Analysis of Censored Life-tables with Covariates by Means of Log-linear Models

A C T A U N I V E R S I T A T I S L O D Z I E N S I S

FOLIA O E C O N O M IC A 175, 2004

A g n i e s z k a R o ssa *

A N A L Y SIS O F C E N S O R E D L IF E -F A B L E S W IT H C O V A R IA T E S BY M E A N S O F L O G -L IN E A R M O D E L S

Abstract. In survival analysis the subject o f observation is duration o f time until som e event called failure event. Often in such studies only partial inform ation on the length o f failure time is available what yields the so-called right-censored observations. The m ain interest in survival analysis is either to estim ate the distribution o f the true failure tim e or to identify the relationship between the true failure time and a set o f som e covariates. A dditional troublesom e poin t o f theory and application o f survival techniques is treatment o f grouped observations (life-tables) alo n g with incorporating covariates.

In the paper a new approach is considered which allow s to treat the censored life-table with qualitative covariates as a standard contingency table. Such a table can be further analysed by m eans o f log-linear m odels or other standard m ultivariate inference techniques.

Key words: survival analysis, censored data, life-tables, log-linear m odels.

1. IN T R O D U C T IO N

T h e usual rep resen ta tio n o f the right-censored ra n d o m sam ple w ith covariates takes th e form

(Tj, Xj), i = L 2, ..., n (1)

where öt = 1 if an i-th individual actually failed at tim e T t, St = 0 if an individual was right-censored at time T t and X t is a p-dim ensio nal vector of know n covariates, fo r exam ple, sex, age and o th er characteristics o f an individual.

N early all the statistical m eth o d s for censored survival d a ta are based °n the assu m p tio n th a t censoring m echanism is n o t related to m echanism causing failures. T h u s th e usu al m odel fo r censored survival analy sis assumes in d ep en d en t ra n d o m censoring. In this m odel v ariab les T t an d ô\ can be defined as follow s

T, = m in (У,, Z (), S, = 1, if T l = Y l (2)

0, if Г,if 7’, = Z,

where У, are independent copies o f a positive random variable У representing true failure tim e with a cum ulative distributio n fun ctio n (cdf) F. Sim ilarly, Z, arc independent copies o f a positive rando m variable Z with a c d f G. It is assum ed th a t variables У an d Z are independent, con d itio n ally on X. T h u s th e observed v aria b les T, rep resen t here in d ep en d e n t copies o f a v a ria b le m in ( y , Z ) w ith a c d f II satisfy in g th e e q u a lity

A special case o f in dependent censoring occurs in studies w here failure time is m easured from entry into the study and one observes th e true failure tim es o f those individuals w ho fail by th e tim e o f analysis and censored tim es fo r th o se in dividuals w ho do no t. In such a case all censoring tim es Z, are know n an d the sequence

instead o f sequence (1) is observed. It is w orth n otin g th a t in th e repeescn- tatio n (3) variables S t are re d u n d an t and therefo re can be om itted .

T h e m ain interest in survival analysis is either to estim ate th e d istrib u tio n o f the tru e failure tim e У represented by F o r the so-called survival function F = 1 — F o r to identify the relationship betw een the tru e failure time У and a set o f covariatcs X. A dditio nal trou blesom e p o in t o f theo ry and application o f survival techniques is trea tm e n t o f g rouped o bserv atio n s (life-tables) along w ith in co rp o ratin g covariates.

S ta n d ard life-tables techniques are the oldest techniques m o st extensively used by actuaries, m edical statisticians and d em o grap hers, startin g from the w ork o f J. G ra u n t in 1662 (cf. D. V. G l a s s (1950), B. B e n j a m i n (1978)).

T h e life-table d a ta arise from a p artitio n o f the range [0,7” ] o f o b ser­ vations into som e tim e intervals = [tk, tt + 1 ) , к = 0, 1, К — 1 where

the en d p o in ts 0 = I0 < < . . . < tK < T* are pre-specified. T he life-table d a ta can be ch a rac te rized by d efining n u m b ers o f in d iv id u a ls alive a t th e beginning o f each tim e interval and by defining nu m b ers o f failures and censored o b serv atio n s in these intervals.

T h e m ain p u rp o se is to estim ate co nditional pro b ab ilities o f failure in the intervals Qt given survival to tk o r to estim ate p ro bab ilities o f survival 11= 1 - ( 1 - F ) ( l - G ) .

(3)

past in +1 for к = 0, 1, К — 1 (sec E. L. K a p l a n and P. M e i e r (1958), C. L. C h i a n g (1968)).

D . R. C o x (1972) gave a First system atic study o f use o f co variates in the analysis o f failure tim e. H e proposed a regression m odel fo r a hazard function and introduced a vector o f unknow n regression p aram eters specifying the effect o f co v ariates on survival. If the covariates are n o t tim e-varying then C o x ’s m odel can be term ed “ p ro p o rtio n al h a z a rd s ” because th e ra tio o f h az ard fu n c tio n s fo r an y tw o individuals is in d e p e n d e n t o f tim e. Sub-sequent p apers by J. D . K a l b f l e i s c h and R. L. P r e n t i c e (1973), N. B r c s l o w and J. C r o w l e y (1973), N. B r e s l o w (1974, 1975), O. O. A a l e n (1978), P. K . A n d e r s e n and R. I). G i l l (1982) are the substantial co n trib u tio n s to this subject.

T. R. H o i f o r d (1976) introduced the p ro p o rtio n al h azard s m od el for life-table d ata. In his m odel the baseline hazard fu nction was assum ed to be c o n sta n t w ithin each tim e interval i l k, w hat im plies piecewise exp on ential distrib u tio n s for failure times.

T his ap p ro ach was fu rth e r developed by T. R. II о 1 f o r d (1980) and N. L a i r d a nd D. O l i v i e r (1981), who discussed ap p licatio n o f log-linear analysis tech niques to life-tables with categorical covariates. T h e ir key result refers to tw o im p o rta n t observations. F irst, log-linear m odel fo r cell m eans o f Poisson contingency tab le d a ta is equivalent to log-linear m o del for a hazard fu nction in piecewise exponential survival m odel. Second, the likelihoods for b o th m odels are equivalent. T h u s, the statistical inference m ethods based on m axim um likelihood for these m odels are also equivalent.

T h e b ro a d survey o f the developm ent o f the survival analysis th ro u g h o u t the tw en tieth century can be found in T . R. F l e m i n g and D. Y. L i n (2000) o r D . O a k e s (2001).

3. LO G-LINEAR M O D E L S FOR LIFE-TA BL ES W ITH CA TEG O R IC A L C O V A R IA T E S

L og-linear m odels provide a flexible and p o p u lar to o l o f tre a tin g the m ultivariate categorical d a ta arran g ed in a m ultidim ension al co ntingency table. Som e o f the m o re attra ctiv e features o f this ap p ro ach are th e easy of m odel specification, flexibility in treating bo th dependent and independent variables and the fact th a t the equivalent m axim um likelihood estim ates o f m odel p aram eters m ay be o btained from different sam pling d istrib u tio n s, such as P oisson, m u ltin o m ial and p roduct m ultin om ial d istrib u tio n s.

As it was p ointed o u t by N. L a i r d and D. O l i v i e r (1981), log-linear techniques can be easily applied in life- tables analysis to identify the relationship betw een th e survival tim e and a set o f categorical covariates.

'Г. R. H o l f o r d (1976) considered the follow ing re p resen ta tio n for the hazard fu n ctio n h(y; X) o f failure time Y

where hk d enotes a c o n sta n t baseline hazard in th e tim e interval í)*, X is a fixed co v ariatc vector and ß is a vector o f unkn ow n p aram eters. N o n ­ p ro p o rtio n al h azards m odel can be reform ulated from (4) by allow ing the baseline h az ard hk, к = 0, 1, К - 1 to depend also on X.

T h e re p resen ta tio n (4) implies th a t, co n dition al on X, h az ard fun ctio n h (y ;X) is a stepwise function o f tim e and failure tim es have piecewise exponential d istrib u tio n s. L og-linear hazard m odel prop osed by L aird and Olivier flows directly from H o lfo rd ’s m odel and takes the form

Let us assum e th a t the vector X specifies the levels o f p categorical c o v a ria te s an d ea ch c o v a ria te X , o f X has / , levels in d ex ed by iv s = l , 2, ..., p. D en o te for sim plicity by i0 the index o f tim e intervals, i0 = 0, 1, 2, ..., К — 1. T h u s, for the given tim e interval Q io and for the fixed set o f co v ariates a t levels (iu i2, ..., ip) the h azard fun ctio n h(y ;X) given in (4) takes a c o n sta n t value, which can be d eno ted by 0ioii if. T h en em ploying the usual log-linear “ u-term s” n o tatio n , intro d u ced by M . W. B i r c h (1963), the m odel (5) can be rew ritten in th e follow ing form

w here param eters {effects} on th e right-hand side o f (6) satisfy the follow ing linear constrain s

T he n o n -p ro p o rtio n a l h az ard s m odel can be in trod uced here by a sim ple generalization o f (6)

K y i X) = /v c x p { X T/0 for y e C l k, fc = 0, 1, K - l (4)

h(y; X) = \nhk + X Tß for y e Q k, к = 0, 1, ..., К - 1 (5)

(

)

T hus, th e problem o f estim atin g survival d istrib u tio n s u n d er the m odel (6) or (7) reduces to estim atin g the u-param etcrs, w hat can be d o n e by m eans o f slightly m odified Iterative P ro p o rtio n al F ittin g R o u tin es (see N. L a i r d a nd I). O l i v i e r (1981) for details).

I'he fo rm u la o f estim ating the log-survival fu nction ln S (i) derived from the piecewise exponential d istrib u tio n is expressed as follows

ln£(i) = -e x p { ú * } _{I ( 0 + 1 - + 0 - i*)exp(t2jt°>}l t e C l k + 1}

(

)

T he m odified log-linear m odel for censored life-table d a ta prop o sed here allows to han dle m an y IP F routines for log-linear m odels w ith o u t any m odification. T h e p ro p o sed m odel is closely related to the on e given in (7), how ever we will assum e th a t 0 , represent pro b ab ilities o f failure in tim e intervals Qio for fixed sets o f covariates at levels (ilt i2, ..., ip), T his approach is based on the extended life-table d a ta and is based on a m eth o d called here “ the co m p letio n m e th o d ” . T h e a p p ro a c h allow s to ap p ly standard inference.

4. E X T E N D E D LIFE-TA BL ES W ITH R IG H T -C O N S O R E D D A TA

F o r sim plicity, let us assum e th a t the covariate vector X is n o t observed. Let T * > 0 be a fixed real n u m b e r such th a t H ( T * ) < 1 . L et ® = fo < h < — < tK = T* < o° co n stitu te a p artitio n o f [0,T*] in to К sub- -intervals o f the form Q k = [t*, í, + J for к = 0, l , . . . , К - I . Let us also assume th a t n A. = [ix , oo).

Let us assum e th a t individuals enter the follow -up study a t ra n d o m time points. F o r an i-th individual we observe a p a ir o f ra n d o m variables f^i> Zj), w here T t an d Z ; are defined in Section 1. T h e o b serv atio n o f m dividuals term in ates w hen for s item s ( 0 2 is a fixed integer) we o b ta in T'i j >T*, j = I, 2, s. Let N s den o te the to tal n u m b er o f individuals observed in the experim ent. T hus, N , is a ran d o m variab le d istrib u te d according to th e negative binom ial distribution w ith p aram eters s and P = 1

-W e will co n sid er a n exten ded life-table d a ta ch a ra c te riz e d by the following statistics

N. Dk = 1 1 ( T i e Slk, Z ^ t k+l ), /= 1 k = 0, 1, .... K - 1 DK = 0, Ok = Í O ( T l eClk, Z l eClk), к i= i = 0, 1, ..., К N. M k = ^ 0(T , ^ i*, Z ; ^ tk + j), 1 = 1 1 о II о II * 5 Wk = Ok + M k, к = 0, 1, ..., К

where 1 (A) den o tes a characteristic function o f a set A. N o tice th a t th ere is W0Ok + M 0 = N , and WK = 0 K.

S tatistics defined in (9) co n stitu te an extended censored life-table and will be em ployed in the p ro cedure called “ a com pletion m e th o d ” .

5. C O M P L E T IO N M E T H O D KOR E X T E N D E D LIFE-TA BL ES

Let us consider a p ro b a b ility qklk defined as follows

qklk = P ( Y e C l k\ Y > t k), к = 0, 1, ..., К - 1 (10)

T his is the p ro b a b ility o f failure in Q*, co nditio nal on survival p ast tk. This prob ab ility will be estim ated by m eans o f the follow ing statistics

^ ' ‘ = л Г ? Т к = 0 ' 1 K ~ l (11)

T h e sim ilar e stim ato r o f qk]k was firstly considered by E. L. K a p l a n and P. M e i e r (1958) for the sam ple with a fixed size. It is usually called the R educed-S am ple E stim a to r (R SE). Let us define a p ro b ab ility qkll as follows

qk[l = P ( Y e a k \ Y ^ t , ) , k = 1, 2, ..., К - i , I = 0, 1, ..., к - 1 (1 2)

T his is the p ro b a b ility o f failure in the interval Q k co nd itio n al on survival past i, for 0 < / < к and can be estim ated from the following recurrent form ula

W — 1

Ük\t = _ j к = I, 2, ..., К — 1, 1 = 0, 1, к —1 (13)

and the estimated number o f failures in the interval П, for О , individuals, who survived past th can be calculated as

Ą . I * O r &II-; l = k , k - 1... 1 , 0 , к = 0, 1, ..., К — 1 (14) Let 6 = Dk + £ ß u , k =0, 1... K - 1 i = ° (15) 0 K = N , -

е Ч

I he sum E f=0Dt / on the rig ht-hand site o f (15) can be trea ted as an estim ated n u m b er o f failures in the interval Clk fo r those item s fo r which Z i < t k + i - T h u s, Ďk is an estim ated to tal num bers o f failures in the intervals 0» for k = 0 , 1 , ..., K.

T he set o f estim ates Dk determ ines a completed version o f an extended censored life-table defined by the statistics (9). T h is co m p letion p ro ced u re !S esplained in details in an exam ple in Section 6.

Generally, we can consider extended life-tables constructed for a categorical covariate vector X fixed a t levels (iu i2, ip) and calculate the estim ated num bers o f failures sim ilarly as in T ab . 2. P roceeding in such a way for each c o m b in a tio n o f levels (i lt i2... ip) o f X we o b ta in as a re su lt a P + 1-dim ensional contingency table with estim ated n u m b ers o f failures ior each co m b in atio n ( ij, i2, ip) and for each tim e interval П,о in its body. Such a table can be next analysed by m ean s o f sta n d a rd log-linear techniques m en tio n ed in Section 3.

6. A N U M E R IC A L EXA M PLE

W e will co n sid er a sam ple o f p atients who have had received a valve •niplantation (b io p ro th esis o r m echanical valve) an d h ad to be reopered because o f som e valve com plications. P atients en ter the stud y a t ra n d o m time points. T h e subject o f observation was the length o f th eir life after feo p e ra tio n (in years). T h e stud y w as term in ated w hen s = 8 p a tie n ts

survived p ast T* = 7 years. T h u s, the length o f life after re o p era tio n is a rando m right-censored variable, for som e o f the patien ts were alive by the end o f the study. T h e to ta l n u m b er o f p a tie n ts N a observed in such an ex p erim en t is a ra n d o m variable. Its re alizatio n observed here was equal to 50.

Let f0 = 0, tj = 1, t 2 = 1 and í í0 = [0, tj ), Q , = [ix, t 2), П2 = [l2, со). We will consider an extended life-table determ ined by the statistics Dk, M k, Ok for к = 0, 1 , 2 (see T ab . 1).

T a b l e 1

The Extended Life-Table

Tim e Dk o k My

Qo = [0, 1) 8 9 41

0 , - [ l . 7) 2 23 10

= [7, oo) 0 8 0

F ro m T ab . 1 we ca n now estim ate to tal n u m b ers o f d ea th s Dk in each interval by m eans o f form ulae given in (15). T hese estim ates co n stitu te “ a com pleted version” o f T a b . 1 (see T ab . 2).

T a b l e 2

Com pleted Version o f Tab. 1

Tim e intervals б к

fto = [0, D 9.8

= [1. 7) 8.0

« 2 = 17, 00] 32.2

N o te, th a t first tw o estim ates in the second colum n o f T ab . 2 represent estim ated values o f Dk for /c = 0, 1, and the last value is calculated as N , - U = 0 Ď k .

7. S O M E TH EOR ETIC A L R E SU L T S

Theorem. T h e estim ato rs Qklk and Qkll defined in (11) and (13) are unbiased estim ato rs o f respective co nditional probabilities qk}k an d qkц.

P roof.

Let us assum e the follow ing n o tatio n

P k - P ( T > x k) for /c = 0, 1, ..., K ,

4k = P ( T > x k, Z > x k+l) for k = 0, 1, K - l .

F o r 0 ^ l < k and l ^ k ^ K — 1 the estim ato r Qkll acco rd in g to (13) equals to

= Wl+lJ- \ Wl+2 - I Wk. t - 1 Wk - 1 Dk

М , - 1 M n i - \ ' M k- 2 - l ' M k- t - l ' M k - l

F o r I = к and 0 ^ k ^ К — 1 we have from (11)

Dk Qklk~ M ^ i '

T he e x p ressio n Dk/ ( M - 1) ca n be also w ritte n e q u iv a le n tly as * ~ (W * + i — l)/(Af* — 1), thus for 0^ l < k and l ^ k ^ K — 1

л Wl + i - 1 Wl +a - 1 W i - i - 1 W i - 1 / Wi+ 1 - 1

M, —

M, +

—

Л/ц _

—

M*-j —

у

M* —

and fo r / = к an d 0 < к < К — 1

ô - i W^ ~ l “ M , - 1

Let us d en o te by A rl, the follow ing expression o f the form

w , * i - i Wi+г - i w t - i - i w ; - i w t+ i — i Ar\i —

M , - 1 M 1+1- l "■ M r _ 2 - 1 M r_ ! - 1 M r - 1

(16) where r > / . N ow the e stim ato r Q*,, expresses as follow s

- _ {Ak- m — A k\t fo r 0 ^ l < k , l ^ k ś K - l

Let us find the expectation o f A rU defined in (16) using th e jo in t d istrib u tio n o f the variables N „ M t, Wl + l , ..., M r_ 2, Wr- X, M r- lt Wr, M T, Wr+i .

N otice, th a t the sam ple size N , is a rand om v ariable w ith a negative binom ial d istribution and th a t given N , = n the variables Wl + l , M r_ 2, Wr- i , M r- i , Wr, M r, Wr+l have a m ultinom ial d istrib u tio n . T h u s the jo in t prob ability d istrib u tio n function is o f the form

P ( N , = n, M , = m „ Wl + 1 = w) + 1, M r _ , = mr _ , , W, = wr , M , = m r, , = w r+ 1) = n - l V n - Л / m , _ S V r m r - l ~ S\ f W' ~ s \ ( m ' ~ S S _ 1 ) \ m l ~ s j \ w l + l — SJ \ W r ~ S ) \ m r - y \ W r + l ~ S ( l - q () " " И'( 9 < -P l + l ) m'” W,+ , • ■ ( P l + 1 - q ,+ 1 • ...(Я г -i - Р г Г ' - ' - №'(Рг - Я гГ '~ т ’(Яг - P r+ l ) " ' " ( P r + l - P K ) W,+í~ ‘PK> where n = s, s + 1, f f l, = s, s + 1, n , w i + i = s , s + 1 , m„ i = /, / + 1 , r, m, = s, s + 1, w, i = / + 1, r.

T h e d istrib u tio n function o f N s, M (, Wi + l , ..., M r- 2, Wr- U i, W,, M r, VFr + 1 can be expressed also equivalently as

P ( N S = n, M , = m„ Wl+1 = Air _! = m ,- ! , W, = wr, M r = m r, Wr+1 = = wr+1) =

n - l V

V

V Wr+1 -

^ p r - qry m^ q ry ^ r ~ P r . Л " ' w' " ^ P r < A " ' 4' / ,_Рг+i- P kY ' * 1 V Pk P r + 1 / \P r + 1 where w r + 1 = 5, s + 1, m, = wl+1, wi + 1 + l , .... i = r, r - 1, / + 1, I, w, = m(, m ,+ l, i = r, r - 1, / + 1, I, n = m(, m, + 1, ... T h u s, th e ex p ectatio n o f A ril is equal to

00 00 00 00 00 Ш — 1 Ul 1 111 I £M ,„> = I E ... I S * r)1 = i ą = w , ( l W|+ ) = m i* i m, = w,+1 n = m, m l * m r - l I M r 1 • p ( w s = n , ..., w ;+ ! = wP+ j) = w f + ! — 1 \ / P r+ 1 Pk\ w ’ * x ~ ‘ ( Рк \ ’ у w r + 1 - 1 / m r — у / " r+ 1 * W Kr+1 I'K 1 / ť í 1 v-1 Wr+ 1 = * \ s — 1 / V P r + l / \ P r + l ) m , . w r + | w r — 1 V w r + 1 1 . | 4 ~ " Wr+ ‘^ r + 1 \ W,+1 _ £ / 4 + , - 1 V pI+ 1 - *,+ i y , ł l ' w,łV ^ t i \ " ,łł £ W,+ 1- 1/ m, — 1 \ W| + 1= m , + 1\ m ( + l — V \ P l + l / \ P ( + l / mi-wi+i m l ~ 1 \ W! + 1 — V i ( " ~ \ \ 1 - ЧГ = & ± i - Ł . . . P t t l . ? ! i i n = m \ m l ~ V <7r 9 r - l 9 i - l <?! P ( T ^ t r+1) P ( T ^ £ r) P ( T ^ t l+2) P ( T > t „ Z > t r+1) P ( T ^ i r _ i , Z ^ i r) P ( T > t l + u Z > t l+2) P ( T > t l+1) P ( T > t[, Z ^ i|+ i)

P ( y > t r+ 1 , Z > t r + ł ) P { Y ^ t „ Z t r) P ( Y ž í|+ 2 i Z ^ íi + 2) P ( Y ž t r, Z ž t r l l ) P ( Y > t r - u Z > t f) P ( Y > t „ u Z > t l + i ) ' P ( Y > t l t l L Z > t l t l ) _ P ( Y ž t i , Z ^ t i +1) = p ( y ^ t r+1) р (У 5 * о P ( ľ > t l+i) P ( y > t , +1) f ( ľ > < r t l ) Р ( У > * , ) P ( Y > t r - \ ) Р ( У > í | +1) P ( ľ > í , ) P ( ľ > t , ) '

F r o m th e r esu lt ju s t o b ta in e d and u s in g th e d e fin itio n (1 7 ) w e h a v e fo r 0 < / < / c ,

U U K - 1 , ч Р ( У > « * ) P ( Y > t k+1) Р ( У е П . ) Я (& к ) - Я ( Л - щ ) - -E(^*tí) = p ( Y p t , ) ~ Р ( У > 0 ~~ P ( Y > t , ) ~ qk>1’ and fo r I = k, O ^ k ^ K - l v r t П А Ф > Ь + 1) P j Y e C K ) _ ( ( M ( *1*) P ( y > í * ) P ( ľ > ý w h a t c o m p le te s th e p r o o f . R EFEREN CES

A a l e n O. O. (1978), N onparam elric Inference For a Fam ily o f C ounting P rocesses, “The A nnals o f Statistics” , 6, 701-726.

A n d e r s e n P. K. , G i l l R. D . (1982), C o x ’s Regression M o d e l For Counting Processes: A Large Sam ple S tu d y, “T he Annals o f Statistics” , 10, 1100-1120.

B e n j a m i n B. (1978), John G raunt, [in:] (eds.) W. H. K r u s k a l , J. M. T a n u r , International Encyclopedia o f Statistics, Vol. 1, Free Press, Macmillan Publishing C o., N ew York, 435-437. B i r c h M. W. (1963), M a x im u m Likelih o o d in T hree-W ay Contingency Tables, J. R oyal Stat.

Soc., Ser. B, 25, 220-233.

B r e s l o w N . (1974), Covariance A na lysis o f C ensored Survival D ata, “ B iom etrics” , 30, 89-99. B r e s l o w N. (1975), A na lysis o f Survival D ata Under the P roportional H a z a rd ’s M odel, Int.

Stat. Rev., 43, 4 5 -5 8 .

B r e s l o w N. , C r o w l e y J. (1973), Large Sam ple S tu d y o f the L ife Table a n d Product L im it E stim ates Under R an d o m Censorship the Annals o f S ta tistics, 2, 437-453.

C h i a n g C. L. (1968), Introdnction to Stochastic Processes in Biostatistics, John W iley, N ew York. C o x D . R. (1972), Regression M o d e l A n d Life-T ables, J. Royal Stat. S oc., Ser. B, 34, 187-220,

(with discussion).

F l e m i n g T. R., L i n D . Y . (2000), Survival A nalysis in C linical Trials: P ast D evelopm ents and Future D irections, “ Biom etrics” , 56, 971-983.

H o i f o r d 'Г. R. (1976), L ife-T a b les with C oncom itant Inform ation, “ Biom etrics” , 32, 587-597. H o i f o r d T. R. (1980), The A na lysis o f R a tes and Survivorship Using Log-linear M o d els,

“Biom etrics” , 36, 299-306.

K a l b f l e i s c h J. D. , P r e n t i c e R. L. (1973), M arginal L ikelihoods B a sed on C o x's Regression and L ife M odel, “ Biom etrika” , 60, 267-278.

K a p l a n E. L., M e i e r P. (1958), N onparam etric E stim ation From Incom plete Observations, J. Amer. Stat. A ssoc., 53, 457-481.

L a i r d , N. , O l i v i e r D . (1981), C ovariance A nalysis o f C ensored Survival D ata Using Lo g -L in ea r A na lysis Techniques, J. Amer. Stat. A ssoc., 76, 231-240.

O a k e s D . (2001), B io m etrika C entenary: Survival A nalysis, “ Biom etrika” , 88, 99-1 4 2 .

A g n ie s z k a R o s s a

A N A L IZ A TA BLIC T R W A N IA ŻYCIA D LA D A N Y C H C E N Z U R O W A N Y C H Z W Y K O R Z Y ST A N IE M M O D E L I L O G A R Y T M IC Z N O -L IN IO W Y C H

W pracy przedstaw iono propozycję analizy tablicy trwania życia dla danych praw ostronnie cenzurowanych. Przedstawiona m etoda pozwala na sprowadzenie takiej tablicy d o w ielo­ wymiarowej tablicy kontyngencyjnej, którą m ożna analizow ać standardowym i technikami wielowymiarowego w nioskow ania statystycznego, np. za pom ocą m odeli logarytm iczno-liniowych.