Bayesian sequential designs in studies with multilevel data.

In many studies in the social and behavioral sciences, the data have a multilevel structure, with subjects nested within clusters. In the design phase of such a study, the number of clusters to achieve a desired power level has to be calculated. This requires a priori estimates of the effect size and intraclass correlation coefficient. If these estimates are incorrect, the study may be under- or overpowered. This may be overcome by using a group-sequential design, where interim tests are done at various points in time of the study. Based on interim test results, a decision is made to either include additional clusters or to reject the null hypothesis and conclude the study. This contribution introduces Bayesian sequential designs as an alternative to group-sequential designs. This approach compares various hypotheses based on the support in the data for each of them. If neither hypothesis receives a sufficient degree of support, additional clusters are included in the study and the Bayes factor is recalculated. This procedure continues until one of the hypotheses receives sufficient support. This paper explains how the Bayes factor is used as a measure of support for a hypothesis and how a Bayesian sequential design is conducted. A simulation study in the setting of a two-group comparison was conducted to study the effects of the minimum and maximum number of clusters per group and the desired degree of support. It is concluded that Bayesian sequential designs are a flexible alternative to the group sequential design.

The effectiveness, efficiency, and acceptability of EMDR vs. EMDR 2.0 vs. the Flash technique in the treatment of patients with PTSD: study protocol for the ENHANCE randomized controlled trial.

Background: Several widely studied therapies have proven to be effective in the treatment of post-traumatic stress disorder (PTSD). However, there is still room for improvement because not all patients benefit from trauma-focused treatments. Improvements in the treatment of PTSD can be achieved by investigating ways to enhance existing therapies, such as eye movement desensitization and reprocessing (EMDR) therapy, as well as exploring novel treatments. The purpose of the current study is to determine the differential effectiveness, efficiency, and acceptability of EMDR therapy, an adaptation of EMDR therapy, referred to as EMDR 2.0, and a novel intervention for PTSD, the so-called Flash technique. The second aim is to identify the moderators of effectiveness for these interventions. This study will be conducted among individuals diagnosed with PTSD using a randomized controlled trial design. Methods: A total of 130 patients diagnosed with (complex) PTSD will be randomly allocated to either six sessions of EMDR therapy, EMDR 2.0, or the Flash technique. The primary outcomes used to determine treatment effectiveness include the presence of a PTSD diagnosis and the severity of PTSD symptoms. The secondary outcomes of effectiveness include symptoms of depression, symptoms of dissociation, general psychiatric symptoms, and experiential avoidance. All patients will be assessed at baseline, at 4-week post-treatment, and at 12-week follow-up. Questionnaires indexing symptoms of PTSD, depression, general psychopathology, and experiential avoidance will also be assessed weekly during treatment and bi-weekly after treatment, until the 12-week follow-up. Efficiency will be assessed by investigating the time it takes both to lose the diagnostic status of PTSD, and to achieve reliable change in PTSD symptoms. Treatment acceptability will be assessed after the first treatment session and after treatment termination. Discussion: This study is the first to investigate EMDR 2.0 therapy and the Flash technique in a sample of participants officially diagnosed with PTSD using a randomized controlled trial design. This study is expected to improve the available treatment options for PTSD and provide therapists with alternative ways to choose a therapy beyond its effectiveness by considering moderators, efficiency, and acceptability. Trial registration: The trial was retrospectively registered in the ISRCTN registry at 10th November 2022 under registration number ISRCTN13100019.

Optimal allocation of clusters in stepped wedge designs with a decaying correlation structure.

The cluster randomized stepped wedge design is a multi-period uni-directional switch design in which all clusters start in the control condition and at the beginning of each new period a random sample of clusters crosses over to the intervention condition. Such designs often use uniform allocation, with an equal number of clusters at each treatment switch. However, the uniform allocation is not necessarily the most efficient. This study derives the optimal allocation of clusters to treatment sequences in the cluster randomized stepped wedge design, for both cohort and cross-sectional designs. The correlation structure is exponential decay, meaning the correlation decreases with the time lag between two measurements. The optimal allocation is shown to depend on the intraclass correlation coefficient, the number of subjects per cluster-period and the cluster and (in the case of a cohort design) individual autocorrelation coefficients. For small to medium values of these autocorrelations those sequences that have their treatment switch earlier or later in the study are allocated a larger proportion of clusters than those clusters that have their treatment switch halfway the study. When the autocorrelation coefficients increase, the clusters become more equally distributed across the treatment sequences. For the cohort design, the optimal allocation is almost equal to the uniform allocation when both autocorrelations approach the value 1. For almost all scenarios that were studied, the efficiency of the uniform allocation is 0.8 or higher. R code to derive the optimal allocation is available online.

The effect of therapist characteristics on the use and outcome of systematic client feedback in outpatient mental healthcare.

OBJECTIVE: Therapist characteristics are known to affect treatment outcome in general and could also influence the use of systematic client feedback (SCF). The current study explores the effect of feedback orientation, regulatory focus, self-efficacy, attitude towards feedback resources and perceived feedback validity on the use and outcome of SCF in outpatient mental healthcare. METHOD: The data of therapists (n = 12) and patients (n = 504) of two outpatient centres offering brief psychological treatment were analysed when SCF, based on the Partners for Change Outcome Management System (PCOMS), was added to treatment as usual. The data of therapists were obtained through a therapist questionnaire composed of relevant characteristics from feedback studies in social and organizational psychology. The effect on the use of SCF was analysed using logistic regression; whereas, the effect on outcome was assessed using a two-level multilevel analysis. Regular use of SCF and the Outcome Questionnaire (OQ-45) were used as outcome variables. DSM-classification, sex and age of each patient were included as covariates. RESULTS: High perceived feedback validity significantly increased the use of SCF. No significant therapist characteristics effects were found on outcome, but high promotion focus was associated with treating more complex patients. CONCLUSIONS: The perceived feedback validity of SCF is likely to have an influence on its use and is probably affected by the changes in the organizational climate.

A comparison of the multilevel MIMIC model to the multilevel regression and mixed ANOVA model for the estimation and testing of a cross-level interaction effect: A simulation study.

When observing data on a patient-reported outcome measure in, for example, clinical trials, the variables observed are often correlated and intended to measure a latent variable. In addition, such data are also often characterized by a hierarchical structure, meaning that the outcome is repeatedly measured within patients. To analyze such data, it is important to use an appropriate statistical model, such as structural equation modeling (SEM). However, researchers may rely on simpler statistical models that are applied to an aggregated data structure. For example, correlated variables are combined into one sum score that approximates a latent variable. This may have implications when, for example, the sum score consists of indicators that relate differently to the latent variable being measured. This study compares three models that can be applied to analyze such data: the multilevel multiple indicators multiple causes (ML-MIMIC) model, a univariate multilevel model, and a mixed analysis of variance (ANOVA) model. The focus is on the estimation of a cross-level interaction effect that presents the difference over time on the patient-reported outcome between two treatment groups. The ML-MIMIC model is an SEM-type model that considers the relationship between the indicators and the latent variable in a multilevel setting, whereas the univariate multilevel and mixed ANOVA model rely on sum scores to approximate the latent variable. In addition, the mixed ANOVA model uses aggregated second-level means as outcome. This study showed that the ML-MIMIC model produced unbiased cross-level interaction effect estimates when the relationships between the indicators and the latent variable being measured varied across indicators. In contrast, under similar conditions, the univariate multilevel and mixed ANOVA model underestimated the cross-level interaction effect.

Optimal placebo-treatment comparisons in trials with an incomplete within-subject design and heterogeneous costs and variances.

The aim of a clinical trial is to compare placebo to one or more treatments. The within-subject design is known to be more efficient than the between-subject design. However, in some trials that implement a within-subject design it is not possible to evaluate the placebo and all treatments within each subject. The design then becomes an incomplete within-subject design. An important question is how many subjects should be allocated to each combination of placebo and treatments. This paper studies optimal allocations of subjects in trials with a placebo and two treatments under heterogenous costs and variances. Two optimality criteria that consider the placebo-treatment contrasts simultaneously are considered, and the design is derived under a budgetary constraint. More subjects are allocated to those combinations with higher variances and lower costs. The optimal allocation is compared to the uniform allocation, which allocates equal number of subjects to each placebo and treatment combination, and to the complete within-subject design, where placebo and all treatments are available in each subject. The methodology is illustrated on the basis of an example on consultation time in primary care. A Shiny app is available to facilitate use of the methodology.

Optimal allocation to treatment sequences in individually randomized stepped-wedge designs with attrition.

BACKGROUND/AIMS: The stepped-wedge design has been extensively studied in the setting of the cluster randomized trial, but less so for the individually randomized trial. This article derives the optimal allocation of individuals to treatment sequences. The focus is on designs where all individuals start in the control condition and at the beginning of each time period some of them cross over to the intervention, so that at the end of the trial all of them receive the intervention. METHODS: The statistical model that takes into account the nesting of repeated measurements within subjects is presented. It is also shown how possible attrition is taken into account. The effect of the intervention is assumed to be sustained so that it does not change after the treatment switch. An exponential decay correlation structure is assumed, implying that the correlation between any two time point decreases with the time lag. Matrix algebra is used to derive the relation between the allocation of units to treatment sequences and the variance of the treatment effect estimator. The optimal allocation is the one that results in smallest variance. RESULTS: Results are presented for three to six treatment sequences. It is shown that the optimal allocation highly depends on the correlation parameter ρ and attrition rate r between any two adjacent time points. The uniform allocation, where each treatment sequence has the same number of individuals, is often not the most efficient. For 0.1≤ρ≤0.9 and r=0,0.05,0.2, its efficiency relative to the optimal allocation is at least 0.8. It is furthermore shown how a constrained optimal allocation can be derived in case the optimal allocation is not feasible from a practical point of view. CONCLUSION: This article provides the methodology for designing individually randomized stepped-wedge designs, taking into account the possibility of attrition. As such it helps researchers to plan their trial in an efficient way. To use the methodology, prior estimates of the degree of attrition and intraclass correlation coefficient are needed. It is advocated that researchers clearly report the estimates of these quantities to help facilitate planning future trials.

Relative Effectiveness of CBT-Components and Sequencing in Indicated Depression Prevention for Adolescents: A Cluster-Randomized Microtrial.

OBJECTIVE: Cognitive Behavioral Therapy (CBT) was dismantled into four modules of three sessions each: cognitive restructuring (Think), behavioral activation (Act), problem solving (Solve) and relaxation (Relax). We investigated the modules' relative effectiveness in indicated depression prevention for adolescents and examined variations in sequencing of these modules. METHOD: We performed a pragmatic cluster-randomized microtrial with four parallel conditions: (1) Think-Act-Relax-Solve (n = 14 clusters, n = 81 participants); (2) Act-Think-Relax-Solve (n = 13, n = 69); (3) Solve-Act-Think-Relax (n = 13, n = 77); and (4) Relax-Solve-Act-Think (n = 12, n = 55). The sample consisted of 282 Dutch adolescents with elevated depressive symptoms (Mage = 13.8; 55.7% girls, 92.9% Dutch). In total 52 treatment groups were randomized as a cluster. Assessments were conducted at baseline, after each module and at 6-month follow-up with depressive symptoms as primary outcome. RESULTS: None of the modules (Think, Act, Solve, Relax) was associated with a significant decrease in depressive symptoms after three sessions and no significant differences in effectiveness were found between the modules. All sequences of modules were associated with a significant decrease in depressive symptoms at post-intervention, except the sequence Relax-Solve-Act-Think. At 6-month follow-up, all sequences showed a significant decrease in depressive symptoms. No significant differences in effectiveness were found between the sequences at post-intervention and 6-month follow-up. CONCLUSIONS: Regardless of the CBT technique provided, one module of three sessions may not be sufficient to reduce depressive symptoms. The sequence in which the CBT components cognitive restructuring, behavioral activation, problem solving and relaxation are offered, does not appear to significantly influence outcomes at post- intervention or 6-month follow-up. ABBREVIATIONS: CDI-2:F: Children's Depression Inventory-2 Full-length version; CDI-2:S: Children's Depression Inventory-2 Short version; STARr: Solve, Think, Act, Relax and repeat.

A review of applications of the Bayes factor in psychological research.

The last 25 years have shown a steady increase in attention for the Bayes factor as a tool for hypothesis evaluation and model selection. The present review highlights the potential of the Bayes factor in psychological research. We discuss six types of applications: Bayesian evaluation of point null, interval, and informative hypotheses, Bayesian evidence synthesis, Bayesian variable selection and model averaging, and Bayesian evaluation of cognitive models. We elaborate what each application entails, give illustrative examples, and provide an overview of key references and software with links to other applications. The article is concluded with a discussion of the opportunities and pitfalls of Bayes factor applications and a sketch of corresponding future research lines. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Power analysis for cluster randomized trials with continuous coprimary endpoints.

Pragmatic trials evaluating health care interventions often adopt cluster randomization due to scientific or logistical considerations. Systematic reviews have shown that coprimary endpoints are not uncommon in pragmatic trials but are seldom recognized in sample size or power calculations. While methods for power analysis based on K ( K ≥ 2 $K\ge 2$ ) binary coprimary endpoints are available for cluster randomized trials (CRTs), to our knowledge, methods for continuous coprimary endpoints are not yet available. Assuming a multivariate linear mixed model (MLMM) that accounts for multiple types of intraclass correlation coefficients among the observations in each cluster, we derive the closed-form joint distribution of K treatment effect estimators to facilitate sample size and power determination with different types of null hypotheses under equal cluster sizes. We characterize the relationship between the power of each test and different types of correlation parameters. We further relax the equal cluster size assumption and approximate the joint distribution of the K treatment effect estimators through the mean and coefficient of variation of cluster sizes. Our simulation studies with a finite number of clusters indicate that the predicted power by our method agrees well with the empirical power, when the parameters in the MLMM are estimated via the expectation-maximization algorithm. An application to a real CRT is presented to illustrate the proposed method.

Sample size determination for Bayesian ANOVAs with informative hypotheses.

Researchers can express their expectations with respect to the group means in an ANOVA model through equality and order constrained hypotheses. This paper introduces the R package SSDbain, which can be used to calculate the sample size required to evaluate (informative) hypotheses using the Approximate Adjusted Fractional Bayes Factor (AAFBF) for one-way ANOVA models as implemented in the R package bain. The sample size is determined such that the probability that the Bayes factor is larger than a threshold value is at least η when either of the hypotheses under consideration is true. The Bayesian ANOVA, Bayesian Welch's ANOVA, and Bayesian robust ANOVA are available. Using the R package SSDbain and/or the tables provided in this paper, researchers in the social and behavioral sciences can easily plan the sample size if they intend to use a Bayesian ANOVA.

Cannabidiol enhancement of exposure therapy in treatment refractory patients with social anxiety disorder and panic disorder with agoraphobia: A randomised controlled trial.

Preclinical research suggests that enhancing CB1 receptor agonism may improve fear extinction. In order to translate this knowledge into a clinical application we examined whether cannabidiol (CBD), a hydrolysis inhibitor of the endogenous CB1 receptor agonist anandamide (AEA), would enhance the effects of exposure therapy in treatment refractory patients with anxiety disorders. Patients with panic disorder with agoraphobia or social anxiety disorder were recruited for a double-blind parallel randomised controlled trial at three mental health care centres in the Netherlands. Eight therapist-assisted exposure in vivo sessions (weekly, outpatient) were augmented with 300 mg oral CBD (n = 39) or placebo (n = 41). The Fear Questionnaire (FQ) was assessed at baseline, mid- and post-treatment, and at 3 and 6 months follow-up. Primary analyses were on an intent-to-treat basis. No differences were found in treatment outcome over time between CBD and placebo on FQ scores, neither across (ß = 0.32, 95% CI [-0.60; 1.25]) nor within diagnosis groups (ß = -0.11, 95% CI [-1.62; 1.40]). In contrast to our hypotheses, CBD augmentation did not enhance early treatment response, within-session fear extinction or extinction learning. Incidence of adverse effects was equal in the CBD (n = 4, 10.3%) and placebo condition (n = 6, 15.4%). In this first clinical trial examining CBD as an adjunctive therapy in anxiety disorders, CBD did not improve treatment outcome. Future clinical trials may investigate different dosage regimens.

Psychopathology and resilience in older adults with posttraumatic stress disorder: a randomized controlled trial comparing narrative exposure therapy and present-centered therapy.

Objective: Using data from a randomized controlled trial on psychotherapy for posttraumatic stress disorder (PTSD) in older adults (aged >55), this study aimed at analysing the efficacy of two psychological interventions in terms of self-reported symptoms, comorbid psychopathology and resilience outcomes. Method: Thirty-three outpatients (age 55-81) with PTSD were randomly assigned to eleven sessions of narrative exposure therapy or present-centered therapy. Self-reported symptom severity of PTSD, depression and general psychopathology, along with measures of resilience (self-efficacy, quality of life and posttraumatic growth cognitions), were target outcomes. Harvard Trauma Questionnaire, Beck Depression Inventory, Brief Symptom Inventory, General Efficacy Scale, World Health Organization Quality of Life Assessment and Meaning of War Scale (personal growth) were assessed pre-treatment, post-treatment and at four months follow-up. Because of variable inter-assessment intervals, a piecewise mixed effects growth model was used to investigate treatment effects. Results: Neither post-treatment, nor at mean follow-up, between-group effects were found. At follow-up, significant medium to large within-group effect sizes were found in the NET-group for psychopathology (self-reported PTSD: Cohen's d = 0.54, p < .01; depression: Cohen's d = 0.51, p = .03; general psychopathology: Cohen's d = 0.74, p = .001), but not so in the PCT-group. Resilience (self-efficacy, quality of life and personal growth cognitions) did not significantly change in either group. Conclusions: In older adults with PTSD, the efficacy of NET extended beyond PTSD, reducing not only self-reported symptoms of PTSD but also comorbid depression and general psychopathology.

Objetivo: Utilizando datos de un ensayo controlado aleatorizado sobre psicoterapia para pacientes con trastorno de estrés postraumático (TEPT) en adultos mayores (> 55 años), este estudio tuvo como objetivo analizar la eficacia de dos intervenciones psicológicas respecto a síntomas autoinformados, psicopatología comorbida, y resultados de resiliencia.Método: Treinta y tres pacientes ambulatorios (de 55 a 81 años) con TEPT fueron asignados al azar a once sesiones de terapia de exposición narrativa (NET en sus siglas en ingles) o terapia centrada en el presente (TCP). Los resultados que se midieron fueron, el autoreporte de la gravedad de síntomas de estrés postraumático, depresión y psicopatología general, junto con medidas de resiliencia (autoeficacia, calidad de vida y cogniciones de crecimiento postraumático). Se evaluaron antes del tratamiento, después del tratamiento y a los cuatro meses de seguimiento con los siguientes cuestionarios: Cuestionario de trauma de Harvard, el Inventario de depresión de Beck, el Inventario breve de síntomas, la Escala de eficacia general, Evaluación de la Calidad de Vida y de Significado de la Guerra de la Organización Mundial de la Salud (crecimiento personal). Debido a los intervalos variables entre evaluaciones, se utilizó un modelo de crecimiento de efectos mixtos por partes para investigar los efectos del tratamiento.Resultados: No se encontraron diferencias entre los grupos ni posteriores al tratamiento ni durante el seguimiento medio. En el seguimiento, se encontraron tamaños de efecto significativos medianos a grandes dentro del grupo NET. para psicopatología (TEPT autoinformado: d de Cohen = 0.54, p < .01; depresión: d de Cohen = 0,51, p = 0,03; psicopatología general: d de Cohen = 0,74, p = 0,001), pero no así en el grupo TCP. La resiliencia (autoeficacia, calidad de vida y cogniciones de crecimiento personal) no tuvieron cambios significativos en ninguno de los grupos.Conclusiones: En adultos mayores con TEPT, la eficacia de la NET se extendió más allá del TEPT, reduciendo no sólo síntomas autoinformados de TEPT, sino también depresión comórbida y psicopatología general.

Power analysis of longitudinal studies with piecewise linear growth and attrition.

In longitudinal research, the development of some outcome variable(s) over time (or age) is studied. Such relations are not necessarily smooth, and piecewise growth models may be used to account for differential growth rates before and after a turning point in time. Such models have been well developed, but the literature on power analysis for these models is scarce. This study investigates the power needed to detect differential growth for linear-linear piecewise growth models in further detail while taking into account the possibility of attrition. Attrition is modeled using the Weibull survival function, which allows for increasing, decreasing or constant attrition across time. Furthermore, this work takes into account the realistic situation where subjects do not necessarily have the same turning point. A multilevel mixed model is used to model the relation between time and outcome, and to derive the relation between sample size and power. The required sample size to achieve a desired power is smallest when the turning points are located halfway through the study and when all subjects have the same turning point. Attrition has a diminishing effect on power, especially when the probability of attrition is largest at the beginning of the study. An example on alcohol use during middle and high school shows how to perform a power analysis. The methodology has been implemented in a Shiny app to facilitate power calculations for future studies.

Enhancing the effect of psychotherapy through systematic client feedback in outpatient mental healthcare: A cluster randomized trial.

Objective: Systematic client feedback (SCF), the regular monitoring and informing of patients' progress during therapy to patient and therapist, has been found to have effects on treatment outcomes varying from very positive to slightly negative. Several prior studies have been biased by researcher allegiance or lack of an independent outcome measure. The current study has taken this into account and aims to clarify the effects of SCF in outpatient psychological treatment.Method: Outpatients (n = 1733) of four centers offering brief psychological treatments were cluster randomized to either treatment as usual (TAU) or TAU with SCF based on the Partners for Change Outcome Management System (PCOMS). Primary outcome measure was the Outcome Questionnaire (OQ-45). Effects of the two treatment conditions on treatment outcome, patient satisfaction, dropout rate, costs, and treatment duration were assessed using a three-level multilevel analysis. DSM-classification, sex, and age of each patient were included as covariates.Results: In both analyses, SCF significantly improved treatment outcome, particularly in the first three months. No significant effects were found on the other outcome variables.Conclusions: Addition of systematic client feedback to treatment as usual, is likely to have a beneficial impact in outpatient psychological treatment. Implementation requires a careful plan of action.Clinical or methodological significance of this article: This study, with large sample size and several independent outcome measures, provides strong evidence that addition of systematic client feedback to outpatient psychological treatment can have a beneficial effect on treatment outcome (symptoms and wellbeing), particularly in the first three months. However, implementation requires a careful plan of action.

Optimal allocation to treatments in a sequential multiple assignment randomized trial.

One of the main questions in the design of a trial is how many subjects should be assigned to each treatment condition. Previous research has shown that equal randomization is not necessarily the best choice. We study the optimal allocation for a novel trial design, the sequential multiple assignment randomized trial, where subjects receive a sequence of treatments across various stages. A subject's randomization probabilities to treatments in the next stage depend on whether he or she responded to treatment in the current stage. We consider a prototypical sequential multiple assignment randomized trial design with two stages. Within such a design, many pairwise comparisons of treatment sequences can be made, and a multiple-objective optimal design strategy is proposed to consider all such comparisons simultaneously. The optimal design is sought under either a fixed total sample size or a fixed budget. A Shiny App is made available to find the optimal allocations and to evaluate the efficiency of competing designs. As the optimal design depends on the response rates to first-stage treatments, maximin optimal design methodology is used to find robust optimal designs. The proposed methodology is illustrated using a sequential multiple assignment randomized trial example on weight loss management.

A multilevel structural equation model for assessing a drug effect on a patient-reported outcome measure in on-demand medication data.

We analyze data from a clinical trial investigating the effect of an on-demand drug for women with low sexual desire. These data consist of a varying number of measurements/events across patients of when the drug was taken, including data on a patient-reported outcome consisting of five items measuring an unobserved construct (latent variable). Traditionally, these data are aggregated prior to analysis by composing one sum score per event and averaging this sum score over all observed events. In this paper, we explain the drawbacks of this aggregating approach. One drawback is that these averages have different standard errors because the variance of the underlying events differs between patients and because the number of events per patient differs. Another drawback is the implicit assumption that all items have equal weight in relation to the latent variable being measured. We propose a multilevel structural equation model, treating the events (level 1) as nested observations within patients (level 2), as alternative analysis method to overcome these drawbacks. The model we apply includes a factor model measuring a latent variable at the level of the event and at the level of the patient. Then, in the same model, the latent variables are regressed on covariates to assess the drug effect. We discuss the inferences obtained about the efficacy of the on-demand drug using our proposed model. We further illustrate how to test for measurement invariance across grouping covariates and levels using the same model.

Bayesian updating: increasing sample size during the course of a study.

BACKGROUND: A priori sample size calculation requires an a priori estimate of the size of the effect. An incorrect estimate may result in a sample size that is too low to detect effects or that is unnecessarily high. An alternative to a priori sample size calculation is Bayesian updating, a procedure that allows increasing sample size during the course of a study until sufficient support for a hypothesis is achieved. This procedure does not require and a priori estimate of the effect size. This paper introduces Bayesian updating to researchers in the biomedical field and presents a simulation study that gives insight in sample sizes that may be expected for two-group comparisons. METHODS: Bayesian updating uses the Bayes factor, which quantifies the degree of support for a hypothesis versus another one given the data. It can be re-calculated each time new subjects are added, without the need to correct for multiple interim analyses. A simulation study was conducted to study what sample size may be expected and how large the error rate is, that is, how often the Bayes factor shows most support for the hypothesis that was not used to generate the data. RESULTS: The results of the simulation study are presented in a Shiny app and summarized in this paper. Lower sample size is expected when the effect size is larger and the required degree of support is lower. However, larger error rates may be observed when a low degree of support is required and/or when the sample size at the start of the study is small. Furthermore, it may occur sufficient support for neither hypothesis is achieved when the sample size is bounded by a maximum. CONCLUSIONS: Bayesian updating is a useful alternative to a priori sample size calculation, especially so in studies where additional subjects can be recruited easily and data become available in a limited amount of time. The results of the simulation study show how large a sample size can be expected and how large the error rate is.

Contamination: How much can an individually randomized trial tolerate?

Cluster randomization results in an increase in sample size compared to individual randomization, referred to as an efficiency loss. This efficiency loss is typically presented under an assumption of no contamination in the individually randomized trial. An alternative comparator is the sample size needed under individual randomization to detect the attenuated treatment effect due to contamination. A general framework is provided for determining the extent of contamination that can be tolerated in an individually randomized trial before a cluster randomized design yields a larger sample size. Results are presented for a variety of cluster trial designs including parallel arm, stepped-wedge and cluster crossover trials. Results reinforce what is expected: individually randomized trials can tolerate a surprisingly large amount of contamination before they become less efficient than cluster designs. We determine the point at which the contamination means an individual randomized design to detect an attenuated effect requires a larger sample size than cluster randomization under a nonattenuated effect. This critical rate is a simple function of the design effect for clustering and the design effect for multiple periods as well as design effects for stratification or repeated measures under individual randomization. These findings are important for pragmatic comparisons between a novel treatment and usual care as any bias due to contamination will only attenuate the true treatment effect. This is a bias that operates in a predictable direction. Yet, cluster randomized designs with post-randomization recruitment without blinding, are at high risk of bias due to the differential recruitment across treatment arms. This sort of bias operates in an unpredictable direction. Thus, with knowledge that cluster randomized trials are generally at a greater risk of biases that can operate in a nonpredictable direction, results presented here suggest that even in situations where there is a risk of contamination, individual randomization might still be the design of choice.

Optimal allocations for two treatment comparisons within the proportional odds cumulative logits model.

This paper studies optimal treatment allocations for two treatment comparisons when the outcome is ordinal and analyzed by a proportional odds cumulative logits model. The variance of the treatment effect estimator is used as optimality criterion. The optimal design is sought so that this variance is minimal for a given total sample size or a given budget, meaning that the power for the test on treatment effect is maximal, or it is sought so that a required power level is achieved at a minimal total sample size or budget. Results are presented for three, five and seven ordered response categories, three treatment effect sizes and a skewed, bell-shaped or polarized distribution of the response probabilities. The optimal proportion subjects in the intervention condition decreases with the number of response categories and the costs for the intervention relative to those for the control. The relation between the optimal proportion and effect size depends on the distribution of the response probabilities. The widely used balanced design is not always the most efficient; its efficiency as compared to the optimal design decreases with increasing cost ratio. The optimal design is highly robust to misspecification of the response probabilities and treatment effect size. The optimal design methodology is illustrated using two pharmaceutical examples. A Shiny app is available to find the optimal treatment allocation, to evaluate the efficiency of the balanced design and to study the relation between budget or sample size and power.