Logistic回帰分析の分散を考える - biostar.co.jp

1 Logistic 1. logit g(x)= ln p/(1-p) .. 1 = .. 1 = .. 2 = 3 2.. 3 .. 3 .. 4 3. 1 .. 7 4. 2 SAS .. 7 1. logit g(x)= ln p/(1-p) Logistic logit g(x)= ln p/(1-p) Var p =pq/n Var ln[p/(1-p)] = Var ln[p/(1-p)] X m 2 f(x) x=m (1) Var{f(x)}={f (m)}2*Var(x) (1) f (p)= logit=ln(p/(1-p)) p=p^ (1) var( f (p))={ f (p^)}2var(p^) (2) f (p)={ ln(p/(1-p))} ={ ln(p)-ln(1-p))} =1/p+1/(1-p)=(p+1-p)/(p(1-p))=1/(pq) (3) var( f (p^))= { 1/(p^q^)}2var(p^)={ 1/(p^q^)}2 p^q^/n=1/np^q^ logit g(x)= ln p/(1-p) var( g(p))= 1/npq n p=n1/n n1 n0 2111111))(log(nnnqnpnpqoddsVar+=+== (4)

2 2 2 a b c d dcbadVarbaVardbaVardbadbaVardcbaVar1111c loglogclog-logcloglogclog-loglog+++= + = = = Var ln[p/(1-p)] ln[p/(1-p)]= 0 Rao(1973) 2 0 2 (6) )5()()0exp(1)0exp(0)())0exp(1ln(0))0exp( 1)0exp(1ln()1())0exp(1)0exp(ln()()0exp(1 )0exp()() ())(1ln()1())(ln()}(ln{)(11111 ===== =+ = + + +++= +==nininininixiyiyiLyiyiyiLxixiyixiyilLp ppp )6()1()0exp(1)0exp(1)0exp(1)0exp()0exp(1 1)0exp(1)0exp()0exp()0exp(11)'0exp()0exp (11')0exp(111')0exp(1)0exp(')0exp(1)0exp (00)(11112121112iiyiLninininininininipp = + + = + + = + = + == + = + =+ = ======== (7) )7(1)0()1(1)0()0()1()0(111qpnVarpiiiIVar iiInini == == = = = ppp pp = (1) Var p =pq/n Var ln[p/(1-p)] =1/npq p p= q= = p= q= = (2) 2 2 (){}dcbaoddsrarioVar1111log+++= a b c d (3) = 2 2.

3 Page34-35 [ ]11)VXX() ( ra v = = = = ) 1( ) 1( ) 1( ) 1( 1 1 1 ) 1( ) 1( ) 1( 1 1 1 VXX2n212211n21iiiiiiiiiiinnxxxixxxxxxppp ppppppppppp [ ]()() = = = 2002000021) 1( 1) 1( 1)8() 1( i) 1( )8() 1( ) 1( ) 1( ) 1( ) 1( 1)(1)VXX( ra vxxxxxxxxVariiiiiiiiiiiiiiiiiiippppppppp ppppppppp 1 x i (1) SAS GENMOD link=logit (link=id g(p)=ln[p/(1-p)] = 0+ 1 ID g(p)=p= 0+ 1 Rao(1973) ====== += + = += += +++= +==ninininininixixiyixixiyiLxiyixiyiLxix iyixixiyiLxiyixiyiLxiyixiyiLxixixiyixiyi lL12222212221111)101()1()10(11)()101(1)1 ()10(100)(101)1(101)(1011)1(1010)()101ln ()1()10ln()(10)() ())(1ln()1())(ln()}(ln{)( ppp n n1 0=p^=n1/n Var( 0 =pq/n p (9) )9()0(1)01(0)0(1)01(1)1()0(100)())

4 0(221222nqpVarqpnqpqpnqnpnnnyiyiLIni ==+=+= += = = = 1 0 1 x i 1 i CHD CHD CHD 0 1 Logit ( 2) ( 4) , Logit 3 0 Odds 1 CHD 2 Logit 95% 3 Logistic 95% 4 =95% 3.)

5 1 X m 2 f(x) x=m (1) Var{f(x|x=m)}={f (m)}2*Var(x) (1) f(x) n f(x) m Taylor Series Expansion x f(x) = f(m) + f (m)(x-m) + f (m)(x-m)2/2+ f(x) f(m) + f (m)(x-m) f(x) x=m Var{f(x)} Var{ f(m)} +Var{ f (m)(x-m)}=0+{f (m)}2*Var(x) 4. 2 SAS * CHD PROC LOGISTIC OUTEST=outt.

6 MODEL CHD(EVENT='1')=AGE; output out=out p=p lcl=lcl ucl=ucl XBETA=XBETA STDXBETA=std; RUN; SYMBOL1 V=star C=RED; SYMBOL2 V=none C=BLUE I=JOIN W=2.

7 SYMBOL3 V=none C=BLUE I=JOIN W=2; * 2 ; data outb; set out; k=1;logit=XBETA;output; k=2;logit= *std;output; k=3; ;logit=XBETA+ *std;output; run; PROC GPLOT DATA=outb.

8 PLOT logit*AGE=k/nolegend; RUN; * 3 data outa; set out.

9 K=1;pp=p;output; k=2;pp=lcl;output; k=3;pp=ucl;output; run; PROC GPLOT DATA=outa; PLOT pp*AGE=k/nolegend.

10 RUN; * CHD proc genmod data=.