Power and sample size calculations for longitudinal studies comparing rates of change with a time-varying exposure

Transcription

1 Reseach Aticle Received 6 Januay 9, Accepted Septembe 9 Published online 6 Novembe 9 in Wiley Intescience ( DOI:./sim.377 Powe and sample size calculations fo longitudinal studies compaing ates of change with a time-vaying exposue X. Basagaña a,b,c and D. Spiegelman d Existing study design fomulas fo longitudinal studies have assumed that the exposue is time-invaiant. We deived sample size fomulas fo studies compaing ates of change by exposue when the exposue vaies with time within a subject, focusing on obsevational studies whee this vaiation is not contolled by the investigato. Two scenaios ae consideed, one assuming that the effect of exposue on the esponse is acute and the othe assuming that it is cumulative. We show that accuate calculations can often be obtained by poviding the intaclass coelation of exposue and the exposue pevalence at each time point. When compaing ates of change, studies with a time-vaying exposue ae, in geneal, less efficient than studies with a time-invaiant one. We povide a public access pogam to pefom the calculations descibed in the pape ( Copyight 9 John Wiley & Sons, Ltd. Keywods: longitudinal study; obsevational study; ate of change; epeated measues; study design; time-dependent. Intoduction Studies involving epeated measuements of a binay exposue typically aim to compae the ate of change in esponse in the two exposue goups, o equivalently, the inteest is in an exposue-by-time inteaction. Study design fomulas fo such studies have mostly been deived in the context of clinical tials, and theefoe assume that the exposue is time-invaiant [--]. When the exposue vaies within a subject, some methods have been developed when the inteest is in the main effect of exposue, fo example in the cossove designs [, 3] and in the multicente clinical tials with andomization at the patient level [4]. Howeve, these methods do not apply to obsevational studies, whee exposue o teatment is not assigned by the investigato. When the inteest is in the exposue-by-time inteaction, no study design fomulas have peviously been developed fo longitudinal studies with a time-vaying exposue, to the best of ou knowledge. In this pape, we deive study design fomulas that ae valid when the aim is to compae ates of change in esponse in elation to a time-vaying exposue. Ou fomulas ae deived fo a binay exposue and fo equidistant time points. The fomulas we deive hee ae motivated by applications in the obsevational studies, whee the exposue is not assigned by design and many exposue pattens may be obseved, with vaiation in the numbe of exposed peiods pe paticipant and changes in the coss-sectional pevalence of the exposue ove time. Howeve, all esults eadily apply to the contolled setting as a simple special case. We then compae the fomulas we deive fo the time-vaying case with the existing ones fo the time-invaiant case, so that the investigatos can anticipate when they will need to ecuit moe o less paticipants because thei exposue is time-vaying instead of time-invaiant. This pape is stuctued as follows. In Section, we intoduce notation and seveal models a Cente fo Reseach in Envionmental Epidemiology (CREAL), Bacelona, Spain b Municipal Institute of Medical Reseach (IMIM-Hospital del Ma), Bacelona, Spain c CIBER Epidemiologia y Salud Publica (CIBERESP), Bacelona, Spain d Depatments of Epidemiology and Biostatistics, Havad School of Public Health, 677 Huntington Avenue, Boston, MA, U.S.A. Coespondence to: D. Spiegelman, Depatments of Epidemiology and Biostatistics, Havad School of Public Health, 677 Huntington Avenue, Boston, MA, U.S.A. stdls@channing.havad.edu Suppoting infomation may be found in the online vesion of this aticle. Contact/gant sponso: NIH; contact/gant numbe: CA66 8 Copyight 9 John Wiley & Sons, Ltd. Statist. Med.,

2 that extend the ates of change compaison to the time-vaying exposue setting. In Section 3, we deive expessions fo the vaiance of the coefficient of inteest fo each model consideed, in ode to obtain the fomula fo the test statistic upon which powe and sample size calculations ae based. In Section 4, we assess the effect of changing some of the assumptions, including time-vaying vs time-invaiant exposue, on sample size. In Section, the methods ae applied to the design of a eal study. In Section 6, the esults and thei implications ae discussed. In addition, we povide public access softwae to pefom all the calculations discussed in this pape ( Notation and models.. Notation Let Y ij be the outcome of inteest fo the measuement taken at the jth (j =,...,) time fo the ith (i =,...,N) paticipant, and E ij epesent the exposue fo the peiod between the measuements of Y i,j and Y ij. Thus, is the numbe of post-baseline measuements of the esponse pe paticipant, o, equivalently, the total numbe of measuements pe paticipant is +. We conside studies that obtain epeated measues evey s time units, as is the usual design in epidemiologic studies. Let t i be the initial time fo paticipant i and V(t ) be its vaiance. When V(t )=, all paticipants have the same time vecto, as when using time since enollment in the study as the time vaiable of inteest. Howeve, when age is the time metamete of inteest, as is often the case in epidemiology, and when, in addition, paticipants ente the study at diffeent ages, V(t )>. We base ou esults on linea models of the fom Y i =X i C (i =,...,N), fo some covaiate matix X i ((+) q), whee q is the numbe of vaiables in the model, and C is a vecto of unknown egession paametes. The (+) (+) esidual covaiance matix is Va(Y i X i )=R i (i =,...,N). Note that R i can be any valid covaiance matix, and can include tems associated with between-subjects vaiability as well as within-subjects vaiation. We base ou development on the genealized least-squaes (GLS) estimato of C, which has the fom Ĉ= ( N i ) ( X i R i X i N i X i R i Y i ) Since the design matix is not known a pioi, following Whittemoe [] and Shieh [6], study design calculations use (/N)R Γ as the vaiance of Ĉ, whee R Γ =(E X [X i R i X i ]) () As long as R i does not depend on the covaiates, () can be fully specified by the fist- and second-ode moments of the covaiate distibution [7]... Extending the model fo the time-vaying exposue When the exposue is time-invaiant, the usual model is E(Y ij X ij )=γ +γ t t ij +γ e E i +γ te (t ij E i ) () 8 whee the inteest is in the exposue-by-time inteaction. Figues (a) (c) illustate some possible tajectoies that could occu when the ate of change depends on the exposue. The left panels show the tajectoies fo a time-invaiant exposue, which occus with a linea exposue-by-time inteaction as descibed in (). The model can be extended to the time-vaying exposue case in at least two ways: one that assumes that the effect of exposue on the esponse is cumulative and the othe that assumes that it is acute and tansient. The new model equations fo these two genealizations ae given below. Fo both options, the esponse tajectoy of paticipants whose exposue status does not vay with time is equal to the tajectoies shown in the left panels of Figues (a) (c), and this tajectoy will be the same whethe we assume that the exposue has a cumulative o an acute effect. The tajectoies fo paticipants with changes in exposue ove time ae diffeent in the two models. Since all possible tajectoies implied by the models ae encompassed by the two exteme tajectoies of those with the time-invaiant exposue, Singe and Willett [8] efeed to these as the envelope tajectoies. When the effect of exposue is cumulative, we define a new vaiable, the cumulative exposue vaiable, Eij,wheeE ij = k j E ik, and we assume that Y ij depends on the exposue only though Eij (see Checkoway et al. [9] fo the motivation fo and examples of cumulative exposue). Note that if a paticipant is exposed fo the entie study peiod, the cumulative exposue, since enteing the study, is popotional to time in the study. We denote the cumulative exposue befoe enteing the study fo subject i as Ei,. Then, E ij =E i, + jk= E ik. Often,Ei, is unknown. The ight panel of Figue (a) shows a possible tajectoy fo one paticipant with Ei, = and exposue E i =(,,,,), which gives E i =(,,,,3). This patten is consistent with a cumulative

3 8 "Envelope" tajectoies Possible patten fo one subject 8 Y 6 Y Unexposed Exposed E= E= E= E= E= (a) Time Time "Envelope" tajectoies Possible patten fo one subject Y Y (b) Unexposed Exposed E= E= E= E= E= Time Time "Envelope" tajectoies Possible patten fo one subject Y Y (c) Unexposed Exposed E= E= E= E= E= Time Time Figue. Envelope tajectoies (tajectoy fo paticipants with time-invaiant exposue) and possible individual pattens accoding to: (a) models (3) and (4); (b) and (c) models () and (6). effect of exposue as well as an independent effect of time fo both the exposed and the unexposed (e.g. due to ageing), and it can be modeled as E(Y ij X i )=γ +γ t t ij +γ e E ij (3) Fo example, in a study of the elationship between changes in lung function and smoking status in a cohot followed fom 3 to 7 yeas of age, it was found that the ate of incease in lung function was smalle in peiods with active smoking [, p. 8]. Since cumulative exposue is, by definition, the sum of the poduct of point exposue by exposue duation, model (3) does not need to include an inteaction tem to be a genealization of model () fo the time-vaying case. A esponse tajectoy is defined fo each possible exposue histoy, E i, o equivalently, fo each cumulative exposue histoy, E i. Model (3) assumes that the within- and between-subject effects of cumulative exposue (and time) ae equal, that is, thee is no confounding by between-subject effects []. If this assumption is uneasonable, one may want to fit the following change model: E(Y ij Y i,j X i )=γ W t +γ W e (E ij E i,j )=γw t +γ W e E ij (4) 83

4 This model esults fom applying the fist diffeence opeato D= to model (3), so that DY i is the vecto with elements Y i,j+ Y ij, j =,...,, and Va(DY i )=DR i D. Fo a multivaiate nomal esponse with known R i, fitting model (4) by GLS is equivalent to fitting model (3) by the conditional linea egession (Web Appendix A). If thee is no confounding by between-subjects deteminants of esponse, ˆγ e will estimate the same paamete as ˆγ W e,othewise not []. In obsevational studies, model (4) is often pefeed, since each paticipant seves as his o he own contol, fully contolling fo confounding by all between-subject (time-invaiant) effects, including cumulative exposue at enty and age at enty. Howeve, the tade-off is that model (4) is less efficient than (3) []. When the effect of exposue is acute, we assume that the esponse depends on exposue only though the exposue in the pevious peiod. The ight panel of Figue (b) shows a possible tajectoy fo one paticipant with exposue E i =(,,,,). This situation can be modeled as E(Y ij X i )=γ +γ t t ij +γ e E ij +γ te (E ij t ij ) () and we ae inteested in the paamete γ te. Note that unde this model, a paticipant shifts tajectoies when exposue changes, the jumps being lage o smalle as time inceases. Although this model is consideed in the liteatue on longitudinal data analysis, the situation it implies may be hade to find in eal life, and vey often models with only a main effect of exposue may be moe appopiate when the effect of exposue is acute. Singe and Willett [8] fit model () in a study on the effect of time since unemployment in elation to the occuence of depession symptoms. The tajectoy implied by thei analysis is illustated in Figue (c). Immediately afte layoff (time ), paticipants had high depession index values. Ove time, they acclimated to thei new status and the values fo the depession index deceased ove time, without eaching the levels of the employed. Once a fomely unemployed individual found a job and kept it, the depession scoe dopped to the level of the employed and emained constant ove time. Like model (3), model () assumes that thee is no between-subject confounding []. As above, the within-subject effects of exposue and time can be estimated by using the model fo change that is obtained afte applying the fist diffeence opeato to model (), E(Y i,j+ Y ij X i )=γ W t +γ W e (E i,j+ E ij )+γ W te [(E i,j+ E ij )t ij +E i,j+ ] (6) Again, unde multivaiate nomality, fitting model (6) by GLS is algebaically equivalent to fitting model (3) by conditional likelihood (Web Appendix A)..3. Powe and sample size fomulas Let γ be the paamete of inteest, which is γ e fo model (3), γ W e fo model (4), γ te fo model () o γw te fo model (6). Let σ be the diagonal element of the matix R Γ, defined in (), that is associated with the paamete γ. The Wald test statistic to test if ˆγ is diffeent fom zeo is T = Nˆγ/ σ [9], and the fomula fo the powe of a study to detect an effect γ is Φ[ N γ / σ z α/ ], whee α is the significance level, and z u and Φ( ) ae the uth quantile and the cumulative density of a standad nomal. The fomula fo the equied sample size to detect an effect γ with powe π is N= σ (z π +z α/ ) / γ. Fo both powe and sample size calculations, we need to deive σ following () and the model of choice fom (3) (6). Note that σ will depend on and on seveal othe paametes associated with the distibutions of the esponse, the exposue and time. 3. Deivation of 3.. Compound symmety of both the esponse and the exposue pocess and othe simplifications 84 We stat by deiving fomulas fo σ when it is assumed that both the esponse and the exposue pocess have a CS covaiance and the exposue pevalence, p e, is constant ove time, in which case a closed-fom fomula fo σ can be deived. Unde CS of the esponse, R i =R has diagonal tems equal to σ,wheeσ =Va(Y ij X ij ) is the esidual vaiance of the esponse given the covaiates, and the off-diagonal tems equal to σ ρ,wheeρ is the coelation between the two measuements fom the same paticipant, known as the eliability coefficient o the intaclass coelation coefficient []. Then, assuming CS of exposue, the covaiance matix of exposue has diagonal elements equal to p e ( p e ) and off-diagonal elements equal to ρ x p e ( p e ), whee ρ x is the common coelation between exposue at diffeent time points. When the pevalence of exposue is constant ove time, the coelation ρ x is equal to the intaclass coelation of exposue, ρ e, defined as the pecentage of vaiation in exposue that is due to between-subject vaiation []. Othewise, one can still deive a elationship between the intaclass coelation, ρ e, and the common coelation unde compound symmety, ρ x (Web

5 . ρ e = 3. ρ e =.8.8 Pob.6.4 Pob E i. E i.. ρ e =. ρ e =.8.8 Pob.6.4 Pob E i. E i. Figue. Distibution of E i fo =3, p e = and diffeent values of ρ e. Appendix B). Because of the equality of ρ e and ρ x in the constant exposue pevalence case, and the fact that ρ e has an intuitive intepetation and well-defined popeties as detailed below, we will paameteize σ in tems of ρ e hencefoth. The intaclass coelation, ρ e, is bounded below by /, and fo binay vaiables, as hee, thee is a coection facto that needs to be added to this bound []. The uppe bound is one when the exposue pevalence is constant ove time, and below one othewise. An expession fo the uppe bound in the latte case was deived in Web Appendix C. These bounds ae calculated by ou pogam and displayed to the use afte and the pevalence at each time point have been povided. Apat fom being the pecentage of between-subject vaiation in exposue and being equal to the common coelation if the exposue pevalence is fixed, the intaclass coelation of exposue can also be egaded as a measue of imbalance in the numbe of exposed peiods pe subject, E i. WhenE i is balanced acoss subjects, then eveyone is exposed fo the same numbe of peiods as, fo example, in some cossove studies. Then, ρ e = /. Convesely, when the exposue is time-invaiant, the imbalance is maximal since E i is eithe zeo with pobability ( p e )o+with pobability p e,andρ e =. In obsevational studies, intemediate values between the bounds ρ e = / (same numbe of exposed peiods fo all paticipants) and ρ e = (time-invaiant exposue) will often be obseved, and when pilot data ae not available, the investigato can assess the sensitivity of the study design ove a ange of plausible values fo ρ e. To help the investigato assess what value of ρ e is appopiate fo his o he exposue, ou pogam povides the distibution of E i once and p e ae fixed and a CS covaiance of exposue is assumed. Examples of distibutions of E i by vaying ρ e ae shown in Figue. Table I shows the expessions fo σ fo models (3) (6). Fo models (3), () and (6), all the paticipants ae assumed to ente the study at the same time (V(t )=). Fo model (3), all the paticipants ae assumed to be unexposed at baseline (Ei, = i). Expessions ae deived in Web Appendix D. In Section 4, we will assess the effects of depatues fom the scenaios assumed in this section that wee used to deive the expessions in Table I. 3.. Geneal covaiance of the esponse and the exposue pocess As shown above, the main difficulty in computing powe and sample size fo longitudinal studies with the time-vaying exposues is chaacteizing σ, the diagonal element of the matix R Γ, defined in (), that is associated with the paamete γ. Its calculation involves the specification of the matix E X [X i R i X i ] and the computation of its invese. A simple geneal expession fo σ is difficult to obtain (Web Appendix D), but it can be easily computed with ou pogam once E X [X i R i X i ] has been defined. So, in this section, we only state the input paametes needed fo each one of the models (3) (6) to pefom such calculations. The paametes defining the covaiance of the esponse need to be povided fo all fou models. Examples of two common covaiance stuctues diffeent fom CS ae discussed in Section 4.. In addition, one needs to povide the exposue pevalence 8

6 Table I. Powe and sample size equations fo models (3) (6) assuming CS covaiance of both esponse and exposue, and constant exposue pevalence. Powe fomula: π=φ[ N γ / σ z α/ ] Sample size fomula: N= σ (z π +z α/ ) / γ Model Paamete of inteest σ (3) γ e σ ( ρ)(+ρ) p e ( p e )(+)(+)[6+( 3)ρ+(4+( )ρ)ρ e ] (4) γ W e σ ( ρ) p e ( p e )s (+)[+( )ρ e ] () γ te σ ( ρ)(+ρ) p e ( p e )s (+)(+)[+ρ ρ( ρ e )] (6) γ W te σ ( ρ) p e ( p e )s (+ρ e )(+) Fo models (3), () and (6), all paticipants ae assumed to ente the study at the same time (V(t )=). Fo model (3), all paticipants ae assumed to be unexposed at baseline (Ei, = i). at each time point, p ej,j=,...,, and the coelation between exposue at the jth and j th measuements, ρ ej,e j j j, fo all fou models. If a cumulative exposue effect is assumed and model (4) is assumed, no additional paametes apat fom those discussed in the pevious paagaph ae needed (Web Appendix D.). Howeve, if model (3) is assumed, age is the time metamete fo the study and the paticipants ente at diffeent ages, i.e. V(t )>, and/o if the paticipants ente the study with diffeent values of cumulative exposue, i.e. V(E )>, then additional paametes ae needed fo an exact calculation of σ (Web Appendix D.). Some of these paametes involve coelations between pais of vaiables, o equivalently, the expected values of poducts of vaiables. These paametes ae: the baseline mean and vaiance of time metamete, E(t )andv(t ); the baseline mean and vaiance of cumulative exposue, E(E )andv(e ); the coelation between the baseline cumulative exposue and the baseline time metamete, o E[E t ]; coelation between the time metamete at baseline and the exposue at all times hencefoth, o E[E j t ] j; and the coelation between the baseline cumulative exposue and the exposue at all subsequent measuement times, o E[E E j] j. These quantities ae difficult to povide a pioi unless longitudinal pilot data ae available; hence, one option is to base the study design on model (4), which estimates only within-subject effects. In that case, none of these additional paametes ae needed and a consevative study design will esult, i.e. moe paticipants than needed will be ecuited. When it is easonable to assume that all paticipants ae unexposed at baseline, i.e. Ei, = i, and time in the study is the time metamete of inteest, i.e. V(t )=, then only the exposue pevalence at each time point, p ej j, and the intaclass coelation of exposue, ρ ej,e j j,j, ae needed, even fo a study designed to fit model (3) (Web Appendix D..). Howeve, in an obsevational study, one still may want to base the study design calculations on model (4), which estimates the cumulative exposue duing the study peiod but contols fo all measued and unmeasued time-invaiant confoundes. If a cumulative effect of exposue is assumed, when age is the time metamete of inteest and paticipants ente the study at diffeent ages, i.e. V(t )>, to exactly compute σ, the investigato would need to specify the following additional paametes fo both models () and (6) (Web Appendix D.3): the vaiance of the time metamete at baseline, V(t ); the coelation between the time metamete at baseline and exposue at all time hencefoth, o E[E j t ] j; the expected value of the cosspoduct of exposue at all pais of time points and baseline time metamete, E[E j E j t ] j,j ; and the expected value of the cosspoduct of exposue at all pais of time points and the baseline time metamete squaed E[E j E j t ] j,j. Clealy, these quantities will be impossible to povide a pioi in most study settings unless longitudinal pilot data ae available; hence, some simplifications need to be implemented. One option is to pefom the calculations fo the case V(t )=, which seemed numeically to always povide consevative estimates, although we wee unable to pove this analytically. When V(t )= is assumed, only p ej j and ρ ej,e j j,j need to be povided (Web Appendix D.3.). 4. Efficiency compaisons 86 In Section 3., we obtained fomulas fo σ unde seveal scenaios likely to be encounteed in pactice. In this section, we compae the equied sample size unde those assumptions with the equied sample size when those assumptions ae not met. We will investigate the diection of the diffeences to identify unde what cicumstances incoect assumptions lead to consevative (ovepoweed) designs, and unde what cicumstances incoect assumptions will lead to undepoweed studies. In pactice, the investigato can also do these compaisons by pefoming the calculations unde vaious scenaios using ou softwae.

7 4.. Effect of depatues fom the assumption of CS esponse covaiance In this section, we conside two esponse covaiance stuctues that ae commonly used and ae moe geneal than CS, but that include CS as a paticula case: damped exponential (DEX) and andom intecepts and slopes (RS). Unde the DEX covaiance stuctue [3], the [j,j ] element of R i =R has the fom σ ρ j j θ, whee the coelation between two measuements decays exponentially as the sepaation between the measuements inceases, but the paamete θ attenuates this decay. Thus, when θ=, the CS covaiance stuctue is obtained, and when θ=, the AR() covaiance stuctue is given. The RS covaiance stuctue is the one that aises in mixed models, when thee is a andom effect associated with the intecept and the one associated with time. The covaiance matix is typically given as R i =Z i DZ i +σ w I,wheeZ i contains a column of ones and the column of times fo paticipant i, andd is the covaiance matix of the andom effects, with elements (σ b,ρ b b σ b σ b,ρ b b σ b σ b,σ b ) []. The RS covaiance stuctue is heteoscedastic and non-stationay []. When all the paticipants ae obseved at the same times, Z i =Z i and, theefoe, R i =R i. When that is not the case, R i depends on the covaiates, and expession () can only be calculated if a distibution fo the time vaiable is assumed. We define the total vaiance at baseline as V(Y i X i )=σ t =σ b +σ w, the intaclass coelation at baseline as ρ t =σ b / σ t, and the slope eliability as ρ b =σ b /( ( ρt )σ t s (+)(+) +σ b which is the pecentage of vaiation in the slopes that is between subjects. It does not depend on the esponse units and it has a closed ange (between zeo and one) [8]. Then, the RS covaiance can be expessed in tems of (σ t,ρ t,ρ b,ρ b b ). When ρ b = the CS covaiance is obtained. The slope eliability needs to be defined fo a paticula. Thoughout this pape, the values of ρ b will be calculated at the value =. Although this choice was abitay, it would be consistent with a typical longitudinal study funded by the U.S. National Institutes of Health. These studies can be funded fo no moe than yeas, and if a measuement was to be taken at the end of each yea of funding, =. Unde the assumption that the effect of exposue is cumulative, we assessed depatues by assuming CS esponse covaiance based on model (4) (simila esults wee obtained using model (3)). The covaiance of the exposue is still assumed to be CS. Though a gid seach, we found that, fo two studies with the same ρ e, depatues fom CS in the esponse, i.e. of θ> if DEX is assumed o ρ b > if RS is assumed, inceased σ and theefoe inceased the equied sample size. Figue 3 illustates this incease when DEX esponse covaiance, CS exposue covaiance and constant exposue pevalence ae assumed, fo seveal values of ρ, θ and ρ e (including independence, i.e. ρ e =, and time-invaiant exposue, i.e. ρ e =). The incease in the equied sample size with θ is lage fo lage values of ρ e and ρ. We pefomed the same assessment when the effect of exposue is acute based on models () and (6), and the effect was simila to that descibed in Figue 3, but with lowe s as the within-subject exposue coelation deceased. 4.. Effect of depatues fom the assumption of CS exposue covaiance ) To assess depatues fom CS in the exposue covaiance, we adopted a slightly diffeent appoach than when assessing depatues fom CS of the esponse covaiance. We pefomed a numeical analysis to evaluate the accuacy of the CS exposue covaiance assumption as an appoximation to σ when the exposue pocess had some othe coelation stuctue, i.e. when the exposue covaiance was misspecified. To compute the tue and misspecified σ, the exposue pevalence vecto and the coelation matix of exposue ae needed. Fo values of equal to, and, we geneated abitay pevalence vectos and coelation matices using a pocess descibed in Web Appendix E. Then, the based upon the misspecified and tue σ was calculated. The esponse covaiance was assumed to be known hee, and the calculations wee epeated fo CS, DEX and RS esponse covaiance to detemine whethe the esults wee dependent upon the assumed esponse covaiance stuctue. The values of the esponse covaiance paametes wee ρ=(.8,,.) fo CS; the same values of ρ and θ=(.,,.8,) fo DEX; and the same values of ρ t that we used fo ρ, andρ b =(.,.,.,,.8) fo RS. Figue 4 summaizes the esults fo some values of the esponse covaiance fo =. Results fo = and = wee simila. Results fom model (3) wee vey simila to those of model (4), and esults fom model (6) wee simila to those of model (). Fo the cumulative exposue effect models (3) and (4), most of the scenaios consideed (aound 9 pe cent) fell into the inteval (.9,.) with the CS esponse. As θ inceased, the esults wee bette than in the CS case. When RS of the esponse was assumed, the esults wee simila fo all combinations of ρ t and ρ b consideed, and less than pe cent of the scenaios had <.9. We ae paticulaly concened with those scenaios with low (<.9), since when is geate than one, consevative designs ae obtained. We obseved that the scenaios with the lowest s wee chaacteized by small, negative coelations fo pais of exposues close in time, and lage, positive coelations fo pais of exposues distant fom each othe. To confim this numeically, we egessed all coelation coefficients fom each of the exposue coelation matices against time sepaation; hence slopes wee obtained. A positive slope indicates that the coelations incease with time sepaation. Then, when the s wee egessed against these slopes, a stong linea negative elationship was obtained fo all esponse covaiances used in Figue 4, with the highe slopes pesenting the smallest s. The R of these egessions anged fom.63 to.79. Fo the acute exposue effect model with CS esponse, the was between.9 and. fo moe than 9 pe cent of the scenaios. Fo DEX esponse, the same patten as with the CS esponse was obseved when θ o ρ wee small, but when both θ and ρ wee lage, a highe pecentage of scenaios with less than.9 wee obseved. Fo example, when the esponse was AR() with ρ=.8, pe cent of the scenaios wee below that value. Similaly, fo RS covaiance, only fo high ρ t and 87

8 3. ρ e = ρ e = θ θ 3. ρ e =.8 ρ e = θ θ Figue 3. Sample size atio (=N θ /N θ= ) compaing the equied sample size of a study with the CS esponse covaiance (i.e. θ=) to a study with the DEX esponse covaiance (i.e. θ>) when the cumulative exposue effect model (4), CS exposue covaiance, constant exposue pevalence and = ae assumed, fo seveal values of ρ and ρ e. Lines indicate: ( ) ρ=., (---)ρ=, ( ) ρ=.8..4 Model (4) (cumulative) CS AR() AR() RS RS ρ b =.8 ρ b =. Model () (acute) RS RS CS AR() AR() ρ b =.8 ρ b = ρ =.8 ρ =.8 ρ =. ρ t =.8 ρ t = ρ =.8 ρ =.8 ρ =. ρ t =.8 ρ t = 88 Figue 4. Box-pecentile plots of the atio of equied sample sizes obtained when incoectly assuming CS covaiance of exposue divided by the equied sample size obtained using the tue exposue covaiance in scenaios geneated to have an abitay exposue covaiance, fo = and fo seveal coelation stuctues of the esponse. Fo RS models, ρ b,b = is assumed. At any height, the width of the iegula box is popotional to the pecentile of that height. Hoizontal lines indicate the th, th, th, 7th and 9th pecentiles. Y-axis on logaithmic scale.

9 ρ b did we find a high pecentage of scenaios with s less than.9 ( pe cent fo ρ t =.8, ρ b =). We obseved that the scenaios with the lowest s wee chaacteized by having high coelations fo pais of exposues that wee both eithe at the beginning o at the end of the study, while the emaining coelations wee negative. This implied a convex quadatic elationship between coelations and the sum of times of each pai. To numeically confim this, we egessed the coelation coefficients fom each of these exposue coelation matices against the sum of times and the sum of times squaed, and obtained coefficients associated with the quadatic tem. Then, when the s wee egessed against these coefficients, a stong linea negative elationship was obtained fo all esponse covaiances used in Figue 4, with the highe coefficients pesenting the lowest s. The R of these egessions anged fom 4 to.77. In conclusion, both fo the cumulative and the acute exposue effect models, the cases with <.9 had exposue coelation matices that ae unlikely to be found in pactice, and assuming CS of exposue esulted in good appoximations of the equied sample size in most cases consideed Effect of ρ e on sample size In this section, we assessed the effect of ρ e on sample size. This will also allow compaing the equied sample size obtained in a study with the time-invaiant exposue (ρ e =) to a study with the time-vaying exposue (ρ e <). This compaison has neve been done, pecluding investigatos to anticipate whethe they will need to ecuit moe o less paticipants if they have a time-vaying exposue instead of a time-invaiant one. When assuming that the effect of exposue is cumulative, the esults wee based on model (4) (esults wee veified to be simila using model (3)). Fo model (4), we show in Web Appendix D.. that if w jj j j,wheew jj is the [j,j ] element of (DR i D ),then σ is minimal when ρ e takes its uppe bound (i.e. ρ e =, the time-invaiant exposue case, if the pevalence is constant ove time) and maximal when ρ e takes its lowe bound, egadless of the fom of the covaiance of the exposue. The condition w jj j j holds when the esponse has CS o DEX covaiance, but it does not necessaily hold fo RS (Web Appendix D..). Thus, unde CS o DEX covaiance of the esponse, and when w jj j j fo RS, efficiency is always lost when the exposue vaies ove time. To illustate this, Figue shows the atio of the equied sample sizes, =N ρe /N ρe =, compaing a study with a time-vaying exposue with CS covaiance to a study with a time-invaiant exposue, fo the case whee the esponse vaiable has RS covaiance. It can be seen that, fo lage and ρ b, a study with a time-vaying exposue can end up being moe efficient than the one with a time-invaiant exposue. In contast, fo DEX esponse, the keeps inceasing as inceases, with a slowe ate than fo CS esponse, which is shown in the fist panel of Figue (ρ b = case). When both the esponse and the exposue pocess can be assumed to follow CS and the pevalence of exposue is constant ove time, we can deive an explicit fomula fo the atio of vaiances, o equivalently, the atio of the equied sample sizes, compaing the time-vaying exposue case to the time invaiant one, =N ρe /N ρe = =(+)/ (+( )ρ e ). The is always geate than o equal to one. So, as discussed above, in this paticula case, efficiency is lost when the exposue vaies ove time. When the exposue is acute, we based ou esults on model () (esults wee veified to be simila using model (6)). Fo model (), Figue 6 shows the atio of the equied sample sizes, =N ρe /N ρe =, compaing a study with a time-vaying exposue with CS covaiance to a study with a time-invaiant exposue, whee both studies have an RS covaiance of the esponse. When the esponse follows CS (ρ b = case), a study with a time-vaying exposue is less efficient than a study with a time-vaying one, although fo lage values of the diffeences become negligible. Fo this paticula case (ρ b =), the has the expession N ρe /N ρe = =(+ρ)/ (+ρ ρ( ρ e )). Thus, as with the cumulative effect model, in this simplest case, efficiency is lost when the exposue vaies ove time, and the loss inceases as the esponse coelation inceases, as the within-subject exposue coelation deceases and as deceases. Howeve, when the esponse has RS covaiance (ρ b >), a study with a time-vaying exposue can become moe efficient than a study with a time-invaiant exposue as inceases, and this gain in efficiency can be substantial. If the esponse follows DEX, esults simila to the RS case (Figue 6) ae obtained. Thus, it is impotant to coectly specify the covaiance of the esponse when assessing the effect of a time-vaying exposue, since assuming CS vs RS o DEX can lead to vey diffeent study designs and studies that can be substantially unde- o ove-poweed.. Example: the MSCM study [4] In this section, we applied ou methods to the setting of a study of the effects of matenal depession on child health (the MSCM study, data set available online at datasets.html [7]), to assess with eal data the pefomance of ou fomulas. Paticipants (N =4 with complete data) wee followed fo +=8 consecutive days. The pevalence of the depession fluctuated fom a maximum of.6 at baseline to a minimum of.6 at the second to the last day, with a mean of p e =.3, and ρ e was.. Although the esponse vaiable in this study (child health) was binay, we consideed the planning of a new study whee child health is measued continuously and wanted to pefom sample size calculations fo it. We pefomed seveal calculations unde the assumption of diffeent scenaios fo the esponse covaiance. As in Section 4., we will illustate, this time with exposue data fom a eal study, that by computing sample size unde an assumed CS exposue covaiance gives a equied sample size that closely appoximates the esult obtained using the obseved exposue covaiance. In addition, we will show that using the fomulas fo a time-invaiant exposue, poo appoximations ae obtained. 89

10 ρ t =.8, ρ b = ρ t =.8, ρ b =. ρ t =.8, ρ b = Figue. Sample size atio (=Nρ e /N ρe =) compaing the equied sample size of a study with the time-vaying exposue to a study with the time-invaiant exposue when the cumulative exposue effect model (4), RS covaiance of the esponse, CS covaiance of exposue, constant exposue pevalence and ρ b,b = ae assumed, fo seveal values of, ρ t, ρ b and ρ e. Y-axis on logaithmic scale. Lines indicate: ( ) ρ e =, (---)ρ e =, ( ) ρ e =. ρ =., ρ b = ρ =., ρ b =. ρ =., ρ b = ρ =.8, ρ b = ρ =.8, ρ b =. ρ =.8, ρ b = Figue 6. Sample size atio (=Nρ e /N ρe =) compaing the equied sample size of a study with time-vaying exposue to a study with the time-invaiant exposue unde the acute exposue effect model (), RS covaiance of the esponse, CS covaiance of exposue, constant exposue pevalence, V(t )= and ρ b,b = ae assumed, fo seveal values of, ρ, θ and ρ e. Y-axis on logaithmic scale. Lines indicate: ( ) ρ e =, (---) ρ e =, ( ) ρ e =. 9

11 Table II shows the atio between the equied sample size obtained when CS o time-invaiant exposue ae used and the equied ( sample size obtained when the obseved exposue pocess is used, i.e. when equation () is eplaced by (/N) ) i X i R i X i using the obseved X i values in the MSCM study. Calculations ae done fo both the cumulative exposue effect model (4) and the acute exposue effect model (). The esults fo models (3) and (6) wee almost identical. In this study, unde the cumulative exposue effect model, incoectly using the sample size fomula fo time-invaiant exposue led to a lage undeestimate of the equied sample size, except when RS was assumed and ρ b was lage. Assuming CS of exposue led to easonably accuate estimates of the equied sample size except when the esponse was RS, in which case slight oveestimations wee obtained. In ode to conside the situation whee fewe epeated measuements ae planned, we epeated these calculations using data only fom the fist thee peiods of the study ( =). In that case, the appoximation of assuming a CS exposue pocess povided accuate calculations fo all the esponse covaiances consideed. Fo the acute exposue effect model (), as the esponse covaiance depated fom CS, the fomulas that assumed a timeinvaiant exposue led to a substantial oveestimation of the equied sample size, in pat due to the lage of the study. Assuming CS covaiance of exposue led to easonably accuate calculations, with undeestimation of sample size no less than pe cent in all the scenaios consideed. As befoe, we epeated the same calculations but only using the fist thee peiods of the study. Then, using the fomulas fo a time-invaiant exposue led to undeestimations of to pe cent, as opposed to the oveestimations obtained when the full study was consideed, while assuming that CS of exposue still povided good appoximations. In conclusion, assuming CS covaiance of exposue gave vey accuate calculations in this example, wheeas the existing fomulas fo a time-invaiant exposue led to seiously flawed calculations. 6. Discussion In this pape, we developed study design fomulas that accommodate a time-vaying exposue fo longitudinal studies designed to compae ates of change in the esponse accoding to exposue. The pape is mainly focused on the design of obsevational studies, whee the within-subject vaiation of exposue is not detemined by design. We defined models fo acute and cumulative effects of exposue and pesented some simplifications so that the only additional paametes needed to calculate the equied sample size when the exposue vaies with time within a subject ae the exposue pevalence and the intaclass coelation of exposue. When studied numeically and applied to a eal-data example, ou methods povided easonable appoximations fo the equied sample size, which in many cases geatly impoved the esults that would have been obtained by using the available methods fo the time-invaiant exposue. The fomulas pesented in this pape ae implemented in a public access pogam that can be downloaded at the link povided in Section (Web Appendix F). We examined whethe studies with a time-vaying exposue equie ecuiting moe o less paticipants than a study with a time-invaiant exposue. Unde the cumulative exposue effect models, a study with a time-vaying exposue equies moe paticipants than the one with a time-invaiant exposue when the esponse covaiance is CS o DEX, and in many cases with RS covaiance. Unde the acute exposue effect models, the esults depend on the covaiance of the esponse. If the esponse has a CS covaiance, we find the same esult as in the cumulative exposue case, and the study with a time-vaying exposue equies moe paticipants. Howeve, in studies with seveal epeated measues and a DEX o RS covaiance of the esponse, a significantly lowe numbe of paticipants may be needed in the time-vaying case compaed with the time-invaiant case. Table II. Ratio of the equied sample sizes obtained by dividing the equied sample size assuming CS o time-invaiant exposue by the equied sample size obtained using the obseved exposue distibution fo the MSCM study ( =7, p e =.3,ρ e =.). Response covaiance ρ=.8 ρ= Exposue covaiance DEX RS RS DEX RS RS assumption CS θ= AR() ρ b =. ρ b = CS θ= AR() ρ b =. ρ b = Cumulative exposue effect (model (4)) Time-invaiant CS Acute exposue effect (model ()) Time-invaiant CS The cumulative exposue effect model (4) o the acute exposue effect model () is assumed. Calculations ae epoted fo diffeent scenaios of the esponse covaiance. The tue pevalence of exposue at each time point is used except fo the time-invaiant exposue case, whee p e is used. Fo model (), V(t )= is assumed. ρ b,b =. 9

12 The influence of dopout in the equied sample size has been studied peviously in studies with the time-invaiant exposue [4,, 8--]. Galbaith and Maschne [8] suggested the method of computing N fo 9 pe cent powe when 8 pe cent powe is intended and Fitzmauice et al. [] suggested inflating N by / ( f), whee f is the anticipated faction of loss to follow-up. The pefomance of these appoaches in longitudinal studies with time-vaying exposues emains to be investigated. Anothe inteesting topic fo futue eseach is to genealize the summay measue appoach [4, ] to the analysis of the longitudinal studies with a time-vaying exposue, and compae the efficiency of that analysis with the GLS appoach used hee. We pesented methods that povide accuate estimates of the equied sample size given the infomation on paametes that can often be obtained o guessed by the investigato a pioi. It is advisable to pefom sensitivity analysis by assessing the effects of diffeent values of the equied paametes befoe detemining the final sample size. We hope that the softwae we povide, which can be downloaded at the link povided in Section and implements all calculations pesented in this pape, will be a helpful tool fo planning study in these unavoidably complex settings. A demonstation of the pogam use can be found in Web Appendix F. Acknowledgements Reseach suppoted, in pat, by NIH gant CA66. Refeences. Schlesselman JJ. Planning a longitudinal study. II. Fequency of measuement and study duation. Jounal of Chonic Diseases 973; 6(9):6-7.. Kiby AJ, Galai N, Munoz A. Sample size estimation using epeated measuements on biomakes as outcomes. Contolled Clinical Tials 994; (3): Fison LJ, Pocock SJ. Linealy divegent teatment effects in clinical tials with epeated measues: efficient analysis using summay statistics. Statistics in Medicine 997; 6(4): Dawson JD. Sample size calculations based on slopes and othe summay statistics. Biometics 998; 4(): Hedeke D, Gibbons RD, Watenaux C. Sample size estimation fo longitudinal designs with attition: compaing time-elated contasts between two goups. Jounal of Educational and Behavioal Statistics 999; 4(): Raudenbush SW, Xiao-Feng L. Effects of study duation, fequency of obsevation, and sample size on powe in studies of goup diffeences in polynomial change. Psychological Methods ; 6(4): Diggle P, Heagety P, Liang KY, Zege S. Analysis of Longitudinal Data (nd edn). Oxfod Statistical Science Seies, vol.. Oxfod Univesity Pess: Oxfod, New Yok,. 8. Galbaith S, Maschne IC. Guidelines fo the design of clinical tials with longitudinal outcomes. Contolled Clinical Tials ; 3(3): Yi Q, Panzaella T. Estimating sample size fo tests on tends acoss epeated measuements with missing data based on the inteaction tem in a mixed model. Contolled Clinical Tials ; 3(): Jung SH, Ahn C. Sample size estimation fo Gee method fo compaing slopes in epeated measuements data. Statistics in Medicine 3; (8):3-3.. Fitzmauice GM, Laid NM, Wae JH. Applied Longitudinal Analysis. Wiley Seies in Pobability and Statistics. Wiley-Intescience: Hoboken, NJ, 4.. Jones B, Kenwad MG. Design and Analysis of Coss-ove Tials (st edn). Monogaphs on Statistics and Applied Pobability, vol. 34. Chapman & Hall: London, New Yok, Julious SA. Sample sizes fo clinical tials with nomal data. Statistics in Medicine 4; 3(): Moebeek M, Van Beukelen JP, Bege MPF. Optimal expeimental designs fo multilevel models with covaiates. Communications in Statistics Theoy and Methods ; 3(): Whittemoe AS. Sample size fo logistic egession with small esponse pobability. Jounal of the Ameican Statistical Association 98; 76(373): Shieh G. On powe and sample size calculations fo likelihood atio tests in genealized linea models. Biometics ; 6(4): Tu XM, Kowalski J, Zhang J, Lynch KG, Cits-Chistoph P. Powe analyses fo longitudinal tials and othe clusteed designs. Statistics in Medicine 4; 3(8): Singe JD, Willett JB. Applied Longitudinal Data Analysis: Modeling Change and Event Occuence. Oxfod Univesity Pess: Oxfod, New Yok, Checkoway H, Peace N, Kiebel D. Reseach Methods in Occupational Epidemiology (nd edn). Monogaphs in Epidemiology and Biostatistics, vol. 34. Oxfod Univesity Pess: Oxfod, New Yok, 4.. Twisk JWR. Applied Longitudinal Data Analysis fo Epidemiology. Cambidge Univesity Pess: Cambidge, U.K., 3.. Neuhaus JM, Kalbfleisch JD. Between- and within-cluste covaiate effects in the analysis of clusteed data. Biometics 998; 4(): Ridout MS, Demetio CG, Fith D. Estimating intaclass coelation fo binay data. Biometics 999; (): Munoz A, Caey V, Schouten JP, Segal M, Rosne B. A paametic family of coelation stuctues fo the analysis of longitudinal data. Biometics 99; 48(3): Alexande CS, Makowitz R. Matenal employment and use of pediatic clinic sevices. Medical Cae 986; 4():