Vignette to package samplingdatacrt

Transcription

1 Vigntt to packag samplingdatacrt Diana Trutschl Contnts 1 Introduction 1 11 Objctiv 1 1 Diffrnt study typs 1 Multivariat normal distributd data for multilvl data 1 Fixd ffcts part Random part 9 3 Manual Dsign matrics 11 3 Covarianc-Varianc-Matrics Sampl data undr a givn study dsign Powr calculations 18 4 Summary 19 1 Introduction 11 Objctiv To valuat for xampl th match of diffrnt statistical modl oftn simulation studis ar usd In simulations th undrlying data can b from a ral study, but also from simulatd data givn a spcific distribution For this purpos w provid a packag to sampling data from normal distribution to mimic data of clustr randomisd trials within diffrnt study dsigns, namly paralll, cross-ovr and stppd wdg dsign Bsids a traditional dsign collcts sampling units within diffrnt groups, which should b compard, in th past yars multilvl dsign, which collct additional units nstd within th original sampling units, bcoms vry popular Masurmnts of diffrnt patints nstd within hospitals, also assignd as clustrs, is on xampl of a two-lvl nstd data Anothr xampl of nstd structurs ar th rpatd masurmnts of patints Furthrmor, thr-lvl nstd data is obtaind for xampl by th combination of both nsting xampls Th complt data of such xampls is thn multivariat distributd Th aim of this packag is to provid a asy implmntation of sampling multivariat normal distributd data for furthr invstigations Additionally, th rquirs powr calculation for a spcial studi dsign, th stppd wdg dsing, is givn 1 Diffrnt study typs Paralll, cross-ovr and stppd wdg dsigns With this packag w provid data sampling within thr common usd study dsign typs: paralll, cross-ovr an stppd wdg dsigns (SWD) Tabl 1 shows xampls of ths kind of typs, ach with C = 6 clustr, followd ovr T = 4 tim points A paralll dsign is prsnt, whn two groups of tratmnts ar givn so that on group rcivs only th first tratmnt whil anothr group rcivs only th scond In contrast to th paralll dsign in a crossovr dsign trail ach xprimntal unit (patint) rcivs diffrnt tratmnts during th diffrnt tim points Hnc, it is 1

2 a rpatd masurmnts dsign An altrnativ and mor popular bcoming dsign is th stppd wdg dsign Hr, th intrvntion is rollout to diffrnt units squntial but random ovr diffrnt tim points A T 1 T T 3 T 4 Cntr Cntr Cntr Cntr Cntr Cntr B T 1 T T 3 T C T 1 T T 3 T Tabl 1: Exampls of diffrnt study dsign typs: A) paralll, B) cross-ovr, and C) stppd wdg dsign Cross-sctional vrsus longitudinal In trials oftn subjcts within clustrs ar followd ovr a priod of tim and masurd to svral masurmnt points Two kinds of data collction is thn possibl: 1) crosssctional data, if at ach tim point th masurmnt units (subjcts) ar diffrnt to th units at anothr timpoints, or ) longitudinal data, th masurmnt units (subjcts) ar th sam to all timpoints (known as rpatd masurmnts) Hnc, if it is a trail with C clustrs and a clustrsiz of N ach, which ar follwd ovr T timpoints, thn th total numbr of includd subjcts is C T N in a cross-sctional and C N in a longitudinal study Multivariat normal distributd data for multilvl data For th situation of a clustr-randomizd trail with T numbr of tim points, C numbr of clustrs and N numbr of patints pr clustr (and tim point, whn it is a cross-sctional) th complt datast can b writtn as th vctor of rsponss Y = {Y ijk } of lngth T C N, which is sampld from a multidimnsional normal distribution: Y N (Zb, V ), whr Zb is thn th fixd ffcts full rank dsign matrix multiplid by th rgrssion fixd ffcts cofficints and V th varianc-covarianc matrix Y ijk is thn th obsrvation in clustr i to tim point k for th subjct j in th cross-sctional cas or th k-th masurmnt of th subjct j in clustr i in th longitudinal cas, rspctivly Th form of th random part (varianc-covarianc matrix) dpnds on th sampling of ithr cross-sctional or longitudinal study dsign, whras th form of th fixd part dpnds on th study dsign typ (paralll, cross-ovr an stppd wdg dsigns) 1 Fixd ffcts part Th rgrssion fixd ffcts cofficints within th providd dsigns ar dfind as b = (, β 1,, β I, θ), whr is th ovrall man, θ is th intrvntion ffct, β k is th fixd tim ffct for tim point k, k = (1,, I) Th dsign matrix X of th modl for such dsigns has th form X = tim point 1 tim point k tim point T clustr 1 x 11 x 1T clustr i clustr C x C1 x CT

3 Th fixd ffcts full rank dsign matrix Z is thn a concatnation of all matrics Z i of all clustrs, which in turn ar a concatnation of N rplications of matrics Z ij (hnc for all j th Z ij is th sam) and which ar cratd out of th dsign matrix X of th SWD modl Thn is Z ij for on subjct in clustr i a column wis bindd matrix of 1 a vctor of ons (th sam for all clustr) a matrix A (th sam for all clustr) 3 a vctor, which is th corrsponding row of th dsign matrix X to clustr i Each row of Z ij corrsponds thn to a idntify ntry of a fixd ffcts in rgrssion fixd ffcts cofficints vctor b Z ij = β 1 β β I 1 θ tim point x i1 0 tim point k 1 xik 1 tim point I x it Hnc, Z i is build by row wis bindd N rplicats of Z ij Z i = subjct 1 subjct N Z ij Z ij and Z by row wis bindd C matrics Z i Z = clustr 1 Z i clustr C Hnc, ach row corrsponds to on subjct j of a clustr i to timpoint k and multiplid with th vctor of rgrssion fixd ffcts cofficints b rsult in th fixd ffct part of th linar quation for this obsrvation Hnc, it is th man vctor of th multivariat normal distribution and it is prformd by th matrix multiplication Zb Paralll dsign For xampl with I = 4 clustr and K = 3 masurmnts, hnc only two clustr for ithr control or tratmnt arm, th dsign matrix X is dfind as X = tim point 1 tim point tim point 3 tim point 4 clustr clustr clustr clustr and th matrix Z i for clustr 1 and is thn Z i (1,) = Z i β 1 β β 3 θ tim point tim point tim point tim point

4 and for clustr 3 and 4 Z i (3,4) = β 1 β β 3 θ tim point tim point tim point tim point thn, if N = subjcts ar within ach clustr th fixd ffcts full rank dsign matrix Z is Z = clustr 1 Z 1 clustr Z clustr 3 Z 3 = clustr 4 Z 4 clustr 1 subjct 1 clustr 1 subjct clustr subjct 1 clustr subjct clustr 3 subjct 1 clustr 3 subjct clustr 4 subjct 1 clustr 4 subjct Z 1j Z 1j Z j Z j Z 3j Z 3j Z 4j Z 4j = β 1 β β 3 θ clustr 1 subjct 1 tim point clustr 1 subjct 1 tim point clustr 1 subjct 1 tim point clustr 1 subjct 1 tim point clustr 1 subjct tim point clustr 1 subjct tim point clustr 1 subjct tim point clustr 1 subjct tim point clustr subjct 1 tim point clustr subjct 1 tim point clustr subjct 1 tim point clustr subjct 1 tim point clustr subjct tim point clustr subjct tim point clustr subjct tim point clustr subjct tim point clustr 3 subjct 1 tim point clustr 3 subjct 1 tim point clustr 3 subjct 1 tim point clustr 3 subjct 1 tim point clustr 3 subjct tim point clustr 3 subjct tim point clustr 3 subjct tim point clustr 3 subjct tim point clustr 4 subjct 1 tim point clustr 4 subjct 1 tim point clustr 4 subjct 1 tim point clustr 4 subjct 1 tim point clustr 4 subjct tim point clustr 4 subjct tim point clustr 4 subjct tim point clustr 4 subjct tim point

5 and th fixd part, hnc th man vctor of th multivariat normal distribution, is thn = Z β 1 β β 3 θ = + β 1 + β + β 3 + β 1 + β + β 3 + β 1 + β + β 3 + β 1 + β + β 3 + β 1 + θ + β + θ + β 3 + θ + θ + β 1 + θ + β + θ + β 3 + θ + θ + β 1 + θ + β + θ + β 3 + θ + θ + β 1 + θ + β + θ + β 3 + θ + θ Cross-ovr dsign For xampl with I = 4 clustr and K = 4 masurmnts, two clustr ach switchs tratmnt and control aftr tim point, th dsign matrix X is dfind as X = tim point 1 tim point tim point 3 tim point 4 clustr clustr clustr clustr and th matrix Z i for clustr 1 and is thn Z i (1,) = β 1 β β 3 θ tim point tim point tim point tim point

6 and for clustr 3 and 4 Z i (3,4) = β 1 β β 3 θ tim point tim point tim point tim point thn, if N = subjcts ar within ach clustr th fixd ffcts full rank dsign matrix Z is Z = clustr 1 Z 1 clustr Z clustr 3 Z 3 = clustr 4 Z 4 clustr 1 subjct 1 clustr 1 subjct clustr subjct 1 clustr subjct clustr 3 subjct 1 clustr 3 subjct clustr 4 subjct 1 clustr 4 subjct Z 1j Z 1j Z j Z j Z 3j Z 3j Z 3j Z 3j = β 1 β β 3 θ clustr 1 subjct 1 tim point clustr 1 subjct 1 tim point clustr 1 subjct 1 tim point clustr 1 subjct 1 tim point clustr 1 subjct tim point clustr 1 subjct tim point clustr 1 subjct tim point clustr 1 subjct tim point clustr subjct 1 tim point clustr subjct 1 tim point clustr subjct 1 tim point clustr subjct 1 tim point clustr subjct tim point clustr subjct tim point clustr subjct tim point clustr subjct tim point clustr 3 subjct 1 tim point clustr 3 subjct 1 tim point clustr 3 subjct 1 tim point clustr 3 subjct 1 tim point clustr 3 subjct tim point clustr 3 subjct tim point clustr 3 subjct tim point clustr 3 subjct tim point clustr 4 subjct 1 tim point clustr 4 subjct 1 tim point clustr 4 subjct 1 tim point clustr 4 subjct 1 tim point clustr 4 subjct tim point clustr 4 subjct tim point clustr 4 subjct tim point clustr 4 subjct tim point

7 and th fixd part, hnc th man vctor of th multivariat normal distribution, is thn = Z β 1 β β 3 θ = + β 1 + β + β 3 + θ + θ + β 1 + β + β 3 + θ + θ + β 1 + β + β 3 + θ + θ + β 1 + β + β 3 + θ + θ + β 1 + θ + β + θ + β 3 + β 1 + θ + β + θ + β 3 + β 1 + θ + β + θ + β 3 + β 1 + θ + β + θ + β 3 Stppd wdg dsign For xampl with I = 3 clustr and K = 4 masurmnts, hnc only on clustr switchs pr timpoint, th dsign matrix X is dfind as X = tim point 1 tim point tim point 3 tim point 4 clustr clustr clustr and th matrix Z i for clustr 1 is thn Z 1 = β 1 β β 3 θ tim point tim point tim point tim point thn, if N = subjcts ar within ach clustr th fixd ffcts full rank dsign matrix Z is 7

8 Z = clustr 1 Z 1 clustr Z = clustr 3 Z 3 clustr 1 subjct 1 clustr 1 subjct clustr subjct 1 clustr subjct clustr 3 subjct 1 clustr 3 subjct Z 1j Z 1j Z j Z j Z 3j Z 3j = β 1 β β 3 θ clustr 1 subjct 1 tim point clustr 1 subjct 1 tim point clustr 1 subjct 1 tim point clustr 1 subjct 1 tim point clustr 1 subjct tim point clustr 1 subjct tim point clustr 1 subjct tim point clustr 1 subjct tim point clustr subjct 1 tim point clustr subjct 1 tim point clustr subjct 1 tim point clustr subjct 1 tim point clustr subjct tim point clustr subjct tim point clustr subjct tim point clustr subjct tim point clustr 3 subjct 1 tim point clustr 3 subjct 1 tim point clustr 3 subjct 1 tim point clustr 3 subjct 1 tim point clustr 3 subjct tim point clustr 3 subjct tim point clustr 3 subjct tim point clustr 3 subjct tim point and th fixd part, hnc th man vctor of th multivariat normal distribution, is thn 8

9 Random part = Z β 1 β β 3 θ = + β 1 + β + θ + β 3 + θ + θ + β 1 + β + θ + β 3 + θ + θ + β 1 + β + β 3 + θ + θ + β 1 + β + β 3 + θ + θ + β 1 + β + β 3 + θ + β 1 + β + β 3 + θ All clustrs ar indpndnt from ach othr, hnc th varianc-covarianc matrix V is a block-diagonal matrix of th matrics V i of all clustrs (and all othrs ar zros), whr for all i th V i ar th sam for all clustr V = clustr 1 clustr C clustr 1 V i clustr C 0 0 V i and V i = subjct 1 subjct N subjct 1 V i1,i1 V i1,i V i1,in V i1,i ViN 1,iN subjct N V i1,in V in 1,iN V in,n Thrfor w dfin V as a submatrix of V ij,i j i for th ntitis corrsponding to th masurmnts of subjct i and th masurmnt of subjct j 9

10 For all two diffrnt subjcts j and j (j j) this submatrix is dfind by V ij,i j = timpoint 1 timpoint T timpoint 1 timpoint T Th diffrnc in th distributions of th obsrvations within a cross-sctional and longitudinal SWD is in th random part of th modl Thus th varianc-covarianc matrix V of th normal distribution N (Zb, V ) and hnc th form of th V i or V ij,ij rspctivly diffr Varianc-Covarianc matrix within a cross-sctional dsign V ij,ij = timpoint 1 timpoint T timpoint 1 α + timpoint T α + For our xampl of (I = clustr,) K = 3 timpoints and N = subjcts ach clustr th Varianc- Covarianc matrix V i for ach clustr is thn subjct 1 subjct tp 1 tp tp T tp 1 tp tp T subjct 1 subjct {}}{{}}{ tp 1 tp tp 3 tp 1 tp tp 3 α + α + α + α + α + α + Varianc-Covarianc matrix within a longitudinal dsign V ij,ij = timpoint 1 timpoint T timpoint 1 α + γ + γ + γ + γ + γ + timpoint T γ + γ + α + γ + For our xampl of (I = clustr,) K = 3 timpoints and N = subjcts ach clustr th Varianc- Covarianc matrix V i for ach clustr is thn subjct 1 subjct tp 1 tp tp 3 tp 1 tp tp 3 subjct 1 { }} { subjct { }} { tim point 1 tim point tim point 3 tim point 1 tim point tim point 3 α + γ + γ + γ + γ + α + γ + γ + γ + γ + γ + α + α + γ + γ + γ + γ + α + γ + γ + γ + γ + α + γ + 10

11 3 Manual Us th following Packag undr GPL and load to th library: #load th packag library(samplingdatacrt) library(lm4) ## Loading rquird packag: Matrix 31 Dsign matrics In ach study a dsign matrix of valus of xplanatory variabls can b usd to dscrib th study typ Hr, ach row rprsnts an study unit (clustr) and th cll ntitis ar th ncoding of rciving th tratmnt or not (zros and ons) Tabl1 shows such dsign matrics or diffrnt study typs In contrast ach row of th dsign matrix of th complt data of th trail rprsnts a masurmnt with th succssiv columns corrsponding to th variabls (ffcts) and thir spcific valus for that Th dsign matrix of th complt data corrsponds to th fixd part of th multivariat normal distribution All matrics could also b implmntd manually using th function matrix(), but, instad of ordring an amount of zros and ons, th providd functions in this packag mak it asy to rciv this complx matrics for simpl study dsigns using only som paramtrs (for balancd data and qual numbr of clustrs pr switch) dsignmatrix Th dsign matrix for th study typ of th thr typs a) paralll, b) cross-ovr, and c) SWD can b prformd by using th function dsignmatrix(), which rquir four paramtrs: th numbr of clustrs within th trail, th numbr of masurmnt tim points, th numbr of clustr, which switch ovr from control to intrvntion at ach tim point and th study typ ( SWD as dfault) I<-6 #numbr of clustr K<-4 #numbr of timcpoints #Dsign matrix for paralll study, s Tabl 1 sw<-3 #numbr of clustr switchs dsignmatrix(nc=i, nt=k, nsw=sw, dsign="paralll") ## [,1] [,] [,3] [,4] ## [1,] ## [,] ## [3,] ## [4,] ## [5,] ## [6,] #Dsign matrix for cross-ovr study, s Tabl 1 dsignmatrix(nc=i, nt=k, nsw=sw, dsign="cross-ovr") ## [,1] [,] [,3] [,4] ## [1,] ## [,] ## [3,] ## [4,] ## [5,] ## [6,]

12 #if swp is st, thn th timpoint of switch is sttd manually dsignmatrix(nc=i, nt=k, nsw=sw, swp=1, dsign="cross-ovr") ## [,1] [,] [,3] [,4] ## [1,] ## [,] ## [3,] ## [4,] ## [5,] ## [6,] #Dsign matrix for SWD study, s Tabl 1 sw<- #numbr of clustr switchs dsignmatrix(nc=i, nt=k, nsw=sw) ## [,1] [,] [,3] [,4] ## [1,] ## [,] ## [3,] ## [4,] ## [5,] ## [6,] compltdatadsignmatrix Th function compltdatadsignmatrix() prforms th dsign matrix for complt data within givn study dsign It rquirs a dsign matrix of a study and th numbr of subjct within ach cll K<-4 #numbr of tim points J<- #numbr of subjcts, ach clustr and timpoint ##### for paralll study ##### I<-4 #numbr of clustr sw<- #numbr of clustr switchs # crat a dsign matrix (X<-dsignMatrix(nC=I, nt=k, nsw=sw, dsign="paralll")) ## [,1] [,] [,3] [,4] ## [1,] ## [,] ## [3,] ## [4,] # crat th corrsponding complt data dsign matrix compltdatadsignmatrix(j, X) ## [,1] [,] [,3] [,4] [,5] ## [1,] ## [,] ## [3,] ## [4,] ## [5,] ## [6,] ## [7,]

13 ## [8,] ## [9,] ## [10,] ## [11,] ## [1,] ## [13,] ## [14,] ## [15,] ## [16,] ## [17,] ## [18,] ## [19,] ## [0,] ## [1,] ## [,] ## [3,] ## [4,] ## [5,] ## [6,] ## [7,] ## [8,] ## [9,] ## [30,] ## [31,] ## [3,] ##### for cross-ovr study ##### # crat a dsign matrix (X<-dsignMatrix(nC=I, nt=k, nsw=sw, dsign="cross-ovr")) ## [,1] [,] [,3] [,4] ## [1,] ## [,] ## [3,] ## [4,] # crat th corrsponding complt data dsign matrix compltdatadsignmatrix(j, X) ## [,1] [,] [,3] [,4] [,5] ## [1,] ## [,] ## [3,] ## [4,] ## [5,] ## [6,] ## [7,] ## [8,] ## [9,] ## [10,] ## [11,] ## [1,] ## [13,] ## [14,]

14 ## [15,] ## [16,] ## [17,] ## [18,] ## [19,] ## [0,] ## [1,] ## [,] ## [3,] ## [4,] ## [5,] ## [6,] ## [7,] ## [8,] ## [9,] ## [30,] ## [31,] ## [3,] ##### for SWD study ##### I<-3 #numbr of clustr # crat a dsign matrix (X<-dsignMatrix(nC=I, nt=k, nsw=1)) ## [,1] [,] [,3] [,4] ## [1,] ## [,] ## [3,] # crat th corrsponding complt data dsign matrix compltdatadsignmatrix(j, X) ## [,1] [,] [,3] [,4] [,5] ## [1,] ## [,] ## [3,] ## [4,] ## [5,] ## [6,] ## [7,] ## [8,] ## [9,] ## [10,] ## [11,] ## [1,] ## [13,] ## [14,] ## [15,] ## [16,] ## [17,] ## [18,] ## [19,] ## [0,] ## [1,]

15 ## [,] ## [3,] ## [4,] Covarianc-Varianc-Matrics Covarianc-Varianc matrix ar ndd bsids th man vctor to spcify th kind of multivariat normal distribution Th form dpnds on th kind of multilvl structur In our xampls of clustr randomizd studis with masurmnts ovr tim thr ar two possibilitis: 1) two-lvl data within cross-sctional studis and ) thr-lvl data within longitudinal studis CovMatDsign Th corrsponding covarianc-varianc matrics can b prformd with th providd CovMatDsign() Th function rquird th dsign paramtr K numbr of timpoints, I numbr of clustrs, J numbr of subjcts within ach clustr to ach timpoint, and also th variancs corrsponding to ach lvl If sigmaq is not givn, thn it a cross-sctional, othrwis a longitudinal dsign is prformd #study dsign paramtr K<-3 #numbr of masurmnt (or timpoints) I<- #numbr of clustr J<- #numbr of subjcts ### for cross-sctional data sigma1<-01 sigma3<-09 CovMatDsign(K, J, I, sigma1q=sigma1, sigma3q=sigma3) ## [,1] [,] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,1] ## [1,] ## [,] ## [3,] ## [4,] ## [5,] ## [6,] ## [7,] ## [8,] ## [9,] ## [10,] ## [11,] ## [1,] ### for longitudinal data sigma1<-01 sigma<-04 sigma3<-09 CovMatDsign(K, J, I, sigma1q=sigma1, sigmaq=sigma, sigma3q=sigma3) ## [,1] [,] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,1] ## [1,] ## [,] ## [3,] ## [4,]

16 ## [5,] ## [6,] ## [7,] ## [8,] ## [9,] ## [10,] ## [11,] ## [1,] Sampl data undr a givn study dsign W provid a function to sampl a complt data st from multivariat normal distribution to mimic data of clustr randomisd trials within diffrnt study dsigns, namly paralll, cross-ovr and stppd wdg dsign and diffrnt typ of longitudinal or cross-sctional data sampldata Thrfor, w provid th sampldata(), whr th man vctor and th covarianc-varianc matrix of th distribution undr such studis has to b givn #dsing paramtr K<-4 #numbr of tim points J<-5 #numbr of subjcts, ach clustr and timpoint #variancs of ach lvl sigma1<-01 sigma<-04 sigma3<-09 #rgrssion paramtrs mu0<-0 thta<-1 btas<-rp(0, K-1) paramtrs<-c(mu0, btas, thta) ##### for paralll study ##### I<-4 #numbr of clustr sw<- #numbr of clustr switchs # crat a dsign matrix X<-dsignMatrix(nC=I, nt=k, nsw=sw, dsign="paralll") # crat th corrsponding complt data dsign matrix D<-compltDataDsignMatrix(J, X) #prform covarianc-varianc matrix for longitudinal dsign V<-CovMatDsign(K, J, I, sigma1q=sigma1, sigmaq=sigma, sigma3q=sigma3) #sampl data within th dsign sampldata<-sampldata(typ = "long", K=K,J=J,I=I, D=D, V=V, paramtrs=paramtrs) #nd th lm4 packag for analysis lmr(val~intrvntion+masurmnt + (1 clustr)+(1 subjct), data=sampldata) ## Linar mixd modl fit by REML ['lmrmod'] ## Formula: val ~ intrvntion + masurmnt + (1 clustr) + (1 subjct) ## Data: sampldata ## REML critrion at convrgnc:

17 ## Random ffcts: ## Groups Nam StdDv ## subjct (Intrcpt) 0588 ## clustr (Intrcpt) ## Rsidual 0306 ## Numbr of obs: 400, groups: subjct, 100; clustr, 4 ## Fixd Effcts: ## (Intrcpt) intrvntion masurmnt masurmnt3 masurmnt4 ## # ##### for cross-ovr study ##### # crat a dsign matrix X<-dsignMatrix(nC=I, nt=k, nsw=sw, dsign="cross-ovr") # crat th corrsponding complt data dsign matrix D<-compltDataDsignMatrix(J, X) #prform covarianc-varianc matrix for longitudinal dsign V<-CovMatDsign(K, J, I, sigma1q=sigma1, sigmaq=sigma, sigma3q=sigma3) #sampl data within th dsign sampldata<-sampldata(typ = "long", K=K,J=J,I=I, D=D, V=V, paramtrs=paramtrs) #analysis of th thr-lvl data lmr(val~intrvntion+masurmnt + (1 clustr)+(1 subjct), data=sampldata) ## Linar mixd modl fit by REML ['lmrmod'] ## Formula: val ~ intrvntion + masurmnt + (1 clustr) + (1 subjct) ## Data: sampldata ## REML critrion at convrgnc: ## Random ffcts: ## Groups Nam StdDv ## subjct (Intrcpt) ## clustr (Intrcpt) 0817 ## Rsidual 0954 ## Numbr of obs: 400, groups: subjct, 100; clustr, 4 ## Fixd Effcts: ## (Intrcpt) intrvntion masurmnt masurmnt3 masurmnt4 ## ##### for SWD study ##### I<-3 #numbr of clustr # crat a dsign matrix X<-dsignMatrix(nC=I, nt=k, nsw=1) # crat th corrsponding complt data dsign matrix D<-compltDataDsignMatrix(J, X) #prform covarianc-varianc matrix for cross-sctional dsign V<-CovMatDsign(K, J, I, sigma1=sigma1, sigma3=sigma3) #sampl data within th dsign sampldata<-sampldata(typ = "cross-sc", K=K,J=J,I=I, D=D, V=V, paramtrs=paramtrs) #analysis of th two-lvldata lmr(val~intrvntion+masurmnt + (1 clustr), data=sampldata) 17

18 ## Linar mixd modl fit by REML ['lmrmod'] ## Formula: val ~ intrvntion + masurmnt + (1 clustr) ## Data: sampldata ## REML critrion at convrgnc: ## Random ffcts: ## Groups Nam StdDv ## clustr (Intrcpt) 1370 ## Rsidual ## Numbr of obs: 300, groups: clustr, 3 ## Fixd Effcts: ## (Intrcpt) intrvntion masurmnt masurmnt3 masurmnt4 ## Powr calculations Powr of tsting th intrvntion ffct is providd for SWD Th function nds th stimatd intrvntion ffct and thir varianc calcpowrswd Powr calculation within stppd wdg dsign modl by Hussy & Hughs 1 for crosssctional and Ho & Kim for longitudinal data nocl<-10 not<-6 switchs<- DM<-dsignMatrix(noCl,noT,switchs) sigma <- sigmaalpha <- #Powr for cross-sctional SWD dsign by formula of Hussy&Hughs calcpowrswd(thtaest=1,dsign=dm, sigmaq=sigma^, tauq=sigmaalpha^, tim=false) ## [1] #Powr for longitudinal SWD dsign by formula of Ho&Kim #Exampl Ho&Kim Tabl 1 ###Tabl 1, 1 row dlta<- 03# tratmnt ffct DMnw<-NULL for(i in 1:dim(DM)[]){ DMnw<-cbind(DMnw,DM[,i], DM[,i]) } DMnw ## [,1] [,] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,1] ## [1,] ## [,] ## [3,] ## [4,] ## [5,] ## [6,] Michal A Hussy and Jams P Hughs,Dsign and analysis of stppd wdg clustr randomizd trials, Contmporary Clinical Trials(8),007 Ho M, Kim N, Rink ML, Wyli-Rostt J, Sampl siz dtrminations for stppd-wdg clinical trials from a thr-lvl data hirarchy prspctiv, Stat Mthods Md Rs,

19 ## [7,] ## [8,] ## [9,] ## [10,] sigma <- sqrt(7/10) sigma <- sqrt(/10) sigmaalpha <- sqrt(1/10 ) K<- 10 #numbr of participants within ach 'cll' calcpowrswd(thtaest=dlta, Dsign=DMnw, tauq=sigmaalpha^, sigmaq=sigma^, sigmaqrror =sigma^, nosub=k, typ="longitudinal") ## [1] Summary dsignmatrix dscription crat dsign matrix for a givn stup of a stppd wdg dsign paramtr nc numbr of clustr nt numbr of timpoints nsw numbr of clustr : within paralll rciv th control (nc-nsw rciv th intrvntion), within cross-ovr rciv th pattrn (0, 1) (nc-nsw rciv th pattrn (1,0)) for narly th sam numbr of tim points, within SWD switchs from control to intrvntion pr tim point swp is th tim point th clustr cross ovr th condition in a cross ovr study, if not givn thn it is narly half of th tim past, param dsign is th study typ (paralll, cross-sctional, stppd wdg) rturn dsign matrix for a givn stup of a stppd wdg dsign implmmatrixswd dscription Crats a implmntation matrix for a givn stppd wdg dsign and grad of intrvntion implmntation pattrn paramtr nc Numbr of clustrs nt Numbr of timpoint nsw numbr of clustrs switchs from control to tratmnt at ach timpoint pattrn a vctor for grad of intrvntion implmntation pattrn, which givs th dviation from 100 prcnt ffctivnss ovr tim rturn Dsign matrix for SWD modl undr a grad of intrvntion implmntation pattrn 19

20 compltdatadsignmatrix dscription crat dsign matrix for complt data within dsign paramtr J numbr of subjcts X givn dsign matrix rturn dsign matrix for complt data within dsign CovMatDsign dscription covarianc matrix fof th normal distribution undr clustr randomizd study typ givn a dsign and a typ paramtr K numbr of timpoints or masurmnts (dsign paramtr) J numbr of subjcts I numbr of clustrs (dsign paramtr) sigma1q varianc of th lowst lvl (rror varianc or within subjct varianc) sigmaq scound lvl varianc (g within clustr and btwn subjct varianc) sigma3q third lvl varianc (g btwn clustr varianc) rturn covarianc matrix sampldata dscription Sampl data (rspons) for givn numbrs of individuals by givn a modl (of a paralll, cross-sctional, stppd wdg dsign study) paramtr typ of th dsign is ithr cross-sctional ( cross-sc ) or longitudinal ( longitudinal ) K numbr of timpoints or masurmnts (dsign paramtr) J numbr of subjcts I numbr of clustrs (dsign paramtr) D a complt data dsign matrix corrsponding to th assumd modl A a complt data dsign matrix corrsponding to th tru data, if A is null, thn A is qual to D V covarianc matrix for th normal distribution paramtrs corrsponding to th modl (rgrssion fixd ffcts cofficints) rturn Data of individuals intnsitis corrsponds to th SWD modl and full modl paramtr information 0

21 calcpowrswd dscription Calculation of powr for a lmm with clustr as random ffct, fixd timpoint ffcts, but st to null, TP numbr of timpoints, I numbr of clustr Th dsign matrix has to b codd by zros and ons paramtr ThtaEst xpctd tratmnt ffct alpha singificanc lvl (by dfault 005) Dsign dsign matrix for a givn SWD modl tauq btwn clustr varianc sigmaq within clustr varianc(btwn subjct varianc) sigmaqrror within subjct varianc/rror varianc nosub numbr of subjcts within ach clustr and ach timpoint (for all an qual siz) typ is of cross-sctional (by dfault) or longitudinal assigns th typ of data ( or 3 lvl nstd structur) tim a logical (FALSE, if no tim trnds ar xpctd, othrwis TRUE) is only rlvant for valuation of cross-sctional data rturn Aproximatd powr of two taild tst, although th dsign matrix is fractionatd, thn powr is not valid, formula usd for cross-sctional data providd by Hussy & Hughs 3, and for longitudinal data by Ho & Kim 4 3 Michal A Hussy and Jams P Hughs,Dsign and analysis of stppd wdg clustr randomizd trials, Contmporary Clinical Trials(8),007 4 Ho M, Kim N, Rink ML, Wyli-Rostt J, Sampl siz dtrminations for stppd-wdg clinical trials from a thr-lvl data hirarchy prspctiv, Stat Mthods Md Rs, 016 1