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Log-Linear model Example and Interpretation 11-1
卵巢癌与口服避孕药之关系:
dm"log;clear;out;clear;";
options nodate nonotes ps=60;
data ovarian;
input v1 v2 v3 v4 wt;
if wt=0 then wt=1e-20;
cards;
1 1 1 1 9
1 1 1 2 7
1 1 2 1 14
1 1 2 2 5
1 1 3 1 5
1 1 3 2 3
1 2 1 1 25
1 2 1 2 6
1 2 2 1 1
1 2 2 2 0
1 2 3 1 1
1 2 3 2 1
2 1 1 1 3
2 1 1 2 10
2 1 2 1 8
2 1 2 2 10
2 1 3 1 3
2 1 3 2 7
2 2 1 1 73
2 2 1 2 68
2 2 2 1 1
2 2 2 2 9
2 2 3 1 1
2 2 3 2 13
/*proc catmod order=data;
weight wt;
model v1*v2*v3*v4=_response_/ml nogls pred;
loglin v1|v2|v3 v2|v3|v4 v1|v4;
run; */
proc catmod order=data;
weight wt;
population v1 v2 v3;
model v4=v1 v2|v3/ml noprofile nodesign nogls pred;
run;
123/234/14:
Parity
Age
OC use
Status
Case
Control
Nonparous
3 years
5(5.10)
3(2.90)
40-59
Never
25(25.42)
6(5.58)
3 years
1(0.44)
0(0.56)
>3 years
1(0.63)
1(1.37)
Parous
3 years
3(2.90)
7(7.10)
40-59
Never
73(72.58)
68(68.42)
3 years
1(1.56)
9(8.44)
>3 years
1(1.37)
13(12.63)
Model
G2(deviance)
df
prob
1
123/234/134/124
1.99
2
NS
2
123/134/124
11.44
4
<.05
3
123/234/134
2.34
3
NS
4
123/234/124
2.33
4
NS
5
123/234
23.96
6
3 v.s. use = 0).
模式:
u24
:
case
control
age3
u34(31) =
u34(32) =0.26
u234:
case
control
use=0
u234(111) =
u234(112) =
age3
u234(131) =
u234(132) =
use=0
u234(211) =
u234(212) =
age40
use3
u234(221) =
u234(222) =
use>3
u234(231) =
u234(232) =
11-4
V4=1
case
V4=2
control
V3=2
(21)
(22)
V3=1
(11)
(12)
age40(V2 =1)
O.R.(usev.s use = 0 ,即V3 = 2 v.s. V3 = 1)
O.R.=
=
=
= = = 2.41
O.R.(usev.s use = 0 ,即V3 = 3 v.s. V3 = 1)
O.R. =
= ==1.35
age40(V2 =2)
O.R.(usev.s use = 0 ,即V3 = 2 v.s. V3 = 1)
O.R.=
=
==0.172
O.R.(usev.s use = 0 ,即V3 = 3 v.s. V3 = 1)
O.R.=
= = = 0.1
故对年轻人而言,OC使用较易致癌;
故对年老者而言,OC使用较不易致癌.
Parity(V1)影响
V4=1
V4=2
V1=2
(21)
(22)
V1=1
(11)
(12)
O.R.= =
== 0.2329
O.R.=
V4=1
V4=2
V1=1
(11)
(12)
V1=2
(21)
(22)
= = = 4.2939
因此非经产妇较易致癌.
11-5
Log-Linear model与Logit model之关系
假设其中一variable可视为response variable,而其他为explanatory variable(或fixed by design),则此时可用logit model,将所有fixed by design之variable及其interaction term加入模式中,有时其他的variable虽不是fixed by design,但为方便起见,我们也可认为是固定的explanatory variable,先从最简单的3-way model,假设fit之log-linear model为12/13/23 model:
3视为response variable,且为dichotomos,令
则 logit model 为
若为12/13 model 则,若为12/23 model 则,
若为13/23 model 则,若为1/23 model 则,
若为2/13 model 则,若为3/12 model 则,
若为123 model 则.
换言之,将-term分成两类,一类与response无关,一类与response有关,例如in 3-way table:
与var3(response)无关,与var3(response)有关.
在loglinear model中,只看与response有关之项,除response外之variable,或interaction即为logit model中所必须采用之项.
例如:4-way table中(var4为response)
,,
,
,
.
11-6
反之,若start with logit model,则所有fixed by design之variable及其interaction均加入模式中,例如3-way table中,若
,
,
,
,
,
同理在4-way table中,若1,2,3 fixed by design(or considered to be fixed),则模式中必须加入123项.
Example:(Cancer-Knowledge Data)Fienberg P.85(25 table,var 5 = response)
Model 为 12/13/15/24/25/345,则logit model 为:,
若将var1(news),var2(radio),var3(reading),var4(lecture),视为影响Cancer-Knowledge之固定之explanatory variables(视为conditioning on these variables)fit loglinear model:
15/25/345/1234
则对应之logit model亦为,Dyke & Patterson(1952 , Biometrics , , 1-12)分析此组数据时,其fit之model为,其所对应之loglinear model为:
15/25/35/45/1234
Bishop(1969 , Biometrics , 388-399)又考虑下列模式:
(C)12/13/14/15/23/24/25/34/35/45(Pseudo-logit)(其所对应之logit模式与前者(B)相同,但d.f增多)
Log-linear:12/13/234/14
Maximum Likelihood Analysis of Variance
Source DF Chi-Square Pr > ChiSq
--------------------------------------------------
v1 1 21.32 <.0001
v2 1 3.26 0.0712
v1*v2 1 25.43 <.0001
v3 2 62.07 <.0001
v1*v3 2 0.55 0.7600
v2*v3 2 54.37 <.0001
v4 1 1.86 0.1732
v2*v4 1 0.99 0.3202
v3*v4 2 4.11 0.1283
v2*v3*v4 2 11.26 0.0036
v1*v4 1 18.98 ChiSq ---------------------------------------------------------------
v1 1 -0.4667 0.1011 21.32 <.0001
v2 2 0.2181 0.1209 3.26 0.0712
v1*v2 3 0.4962 0.0984 25.43 <.0001
v3 4 1.0319 0.1311 61.96 <.0001
5 -0.4014 0.1723 5.43 0.0198
v1*v3 6 0.0959 0.1364 0.49 0.4820
7 -0.0677 0.1472 0.21 0.6459
v2*v3 8 -0.9605 0.1344 51.05 <.0001
9 0.7196 0.1724 17.43 <.0001
v4 10 -0.1561 0.1146 1.86 0.1732
v2*v4 11 0.1148 0.1155 0.99 0.3202
v3*v4 12 0.2388 0.1276 3.50 0.0613
13 0.0164 0.1676 0.01 0.9221
v2*v3*v4 14 -0.4139 0.1238 11.18 0.0008
15 0.2293 0.1642 1.95 0.1626
v1*v4 16 0.3521 0.0808 18.98 ChiSq
-------------------------------------------------- Intercept 1 1.89 0.1689
v1 1 19.00 ChiSq ---------------------------------------------------------------
Intercept 1 -0.3163 0.2299 1.89 0.1689
v1 2 0.7286 0.1671 19.00 <.0001
v2 3 0.2402 0.2311 1.08 0.2986
v3 4 0.4718 0.2564 3.39 0.0658
5 0.0426 0.3383 0.02 0.8998
v2*v3 6 -0.8723 0.2580 11.43 0.0007
7 0.4473 0.3380 1.75 0.1857
卵巢癌与口服避孕药之数据补充 11-7
离散分析习题九
Ex9.下面的资料是根据1991年社会调查所得数据,有四个变数,
R(religious attendance宗教信仰)
1:不常上教会 , 2:常上教会.
P(political attitude政治观点)
1:自由开放 , 2:中立 , 3:保守.
B(birth control节育)
1:赞成节育 , 2:反对节育.
S(premarital sex 婚前性行为)
1:可容忍婚前性行为 , 2:无法容忍婚前性行为.
S
1
2
R
1
2
1
2
B
1
2
1
2
1
2
1
2
1
99
15
73
25
8
4
24
22
P
2
73
20
87
37
20
13
50
60
3
51
19
51
36
6
12
33
88
将此四个变数均视为反应变数,利用常态机率图,配适一个对数线性模式,并解释此数据的意义.
81421
·上一篇：卵巢癌的治疗卵巢癌的治疗
·下一篇：白细胞介素21对卵巢上皮细胞及卵巢癌细胞
卵巢癌与口服避孕药之关系:
dm"log;clear;out;clear;";
options nodate nonotes ps=60;
data ovarian;
input v1 v2 v3 v4 wt;
if wt=0 then wt=1e-20;
cards;
1 1 1 1 9
1 1 1 2 7
1 1 2 1 14
1 1 2 2 5
1 1 3 1 5
1 1 3 2 3
1 2 1 1 25
1 2 1 2 6
1 2 2 1 1
1 2 2 2 0
1 2 3 1 1
1 2 3 2 1
2 1 1 1 3
2 1 1 2 10
2 1 2 1 8
2 1 2 2 10
2 1 3 1 3
2 1 3 2 7
2 2 1 1 73
2 2 1 2 68
2 2 2 1 1
2 2 2 2 9
2 2 3 1 1
2 2 3 2 13
/*proc catmod order=data;
weight wt;
model v1*v2*v3*v4=_response_/ml nogls pred;
loglin v1|v2|v3 v2|v3|v4 v1|v4;
run; */
proc catmod order=data;
weight wt;
population v1 v2 v3;
model v4=v1 v2|v3/ml noprofile nodesign nogls pred;
run;
123/234/14:
Parity
Age
OC use
Status
Case
Control
Nonparous
3 years
5(5.10)
3(2.90)
40-59
Never
25(25.42)
6(5.58)
3 years
1(0.44)
0(0.56)
>3 years
1(0.63)
1(1.37)
Parous
3 years
3(2.90)
7(7.10)
40-59
Never
73(72.58)
68(68.42)
3 years
1(1.56)
9(8.44)
>3 years
1(1.37)
13(12.63)
Model
G2(deviance)
df
prob
1
123/234/134/124
1.99
2
NS
2
123/134/124
11.44
4
<.05
3
123/234/134
2.34
3
NS
4
123/234/124
2.33
4
NS
5
123/234
23.96
6
3 v.s. use = 0).
模式:
u24
:
case
control
age3
u34(31) =
u34(32) =0.26
u234:
case
control
use=0
u234(111) =
u234(112) =
age3
u234(131) =
u234(132) =
use=0
u234(211) =
u234(212) =
age40
use3
u234(221) =
u234(222) =
use>3
u234(231) =
u234(232) =
11-4
V4=1
case
V4=2
control
V3=2
(21)
(22)
V3=1
(11)
(12)
age40(V2 =1)
O.R.(usev.s use = 0 ,即V3 = 2 v.s. V3 = 1)
O.R.=
=
=
= = = 2.41
O.R.(usev.s use = 0 ,即V3 = 3 v.s. V3 = 1)
O.R. =
= ==1.35
age40(V2 =2)
O.R.(usev.s use = 0 ,即V3 = 2 v.s. V3 = 1)
O.R.=
=
==0.172
O.R.(usev.s use = 0 ,即V3 = 3 v.s. V3 = 1)
O.R.=
= = = 0.1
故对年轻人而言,OC使用较易致癌;
故对年老者而言,OC使用较不易致癌.
Parity(V1)影响
V4=1
V4=2
V1=2
(21)
(22)
V1=1
(11)
(12)
O.R.= =
== 0.2329
O.R.=
V4=1
V4=2
V1=1
(11)
(12)
V1=2
(21)
(22)
= = = 4.2939
因此非经产妇较易致癌.
11-5
Log-Linear model与Logit model之关系
假设其中一variable可视为response variable,而其他为explanatory variable(或fixed by design),则此时可用logit model,将所有fixed by design之variable及其interaction term加入模式中,有时其他的variable虽不是fixed by design,但为方便起见,我们也可认为是固定的explanatory variable,先从最简单的3-way model,假设fit之log-linear model为12/13/23 model:
3视为response variable,且为dichotomos,令
则 logit model 为
若为12/13 model 则,若为12/23 model 则,
若为13/23 model 则,若为1/23 model 则,
若为2/13 model 则,若为3/12 model 则,
若为123 model 则.
换言之,将-term分成两类,一类与response无关,一类与response有关,例如in 3-way table:
与var3(response)无关,与var3(response)有关.
在loglinear model中,只看与response有关之项,除response外之variable,或interaction即为logit model中所必须采用之项.
例如:4-way table中(var4为response)
,,
,
,
.
11-6
反之,若start with logit model,则所有fixed by design之variable及其interaction均加入模式中,例如3-way table中,若
,
,
,
,
,
同理在4-way table中,若1,2,3 fixed by design(or considered to be fixed),则模式中必须加入123项.
Example:(Cancer-Knowledge Data)Fienberg P.85(25 table,var 5 = response)
Model 为 12/13/15/24/25/345,则logit model 为:,
若将var1(news),var2(radio),var3(reading),var4(lecture),视为影响Cancer-Knowledge之固定之explanatory variables(视为conditioning on these variables)fit loglinear model:
15/25/345/1234
则对应之logit model亦为,Dyke & Patterson(1952 , Biometrics , , 1-12)分析此组数据时,其fit之model为,其所对应之loglinear model为:
15/25/35/45/1234
Bishop(1969 , Biometrics , 388-399)又考虑下列模式:
(C)12/13/14/15/23/24/25/34/35/45(Pseudo-logit)(其所对应之logit模式与前者(B)相同,但d.f增多)
Log-linear:12/13/234/14
Maximum Likelihood Analysis of Variance
Source DF Chi-Square Pr > ChiSq
--------------------------------------------------
v1 1 21.32 <.0001
v2 1 3.26 0.0712
v1*v2 1 25.43 <.0001
v3 2 62.07 <.0001
v1*v3 2 0.55 0.7600
v2*v3 2 54.37 <.0001
v4 1 1.86 0.1732
v2*v4 1 0.99 0.3202
v3*v4 2 4.11 0.1283
v2*v3*v4 2 11.26 0.0036
v1*v4 1 18.98 ChiSq ---------------------------------------------------------------
v1 1 -0.4667 0.1011 21.32 <.0001
v2 2 0.2181 0.1209 3.26 0.0712
v1*v2 3 0.4962 0.0984 25.43 <.0001
v3 4 1.0319 0.1311 61.96 <.0001
5 -0.4014 0.1723 5.43 0.0198
v1*v3 6 0.0959 0.1364 0.49 0.4820
7 -0.0677 0.1472 0.21 0.6459
v2*v3 8 -0.9605 0.1344 51.05 <.0001
9 0.7196 0.1724 17.43 <.0001
v4 10 -0.1561 0.1146 1.86 0.1732
v2*v4 11 0.1148 0.1155 0.99 0.3202
v3*v4 12 0.2388 0.1276 3.50 0.0613
13 0.0164 0.1676 0.01 0.9221
v2*v3*v4 14 -0.4139 0.1238 11.18 0.0008
15 0.2293 0.1642 1.95 0.1626
v1*v4 16 0.3521 0.0808 18.98 ChiSq
-------------------------------------------------- Intercept 1 1.89 0.1689
v1 1 19.00 ChiSq ---------------------------------------------------------------
Intercept 1 -0.3163 0.2299 1.89 0.1689
v1 2 0.7286 0.1671 19.00 <.0001
v2 3 0.2402 0.2311 1.08 0.2986
v3 4 0.4718 0.2564 3.39 0.0658
5 0.0426 0.3383 0.02 0.8998
v2*v3 6 -0.8723 0.2580 11.43 0.0007
7 0.4473 0.3380 1.75 0.1857
卵巢癌与口服避孕药之数据补充 11-7
离散分析习题九
Ex9.下面的资料是根据1991年社会调查所得数据,有四个变数,
R(religious attendance宗教信仰)
1:不常上教会 , 2:常上教会.
P(political attitude政治观点)
1:自由开放 , 2:中立 , 3:保守.
B(birth control节育)
1:赞成节育 , 2:反对节育.
S(premarital sex 婚前性行为)
1:可容忍婚前性行为 , 2:无法容忍婚前性行为.
S
1
2
R
1
2
1
2
B
1
2
1
2
1
2
1
2
1
99
15
73
25
8
4
24
22
P
2
73
20
87
37
20
13
50
60
3
51
19
51
36
6
12
33
88
将此四个变数均视为反应变数,利用常态机率图,配适一个对数线性模式,并解释此数据的意义.
81421
·上一篇：卵巢癌的治疗卵巢癌的治疗
·下一篇：白细胞介素21对卵巢上皮细胞及卵巢癌细胞
