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」 2016/5/16 @kenmatsu4

3. https://twitter.com/_inundata/status/616658949761302528

5. http://www.slideshare.net/matsukenbook

9. p11 p01
= p11 p01 =
n11
n11 + n10
n01
n01 + n00

11. RR =
p11
p01
=
n11
N1·
/
n01
N0·
=
n11N0·
n01N1·
0
RR < 1
RR = 1
1 < RR

12. OR =
p11/p10
p01/p00
=
(n11/N1·)/(n10/N1·)
(n01/N0·)/(n00/N0·)
=
(n11/N·1)/(n10/N·1)
(n01/N·0)/(n00/N0·0)
=
n11
p10
/
p01
p00
OR < 1
OR = 1
1 < OR

13. 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]);
}
}
}

15.
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

17. q =
1
1 ⇢2
"✓
x1 µ1
1
◆2
2⇢
✓
x1 µ1
1
◆ ✓
x2 µ2
2
◆
+
✓
x2 µ2
2
◆2
#
f(x1, x2|µ1, µ2, 2
1, 2
2) =
1
2⇡ 1 2
p
1 ⇢
e q/2

18. ⇢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

19. 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);
}

21.
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

23. ⇢11(= 1.00)
⇢22(= 1.00)
⇢33(= 1.00)
⇢21
⇢31 ⇢32

24. ⇢2
= ⇢32 ⇢21

25. 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);
}

27.
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

28. (t)
⇢2
= g(⇢21 ⇢32) = ⇢
(t)
32 ⇢
(t)
21
u
(t)
⇢2 >0
(
1
(t)
⇢2 > 0
0 otherwise

30.
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

33. x
y
x1 x2
y
Nx = 500
Nx1
= 378
Ny = 122

34. Ny
Y
i=1
p(x2i, yi|µx2
, µy, 2
x2
, 2
y, 2
x2y)
x2
y
for(i
in
1:Ny){
y[i]
~
multi_normal(mu2,
S2);
}

35. Ny
Y
i=1
p(x2i, yi|µx2
, µy, 2
x2
, 2
y, 2
x2y)
Ny
Y
i=1
p(x2i, yi|µx, µy, 2
x, 2
y, 2
xy)
Nx1Y
j=1
p(x1j|µx, 2
x)

36. 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
Ny
Y
i=1
p(x2i, yi|µx, µy, 2
x, 2
y, 2
xy)
Nx1Y
j=1
p(x1j|µx, 2
x)

38.
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
2
xy

39. x
y
Nx = 500
Ny = 500

40.
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

43. xij = µk + ↵ki + kj + ekij
i
j
k r m
↵ki = µki µk

44. xij = µr + ↵ri + rj + erij
↵ri = µri µk
erij ⇠ N(0, 2
er)
↵r ⇠ N(0, 2
↵r)
r ⇠ N(0, 2
r)
2
x = 2
↵r + 2
r + 2
er
xij ⇠ N(µr + ↵ri + rj, 2
x)

46. 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

47. 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));
}

49.
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

50. xij = µm + ↵mi + mj + emij
↵mi = µmi µm
emij ⇠ N(0, 2
em)
↵m ⇠ N(0, 2
↵m)
m ⇠ N(0, 2
m)
2
x = 2
↵m + 2
em
xij ⇠ N(µm + ↵mi + mj, 2
x)

51. 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

52. 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));
}

54.
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

57. ICC(3, J0
)(t)
= g( 2(t)
↵m , 2(t)
em ) =
2(t)
↵m
2(t)
↵m +
2(t)
em /J0
ICC(2, J0
)(t)
= g( 2(t)
↵r ,
2(t)
r , 2(t)
er ) =
2(t)
↵r
2(t)
↵r + (
2(t)
r +
2(t)
er )/J0
u
(t)
ICC(2,J0) = g( 2(t)
↵r ,
2(t)
r , 2(t)
er ) =
(
1 ICC(2, J0
)(t)
> 0.9
0 otherwise
u
(t)
ICC(3,J0) = g( 2(t)
↵, , 2(t)
em ) =
(
1 ICC(3, J0
)(t)
> 0.9
0 otherwise

58. 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);
}

59.
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