Design and analysis of stratified clinical trials in the presence of bias (2024)

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Supplementary Materials

Abstract

Keywords: Multi-center clinical trial, weighted t test, sample size, stratified randomization, type I error probability, selection bias, time-trend bias

1 Introduction

Large clinical trials often stratify the randomization on a small collection ofcovariates that may introduce heterogeneity into the patient stream. An importantcovariable in multi-center trials is often the clinical center, as different studypersonnel, clinical settings, and patient populations may result in differentialstudy outcomes.¹ A stratified population-based analysis can be performed with or withoutstratification in the design. Less is known about the impact of stratification whenthere is a bias in the clinical trial. In this paper, we explore this issue both forselection bias and chronological bias, and we demonstrate the impact of theseanalyses on a weighted stratified analysis. In so doing, we explore the role ofspecific stratified randomization procedures (RPs) and how certain procedures maymitigate the effects of bias. The recognition of the role of RPs in mitigating biashas been explored in prior research for unstratified trials.^2–6 But because stratification intoK strata creates K different independentrandomized clinical trials, and a stratified test combines Kindependent tests, the impact of bias can be more pronounced.

The paper is organized as follows: In Section 2, we describe different stratified RPsand discuss aspects to formulate stratified versions of RPs. In Section 3, we derivea formulation of Fleiss¹ stratified test statistic preserving the allocation sequence and derive thedistribution of the test statistic taking bias into account but ignoring bias in theanalysis and mention some sample size considerations for the stratified test. InSection 4, we specify the bias model in the form of an additive combination ofstrata-specific selection bias and strata-specific time-trend bias linked tostratified allocation sequence. The criterion introduced in Section 5 is used tosummarize the impact of the allocation sequence-specific bias on the type I errorprobability over the range of all sequences induced by a specific RP. Consequently,an assessment of different RPs is enabled which guides the choice of an RPs forapplication in a particular clinical trial setting. The methodology is applied tosome-specific scenarios in Section 6 to illustrate the effects. We discuss thefindings in Section 7 and draw conclusions in Section 8.

2 Stratified RPs

RPs for clinical trials for two treatments are well described in literature.² In principle, any RP used for two-treatment clinical trials can be employedwithin strata in a stratified randomization. A comprehensive review is given in Rosenberger.² Complete randomization in which patients are assigned to treatments withprobability 1/2 is rarely used in stratified clinical trials. Rather, some form ofrestricted randomization is employed in an effort to balance treatments withinstrata. Hilgers⁶ categorized restricted RPs that force balance in probability, force balanceusing a maximal tolerated imbalance, or force terminal balance. A selective list ofrestricted RPs is given as follows²:

• Efron's biased coin design (EBC(p)),which consists of flipping a biased coin with probabilityp≥0.5 in favor of the treatment which has been allocatedless frequently and a fair coin in case of equal numbers of treatment assignments,⁷

• Big stick design (BSD(a)), which can beimplemented via complete randomization with a forced deterministicassignment when a maximal tolerated imbalance a isreached during the enrollment,⁸

• Random allocation rule (RAR), which assigns half thepatients to E and C randomly,⁹

• Permuted block randomization (PBR(b))with block size b uses RAR within blocks ofb patients, for b even,¹⁰

• Maximal procedure (MP(a)) which usesthe allocation sequences of RAR by additionally imposing a maximaltolerated imbalance (a) and assigning equal probabilityto all such sequences.¹¹

Note that EBC(p) may be classified as a restricted RPforcing balance in probability. BSD(a) forces balance by maximaltolerated imbalance a during the allocation process but does notforce terminal balance. Restricted RPs with a maximal tolerable imbalance andterminal balance are PBR(b) and MP(a).

The International Council of Harmonization stated in the E9 recommendation (ICH E9)

It is advisable to have a separate random scheme for each centre, i.e., tostratify by centre or to allocate several whole blocks to each centre.

The European Medicines Agency “Guideline on Clinical Trials in Small Populations”recommends stratified randomization to improve power. Using permuted blocks withineach stratum is the most popular method of stratified randomization, and this isoften called the stratified block design. Blocks can be selectedwith a fixed size or with variable sizes. However, blocking is not the only methodto use within strata. The ICH E9¹² guidelines also state that “differenttrial designs will require different procedures for generating randomizationschedules.” We now define stratified randomization more formally.

Consider the allocation zji∈{0,1} of patients i=1,…,nj either to the treatment E if z_ji = 1 or C if z_ji = 0 in stratum j. An RP is implemented by assigningprobabilities P(Zj=zj|zj∈{0,1}nj) to the possible allocations Zj=(Zj1,…,Zjnj) in stratum j. A stratifiedrandomization is implemented by creating independent randomizationlists zj∈{0,1}nj for each stratum 1≤j≤K. Denote the possible allocations by z=(z1,…,zK)∈×j=1K{0,1}nj={0,1}N with N=∑j=1Knj. Then a stratified version of an RP is implemented by assigningprobabilities P(Z=z|z∈{0,1}N)⋮=Πj=1KP(Zj=zj|zj∈{0,1}nj) to the possible allocations z∈{0,1}N. Of course, when implementing complete randomization, thestratified and unstratified RPs result in the same set if randomization sequenceswith the same probabilities because assignments are independent and equiprobable,i.e.

However, when implementing a stratified restricted RP, this observation generallydoes not hold and some further definitions are necessary. Even in the very simpleRAR, the set of possible randomization sequences is reduced considerable and theprobability for the stratified allocation sequence becomes

Another important aspect concerns the “balancing behavior” of restricted RPs. Theterm restricted refers to the fact that conditions on the randomization process areintroduced to control the potential imbalance in the frequency of treatmentallocations. Let s,1≤s≤N, denotes the patient's number preserving the appearance of patientsin the trial so that s = 1 denotes the first patient ands = N the last enrolled patient.njE(s) and njC(s) denote the number of patients allocated to treatmentsE and C in stratum j until atotal of s patients are recruited in the trial so far. Then, theimbalance in the number of allocations to treatment E andC in stratum j until a total ofs patients are recruited is measured by

Three definitions of imbalance are used in the following:

1. An RP shows overall final balance, if d(N)=def∑j=1Kdj(N)=0

2. An RP controls the final balance within strata, ifdj(N)=0 for 1≤j≤K

3. An RP controls the maximal tolerated imbalance, if-a≤dj(s)≤a for all 1≤j≤K,1≤s≤N.

Of course controlling the overall final balance within strata does not imply tocontrol final balance within stratum, i.e. dj(s)=0. Simply controlling the overall final balance may result in onestratum assigning patients only to E and another stratum assigningthe same number of patients to C only, a case which invalidates theestimation of treatment difference within strata, presumably one issue in the ICHguidance. On the other hand, final balance within strata (dj(N)=0) implies overall final balance d(N)=0. In the following, we deal with stratified RPs and deriveadditional restrictions for meaningful definitions. With RAR, stratification andconsequently final balance within strata require even samples sizes within eachstratum and two treatment arms. The requirement of final balance within strataimplies in the case of the stratified block design that the block sizes are divisorsof the stratum sample sizes nj,1≤j≤K. Note that in stratified trials with a larger number of centers,usually smaller sample sizes in centers occur and thus final balance within strataforces the block sizes to be small, which will increase the potential for selectionbias. Of course, center-specific block sizes are possible but rather uncommon. Wewill consider common block sizes in the following.

It should also be noted that the stratified RAR procedure in general cannot beconsidered as an unstratified PBR with block sizes nj,1≤j≤K, because enrollment of patients in the trial is parallel in strata,so that in general d(∑j=1knj)=0,1≤k≤K.

Similar problems arise, when controlling the maximal tolerated imbalance with margina, which results in an upper overall bound of ∑j=1K|dj(s)|≤Ka for all 1≤s≤N. Thus, controlling the maximal tolerated imbalance across stratacould be accomplished by having a different imbalance level in each stratum. Twovery straightforward simple settings are uniform spread |dj(s)|≤a/K,1≤j≤K for all 1≤s≤N resulting ∑j=1K|dj(s)|≤∑j=1Ka/K=a for all 1≤s≤N or proportional spread with |dj(s)|≤anj/N,1≤j≤K for all 1≤s≤N resulting ∑j=1K|dj(s)|≤∑j=1Kanj/N=a for all 1≤s≤N. Hilgers⁶ suggests defining a in relation to loss of power, and thisimplies stratum-specific maximal tolerated imbalance according to the rulesabove.

3 Stratified analysis

As mentioned in Section 1, a stratified randomization requires a stratified analysis,although a stratified analysis can be performed whether or not the randomization wasstratified. In this section, we examine the distributional properties of a teststatistic introduced by Fleiss¹ (page 268, formulas 1 and 2) based on a weighted t statisticfor the analysis of stratified clinical trials. While we do not considerrandomization tests in this paper, clearly randomization-based inference is anattractive alternative, see Rosenberger.² The reason for using a parametric t test is that itfacilitates our goal of determining the effect of bias on inference, since we canderive the distribution of the test statistic under various forms of bias. Inparticular, in this section, we derive the non-centrality parameter for thedistribution of the test statistic under alternative hypotheses and comment on howit can be used for sample size considerations. In the sequel, we are interested inthe role of the RP in the analysis of stratified trials. Because Fleiss wrotespecifically about centers rather than strata, we use both interchangeably; itshould be clear that stratification can be done on variables other than centerhowever.

We will consider a two-arm parallel group clinical trial stratified byK centers with no interim analysis. The response to thetreatments E and C respectively is measured withthe continuous normally distributed endpoint yji,1≤i≤nj=njE+njC,1≤j≤K, on n_jE patients in the experimental group (E) andn_jC patients in the control group (C) in centersj. The total sample size is denoted by N=∑j=1Knj.

We use the allocation sequence notation of the statistical modelassuming no treatment by center interaction by

where εjiN(0,σ2),1≤i≤nj=njE+njC,1≤j≤K. The expected treatment effects under E andC are denoted by μ_E and μ_C, respectively. The Z_ji denotes the allocation sequence indicator with Z_ji = 1 if patient i in center j is allocatedto treatment E and Z_ji = 0 if patient i in center j is allocatedto treatment C. Here and in what follows the notationsnjE=∑i=1njZji and njC=∑i=1nj(1-Zji) are used. Furthermore, τ_ji denotes the fixed “bias” effect acting on the response of patienti in center j. Without loss of generality, weassume τji>0.

Fleiss's statistic to test the hypothesis (H0⋮μE=μC) of no treatment effect across centers becomes

where Dj=y∼jE-y∼jC are the mean treatment differences with y∼jE=1njE∑i=1njyjiZji and y∼jC=1njC∑i=1njyji(1-Zji). Furthermore, w_j are weights associated with center j, wj*=njE×njCnjE+njC and s_p is the pooled variance given by

Here sjℓ2 denotes the variance of treatment group ℓ in centerj. To derive the distribution of equation (3) under model (2),the distributions of the numerator as well denominator must be calculated. Note thatthe variance is given by

so that the numerator of equation (3) has variance 1. Of course,D_j is normally distributed via the distribution of y_ij and thus the expectation of the denominator equals

where Z=(Z11,…,Z1n1,…,ZK1,…,ZKnK)t is the observed allocation vector, τ∼jE=1njE∑i=1njτjEZji and τ∼jC=1njC∑i=1njτjC(1-Zji). In summary, the numerator is i.i.d. normally distributed withexpectation δ(Z) and variance 1.

Next, we calculate the distribution of the denominator sp/σ using the allocation sequence notation

Note that Var(yji/σ)=1 and E(yji/σ)=(μE+τji)/σ for Z_ji = 1 and or E(yji/σ)=(μC+τji)/σ for Z_ji = 0 are i.i.d. normally distributed for all 1≤i≤nj and 1≤j≤K. Following the arguments in Johnson and Kotz,¹³ the ∑i=1njZji(yji-y∼jE)2/σ2 for group E, i.e. Z_ij = 1 and the ∑i=1nj(1-Zji)(yji-y∼jC)2/σ2 for group C, i.e. Z_ij = 0 are χ2 distributed with njE-1 and njC-1 degrees of freedom respectively and non-centrality parameters

Applying that the sum of independent non-central χνj2(λj) distributions is non-central χ2 with ∑νj degrees of freedom and non-centrality parameter ∑λj, it follows that the distribution of (∑j=1K∑ℓ=E,C(njℓ-1))sp2/σ2 is non-central χ2 with non-centrality parameter

and

degrees of freedom. Finally, we have to show the independence of thenumerator

and denominator

as random variables. Here, Theorem 3 of Searle¹⁴ is used, stating that two random variables that can be expressed asxtAx and Bx, where x∼N(μ,V) is independent, if BVA=0. First, note that V=σ2I holds in our case.

For enabling the matrix notation of the above expressions, a usual design matrixX can be defined which includes two columns for the allocationindicator variables and N rows. Rearrangement of the design matrixby center and treatment group so that the first n1E observations belong to treatment E and thepreceding n1C observations belong to C in center 1 and so oncan be implemented by a suitable permutation matrix P. This simplifies the matrix notation of the above numerator anddenominator in terms of B and A by reshuffling the allocation sequence Z=(Z11,…,Zknk)t using a suitable permutation matrix P. This permutation matrix does not affect the matrix equation.Furthermore, note that it is sufficient to show the matrix equation for a particularcenter j because of the block structure implied by the independentobservations in different centers. With this reshuffling, the notation for centerj corresponding to Theorem 3 is

and with Hij=Inij-1nij1nij×nij the matrix

so that σ2BjInjE+njCAj=0 for all 1≤j≤K, which shows the independence. In summary, the distribution of thestatistic in equation (3) is doubly non-central t,¹³ with non-centrality parameter

In the case sampling is “stratified” by center and the objective is to estimate theoverall treatment effect accounting for center, Fleiss¹ proposed the weights wj=wj*=njE×njCnjE+njC resulting in the test statistic (3)

Of course, equation (5) implies that δ(Z) depends on the weights only and becomes

In the case sampling is “stratified” by center and the objective is to estimate theoverall treatment effect, Fleiss¹ proposed the weights w_j = 1 so that equation (3) becomes

whereas the first non-centrality parameter δ(z) equals

Weighting centers in the absence and presence of center-by-treatment interaction hasdiscussed in detail by other authors.¹⁵

4 Stratification in the presence of bias

We now turn to the question of bias. Two common forms of bias encountered in clinicaltrials are chronological bias due to time trends in patient outcomes,¹⁶ and selection bias, which can result in covariate imbalancesand inflation of type I error rates.³ By definition, selection bias arises from the conscious or unconsciousguessing of treatment assignments so that patients have a higher chance ofassignment to the investigator's treatment of choice for those patients. Whiledouble-blinded studies, and multi-center studies with a central randomization unitmitigate the possibility of selection bias, Berger¹⁷ gives numerous examples of when selection bias has arisen in practice. As theICH E9 Guidelines note,¹²

It is important to identify potential sources of bias as completely aspossible so that attempts to limit such bias may be made…. The treatmenteffect and treatment comparisons should involve consideration of thepotential contribution of bias to the p-value.

A recent paper provides a template on assessing the potential forchronological or selection bias and gives guidance on how to choose an appropriateRP and test statistic to account for that possibility.⁶ Here, we use a similar model to determine the impact on Fleiss's test in thepresence of such bias.

We first specify a compound bias vector τ_ji for stratum j and patient i that is alinear combination of a metric of chronological bias and selection bias. Taking intoaccount the stratified randomization, we explore a linear time-trend¹⁶ model per stratum similar to Hilgers⁶ given by

Hereby, the magnitude θ_j of the linear time trend varies between centers. Note that Hilgers⁶ proposed to formulate θ_j as fraction of the variance σ2. The second term generalizes the biasing policy first introduced by Proschan³ for the Gauss test and later investigated by Hilgers⁶ for the t test. The amount of selection bias ηj≥0 is allowed to vary between centers. The biasing policy in equation(10) “favors” or biases the expected response towards treatment Eassuming if the less frequent treatment allocated so far is Eassuming E will be allocated next. The direction ηj≥0 corresponds to favoring E. Other metrics havebeen used to define the selection bias metric, including just the sign ofnjE(i-1)-njC(i-1). We chose our metric so that it is roughly the same scale as thechronological bias metric.

5 Evaluation criterion

In our numerical evaluation study, we enumerate all possible randomization sequencesfor four different procedures and compare the bias to the type I error rate viacomputing the proportion of sequences that preserve the type I error rate at thenominal (0.05) level. If there is no bias (e.g. ηj=θj=0), 100% of sequences will preserve the type I error rate,regardless of the procedure used. To be more formal, denote the bias vectorτ=(τ11,…,τ1nj,…,τK1,…,τKnj) and the set of all sequences z generated by the RP by Ω_RP. The test statistic t(z) depends on the randomization sequence is centralt distributed with N-2k degrees of freedom under the null hypotheses and no biasτ=0, i.e. the null hypotheses H₀ will berejected at the α level if |t(z)|≥tN-2k(1-α/2). Then, the evaluation criterion can be expressed by using ourdistributional result above including the non-centrality parameter (7)

where FN-2k,δ,λ denotes the distribution function of the doubly non-centralt-distribution with N-2K degrees of freedom and non-centrality parameters δ(Z) and λ(Z). In the ideal case, the probability should be 1, meaning that the5% level is maintained by all allocation sequences. A value below 1 indicates thatthe actual type I error rate is higher than the target level of 5%. Note that thisquantity summarizes the impact of bias over all randomization sequences anddemonstrates the clinical consequences as well as the “go/no-go” decision of theregulator directly.

6 Numerical evaluation study

The objective of the following numerical evaluation study is to illustrate effects ofstratification in both the randomization and the test statistic. It is not intendedto conduct a comprehensive simulation study, recognizing that the specification ofthe sample size as well as θ_j and η_j depends on the practical situation. To be more specific, we start with aK = 2 center clinical trial and use a total sample size of 80patients with common θ_j and η_j in all centers. The following reasoning leads to the specification ofθ_j and η_j. Concerning the linear time trend θ_j, it should be noted that although the θ_j are defined within each center, the maximal extent of the time trend shouldnot exceed σ. In contrast, although the magnitude of the selectionbias effect η_j may vary between centers, it is like a population effect within center and nomaximal extent restriction may apply. To relate the total sample size of 80 in aK = 2 center clinical trial to the effect size, formula (9) isused. The hypothesis H0⋮μE=μC vs. H∼1⋮μE-μC=Δ should be tested with the (optimal) weighted ttest (equation (3)) assuming common variance σ = 1 and intendedallocation ratio of 1:1 at the 5% significance level with a power of 80%. Thisresults in a uniform effect size of Δ=0.635. With this effect size, the allocation ratio r isvaried so that the loss in power does not exceed 2%. This yields an allocation ratioof r = 0.608 which translates to sample size of 31:49 correspondingto a maximal tolerable imbalance of 18. With the uniform or proportional spread,this results in a maximal tolerated imbalance by center of 4 and 5,respectively.

For illustration purposes, we will compare the stratified and unstratified versionsof CR, BSD(9), PBR(4), and EBC(2/3). These four procedures represent completerandomization and the three types of restricted randomization mentioned earlier.These procedures were evaluated for two different splits of the total sample size(n1=n2=40 and n1=60,n2=20) and the combinations of selection and time-trend bias as(η,θ)=(0,0.2),(0.2,0),(0.2,0.2),(0,0.05),(0.05,0),(0.05,0.05). The evaluation criterion was the number of sequences protectingthe 5% level for stratified and unstratified randomization as well as stratified(wj=1,wj*) and unstratified (us) test statistic and RP (seeSupplementary Material). Note that unstratified randomization and test statisticcorrespond to the case presented in Hilgers.⁶ The results for (0,0.05), (0.05,0), (0.05,0.05) are given in Table 1 as well as for(0,0.2), (0.2,0), (0.2,0.2) in Table 2. In an additional evaluation, the number of centersK is increased from 2 to 8 while splitting the total sampleuniformly to the centers to show, whether there is a different influence on the typeI error rate. We used an R software script to conduct the analysis, seeSupplementary Material.

In the case where both biases are present, the stratified randomization withstratified analysis performs worse than unstratified analysis scenarios. Themagnitude does not depend on the balancing of sample sizes between centers(20⋮60 vs. 40⋮40; Table1). Using the favored weighted test statistic following a stratifiedanalysis, it appears that BSD and CR perform much better than all other RPs in theboth biased scenarios. However, the effect depends markedly on the type of bias. Inthe case of only time trend in the data, the final balance procedures (EBC(0.67),PBR) perform better than BSD or CR as well as with the unstratified analysisfollowing unstratified randomization. Weightig with wj* performs uniformly better than weighting with w_j=1.

7 Discussion

The approach presented in this paper for multi-center trials follows the ideas of theevaluation of randomization procedures for design optimization (ERDO)⁶ framework. However as outlined, many aspects need to be addressed todemonstrate the contribution of randomization in mitigating bias during the planningphase of a multi-center trial.

Although Kraemer¹⁸ discussed various RPs in clinical trials including stratification, the mostcommon choice of stratified randomization is PBR with common block size.^19–22 We have presented new aspectsto formulate RPs, whether unrestricted or restricted, in order to induce the finalbalance or maximal tolerated imbalance including PBR in a stratified form. We havediscussed the formulation of stratified unrestricted and restricted proceduresforcing balance in probability, forcing balance by maximal tolerable imbalance, andforcing terminal balance as three subclassifications of restricted RPs.

There are several limitations of this study. First, our compound criterion forselection bias and chronological bias imposes similar scaling, but it is difficultor impossible to scale them identically. Second, the weighting of the two criteriais subjective and may be adjusted to account for the different scaling. Although ourstatistical test assumes hom*ogeneous variances across centers, the methodology canbe used with standardized observation in the case of known heterogeneous variancesacross centers.

Our proposed approach is demonstrated in a numerical evaluation study. Here, we usevery specific settings, e.g. common selection bias and time-trend effects acrosscenters, limited sample sizes corresponding to a particular effect size. We areaware that this evaluation study does not mirror all practical situations. However,specific practical situations of the multi-center clinical trial to be planned canbe embedded easily into the evaluation study to demonstrate the correspondingeffects. Moreover, the corresponding results for different evaluation metrics, e.g.mean type I error probability, are supplemented in tables. We used the supplementedR code for all computations.

We have chosen to use a parametric t test as our evaluationstatistic rather than the more natural randomization test.²³ Randomization tests can be computed easily through the Monte Carlore-randomization methodology, although power considerations are computationallyintensive. They tend to preserve type I error rates under time trends and have nodistributional assumptions.² Randomization tests can be formulated easily incorporating stratification,but the theoretical results we have derived herein would be impossible for exactrandomization tests or Monte Carlo re-randomization tests.

Our theoretical derivation could be applied to a general class of weightsw_j including, in particular, the inverse variance approach, although we focusour numerical evaluation study to the weights w_j = 1 or wj=wj*, see Lin.¹⁵ Lin stated that many statisticians as well as the U.S. Food and DrugAdministration recommend the unweighted w_j = 1 analysis.

Sample size considerations are presented by various authors. Whereas Ruvuna²⁴ and Vierron and Giraudeau²⁵ used the normal approximation formula, Lin's¹⁵ approach is based on the t statistic. We presented a generalsample size formula for the weighted t test with Kcenters which generalizes Lin's approach for the two center case and the weighted(wj*) and unweighted w_j = 1 evaluation. Among others, our results can be used to demonstrate theeffect on the power when adding centers during progress of the trial, which seem tobe common practice to increase recruitment. Furthermore, our formulas can be usedfor power considerations, when imbalance in sample sizes between centers isassumed.^24,25 Although it was not discussed in here, the approach can beextended to the case of random center size by using the corrected variance formulasof Ganju and Mehrotra.²⁶

Although some authors mention that randomization is used to avoid bias, bias is quitelikely to occur when the PBR is used, particularly when the block size is small. Wepresent a general formal analytical approach to show how RPs are able to limit theimpact of selection and chronological bias on the test decision.

The idea behind the selection bias used originates from a natural preference for oneof the treatments. Furthermore, it seems to be very common, assuming that theallocation process tends to produce a balanced allocation ratio at least at the end,that investigators would believe that the treatment used most frequently thus far isless likely to appear next. Combining these two arguments, it may be reasonable,that in the situation of knowledge or best guessing what the next allocation wouldprobably be, to choose the next patient according to the expected next treatment.This is also in line with the patient's hope to be assigned to the better treatment.Summarizing, it has to be stated that this process is unconscious or subconscious.The question is not whether selection bias occurs or not, but rather how much impactof bias one is willing to accept. This can be investigated with the proposedsensitivity analysis approach even in the planning phase. With this consideration, aunique approach is presented to link the randomization process of unrestricted orrestricted procedures with the trial outcome.

Of course, other biases for time trend, e.g. log-time trend and step time trend¹⁶ or attrition bias could be easily implemented in the modeling and then usedin a numerical evaluation study. For instance, attrition bias could be modeled by avariable taking 0 or 1 on missingness, which offers opportunities, to studymechanism like missing at random.

Within this paper, we formulate a biasing policy for selection and chronological biasfor a two-arm, parallel group, multi-center trial, according to the weightedstratified t test procedure proposed by Fleiss.¹ We further derive the distribution of the stratified weighted test statisticto calculate the impact on the type I error rate. Finally, theimpact of the combined additive bias in multi-center trials using the unstratifiedt test compared to the weighted stratified ttest is demonstrated in a simulation study.

8 Conclusion

Stratification in the randomization process makes the analysis sensitive to bias,i.e. results in type I error inflation. Procedures forcing terminalbalance are worse in the cases where the study is prone to selection bias,irrespective if time trend is present additionally. Unbalanced sample size betweencenters does not affect the results. This leads to the conclusion thatstratification in the randomization should be considered carefully if bias issupposed to be present. In summary, the presented approach contributes to optimizingthe design of clinical trials stratified by center with respect to improve thederived level of evidence.

Supplemental Material

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to theresearch, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship, and/or publication of this article: This research was supported by theIDeAl project funded from the European Union Seventh Framework Programme (FP72007-2013) under grant agreement No. 602552. RDH received funding from the EuropeanJoint Programme on Rare Diseases within European Union’s Horizon 2020 research andinnovation program under grant agreement No. 825575. Part of the work was done whileRDH joined 2018 workshop on Design of Experiments: New Challenges at CIRM Luminy,France. RDH was granted by RWTH Aachen University under project rwth0334 withcomputing resources for simulations.

Supplemental material

Supplemental material is available for this article online.

References