Power Calculations for Two-Wave, Change from Baseline to Follow-Up Study Designs

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1 International Journal of Statitic in Medical Reearch, 01, 1, Power Calculation for Two-Wave, Change from Baeline to Follow-Up Study Deign M. Colin Ard 1 and Steven D. Edland 1, * 1 Department of Neurocience; Department of Family and Preventive Medicine Diviion of Biotatitic, Univerity of California San Diego, La Jolla, California, USA Abtract: Change in a quantitative trait i commonly employed a an endpoint in two-wave longitudinal tudie. For example, early phae clinical trial often ue two-wave deign with biomarker endpoint to confirm that a treatment affect the putative target treatment pathway before proceeding to larger cale clinical efficacy trial. Power calculation for uch deign are traightforward if pilot data from longitudinal invetigation of imilar duration to the propoed tudy are available. Often longitudinal pilot data of imilar duration are not available, and implifying aumption are ued to calculate ample ize from cro-ectional data, one tandard approach being to ue a formula baed on variance etimated from cro ectional data and correlation etimate abtracted from the literature or inferred from experience with imilar endpoint. An implicit aumption of thi tandard approach i that the variance of the quantitative trait i the ame at baeline and follow-up. In practice, thi aumption rarely hold, and ample ize etimate by thi tandard formula can be dramatically anti-conervative. Even when longitudinal pilot data for etimating parameter required in ample ize calculation are available, ample ize calculation will be biaed if the interval from baeline to follow-up i not of imilar duration to that propoed for the tudy being deigned. In thi paper we characterize the magnitude of bia in ample ize etimate when formula aumption do not hold and derive alternative conervative formula for ample ize required to achieve nominal power. Keyword: Sample Size, Phae, Phae II, Clinical Trial, Rate of Change, Compound Symmetry. 1. INTRODUCTION Quantitative trait are commonly employed a endpoint in phae II clinical trial. In Alzheimer dieae, recent effort have focued on change in biomarker a a mean of tudying dieae progreion and drug efficacy, and a an aid to diagnoi and early deciion-making in the evaluation of clinical intervention for chronic, debilitating and degenerative dieae [1-4]. The ongoing proce of identifying and validating biomarker for ue a urrogate endpoint, and more generally the quality and informativene of reearch outcome derived from the analyi of uch endpoint, require that thee trial be adequately powered. Power and ample ize calculation for two-wave (baeline and one follow-up) deign uing quantitative endpoint are traightforward if pilot data from longitudinal invetigation of imilar duration to the propoed tudy are available. In thi cae the variance of change core, d(t), can be etimated from pilot *Addre correpondence to thi author at the Biotatitic Diviion, Dept. of Family and Preventive Medicine, Profeor, Dept. of Neurocience, Univerity of California, San Diego, 8900 Gilman Dr., La Jolla, CA 9098, USA; Tel: ; Fax: ; edland@ucd.edu Data ued in preparation of thi article were obtained from the Alzheimer Dieae Neuroimaging Initiative (ADNI) databae (adni.loni.ucla.edu). A uch, the invetigator within the ADNI contributed to the deign and implementation of ADNI and/or provided data but did not participate in analyi or writing of thi report. A complete liting of ADNI invetigator can be found at: data, and upplied to the uual two-ample normal approximation formula for calculating ample ize requirement for a t-tet of no treatment effect againt a non-directional alternative hypothei: n t = d(t ) 1 ( )+ 1 ( ) (t ) where n t i the number of ubject required per tudy arm for a trial of duration t, d(t) i the within-group variance of the change in meaurement from baeline to follow-up for a tudy of duration time t, -1 (p) i the quantile function of the tandard normal ditribution, and are the Type I and Type II error rate, and (t) i the magnitude of the between-group difference in mean change from baeline to follow-up time t that the invetigator wihe to be able to detect with probability 1 -. Longitudinal pilot data are required to etimate the variance term, d(t), in equation (1). Lacking longitudinal pilot data or publihed etimate, it may be poible to abtract related parameter from the literature for etimating d(t). For example, value of variance at baeline and at follow-up, and the correlation of baeline to follow-up, may be available in publihed report. Calling the variance at baeline (0), the variance at follow-up (t), and the correlation between baeline and follow-up (0t), d(t) i then calculated a: (1) E-ISSN: /1 01 Lifecience Global

2 46 International Journal of Statitic in Medical Reearch, 01 Vol. 1, No. 1 Ard and Edland dt () = ( 0) + () t ( 0t) ( 0) () t () and the value obtained from equation () can then be ued in place of d(t) in equation (1). If (t) = (0), that i, if there i compound ymmetry of the covariance matrix of the baeline and follow-up (time t) meaurement, then equation () reduce to: dt () ( ) ( 0) = 1 ( 0t) A wa the cae with equation (), the reult of equation can be ubtituted directly into equation (1), but thi i valid only when (t) = (0). Equation i preented, for example, in Meinert [5] (equation 9.14), Lachin [6], Overall and Doyle [7], and Overall & Starbuck [8]. In all cae the formula i preented with no dicuion of what the aumption that (t) = (0) implie about the data generating proce, or of the conequence of making thi aumption in error. Thi i potentially problematic, becaue applying equation when compound ymmetry doe not hold can lead to dramatic underetimation of the required ample ize. For chronic progreive dieae in particular, peron tend to progre at different rate, o that trajectorie of progreion tend to fan apart and the variance of meaurement of dieae everity increae a the cohort i followed over time [9]. If the interval from baeline to follow-up i ufficient, there may be a ubtantial increae in variance from baeline to followup and a ubtantial underetimation of required ample ize by equation. A imilar danger arie when the reearcher ha direct acce to longitudinal pilot data or i able to abtract relevant parameter from a publihed longitudinal tudy, but the duration t of the planned trial exceed the duration, ay, of the available pilot data. Here, again, if longitudinal trajectorie fan apart, then d() < d(t) whenever < t, and ample ize etimate obtained by naïve ue of d() in place of d(t) will be anti-conervative. In the abence of a model decribing the functional dependence of the per-occaion variance and between-occaion correlation on the meaurement time, no ample ize formula i available in thi cenario. In thi paper we illutrate analytically and by way of example the potential magnitude of underetimation in ample ize calculation by naively applying equation when the aumption that (t) = (0) doe not hold (Section ). We alo demontrate that even when longitudinal pilot data are available for etimating parameter ued in ample ize calculation, thee etimate and reulting ample ize calculation can be biaed if the interval from baeline to follow-up i not of imilar duration to that propoed for the trial (Section 3). Section 3 will alo derive conervative upper bound on required ample ize ueful for enitivity analye when adequate pilot data are not available for planning a future trial.. BIAS IN SAMPLE SIZE CALCULATIONS WHEN THE (t) = (0) ASSUMPTION DOES NOT HOLD Letting n t repreent the true required ample ize per arm for a tudy of length t and denoting the etimate obtained uing equation a n t, for a given effect ize and Type I and Type II error probabilitie, the extent to which n t underetimate n t, expreed a a percentage of the true required ample ize i given by: %Underetimation = 100% n t n ( 3) t () t ( ( 0) ) t 0 ( ) = 100% n t () ( ( 0t) 1 ) ( 0) + () t ( 0t) ( 0) () t Note that (0t) 1 1, o that the numerator term in equation (4) are alway greater than zero, and hence n t underetimate the true required ample ize, whenever (t) > (0). Example To illutrate, we preent example of ample ize calculation for a hypothetical phae II clinical trial in Alzheimer dieae. Longitudinal data excerpted from the Alzheimer Dieae Neuroimaging Initiative (ADNI) cohort tudy are ued to inform power calculation. ADNI i a joint government, private pharmaceutical, and non-profit initiative to determine promiing marker of early AD progreion and aid reearcher and clinician in developing effective therapie. The ADNI cohort i a longitudinal obervational tudy that i by deign repreentative of ubject eligible for recruitment to Alzheimer dieae treatment trial [10]. Baeline and 1-month follow-up data were acceed from thi publically available dataet [10] on September 9, 009. Etimate of (0), (t), and (0t) for lateral inferior ventricular volume and for the Alzheimer Dieae Aement Scale cognitive ubcale (ADAS- (4)

3 Power Calculation for Two-Wave, Change from Baeline International Journal of Statitic in Medical Reearch, 01 Vol. 1, No cog [11]), two potential endpoint for a phae II trial of an Alzheimer dieae treatment, are ummarized in Table 1. Subject with an Alzheimer diagnoi at baeline and data at both baeline and 1-month time point were included in the calculation. The variance i larger at follow-up than at baeline for both outcome, a expected for longitudinal meaurement of a chronic progreive dieae. The parameter etimate in Table 1 were ued to calculate ample ize to obtain 80% power to detect (t) equal to a 5% reduction of ventricular enlargement relative to that oberved in the ADNI cohort placebo condition pilot data, and likewie to calculate ample ize to obtain 80% power to detect a 5% lowing of rate of progreion on the ADAS-cog relative to placebo, uing both the naïve but biaed etimate of d(t) (equation ) and the appropriate etimate of d(t) (equation ()). Sample ize etimate reulting from naively uing equation to calculate d(t) are approximately 50% maller than the ample ize required to obtain nominal power a calculated uing equation () (Table 1). For example, naïve ue of equation to etimate d(t) in equation (1) would lead u to conclude that a ample ize of 96 per arm would be ufficient to enure power = 0.8 to detect a 5% reduction in the rate of ventricular volume increae, wherea the actual required ample ize, derived uing equation (), would be 195 ubject per arm. For the ADAS-cog data, thee value are 356 and 719, repectively. Thee ample calculation are for illutrative purpoe only. Power calculation for an Alzheimer treatment trial involve additional conideration, including but not limited to calculating boottrap confidence interval about ample ize etimate acknowledging the uncertainty in population parameter upplied to the power formula [1, 13]. 3. POTENTIAL BIAS WHEN PILOT DATA ARE INCONSISTENT WITH PLANNED TRIAL DESIGN If reearcher have acce to pilot data or ummary tatitic from a tudy of comparable duration to the planned trial, a valid etimate of the required ample ize can be determined uing equation () to calculate d(t). Often, however, pilot data are from a tudy of duration, with < t. Thi mimatch in trial duration i ignificant becaue typically () < (t) when i horter than t, and therefore ample ize calculation [] tend to be anti-conervative. Letting n t denote the etimated required ample ize per arm calculated from equation (1) with d() ued in place of d(t), the magnitude of underetimation expreed a a percentage of actual required ample ize n t i given by: %Underetimation = 100% n t n [] t = 100% n t () t ( ) () ( 0) ( 0t) () t ( 0) () ( ) + () t ( 0t) ( 0) () t 0 While the pecification of condition that are both neceary and ufficient for equation (5) to take value greater than zero i omewhat more involved than wa the cae with equation (4), it can be hown, for [] example, that n t will be anti-conervative for data (5) Table 1: Parameter and Sample Size Requirement Etimate for ADAS-cog and Ventricular Volume Endpoint, a Calculated from ADNI Pilot Data Statitic ADAS-cog Endpoint Inf. Lat. Ventricle Pilot Data Sample Size Baeline Mean Month Change (0) (1) (0,1) n n %Underetimation 50.5% 51.1% Note: Inf. Lat. Ventricle i the um of the volume of the left and right inferior lateral ventricle in cm 3 ; (0) = Baeline variance; (1) = Month 1 variance; (0,1) = Baeline-to-Month 1 correlation; n 1 = required ample ize per arm to detect a 5% reduction in 1-Month Change with power = 0.8 and two-tailed ignificance level = 0.05; n 1 = etimated ample ize per arm to detect a 5% reduction in 1-Month Change with power = 0.8 and two-tailed ignificance level = 0.05 a calculated by equation ; %Underetimation = underetimation of the required ample ize per arm by ue of equation, expreed a a percent.

4 48 International Journal of Statitic in Medical Reearch, 01 Vol. 1, No. 1 Ard and Edland generated by the familiar mixed effect model with linear trajectorie fanning apart over time and i.i.d. reidual error, or whenever both: 1) (0) < () < (t) and ) 0 (0t) (0) (Appendix). Lacking pilot data conitent with the planned trial, there are everal alternative approache. If longitudinal data with more than two wave of obervation are available, then power calculation can be performed under the aumption of a linear mixed effect model with correlated random intercept and lope [1], a flexible yet parimoniou analytic framework for longitudinal data that i well-uited to ituation where variance are expected to increae (and correlation are expected to decreae), a the tudy duration i extended. However, if only two-wave pilot data are available and the duration of pilot data obervation i different (typically horter than), the duration propoed for the planned trial, the variance component from the correlated random intercept and lope variant of the linear mixed effect model are not identifiable, and calculation of the required ample ize per arm i not poible. Nonethele, the linear mixed effect model can till be relied upon to motivate olution for the required ample ize per arm that are guaranteed to be conervative under certain condition. We offer here two power formula that provide conervative overetimate of required ample ize when the duration of pilot data i le than the duration of the planned trial. Conervative Formula 1 Under the linear mixed effect model, the variance of change from baeline to follow-up d(t) for a tudy of length t can be expreed a a function of the variance of change from baeline to follow-up d() for a tudy of length and the mixed model reidual error variance e a follow (Appendix): dt () = t d() t e For t >, the econd term i greater than or equal to zero, and therefore: dt () = t d() i an overetimate of d(t), and power calculation by equation (1) with d(t) ued in place of d(t) will be conervative overetimate of required ample ize to achieve power (1 ). (6) (7) Conervative Formula Alternatively, under the linear mixed effect model, the variance of change from baeline to follow-up d(t) for a tudy of length t can be expreed a a function of d(), () = the variance of the quantitative meaure at follow-up time, (0) = the variance at the baeline obervation, and ab = the covariance of the random intercept and lope coefficient (Appendix): dt () = d() + t () ( ( 0) ) t ab (8) For ab > 0 and t > the final term i greater than or equal to zero, and therefore: dt () = d() + t () ( ( 0) ) (9) i an overetimate of d(t), and power calculation by equation (1) with d(t) ued in place of d(t) will yield conervative overetimate of required ample ize to achieve power (1 ). Note that the olution i exact, i.e., d(t) = d(t), when ab = 0. Since both equation (7) and equation (9) yield exact or conervative olution for required ample ize under the linear mixed effect model aumption when the covariance of the random intercept and lope coefficient i non-negative, the mallet ample ize etimate of the two calculated ample ize, i.e., the one uing min [ d(t), d(t)], i to be preferred under thee condition. 4. DISCUSSION Coniderable cot can be incurred when time and reource are allocated to an otherwie well conceived but underpowered tudy. Invetigator may therefore wih intead to opt for a conervative olution to the problem at hand, i.e., one that reult in a trial with greater than nominal power. In the abence of longitudinal pilot data, there i little information and little recoure for powering longitudinal trial. We recommend againt uing equation however, and rather ugget uing equation () to etimate the variance for ample ize calculation when balanced two-wave pilot data i available, becaue equation () require you to explicitly acknowledge the unknown parameter and therefore provide etimate for thee parameter to the ample ize formula. For the cae where longitudinal pilot data of length < t are available, we have provided conervative formula for

5 Power Calculation for Two-Wave, Change from Baeline International Journal of Statitic in Medical Reearch, 01 Vol. 1, No ample ize calculation for trial of length t that are valid under the modet aumption that the data derive from a mixed effect model with linear longitudinal trajectorie that are fanning apart. Anti-conervative trial that ignore thee concern rik unneceary fale negative finding and lot opportunitie for drug development. ACKNOWLEDGEMENTS The author were upported by NIA grant AG034439, AG005131, and AG Data collection and haring for thi project wa funded by the Alzheimer Dieae Neuroimaging Initiative (ADNI) (NIH U01 AG04904). ADNI i funded by the National Intitute on Aging, the National Intitute of Biomedical Imaging and Bioengineering, and through generou contribution from the following: Abbott; Alzheimer Aociation; Alzheimer Drug Dicovery Foundation; Amorfix Life Science Ltd.; AtraZeneca; Bayer HealthCare; BioClinica, Inc.; Biogen Idec Inc.; Britol- Myer Squibb Company; Eiai Inc.; Elan Pharmaceutical Inc.; Eli Lilly and Company; F. Hoffmann-La Roche Ltd and it affiliated company Genentech, Inc.; GE Healthcare; Innogenetic, N.V.; IXICO Ltd.; Janen Alzheimer Immunotherapy Reearch & Development, LLC.; Johnon & Johnon Pharmaceutical Reearch & Development LLC.; Medpace, Inc.; Merck & Co., Inc.; Meo Scale Diagnotic, LLC.; Novarti Pharmaceutical Corporation; Pfizer Inc.; Servier; Synarc Inc.; and Takeda Pharmaceutical Company. The Canadian Intitute of Health Reearch i providing fund to upport ADNI clinical ite in Canada. Private ector contribution are facilitated by the Foundation for the National Intitute of Health Rev March 6, 01 ( The grantee organization i the Northern California Intitute for Reearch and Education, and the tudy i coordinated by the Alzheimer' Dieae Cooperative Study at the Univerity of California, San Diego. ADNI data are dieminated by the Laboratory for Neuro Imaging at the Univerity of California, Lo Angele. APPENDIX Condition Under which d() Lead to Anti- Conervative Sample Size Etimation When Data are Generated by a Linear Mixed Effect Model We begin by introducing notation for two-wave data generated by the mixed effect model commonly applied to longitudinal data from clinical trial and cohort tudie. Let y i be obervation on ome random variable of interet that follow the linear mixed effect model with bivariate normal random intercept and lope coefficient and i.i.d. reidual error: y i = + + a i + b i + e i a i b i (A1) ~ N 0, a ab ab, e i ~ N(0, e ) (A) b Here and repreent the fixed intercept and lope term (rather than the Type I and Type II error probabilitie), repectively, a i and b i are their random counterpart for ubject i, and the e i are i.i.d. reidual error term pecific to ubject i at time. For two-wave data, will typically only take on two value by deign, e.g., zero at baeline and t at follow-up for a trial of length t, or zero at baeline and at follow-up for a trial of length. It follow from equation (A1) and (A) that for any : ( ) = Var { y i } = Var { a i + b i + e i } = a + b + ab + e Alo, for any 0: d( ) = Var { y i y i0 } { } = Var b i + e i e i0 = b + e (A3) (A4) To prove that d() lead to anti-conervative ample ize etimate under thi model, imply note that by (A4) d() < [] d(t), and therefore n t calculated uing d() in equation (1) will underetimate n t when < t and b > 0. The b > 0 condition imply mean that trajectorie are fanning apart, a i typically oberved in longitudinal tudie of chronic, progreive dieae. When (0) < () < (t) and 0 (0t) (0) To how that equation (5) i greater than zero, and [] hence that n t i anti-conervative, under the general condition that 1) (0) < () < (t) and ) 0 (0t) (0), we need to prove that the numerator of equation (5) i poitive under thee condition, which i equivalent to howing that: () + () t ( 0t) () t ( 0) () > (A5) ( 0) () t ()

6 50 International Journal of Statitic in Medical Reearch, 01 Vol. 1, No. 1 Ard and Edland The proof follow immediately by noting that, given the aumption, the left-hand ide of inequality (A5) i > 1, and the right-hand ide of (A5) i 1. Derivation of Equation (6) Uing the model and notation above, equation (6) follow by writing: dt () t d() = t b + e t = t e Derivation of Equation (8) dt ( ) b + e Similarly, equation (8) follow by noting that: () d() t () ( ( 0) ) = t b + e ( b + e ) t a + ( b + ab + e ) ( a + e ) = ( t ) b t ( b + ab ) = t ab REFERENCES (A6) (A7) [1] Hampel H, Mitchell A, Blennow K, et al. Core biological marker candidate of Alzheimer' dieae - perpective for diagnoi, prediction of outcome and reflection of biological activity. J Neural Tran 004; 111: [] Matthew B, Siemer ER, Mozley PD. Imaging-baed meaure of dieae progreion in clinical trial of dieae modifying drug for Alzheimer dieae. Am J Geriat Pychiatr 003;11: [3] Moh RC, Kawa C, Carrillo MC. Optimal deign of clinical trial for drug deigned to low the coure of Alzheimer' dieae. Alzheimer Dement 006; : [4] Thal LJ, Kantarci K, Reiman EM, et al. The role of biomarker in clinical trial for Alzheimer dieae. Alz Di Aoc Di 006; 0: [5] Meinert CL. Clinical trial: deign, conduct, and analyi. nd ed. New York: Oxford Univerity Pre 01. [6] Lachin JM. Introduction to ample-ize determination and power analyi for clinical trial. Control Clin Trial 1981; : [7] Overall JE, Doyle SR. Etimating ample ize for repeated meaurement deign. Control Clin Trial 1994; 15: [8] Overall JE, Starbuck RR. Sample-ize etimation for randomized pre-pot deign. J Pychiatr Re 1979; 15: [9] Edland SD. Blomqvit reviited: how and when to tet the relationhip between level and longitudinal rate of change. Stat Med 000; 19: [10] Peteren RC, Aien PS, Beckett LA, et al. Alzheimer' Dieae Neuroimaging Initiative (ADNI): Clinical characterization. Neurol 010; 74: [11] Moh RC, Knopman D, Peteren RC, et al. Development of cognitive intrument for ue in clinical trial of antidementia drug: Addition to the Alzheimer' dieae aement cale that broaden it cope. Alz Di Aoc Di 1997; 11: S13-S1. [1] Ard MC, Edland SD. Power calculation for clinical trial in Alzheimer' dieae. J Alzheimer Di 011; 6: [13] McEvoy LK, Edland SD, Holland D, et al. Neuroimaging enrichment trategy for econdary prevention trial in Alzheimer dieae. Alz Di Aoc Di 010; 4: Received on Accepted on Publihed on Ard and Edland; Licenee Lifecience Global. Thi i an open acce article licened under the term of the Creative Common Attribution Non-Commercial Licene ( which permit unretricted, non-commercial ue, ditribution and reproduction in any medium, provided the work i properly cited.