Logistic E xpon e n tial Model Model and Apalysis

For i ■ 1, . . . » n assume that is exp on entially distributed with parameter X^. S p lit the time axis into u n it in te rv a ls and for deaths l e t T j represent the in te rv a l in which in d ivid u al i d ies. For censorings le t t ^ denote the l a s t complete interval in which individual i was observed to have not yet died i . e . approximate censorings to have occurred at the beginning o f the corresponding unit in te rv a l. Then i f ■ e ** , the p ro b a b ility o f in d iv id u a l i surviving a unit in te rv a l conditional on en tering i t , the lo g lik e lih oo d function o f X* ■ ( X j , . . . . Xq) i 8 given

6,_i i f ti is a death i f t^ is a censoring. appropriate section. tyers e t . a l . (1973) propose a i the independent v a r ia b l e s , i . e . i » 1, 3.1 I t follow s that

Substitution o f these expressions in l(X ) y ie ld the lo g likelih ood function t (6 © ,j0 . The above authors indicate procedures fo r the maximum lik e lih oo d estimation o f 6Q, ■ ( $ ! , . • • , Bp) .

where lo g Qik • - lo g j l ♦ exp (B(

-5 2 -

P ■ . A

i * 1 j - i * j1 * i - i 1 y tk >rj Mantel and Hankey develop procedures f o r e s t is ia t in g the parameters in the model and discuss an a p p lic a tio n u sin g th e d a ta given in tye rs e t . e l . Estimation o f covariance m atrix

% * r i e t . a l . in c a lc u la t in g the c o v a ria n c e m atrix o f • „ , £ ( and tf) in the model at

3.1

3.2)

A lte rn a tiv e ly , the expected v a lu e s, may be evaluated by n o tin g that under the random censorship model, with c e n s o rin g d is tr ib u t io n H „ (y ),

r -x jt

E («i> - I A. a 1 (1 - Hy ( t ) > dt

#0

* ( . , ) - J u / . X<t [ » u - H <t>> . 0 > l H1 < « . ! ) ] M .

u—1 u—1

Assuming a s p e c if ic form f o r H y(y) the ex p e c te d values o f t . and 6^ may be substituted in place o f x^ and 6^ in the second p a r t ia l d e riv a tiv e s o f the lo g lik e lih o o d to y i e l d the covariance m a trix o f parameter e stim ators. (The second de riv a tiv e s a re simple li n e a r fu n c tion s o f Xj and 6 ^ ). The introduction o f the fun ction g makes th e use o f th is technique fo r the model at

3.3

-5 3 - 1 3 » 3 . F r o p ° r V i ^ ‘ V Tha. cpa.J3R.a~A C o x ( 1 9 T 2 ) p r o p o a e » a m o d e l i n w h ic h . f o r i » 1, . . . . n, the hazard f u n c t i o n » ¿ ( O f o r t h e i * t h i n d i v i d u a l i a g iv e n by » D E L X s X£ < t > - X0 ( t ) e x p ( £ ’ * ± )

w h e r e ¿ ’ - C » x . - - - . S p ) i s » v e c t o r o f unknown parameters and * Q( t ) U e n u n k n o w n f u n c t i o n o f t i m e . N o t e t h a t f o r any two in d iv id u al» i , j , i f j ,

X t ( % ) - X j t t ) e x p < X * ( ^ i - J g > >

a o t h a t t h e m o d e l i a o f t h e p r o p o r t i o n a l ha zard a type. An a t t r a c tiv e f e a t u r e o f m o d e l I i a t h a t t h e f u n c t i o n AQ( t ) , termed the w iderlyin g h a z a r d f u n c t i o n , i a l e f t a r b i t r a r y . I n th e next »e c tio n models in which XQ ( t ) t a k e s a a p e c i f i e f o r m w i l l b e c o n s id e r e d .

T h e E x p o n e n t i a l a n d W e i b u l l « a u a c is

P r e n t i c e ( 1 9 T 3 ) h a s c o n s i d e r e d tw o models in which the fun ction X ( t ) t a k e s a s p e c i f i c f o r m . e - 1 « 1 X then <*,X>0 o d e l Z th e n I f Xo ( t ) - Xat », / * > - X .t" - 1 .XI ( f Z l ) . MODEL I I I s I f Xo ( t ) - X in l X±( t ) - X exp ( £ , '

N o t e t h a t i f a - 1 , m o d e l I I r e d u c e s t o I I I . The ran dee v a ria b le T j , r e p r e s e n t i n g s u r v i v a l t i m e f o r t h e i ’ t h i n d iv id u a l ia exponential under m o d e l I I I a n d W e i b u l l u n d e r m o d e l I I .

In th e two group case

group 1 members,