User's guide for R package simMSM
Contents
1. setwd C BayesX commandline output log bhr rep index beta rep index vector list 6 log bhr rep index 1 read table simmsm f exit logbaseline res header T log bhr rep index 2 read table simmsm f 2 exit logbaseline res header T log bhr rep index l 3 read table simmsm f 3 exit logbaseline res header T log bhr rep index 4 lt read table simmsm f 4 exit logbaseline res header T log bhr rep index 5 lt read table simmsm f 5 exit logbaseline res header T log bhr rep index l 6 read table simmsm f 6 exit logbaseline res header T beta rep index 1 read table simmsm FixedEffects res header T beta rep index 2 read table simmsm FixedEffects 2 res header T beta rep index l 3 lt read table simmsm FixedEffects 3 res header T beta rep index 4 read table simmsm FixedEffects 4 res header T beta rep index l 5 lt read table simmsm FixedEffects 5 res header T beta rep index 6 lt read table simmsm FixedEffects 6 res header T 3 3 1 Baseline hazard rates gt plot baseline hazard rates gt main titles lt c 1 gt 2 2 gt 1 2 gt 3 3 gt 2 3 4 4 gt 3 gt par mfrow c 3 2 mar c 2 2 2 1 oma c 0 0 0 0 gt for i in 1 6 1 do bob o to bo plot 1 1 ylim c 0 3 xlim c 0 10 type n ylab Baseline hazard rate xlab Time mai2. remlreg simmsm simmsm mregress t12 exit baseline gridchoice a11 nrknots 30 lambdastart 5 x1 x2 t21 exit baseline gridchoice all nrknots 30 lambdastart 5 1x11x2 t23 exit baseline gridchoice all nrknots 30 lambdastart 5 1x11x2 t32 exit baseline gridchoice all nrknots 30 lambdastart 5 1x11x2 t34 exit baseline gridchoice all nrknots 30 lambdastart 5 x1 x2 t43 exit baseline gridchoice all nrknots 30 lambdastart 5 x1 x2 family multistate lefttrunc entry state from maxit 1000 using dat delimiter return ok 4 V VV Vv vv v max time lt 10 N 100 replicates 10 set seed 13 log bhr beta vector list replicates for rep index in 1 replicates d sim event histories n N all to all bhr all beta max time max time d ti2 as integer d from 1 amp d to 2 d t21 as integer d from 2 amp d to 1 d t23 as integer d from 2 amp d to 3 d t32 as integer d from 3 amp d to 2 d t34 as integer d from 3 amp d to 4 d t43 as integer d from 4 amp d to 3 setwd P home research 06 project sim package shortnote checking results write table d row names FALSE quote FALSE Vv vv v v 10 bb b b cock 4 oco file paste dat rep index raw sep BayesX setwd C BayesX commandline writeLines paste p 1 rep index p 2 sep con prg system bayesx prg
3. 2 3 T tra 2 1 tra 3 2 T tra 3 4 tra 4 3 T etm res vector list 10 d read table dati raw header T etm res 1 etm d tra tra state d read table dat2 raw header T etm res 2 etm d tra tra state d read table dat3 raw header T etm res 3 etm d tra tra state d read table dat4 raw header T etm res 4 etm d tra tra state d read table dat5 raw header T etm res 5 etm d tra tra state 13 names 1 names 1 names 1 names 1 names 1 cens cens cens cens cens name NULL name NULL name NULL name NULL name NULL s 0 s 0 s 0 s 0 s 0 gt VV V V V V V V V b b bk R V V V V V V V V V V V V Vv v v d lt read table dat6 raw header T etm res 6 etm d tra tra state names 1 4 cens name NULL s 0 d read table dat7 raw header T etm res 7 etm d tra tra state names 1 4 cens name NULL s 0 d read table dat8 raw header T etm res 8 etm d tra tra state names 1 4 cens name NULL s 0 d read table dat9 raw header T etm res 9 etm d tra tra state names 1 4 cens name NULL s 0 d read table dat10 raw header T etm res 10 etm d tra tra state names 1 4 cens name NULL s 0 theoretical transition matrix P lt matrix nrow 4 ncol 4 data 0 gridlength 200
4. 5 1 sin t exp 0 5 zj 1 zai A34 t A t exp E De Tai 0 5 1 sin t exp 0 5 zi 1 29 11 Xa3 t AQ t exp FA Zii Bi 22 0 5 exp 0 5 zi 1 25i In the current implementation of simMSM covariates are drawn from uniform distribu tions on 1 1 3 1 Parameterization bhr 11 lt function t freturn 04t bhr 12 lt function t return 0 5 bhr 13 lt function t freturn 04t bhr 14 lt function t freturn 04t bhr 21 lt function t return 0 5 sin t 0 5 bhr 22 lt function t freturn 04t bhr 23 lt function t return 0 5 sin t 0 5 bhr 24 lt function t freturn 04t bhr 31 lt function t freturn 04t bhr 32 function t return 0 5 sin t 0 5 bhr 33 lt function t freturn 04t bhr 34 lt function t freturn 0 5xsin t 10 5 bhr 41 lt function t freturn 04t bhr 42 lt function t freturn 04t bhr 43 lt function t return 0 5 V V VV V V V V V V V V Vv v v bhr 44 lt function t freturn 0xt all beta list to all to all bhr all beta list from i list from lt 1 all to all all all all all all all all all all all all all all all all all all all all all all all to to to to to to to to to to to to to to to to to to to to to to to all all all all all all all all all all all all al
5. for i in 1 n 4 history i lt sim single history first entry 0 first from all first from i max time all to all bhr all beta x i all x i history i lt cbind history i rep i nrow history i for x index in 1 p 4 history i cbind history i rep all x i x index nrow history i T histories lt rbind histories history i rm history i J histories as list lt list id histories 5 entry histories 1 exit lt histories 2 from lt histories 3 to lt histories 41 for x index in 1 p 4 histories as list 5 x index lt histories 5 x index names histories as list 5 x index lt paste x x index sep J histories lt data frame histories as list rm histories as list return histories lt environment namespace simMSM gt 3 and how we can use it We will simulate data from 4 state multi state model in the following Only transitions between adjacent states as illustrated in the following figure are possible Po S Se Ds uf We assume two covariates affecting each of the transition specific hazard rates as follows A1 t AQ t exp BL Tri Bos 12 0 5 exp 0 5 z1 1 224 dar t AS t exp 821 Zii Be 29 0 5 1 sin t Aaa t A t exp 28 Lue Be Tai 0 5 1 sin t exp 0 5 ay 1 x9 exp 0 5 zi 1 zoi T1 Ago t Aj t exp 83 Mit na 22 0
6. lt 1 x 2 lt 2 x 1 lt 2 x 2 lt 3 x 1 lt 3 x 2 lt 4 x 1 lt 4 x 2 lt 1 x 1 lt 1 x 2 lt 2 x 1 lt 2 x 2 lt 3 x 1 lt 3 x 2 lt 4 x 1 lt NULL NULL NULL NULL e02 list bhr 11 bhr 12 bhr 13 bhr 14 all beta c 1 3 list bhr 21 bhr 22 bhr 23 bhr 24 all beta c 2 4 list bhr 31 bhr 32 bhr 33 bhr 34 all beta c 3 list bhr 41 bhr 42 bhr 43 bhr 44 all beta function x i return 0 x function x i return 0 x function x freturn 0 5xx function x freturn 1xx function x freturn 0xx function x freturn 0xx function x freturn 0xx function x freturn 0xx function x freturn 0 5xx function x freturn 1xx function x freturn 0xx function x freturn 0xx function x freturn 0 5xx function x freturn 1xx function x i return 0 x function x i return 0 x function x freturn 0xx function x freturn 0xx function x freturn 0 5xx function x freturn 1xx function x freturn 0xx function x freturn 0xx function x freturn 0 5xx V V VV V V Vv v v al1 al1 al1 al1 al1 al1 al1 al1 al1 to to to to to to to to to all all all all all all all all all bhr bhr bhr bhr bhr bhr bhr bhr bhr 3 2 Simulation gt max time 10 gt N lt 100 gt set seed 13 gt d lt sim event gt head d n 30 O 0 NO Cd BK 0 NE H a Be RO NNNNNEBGEE
7. times seq 0 15 length gridlength A P t vector list gridlength for i in 1 gridlength AT 1 lt P A i 1 1 lt 0 5x times i A i 1 2 lt 0 5 times i A i 2 1 lt O 5 times i 0 5 cos times i A i 2 3 lt O 5 times i 0 5 cos times i A i 2 2 lt A i 2 1 A i 2 3 1 A i 3 2 lt O 5 times i 0 5 cos times i A i 3 4 lt O 5 times i 0 5 cos times i A i 3 3 lt A i 3 2 A i 8 4 1 A ill 4 3 lt 0 5 times i A ill 4 4 lt 0 5 times i P t 1 lt diag 4 for i in 2 gridlength P t i lt P t i 1 4 diag 4 A i A i 1 hf lt function mat k l return mat k 1 P 12 lt do call c lapply P t FUN hf k 1 1 2 P 21 lt do call c lapply P t FUN hf k 2 1 1 P 23 lt do call c lapply P t FUN hf k 2 1 3 P 32 lt do call c lapply P t FUN hf k 3 1 2 P 34 lt do call c lapply P t FUN hf k 3 1 4 t t t P 43 lt do call c lapply P t FUN hf k 4 1 3 Plot empirical transition probabilities v V V v par mfrowzc 3 2 mar c 2 2 2 1 oma c 0 0 O 0 plot 1 1 type n xlim c 0 15 ylim c 0 1 main 1 gt 2 for i in c 1 10 lines etm res i tr choice 1 2 rgb 0 5 0 8 0 8 alpha 0 5 lwd 2 lines times P 12 14 V V V V V V V V V V V V V V V plot 1 1 type n xlim c 0 15 ylim c 0 1 main 2 gt 1 for i
8. BEE BEE OH BU MH O 0 0 NO 0 BON W C WU V V IO V V V V VV V V U H H H RIO POR H RP BE BB N a O OO 0 0 0 0 VO OD 0 V V O NO OD O U V H ROO entry 000000000 002185974 833642846 891975644 136696669 667099687 726154920 360039839 601822157 613853327 047962219 000000000 224473254 326072233 707245122 795029415 957340396 899518311 131079152 549863420 612540824 267444034 000000000 082162733 398982767 all all all all J 0 O 0 V WU H H 1 o co 0 O NO OM W N ur Pp WU O beta from beta from all all beta from beta from beta from all all all beta from beta from beta from beta from 3 all 4 all 4 all 4 all 4 all 4 all 4 all 4 all 4 all beta to beta to beta to beta to beta to beta to beta to beta to beta to histories n N all to exit from to 002185974 833642846 891975644 136696669 667099687 726154920 360039839 601822157 613853327 047962219 730551231 224473254 326072233 707245122 795029415 957340396 899518311 131079152 549863420 612540824 267444034 957285490 082162733 398982767 307642783 w BUM U BM 0OBO0OBO0OBO0ONEENENENEMNO NM U B VB OBO0OBOBO0OBODNENENDENENONDN 4 x 1 x 1 x 2 x 2 x 3 x 3 x 4 x 4 x lon MH N HK V HN all bhr all O O O O O O O O O O O O O O O O O O O O O O O O O
9. IHEIEE ERE TEHEHHHBRHRHHRR I RE gt simulate new exit time using numerical inversion sampling f gt PRE IEEE TE ETE HEHHHRRHHHHHHE I RE gt f for uniroot function exit u entry all bhr eta ij eta ij 1 return cumallcausehr entry exit all bhr eta ij eta ij log 1 u environment namespace simMSM gt gt GHIHHHHEHHHHHHHHHHHHHHHHHHHBHHBWBHEE gt simulate new final state gt GHIHHHHEHHHHHHHHHHHHHHHHHHBHBHHBWHEIE gt sim to function entry ij from ij all to all bhr all beta eta ij eta ij 4 exit ij lt sim exit entry ij all to all bhr all beta from ij all bhr eta ij eta ij new exit hr at exit ij lt rep NA length eta ij for hi in all to all bhr all beta from ijl a11 to 4 hr at exit ij hi lt hr all to all bhr all beta from ij all bhr hil exit ij eta ij hil J hr at exit ij lt hr at exit ij is na hr at exit ij if length hr at exit ij gt 1 5 1 probs lt hr at exit ij sum hr at exit ij to ij lt sample all to all bhr all beta from ij a11 to size 1 prob probs J else to ij lt as numeric all to all bhr all beta from ij all to J return list entry ij entry ij exit ij exit ij from ij from ij to ij lt to ij I environment namespace simMSM gt gt GHEEHEHEEHEHEIHEIHEIHIHIEHEEE EE ETEEETHHHHBEHRRHHHRHERE gt simulate one individual event history gt TREE EH EH HHH HE gt sim single history funct
10. User s guide for R package simMSM Holger Reulen Chair of Statistics University of Goettingen June 5 2013 The Package simMSM provides R functions to simulate event histories The underlying multi state model is parameterized by possibly nonlinear baseline hazard rate functions as well as possibly nonlinear covariate effect functions Sampling of event histories is then performed using inversion sampling on all cause hazard rate functions This guide gives some background theory in section 1 prints the imple mented source code in section 2 and shows the results of 10 simulation repli cations of an event history dataset in section 3 The simulation is based on a 4 state multi state model with 6 possible transitions and 2 covariates The results are verified using software BayesX and R package etm in subsections 3 3 and 3 4 Contents 1 The theory behind simMSM in a nutshell 2 2 Implementation of simMSM 3 3 and how we can use it 7 31 ParameterizatlON 2 30100404 874 a A a ek zda 7 32 Simulation s A E ehe e Nod decido oi dod ek rs dr 9 3 3 Check the results using BayesX 10 3 3 1 Baseline hazard rates 11 3 9 2 Covariate effects i e ox soo dk a a 12 3 4 Check the results using etm 13 1 The theory behind simMSM in a nutshell simMSM simulates event histories using inversion sampling which was outlined before for general multi state models in section 7 of Kneib and H
11. ennerfeind 2008 and described more explicitly for a single competing risks scenario in section 3 2 of Beyersmann et al 2012 In brief inversion sampling is performed in the following way 1 The multi state model and its stochastic behaviour is fully determined by the transition specific hazard rate functions Az t So for a specific initial state k the integral from 0 to over the all cause hazard rate function A t gt yz Aui t the cumulative all cause hazard rate function A t specifies the distribution of sojourn times Ty in state k We therefore perform simulation of sojourn times T based on the cumulative all cause hazard rate functions 2 For given Ty we can draw the final state j for our current transition using a one trial multinomial distribution with probabilities pg calculated as As Aj t s yt Akl t Step 1 includes the inversion sampling step which maybe needs more explanantion As the hazard rate function Ax t are positive functions the integral from 0 to t over the sums of these functions are strictly increasing and invertible As already stated above the cumulative all cause hazard rate function A t specifies the probability distribution function of sojourn times Tk by F t P Tk lt t 1 exp As t j 1 q and j k Consequently the probability distribution function F is invertible as well Since I don t want to reinvent the wheel here and I could impossibly
12. in c 1 10 lines etm res i tr choice 2 1 rgb 0 5 0 8 0 8 alpha 0 5 lwd 2 lines times P 21 plot 1 1 type n xlim c 0 15 ylim c 0 1 main 2 gt 3 for i in c 1 10 flinesletm resl i tr choice 2 3 rgb 0 5 0 8 0 8 alpha 0 5 lwd 2 lines times P 23 plot 1 1 type n xlim c 0 15 ylim c 0 1 main 3 gt 2 for i in c 1 10 lines etm res i tr choice 3 2 rgb 0 5 0 8 0 8 alpha 0 5 lwd 2 lines times P 32 plot 1 1 type n xlim c 0 15 ylim c 0 1 main 3 gt 4 for i in c 1 10 lines etm res i tr choice 3 4 rgb 0 5 0 8 0 8 alpha 0 5 lwd 2 lines times P 34 plot 1 1 type n xlim c 0 15 ylim c 0 1 main 4 gt 3 for i in c 1 10 lines etm res i tr choice 4 3 rgb 0 5 0 8 0 8 alpha 0 5 1wd 2 lines times P 43 15 2 gt 1 0L 80 90 3 22 0L 80 90 vo 20 00 4 23 Ok 80 90 vo 20 00 1 gt 2 15 10 2 gt 3 3 gt 4 80 90 vo 20 00 0L 80 90 vo 20 00 16
13. ion first entry 0 first from max time all to all bhr all beta x i 1 current exit lt current entry lt first entry current to lt current from lt first from history i lt NULL p lt length x i while current entry lt max time 1 f x lt all to all bhr all beta current from all beta eta ij lt rep 0 length f x for oi in 1 length f x 4 for ii in 1 length f x oill 4 f now lt f x oill iil eta ij oi lt eta ij oi f now x i ii current sim lt sim to current entry current from all to all bhr all beta eta ij eta ij history i lt rbind history i c current sim entry ij current sim exit ij current sim from ij current sim to ij current entry lt current sim exit ij current from lt current sim to ij return history i I environment namespace simMSM gt gt GHHIHHHHEHHHHHHHHHHBHHHHBHHBHHHBHHHBWHHHHEE gt simulate n event histories gt HHHBHHHEHHHHHHHHHHBHHHBHHHHHHHHHHHBHHEE gt sim event histories function n all to all bhr all beta max time 10 1 hf function x k 4 return x k 1 all possible from states lt as numeric do call c lapply all to all bhr all beta FUN hf k 1 all first from lt sample all possible from states size n replace T p lt length all to all bhr all beta 1 all beta 1 all x lt runif n n p min 1 max 1 all x lt matrix nrow n ncol p data all x histories lt NULL
14. l all all all all all all all all all all bhr bhr bhr bhr bhr bhr bhr bhr bhr bhr bhr bhr bhr bhr bhr bhr bhr bhr bhr bhr bhr bhr bhr to to to al1 al1 al1 al1 al1 all all all all all all all all all all all all all all all all all all 1 list x 1 2 list x 1 3 list x 1 4 list x 1 beta from beta from beta from beta from beta from beta from beta from beta from beta from beta from beta from beta from beta from beta from beta from beta from beta from beta from beta from beta from beta from beta from beta from 1 all 1 all 1 all 1 all 1 all 1 all 1 all 1 all 2 all 2 all 2 all 2 all 2 all 2 all 2 all 2 all 3 all 3 all 3 all 3 all 3 all 3 all 3 all NULL NULL NULL NULL beta to beta to beta to beta to beta to beta to beta to beta to beta to beta to beta to beta to beta to beta to beta to beta to beta to beta to beta to beta to beta to beta to beta to x x x x wan eae 2 2 all bhr all beta from 2 list from 2 all to all bhr all beta from 3 lt list from 3 all to all bhr all beta from 4 list from 4 all to all bhr all beta 1 x 1 1 x 2 2 x 1 lt 2 x 2 lt 3 x 1 lt 3 x 2 lt 4 x 1 lt 4 x 2 lt 1 x 1
15. n main titlesli for rep index in 1 replicates hilf lt betal rep index i intercept lt hilf pmode 1 hilf lt log bhr rep index i lines hilf exit exp hilf pmode intercept col rgb 0 5 0 8 0 8 alpha 0 5 lwd 2 x lt seg 0 10 length 100 if i 1 lines x apply as matrix x MAR 1 FUN bhr 12 if i 2 flines x bhr 21 x 11 1 0 2 0 3 0 0 0 1 0 2 0 3 0 0 0 if i if i if i if i 3 1flines x 4 lines x 5 lines x 6 lines x 1 gt 2 bhr 23 x bhr 32 x bhr 34 x apply as matrix x MAR 1 FUN bhr 43 2 gt 1 1 0 2 0 3 0 0 0 3 0 2 0 1 0 0 0 2 0 3 0 1 0 0 0 3 3 2 Covariate effects V V V V V betas lt c beta 11 1 11 1 pmode 2 3 beta 111 211 pmode 2 3 beta 1 3 pmode 2 3 beta 11 4 pmode 2 3 par mfrow c 1 1 mar c 2 2 2 1 oma c 0 0 0 0 plot 1 1 type n abline h c 1 0 5 for i in 1 10 1 betas lt c betal i 1 pmode 2 3 beta i l 2 pmode 2 3 beta i 3 pmode 2 3 betal i 4 pmode 2 3 points 1 8 betas pch 16 col rgb 0 5 0 8 0 8 alpha 0 5 0 5 1 12 xlim c 1 8 ylim c 2 2 xaxt n 2 3 4 Check the results using etm 2 2 2 2 2 V V V V V V V V Vv v v library etm tra lt matrix nrow 4 ncol 4 F tra 1 2 tra
16. red Zo e Tpi g may be a linear function a xi BEL x as well as any other function e g piecewise constant piecewise linear trigonometrical polynomial or any mixture of these function classes The same applies for baseline hazard rate functions with the additional reguirement that they have to be positive 2 Implementation of simMSM gt setwd P home research 06 project sim package gt install packages simMSM 1 0 tar gz type source gt library simMSM gt PRE EH EEG HEH EE AR ART TRAR gt function for cox type proportional hazards parameterization gt PREHRA AAA gt hr function bhr t eta ij 4 return bhr t x exp eta ij J environment namespace simMSM gt gt PRHE HHHHHH HE gt function calculating possibly all cause hazard rate gt PRE HHHHHHH HE gt allcausehr function t all bhr eta ij 1 res lt 0 p lt length eta ij for hi in 1 p 4 res lt res hr all bhr hi t eta ij hil J return res J environment namespace simMSM gt gt PRE EEG EEA gt function calculating possibly all cause cumulative hazard rate gt PRE EH EEG EEA gt cumallcausehr function entry exit all bhr eta ij eta ij 1 return integrate allcausehr lower entry upper exit all bhr eta ij eta ij subdivisions 10000 value environment namespace simMSM gt gt GHEHHEEHEBBIBIBIBEHHEHEEEEEE ERE TE HERRBIRIRIHIH
17. write it down any better next up a slightly adapted quotation from Beyersmann et al 2012 p 47 that describes the remainder of the inversion sampling procedure We write FL for the inverse of F and A t for the inverse of A t Consider the transformed sojourn time Fx Tk The key of the inversion method is that Fy 75 is uniformly distributed on 0 1 P F T lt u P T lt F u F F u u u 0 1 Hence if U is a random variable with uniform distribution on 0 1 then Fx U has the same distribution as T The inversion method works as fol lows Kneib T Hennerfeind A 2008 Bayesian semi parametric multi state models Statistical Modelling vol 8 no 2 p 169 198 ZBeyersmann J Allignol A Schumacher M 2012 Competing Risks and Multistate Models with R Series Use R Springer Berlin 1 Compute F u Az log 1 U U 0 1 2 Choose a realization u that is uniformly distributed on 0 1 3 Ft u is the desired replicate of Ty Sometimes the Ax t are chosen in such a manner that we do not find an explicit expression for ACT We may then use numerical inversion We will use a Cox type parameterization of the transition specific hazard rate functions in the following Ag ti AC ti exp ut with transition specific baseline hazard rates AD t and transition as well as individual specific linear predictors nf e zii
18. x1 2152461 2152461 2152461 2152461 2152461 2152461 2152461 2152461 2152461 2152461 2152461 2073390 2073390 2073390 2073390 2073390 2073390 2073390 2073390 2073390 2073390 2073390 5500660 5500660 5500660 O O S O S O O O O O O function x i return 1 x function x freturn 0xx function x freturn 0xx function x freturn 0xx function x freturn 0xx function x freturn 0 5xx function x freturn 1xx function x freturn 0xx function x freturn 0xx beta max time max time x2 2880010 2880010 2880010 2880010 2880010 2880010 2880010 2880010 2880010 2880010 2880010 9572130 9572130 9572130 9572130 9572130 9572130 9572130 9572130 9572130 9572130 9572130 4591297 4591297 4591297 26 3 3 307642783 8 378749765 27 3 8 378749765 8 574201714 28 3 8 574201714 9 721422824 29 3 9 721422824 15 494700419 30 4 0 000000000 0 115538953 2 0 5500660 0 4591297 3 0 5500660 0 4591297 4 0 5500660 0 4591297 3 0 5500660 0 4591297 2 0 2447418 0 1778753 H B 0 NO 3 3 Check the results using BayesX library BayesX library R2BayesX THEHIHEEHIHERHHETHHHHEHHBHEHHHBHEHHHHBHEE code for bayesX program THEHIHEEHHEHHETHBHHEHHHEHHHBHHHHHBHEE teile des bayesx programms p 1 lt delimiter lt dataset dat dat infile using P home research 06 project sim package shortnote checking results dal p 2 lt raw
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