基礎からのベイズ統計学 輪読会資料 第8章 「比率・相関・信頼性」
TRANSCRIPT
https://twitter.com/_inundata/status/616658949761302528
�p11�p01 = p11 � p01 =n11
n11 + n10� n01
n01 + n00
RR =p11p01
=n11
N1·/n01
N0·=
n11N0·n01N1·
� 0
RR < 1
RR = 1
1 < RR
OR =p11/p10p01/p00
=(n11/N1·)/(n10/N1·)
(n01/N0·)/(n00/N0·)=
(n11/N·1)/(n10/N·1)
(n01/N·0)/(n00/N0·0)=
n11
p10/p01p00
OR < 1
OR = 1
1 < OR
n11 ⇠ Bin(p11, N1·)
n10 ⇠ Bin(p10, N1·)
n01 ⇠ Bin(p01, N0·)
n00 ⇠ Bin(p00, N0·)
model{ for(i in 1:2){ for(j in 1:2){ n[i,j] ~ binomial(N[j], p[j][i]); } } }
mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat p[0,0] 0.58 5.0e-‐4 0.02 0.53 0.56 0.58 0.59 0.63 2429 1.0 p[1,0] 0.42 4.9e-‐4 0.03 0.37 0.41 0.42 0.44 0.47 2557 1.0 p[0,1] 0.42 5.0e-‐4 0.02 0.37 0.41 0.42 0.44 0.47 2429 1.0 p[1,1] 0.58 4.9e-‐4 0.03 0.53 0.56 0.58 0.59 0.63 2557 1.0 d 0.15 7.0e-‐4 0.04 0.08 0.13 0.15 0.18 0.22 2513 1.0 delta_over 1.0 0.0 0.0 1.0 1.0 1.0 1.0 1.0 10000 nan p11 0.58 5.0e-‐4 0.02 0.53 0.56 0.58 0.59 0.63 2429 1.0 p10 0.42 5.0e-‐4 0.02 0.37 0.41 0.42 0.44 0.47 2429 1.0 p01 0.42 4.9e-‐4 0.03 0.37 0.41 0.42 0.44 0.47 2557 1.0 p00 0.58 4.9e-‐4 0.03 0.53 0.56 0.58 0.59 0.63 2557 1.0 RR 1.37 2.0e-‐3 0.1 1.19 1.3 1.37 1.43 1.58 2510 1.0 OR 1.89 5.6e-‐3 0.28 1.41 1.69 1.86 2.06 2.49 2466 1.0 lp__ -‐548.7 0.03 1.05 -‐551.6 -‐549.0 -‐548.3 -‐547.9 -‐547.7 1586 1.0
q =1
1� ⇢
2
"✓x1 � µ1
�1
◆2
� 2⇢
✓x1 � µ1
�1
◆✓x2 � µ2
�2
◆+
✓x2 � µ2
�2
◆2#
f(x1, x2|µ1, µ2,�21 ,�
22) =
1
2⇡�1�2p1� ⇢
e
�q/2
⇢A, ⇢B
�(t)⇢ = �(t)⇢B�⇢A= g(⇢(t)A , ⇢(t)B ) = ⇢(t)B � ⇢(t)A
u(t)�⇢>0 = g(⇢(t)A , ⇢(t)B )
(1 �(t)⇢ > 0
0 otherwise
transformed parameters{ SigmaA[1,2] <-‐ sigmaA[1]*sigmaA[2]*rhoA; SigmaA[2,1] <-‐ sigmaA[1]*sigmaA[2]*rhoA; SigmaB[1,2] <-‐ sigmaB[1]*sigmaB[2]*rhoB; SigmaB[2,1] <-‐ sigmaB[1]*sigmaB[2]*rhoB; } model{ for(i in 1:N){ xA[i] ~ multi_normal(muA, SigmaA); xB[i] ~ multi_normal(muB, SigmaB); } } generated quantities{ real delta_r; real delta_r_over; delta_r <-‐ rhoB -‐ rhoA; delta_r_over <-‐ step(delta_r); }
mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat rhoA 0.63 8.1e-‐4 0.04 0.54 0.6 0.63 0.66 0.71 2876 1.0 rhoB 0.72 6.7e-‐4 0.03 0.65 0.7 0.73 0.75 0.79 2626 1.0 delta_r 0.1 1.0e-‐3 0.06-‐9.0e-‐3 0.06 0.1 0.14 0.21 2790 1.0 delta_r_over 0.96 3.6e-‐3 0.19 0.0 1.0 1.0 1.0 1.0 2814 1.0
⇢11(= 1.00)
⇢22(= 1.00)
⇢33(= 1.00)
⇢21
⇢31 ⇢32
�⇢2 = ⇢32 � ⇢21
transformed parameters{ vector<lower=0>[3] sig2; matrix[3,3] Sigma; for(i in 1:3){ sig2[i] <-‐ pow(sigma[i],2); } Sigma <-‐ diag_matrix(sigma) * rho * diag_matrix(sigma); } model{ for(i in 1:N){ x[i] ~ multi_normal(mu,Sigma); } } generated quantities{ rho_21 <-‐ rho[2,1]; rho_31 <-‐ rho[3,1]; rho_32 <-‐ rho[3,2]; delta_r2 <-‐ rho[3,2] -‐ rho[2,1]; delta_r2_over <-‐ step(delta_r2); }
mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat rho_21 0.45 1.3e-‐3 0.06 0.33 0.41 0.45 0.49 0.57 2358 1.0 rho_31 0.62 9.7e-‐4 0.05 0.52 0.59 0.62 0.65 0.7 2406 1.0 rho_32 0.75 6.8e-‐4 0.03 0.67 0.73 0.75 0.77 0.81 2552 1.0
�(t)⇢2= g(⇢21 � ⇢32) = ⇢(t)32 � ⇢(t)21
u(t)�⇢2>0
(1 �(t)⇢2 > 0
0 otherwise
mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat delta_r2 0.3 1.1e-‐3 0.05 0.19 0.26 0.29 0.33 0.41 2562 1.0 delta_r2_over 1.0 0.0 0.0 1.0 1.0 1.0 1.0 1.0 10000 nan
x
y
x1 x2
y
Nx
= 500
Nx1 = 378
Ny = 122
NyY
i=1
p(x2i,yi|µx2 , µy ,�2x2
,�2y
,�2x2y
)
x2
y
for(i in 1:Ny){ y[i] ~ multi_normal(mu2, S2); }
NyY
i=1
p(x2i,yi|µx2 , µy ,�2x2
,�2y
,�2x2y
)
NyY
i=1
p(x2i,yi|µx, µy ,�2x
,�2y
,�2xy
)
Nx1Y
j=1
p(x1j |µx,�2x
)
x2
y
for(i in 1:Ny){ y[i] ~ multi_normal(mu, Sigma); } for(i in 1:Nx){ x[i] ~ normal(mu[1], sqrt(sigma[1])); }
x
NyY
i=1
p(x2i,yi|µx, µy ,�2x
,�2y
,�2xy
)
Nx1Y
j=1
p(x1j |µx,�2x
)
mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat rho_truncated 0.6 1.1e-‐3 0.06 0.47 0.56 0.6 0.64 0.7 2824 1.0 rho_corrected 0.81 8.8e-‐4 0.04 0.71 0.79 0.82 0.84 0.88 2182 1.0
�2xy
x
y
Nx
= 500
Ny = 500
mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat rho_truncated 0.6 1.1e-‐3 0.06 0.47 0.56 0.6 0.64 0.7 2824 1.0 rho_corrected 0.81 8.8e-‐4 0.04 0.71 0.79 0.82 0.84 0.88 2182 1.0 rho_complete 0.83 2.9e-‐4 0.01 0.8 0.82 0.83 0.84 0.86 2324 1.0
xij = µk + ↵ki + �kj + ekij
i j k r m
↵ki = µki � µk
xij = µr + ↵ri + �rj + erij
↵ri = µri � µk
erij ⇠ N(0,�2er)
↵r ⇠ N(0,�2↵r)
�r ⇠ N(0,�2�r)
�2x
= �2↵r
+ �2�r
+ �2er
x
ij
⇠ N(µr
+ ↵
ri
+ �
rj
,�
2x
)
ICC(2, 1)
ICC(2, 1)(t) = g(�2(t)↵r ,�2(t)
�r ,�2(t)er ) =
�2(t)↵r
�2(t)↵r + �2(t)
�r + �2(t)er
ICC(2, j)
ICC(2, j)(t) = g(�2(t)↵r ,�2(t)
�r ,�2(t)er ) =
�2(t)↵r
�2(t)↵r + (�2(t)
�r + �2(t)er )/j
model{ mu ~ normal(0, 1000); for(s in 1:S){ alpha[s] ~ normal(0, tauSubject); } for(r in 1:R){ beta[r] ~ normal(0, tauRater); } for(s in 1:S) { for(r in 1:R) { nu <-‐ mu + alpha[s] + beta[r]; Score[s,r] ~ normal(nu, tauWithin); } } } generated quantities{ ICC21 <-‐ sig2subject / (sig2subject + sig2rater + sig2within); ICC24 <-‐ sig2subject / (sig2subject + ((sig2rater + sig2within)/4)); }
mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat sig2subject 8.8 0.06 4.47 3.44 5.9 7.8 10.48 20.28 6130 1.0 sig2rater 4.29 2.33 61.98 1.7e-‐3 0.03 0.16 0.58 12.73 705 1.0 sig2within 3.65 0.03 0.84 2.37 3.03 3.52 4.12 5.57 683 1.0 ICC21 0.64 4.9e-‐3 0.14 0.28 0.57 0.66 0.74 0.85 834 1.0 ICC24 0.86 5.0e-‐3 0.11 0.61 0.84 0.89 0.92 0.96 446 1.0
xij = µm + ↵mi + �mj + emij
↵mi = µmi � µm
emij ⇠ N(0,�2em)
↵m ⇠ N(0,�2↵m)
�m ⇠ N(0,�2�m)
�2x
= �2↵m
+ �2em
x
ij
⇠ N(µm
+ ↵
mi
+ �
mj
,�
2x
)
ICC(2, j)
ICC(3, 1)
ICC(3, 1)(t) = g(�2(t)↵m ,�2(t)
em ) =�2(t)↵m
�2(t)↵m + �2(t)
em
ICC(3, j)(t) = g(�2(t)↵m ,�2(t)
em ) =�2(t)↵m
�2(t)↵m + �2(t)
em /j
model{ mu ~ normal(0, 1000); for(s in 1:S){ alpha[s] ~ normal(0, tauSubject); } for(s in 1:S) { for(r in 1:R) { nu <-‐ mu + alpha[s] + beta[r]; Score[s,r] ~ normal(nu, tauWithin); } } } generated quantities{ ICC31 <-‐ sig2subject / (sig2subject + sig2within); ICC34 <-‐ sig2subject / (sig2subject + (sig2within/R)); }
mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat sig2subject 8.82 0.05 4.48 3.39 5.8 7.79 10.67 20.29 9199 1.0 sig2within 3.76 9.6e-‐3 0.89 2.41 3.13 3.63 4.25 5.89 8580 1.0 ICC31 0.67 1.1e-‐3 0.11 0.44 0.6 0.68 0.75 0.86 9291 1.0 ICC34 0.89 5.5e-‐4 0.05 0.76 0.86 0.9 0.92 0.96 9239 1.0
ICC(3, J 0)(t) = g(�2(t)↵m ,�2(t)
em ) =�2(t)↵m
�2(t)↵m + �2(t)
em /J 0
ICC(2, J 0)(t) = g(�2(t)↵r ,�2(t)
�r ,�2(t)er ) =
�2(t)↵r
�2(t)↵r + (�2(t)
�r + �2(t)er )/J 0
u(t)ICC(2,J 0) = g(�2(t)
↵r ,�2(t)�r ,�2(t)
er ) =
(1 ICC(2, J 0
)
(t) > 0.9
0 otherwise
u(t)ICC(3,J0) = g(�2(t)
↵, ,�2(t)em ) =
(1 ICC(3, J 0
)
(t) > 0.9
0 otherwise
generated quantities{ ICC25 <-‐ sig2subject / (sig2subject + ((sig2rater + sig2within)/5)); ICC26 <-‐ sig2subject / (sig2subject + ((sig2rater + sig2within)/6)); nine6 <-‐ step(rho6 -‐ 0.9); }
generated quantities{ ICC34 <-‐ sig2subject / (sig2subject + (sig2within/R)); ICC35 <-‐ sig2subject / (sig2subject + (sig2within/5)); nine <-‐ step(ICC35 -‐ 0.9); }
mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat ICC25 0.89 4.8e-‐3 0.1 0.66 0.87 0.91 0.93 0.97 411 1.0 ICC26 0.9 4.7e-‐3 0.09 0.7 0.89 0.92 0.94 0.97 388 1.0 nine6 0.69 0.01 0.46 0.0 0.0 1.0 1.0 1.0 1375 1.0
mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat ICC34 0.89 5.5e-‐4 0.05 0.76 0.86 0.9 0.92 0.96 9239 1.0 ICC35 0.91 4.7e-‐4 0.05 0.8 0.88 0.92 0.94 0.97 9231 1.0 nine 0.64 4.8e-‐3 0.48 0.0 0.0 1.0 1.0 1.0 9981 1.0
