10.4 Symmetry, Quasi-symmetry and Quasi-independence

1STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

10.4 Symmetry, Quasi-symmetry and Quasi-independence

2STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

3STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

4STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

5STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

6STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

7STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

8STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

9STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

SAS code:data migrate;input then $ now $ count symm qi;datalines; ne ne 11607 1 1 ne mw 100 2 5 ne s 366 3 5 ne w 124 4 5 mw ne 87 2 5 mw mw 13677 5 2 mw s 515 6 5 mw w 302 7 5 s ne 172 3 5 s mw 225 6 5 s s 17819 8 3 s w 270 9 5 w ne 63 4 5 w mw 176 7 5 w s 286 9 5 w w 10192 10 4

;

symm - symmetry

qi - quasi indep.

10STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

SAS modelingproc genmod; class then now;

model count =then now / dist=poi link=log; run;/*independence DF=(4-1)+(4-1)=6, residual DF=9 */

proc genmod; class symm;

model count = symm / dist=poi link=log; * symmetry DF=4-1+4*(4-1)/2=9, residual DF=6 ;

proc genmod; class then now qi;

model count = then now qi / dist=poi link=log;

* quasi indep DF=(4-1)+(4-1)+(5-1)=10, residual DF=5 ;

proc genmod; class then now symm;

model count = symm then now / dist=poi link=log;

*quasi symmetry DF=(4-1+4*(4-1)/2)+(4-1)=12, residual DF=3 ;

11STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

Symmetry model – predicted values

data

12STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

Quasi Indep. model – predicted values

data

13STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

14STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

Quasi Symmetry model

data

15STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

Quasi symmetry

16STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

17STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

18STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

19STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

20STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

21STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

10.4.7 Premarital and extramarital sex example revisited

symm

qi - quasi indep.

22STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairsproc genmod data=sex; class symm;

model count = symm / dist=poi link=log; * symmetry;

proc genmod data=sex; class extramar premar symm;

model count = symm extramar premar / dist=poi link=log; *QS;

proc genmod data=sex; class symm;

model count = symm extramar premar / dist=poi link=log; * ordinal QS;

proc genmod data=sex; class extramar premar qi;

model count = extramar premar qi / dist=poi link=log; * quasi indep;

proc genmod data=sex; class extramar premar;

model count = extramar premar unif / dist=poi link=log; run;/*linear-linear association*/

proc genmod data=sex; class symm;

model count = symm tao/ dist=poi link=log; * conditional symmetry;

proc genmod data=sex; class extramar premar qi;

model count = extramar premar unif qi/ dist=poi link=log;*quasi uniform association;

run;

G2=402.2 DF=6

G2=1.36 DF=3

G2=2.09 DF=5

G2=7.04 DF=5

G2=8.32 DF=8

G2=15.5 DF=5

G2=1.43 DF=4

23STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

24STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

proc genmod data=sex; class symm;

model count = symm extramar premar / dist=poi link=log; * ordinal QS;

25STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

data sex1; set sex;

ub_a=extramar-premar;

if extramar<premar then nab=count; else nab=0; run;

proc sql;

create table aa as

select symm, sum(count) as tcount, max(nab) as n_ab,max(ub_a) as ub_ua, count(symm) as nsumm

from sex1 group by symm having nsumm=2;

proc logistic data=aa;

model n_ab/tcount=ub_ua/ noint; run;

26STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

Data aa

proc genmod data=aa;

model n_ab/tcount=/dist=bin link=logit noint; run; /*equivalent to symmetry*/

proc genmod data=aa;

model n_ab/tcount=/dist=bin link=logit; run; /*equivalent to conditional symmetry*/

27STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

28STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

29STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

30STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

ORDINAL MODELS:

proc genmod data=sex; class symm;

model count = symm extramar premar / dist=poi link=log; * ordinal QS;

proc genmod data=sex; class extramar premar;

model count = extramar premar unif / dist=poi link=log; run;/*linear-linear association*/

proc genmod data=sex; class symm;

model count = symm tao/ dist=poi link=log; * conditional symmetry;

proc genmod data=sex; class extramar premar qi;

model count = extramar premar unif qi/ dist=poi link=log;*quasi uniform association;

run;

G2=8.32 DF=8

G2=15.5 DF=5

G2=1.43 DF=4

G2=2.09 DF=5

31STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

32STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

Models Summary of IXI table

Define two variables, such as

symm - symmetry

qi - quasi indepedence

33STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

proc genmod data=sex; class symm;

model count = symm / dist=poi link=log; * symmetry;

proc genmod data=sex; class extramar premar symm;

model count = symm extramar premar / dist=poi link=log; *QS;

proc genmod data=sex; class extramar premar qi;

model count = extramar premar qi / dist=poi link=log; * quasi indep;

34STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

proc genmod data=sex; class symm;

model count = symm extramar premar / dist=poi link=log; * ordinal QS;

proc genmod data=sex; class extramar premar;

model count = extramar premar unif / dist=poi link=log; run;/*linear-linear association*/

proc genmod data=sex; class extramar premar qi;

model count = extramar premar unif qi/ dist=poi link=log;*quasi uniform association;

run;

35STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

proc genmod data=sex; class symm;

model count = symm tao/ dist=poi link=log; * conditional symmetry;

Note:

symm - symmetry

36STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

Symmetry, Ordinal Quasi-symmetry, conditional symmetry

Logistic form

37STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairsdata sex1; set sex;

ub_a=extramar-premar;

if extramar<premar then nab=count; else nab=0; run;

proc sql;

create table aa as

select symm, sum(count) as tcount, max(nab) as n_ab,max(ub_a) as ub_ua, count(symm) as nsumm

from sex1 group by symm having nsumm=2;

proc logistic data=aa;

model n_ab/tcount=ub_ua/ noint; run; /*ordinal QS*/

proc genmod data=aa;

model n_ab/tcount=/dist=bin link=logit; run; /*conditional symmetry*/

proc genmod data=aa;

model n_ab/tcount=/dist=bin link=logit noint; run; /*equivalent to symmetry*/

38STA 617 – Chp10 STA 617 – Chp10 Models for matched pairsModels for matched pairs

Log-linear model form