Duration models. Jean-Marie Le Goff Pavie-Unil

Transcription

1 Duraion models Jean-Marie Le Goff Pavie-Unil

2 Oher erms for duraion models Duraion models: economery Survival analysis : medical sciences, demography Even hisory analysis, ransiion analysis : social sciences

3 Ouline Aim of even hisory analysis How o prepare daa? Rule of precauions Censoring To invesigae daa The life able command in SPSS Elemens of hazard regression models The choice of a model (Cox model, ime discree logi model) The person-period file for discree ime models Esimaions Inerpreaion of resuls Example of Inerne adopion on panel daa

4 Aims of even hisory analysis Sae (Y) Even = A ransiion or a swich from a sae 0 o a sae 1 «A change in a variable» (Tuma & Hannan 1984) 1 0 Duraion a risk o experience he even l ime -Esimaion of he disribuion of he risk or he hazard o experimen he even along he ime - Influences of individual and conexual characerisics on he hazard

5 Noaions Discree ime Coninuous ime P( l ) = P( T = l T l ) PT ( < +Δ T ) h ( ) = lim Δ 0 Δ The hazard corresponds o he condiional probabiliy o experience he even = Probabiliy o experience he even a ime given ha people did no experience i before

6 Before running an even hisory analysis Do no deermine he fuure of individuals The risk of occurrence of an even depends on he pas and he presen of individuals and no of heir fuure A second marriage canno explain a firs divorce The rule is o follow individuals hrough ime

7 To operaionalize an analyze Definiion of he analyze Definiion of he populaion submied o he risk Time 0 and clock Independan covariaes To define censors Fixed covariaes and ime dependan covariaes To have experience he even or no Example Firs union formaion Individual who never cohabied wih a parner Age from 16 years before To ge in a firs union Religion, pracice. Ec. People who did no ge ino a union

8 Censoring Lef censors Righ censors o max

9 A couple of wo dependen Age from 16 years old o : variables If a firs union : age of his firs union formaion If no : age a he momen of he survey Censors: Everybody is considered o be submied o he risk unil he disappearance of he populaion (because of union formaion, or because of he momen of he survey) 0 : people who did no ge ino a union 1: people who ge ino a cohabiing union 2: people who ge ino a direc marriage

10 How o invesigae daa. The life able mehod Hazard o experience he even in discree ime: condiional probabiliy o experience he even P( l ) = Number Number of of evens occured beween i me l and ime l + 1 persons who did no experience he even a ime l -E l = Number of evens in l -C l = Numbers of censoring spells during he inerval [ l, l+1 ] -R l = number of persons who did no experience he even in l, = number of persons who have an observed or censored duraion greaer or equal o l P( l ) = R l El 1 C 2 l

11 Probabiliy of survival (probabiliy o no have experienced he even) S( ) = (1 P(0))(1 P(1))...(1 P( )) S( ) = u= Π (1 u= o P( u))

12 Hazard rae (Hazard in case of coninuous ime) h = Number of evens during an inerval of ime Number of person -years submiing o experience he even -E l = Number of evens in l -C l = Numbers of censoring spells during he inerval [ l, l+1 ] -R l = number of persons who did no experience he even in l, = number of persons who have an observed or censored duraion greaer or equal o l h ( l ) = R l 1 2 E l ( E C ) l l

13 How o choose beween a Cox model and discree ime logisic Cox model Logisique Coninuous ime Discree ime A small uni of ime A large uni of ime Less han 5% of individuals experience he even during a ime inerval More han 5%

14 Coninuous ime vs discree ime Classic models (Cox models) are based on a coninuous ime (ime in days in medical research) Two kinds of discree ime: «rue» discree process of daa generaion (acces ino higher degree for a populaion of sudens) Coninuous process of daa generaion bu long inerval beween measures

15 Cox model h h (, x ) = h0 ( ) exp( β ) x = h ( ) exp i (, x ) 0 x i β i Où β1 β2 β = β 3... β n coefficiens o be esimaed -Non-parameric composan of he model : risk in he case of individuals who have all heir characerisic x =0 (individual of reference)

16 Time discree logisic model α : foncion of ime ( ) [ ] x x P x P x x P β α β α + = + + = ), ( 1 ), ( log exp 1 1 ), (

17 Alernaives Discree ime logi models should be used in case of a «rue» discree process of daa generaion Alernaive 1: discree ime probi models Alernaive 2: discree ime complemenary log-log models. Theoriically more adequae if he process of daa generaion is coninuous like in he case of he panel Logi model remains he more diffused and developped in he lieraure because of is simpliciy

18 Preparaion of a daabase (Allison, 1982) The equaion of log-likelihood for discree ime models can be simplified in an equaion of log-likelihood of a dichoomic covariae Which means ha discree models can be esimaed on a «person wave» daabase (person-period daabase) An individual is represened by a number of lines equal o he number of waves he is presen before o experimen he even or o leave he observaion The dichoomic dependan variable is equal o 0 in all lines excep in he las one where i is 0 or 1 (censored or even) Remains rue when several levels in he daa (Barber e al, 2000).

19 File person-year ID Age Censor

20 Noes on covariaes x Fixed covariaes (non-ime dependen) Saus a he birh Saus reached before he beginning of he observaion Time dependan covariaes Predefined covariaes (clocks) Auxiliary covariaes (conex) Inernal covariaes (inerdependencies, linked lives)

21 Example of inerne adopion Even hisory can also be used o analyse processes of diffusion of an innovaion, a behavior, a rumor, (Dieckmann; 1989, Srang and Tuma, 1993) Influence beween persons in a household Does a firs user in an household influence ohers? If yes, increase in he probabiliy o adop inerne when a firs person adoped i. Limied here o parners Does he adopion of inerne by he man (he woman) increase he risk of adopion of his (her) parner?

22 Use of Inerne % Source: BFS (MA_ne,Ne- Marix base) and Swisspanel Resrained circle Exended circle Panel

23 In he presen case Selecion of couples saring o be inerviewed in 1999 and who did no declare o use inerne in Couples wih no missing values on he use of inerne Conrol covariaes (educaion, language, ec).

24 File +1 ID IDHOUS NUMBER INTERNET COVARIATES IDPART INTERNET PARTENAIRE

25 RESULTS (esimaed coefficiens) Men Women Model 1 Model 2 Model 3 Model 1 Model 2 Model 3 Consan *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** * *** *** *** *** *** *** *** *** *** *** *** ** *** *** *** *** *** ** *** *** *** *** *** ** *** *** *** *** *** * Children 0.72 *** *** Compuer 1.34 *** 1.13 *** 0.95 *** 0.79 *** Parner already use *** 0.53 *** Before _ *** 0.95 *** 1950_ *** 1.84 *** 1960_ *** 1.90 *** 1970 and afer 1.50 *** 2.32 *** Level1 0 0 Level Level *** 0.98 *** German 0 0 French * Ialian Oher *** Non-Swiss *:5%,**:1%,***,0,1%.

26 References Blossfeld H.P. and Rohwer G (1995) Techniques of even hisory modelling. Mahwah: Lawrence Erlbaum. Box-Sephensmeier J. and Jones B. (2004). Even Hisory Modeling: A Guide for Social Scieniss. Cambridge: Cambridge Universiy Press. Singer J.D. and Wille J.B. (2003). Applied Longiudinal Daa Analysis. Modelling change and Even Occurrence. Oxford. Oxford Universiy Press.