 = E[X11 X+1 = n+1, ψc] ˆ (24 ( ( u ψc ˆ u E[X11 X+1 = n+1, ψc] ˆ = ( ( (25 ψc ˆ u ΩX = {u Z+ max(0, n+1 n2+ u min(n1+, n+1} (23 21 / 39 5 I Mantel-Haenszel (1959 R, Haenszel, et al (1954 R1, Wynder, et al (1954 R2, R3 R4 A = X11 B = n1+ X11 C = n+1 X11 D = n2+ n+1 X11 AD/n+1 R = BC/n+2 / A D E[A] E[D] R1 = B C E[B] E[C] A (n1+ D n2+ R2 = B (n1+ C n2+ (26 (27 22 / 39 5 II (n2+ A n1+ R3 = D (n2+ B n1+ C (n++ A n1+ R4 = (n++ D (n++ B n1+ (n++ C n2+ n2+ R, R1 X11 = E[X11 n+1, ψ = 1] 1 (28 R1, 1 2 2, H0 1 R1 ψ = 1 R4, R2, R3,? R2 n2+, R3 n1+, R4 0
27 / 39 28 / 39 Mantel-Haenszel (1959 ψc = 1 H0 : ψ1 = ψ2 = = ψ = ψ = 1 (29 H1 : ψ1 = ψ2 = = ψ = ψ χ 2 MH = ( X11 E[X11] c 2 χ 2 (1 (30 V[X11] E[X11] = n1+n+1 n++ V[X11] = n1+n2+n+1n+2 (31 n 2 ++ (n++ 1 Cochran-Mantel-Haenszel Cochran (1954; pp443 446 [4]?, Mantel-Haenszel R ( np(1 p Cochran (1954 V C [X11] = n1+n2+n+1n+2 n 3 ++ (32 χ 2 (,, Cochran-Armitage trend test 25 / 39 1:1 Case Control 2 1, M(z++, π11, π12, π21 : 1:1 Control π11 π12 p1 Case π21 π22 1 p1 p2 1 p2 1 26 / 39 1:1 Mantel-Haenszel 2 2 (= Z11 + Z12 + Z21 + Z22, ψ 2 [3, 7] Case 1 0 Z11 Case 1 0 Z12 Control 1 0 Control 0 1 Control Z11 Z12 Case Z21 Z22 z++ Case 0 1 Z21 Case 0 1 Z22 Control 1 0 Control 0 1
1:1 Mantel-Haenszel Mantel-Haenszel R X11(n2+ n+1 + X11/n+1 R = = Z12 (n1+ X11(n+1 X11/n+2 Z21 (34 1:1 R = ψc ˆ ( ( u ψc ˆ u uψc ˆ u x11 = ( ( = ψc ˆ u ψc ˆ u ψc ˆ ψc ˆ Z11 + Z12 + 0 + 0 = Z11 + Z12 ψc ˆ = Z12 Z21 1 + ψc ˆ + Z21 1 + ψc ˆ + 0 (33 (34 (35 29 / 39 1:1 Mantel-Haenszel Mantel-Haenszel ( χ 2 X11 2 n1+n+1 MH = n++ n1+n2+n+1n+2 n 2 ++ (n++ 1 = 2Z11+Z12+Z21+0 (Z11 + Z12 + 0 + 0 2 2 0+Z12+Z21+0 4 (Z12 Z212 = Z12 + Z21 (36 30 / 39 McNemar [6] 2 2 (marginal homogenity 2 2, d H0 : p1 = p2 (d = π12 π21 = 0 H1 : p1 p2 (d 0 XMc 2 = ˆd 2 ( Z12 Z21 c2 = ˆV[ ˆd H0] Z12 + Z21 Z12 Z21 ˆd = ˆπ12 ˆπ21 = z++ V[Z12] + V[Z21] 2Cov[Z12, Z21] V[ ˆd] = z 2 ++ ˆV[ ˆd H0] = ˆπ12 + ˆπ21 Z12 + Z21 = z++ z 2 ++ (37 (38 (39 31 / 39 i j B1 B2 B j A1 π11 π12 π1 j 1 A2 π21 π22 π2 j 1 Ai πi1 πi2 πi j 1 B1 B2 B j c1 c2 c j A1 r1 X11 X12 X1 j n1+ A2 r2 X21 X22 X2 j n2+ Ai ri Xi1 Xi2 Xi j ni+ n++ 32 / 39
35 / 39 36 / 39 ni+! xi j Pr(Xi j = xi j = pi j (40 i j xi j! j pi j p11 ψi j = (41 pi1 p1 j ( Pr(Xi j = xi j = c(ψ, θh(x exp xij log ψij + x+ jθ j ( p1 j θ j = log p11, c(ψ, θ = i=2 j=2 ( 1 + i j=2 ni+! h(x = i j xi j! j=2 exp(log ψi j exp(θ j ni+ (42 (43 33 / 39 ( Pr(Xi j = xi j = c(ψ, θh(x exp xij(cj c1βi + x+ jθ j i j i=2 ( c(β, θ = 1 + exp[log{(cj c1βi}] exp(θ j j=2 j=2 ni+ (44 (45 βi Wi = j xi j(c j c1 θ j X+ j h(x exp( i=2 Wiβi Pr(Xi j = xi j X+ j = n+ j = j h(u exp( (46 i=2 Wiβi [8] (r1 r2 ri (c1 c2 c j, log ψi j = (c j c1βi Mantel, log ψi j = (ri r1(c j c1β 2, c j = c j Mantel [5] log ψ ij c 1 c 2 β i β 3 β 2 c j 34 / 39 Mantel 2, (c j = c j, c1 = 0, W = j X2 jc j 2 1, H0 : β = 0 (47 H1 : β 0 ( n+ j j x2 j Pr(Xi j = xi j X+ j = n+ j; β = 0 = ( n++ (48 n2+ E[W X+ j = n+ j; β = 0] = c jn+ j n2+ j n++ V[W X+ j = n+ j; β = 0] = { n1+n2+ n++(n++ 1 n++ j c 2 j n+ j ( j c jn+ j 2} (49 χ 2 (W E[W X+ j = n+ j; β = 0]2 EMH = χ 2 (1 V[W X+ j = n+ j; β = 0] (50
39 / 39 [1] Agresti A A survey of exact inference for contingency tables Statistical Science 1992; 7(1: 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; 68(1: 73 84 [4] Cochran WG Some methods for strengthening the common χ 2 tests Biometrics 1954; 10(4: 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; 58(303: 690 700 [6] McNemar Q Note on the sampling error of the difference between correlated proportions or percentages Psychometria 1947; 12(2: 153 157 [7],,, Mantel-Haenszel 2 2 1998; 46(1: 153 177 [8] 1986 E[X11 n+1, ψ = 1] E[X11] ( ( u E[X11] = u Pr[u] = ( ( ( n++ 1 = ( 1 ( n1+ n1+(n1+ 1! n2+ n+1 1 = ( = n1+n+1 n++ (u 1! (n1+ u! n+1 u n++ n++ n+1 n+1 (51 37 / 39 t ( n ( m ( u=0 u t u = n+m t u+u 38 / 39 V[X11] = E[X11 2 ] {E[X11]}2 = u(u 1 Pr[u] + u Pr[u] {E[X11]} 2 n1+(n1+ 1n+1(n+1 1 = + n1+n+1 n++(n++ 1 n++ n1+(n++ n1+n+1(n++ n+1 = n 2 ++(n++ 1 n2 1+ n2 +1 n 2 ++ (52,