Statistical Analysis in the You should use you preferred R - PDF document

About the practicals Statistical Analysis in the ◮ You should use you preferred R -environment. Lexis Diagram: ◮ Epi -package for R is needed, check that you have version 2.35 ◮ Data are all on the course website. Age-Period-Cohort models ◮ Try to make a text version of the answers to the exercises — — and some cousins it is more rewarding than just looking at output. The latter is soon forgotten — Rmd is a possibility. ◮ An opportunity to learn emacs , ESS and Sweave ? Bendix Carstensen Steno Diabetes Center Copenhagen, Gentofte, Denmark http://BendixCarstensen.com European Doctoral School of Demography, Odense, April 2019 Monday 1 st April, 2019, 13:12 From /home/bendix/teach/APC/EDSD.2019/slides/slides.tex 1/ 332 Introduction ( intro ) 5/ 332 Introduction Rates and Survival Bendix Carstensen Bendix Carstensen Statistical Analysis in the Statistical Analysis in the Lexis Diagram: Lexis Diagram: Age-Period-Cohort models Age-Period-Cohort models — and some cousins — and some cousins European Doctoral School of Demography, Odense,April 2019 European Doctoral School of Demography, Odense,April 2019 http://BendixCarstensen/APC/EDSD-2019 http://BendixCarstensen/APC/EDSD-2019 intro surv-rate Welcome Survival data ◮ Purpose of the course: ◮ Persons enter the study at some date. ◮ knowledge about APC-models ◮ Persons exit at a later date, either dead or alive. ◮ technical knowledge of handling them ◮ Observation: ◮ insight in the basic concepts of analysis of rates ◮ handling observation in the Lexis diagram ◮ Actual time span to death ( “event” ) ◮ Remedies of the course: ◮ . . . or . . . ◮ Lectures with handouts (BxC) ◮ Some time alive ( “at least this long” ) ◮ Practicals with suggested solutions (BxC) ◮ Assignment for Thursday Introduction ( intro ) 2/ 332 Rates and Survival ( surv-rate ) 6/ 332 Scope of the course Examples of time-to-event measurements ◮ Rates as observed in populations ◮ Time from diagnosis of cancer to death. — disease registers for example. ◮ Time from randomization to death in a cancer clinical trial ◮ Understanding of survival analysis (statistical analysis of rates) ◮ Time from HIV infection to AIDS. — this is the content of much of the first day. ◮ Time from marriage to 1st child birth. ◮ Besides concepts, practical understanding of the actual ◮ Time from marriage to divorce. computations (in R ) are emphasized. ◮ Time from jail release to re-offending ◮ There is a section in the practicals: “Basic concepts of rates and survival” — read it; use it as reference. ◮ If you are not quite familiar with matrix algebra in R , there is an intro on the course homepage. Introduction ( intro ) 3/ 332 Rates and Survival ( surv-rate ) 7/ 332 About the lectures ● Each line a person ● ◮ Please interrupt: ● Most likely I did a mistake or left out a crucial argument. ● ● ● Each blob a death ● ● ◮ The handouts are not perfect ● ● ● — please comment on them, Study ended at 31 ● prospective students would benefit from it. Dec. 2003 ● ● ● ◮ Time-schedule: ● ● ● ● Two lectures ( ≈ 2 hrs) ● one practical ( ≈ 1 hr) ● ● ● ● ● ● ● ● 1993 1995 1997 1999 2001 2003 Calendar time Introduction ( intro ) 4/ 332 Rates and Survival ( surv-rate ) 8/ 332

Survival after Cervix cancer Ordered by date of ● Stage I Stage II entry ● ● Year N D L N D L ● Most likely the 1 110 5 5 234 24 3 ● 2 100 7 7 207 27 11 order in your ● 3 86 7 7 169 31 9 ● database. 4 72 3 8 129 17 7 ● ● 5 61 0 7 105 7 13 ● ● ● ● 6 54 2 10 85 6 6 ● ● 7 42 3 6 73 5 6 8 33 0 5 62 3 10 ● ● ● ● 9 28 0 4 49 2 13 ● ● ● ● 10 24 1 8 34 4 6 ● ● Estimated risk in year 1 for Stage I women is 5 / 107 . 5 = 0 . 0465 ● ● ● Estimated 1 year survival is 1 − 0 . 0465 = 0 . 9535 — Life-table estimator. 1993 1995 1997 1999 2001 2003 Calendar time Rates and Survival ( surv-rate ) 9/ 332 Rates and Survival ( surv-rate ) 14/ 332 Survival function Timescale changed Persons enter at time 0 : ● to ● ● Date of birth “Time since Date of randomization ● diagnosis” . ● Date of diagnosis. ● ● How long they survive, survival time T — a stochastic variable. ● ● ● ● ● ● Distribution is characterized by the survival function: ● ● ● ● ● S ( t ) = P { survival at least till t } ● ● ● ● ● ● = P { T > t } = 1 − P { T ≤ t } = 1 − F ( t ) ● ● ● ● 0 2 4 6 8 10 Time since diagnosis Rates and Survival ( surv-rate ) 10/ 332 Rates and Survival ( surv-rate ) 15/ 332 Intensity or rate ● ● Patients ordered by ● ● ● ● λ ( t ) = P { event in ( t , t + h ] | alive at t } / h ● ● survival time. ● ● ● ● ● ● ● ● = F ( t + h ) − F ( t ) ● ● ● ● S ( t ) × h ● ● ● = − S ( t + h ) − S ( t ) ● h → 0 − dlog S ( t ) − → ● S ( t ) h d t ● ● This is the intensity or hazard function for the distribution. ● Characterizes the survival distribution as does f or F . Theoretical counterpart of a rate . 0 2 4 6 8 10 Time since diagnosis Rates and Survival ( surv-rate ) 11/ 332 Rates and Survival ( surv-rate ) 16/ 332 Empirical rates for individuals ● ● Survival times ● ● ● ◮ At the individual level we introduce the ● ● grouped into bands ● ● ● empirical rate: ( d , y ) , ● ● of survival. ● ● ● — no. of events ( d ∈ { 0 , 1 } ) during y risk time ● ● ● ● ● ◮ Each person may contribute several empirical ( d , y ) ● ● ◮ Empirical rates are responses in survival analysis ● ● ◮ The timescale is a covariate : ● ● — that varies between empirical rates from one individual: ● Age, calendar time, time since diagnosis ◮ Do not confuse timescale with ● y — risk time (called exposure in demography) a difference between two points on any timescale 1 2 3 4 5 6 7 8 9 10 Year of follow−up Rates and Survival ( surv-rate ) 12/ 332 Rates and Survival ( surv-rate ) 17/ 332 ● ● ● ● ● Patients ordered by Empirical rates by ● ● ● ● ● ● ● ● ● survival status calendar time. ● ● ● within each band. ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1 2 3 4 5 6 7 8 9 10 1993 1995 1997 1999 2001 2003 Year of follow−up Calendar time Rates and Survival ( surv-rate ) 13/ 332 Rates and Survival ( surv-rate ) 18/ 332