Correction to: Sample size estimation for randomised controlled trials with repeated assessment of patient-reported outcomes: what correlation between baseline and follow-up outcomes should we assume?

Following publication of the original article [1], we have been notified that one of an error in the Conclusions section of the Abstract.

Sample size estimation for randomised controlled trials with repeated assessment of patient-reported outcomes: what correlation between baseline and follow-up outcomes should we assume?

BACKGROUND: Patient-reported outcome measures (PROMs) are now frequently used in randomised controlled trials (RCTs) as primary endpoints. RCTs are longitudinal, and many have a baseline (PRE) assessment of the outcome and one or more post-randomisation assessments of outcome (POST). With such pre-test post-test RCT designs there are several ways of estimating the sample size and analysing the outcome data: analysis of post-randomisation treatment means (POST); analysis of mean changes from pre- to post-randomisation (CHANGE); analysis of covariance (ANCOVA). Sample size estimation using the CHANGE and ANCOVA methods requires specification of the correlation between the baseline and follow-up measurements. Other parameters in the sample size estimation method being unchanged, an assumed correlation of 0.70 (between baseline and follow-up outcomes) means that we can halve the required sample size at the study design stage if we used an ANCOVA method compared to a comparison of POST treatment means method. So what correlation (between baseline and follow-up outcomes) should be assumed and used in the sample size calculation? The aim of this paper is to estimate the correlations between baseline and follow-up PROMs in RCTs. METHODS: The Pearson correlation coefficients between the baseline and repeated PROM assessments from 20 RCTs (with 7173 participants at baseline) were calculated and summarised. RESULTS: The 20 reviewed RCTs had sample sizes, at baseline, ranging from 49 to 2659 participants. The time points for the post-randomisation follow-up assessments ranged from 7 days to 24 months; 464 correlations, between baseline and follow-up, were estimated; the mean correlation was 0.50 (median 0.51; standard deviation 0.15; range - 0.13 to 0.91). CONCLUSIONS: There is a general consistency in the correlations between the repeated PROMs, with the majority being in the range of 0.4 to -0.6. The implications are that we can reduce the sample size in an RCT by 25% if we use an ANCOVA model, with a correlation of 0.50, for the design and analysis. There is a decline in correlation amongst more distant pairs of time points.

Statistical design and analysis in trials of proportionate interventions: a systematic review.

BACKGROUND: In proportionate or adaptive interventions, the dose or intensity can be adjusted based on individual need at predefined decision stages during the delivery of the intervention. The development of such interventions may require an evaluation of the effectiveness of the individual stages in addition to the whole intervention. However, evaluating individual stages of an intervention has various challenges, particularly the statistical design and analysis. This review aimed to identify the use of trials of proportionate interventions and how they are being designed and analysed in current practice. METHODS: We searched MEDLINE, Web of Science and PsycINFO for articles published between 2010 and 2015 inclusive. We considered trials of proportionate interventions in all fields of research. For each trial, its aims, design and analysis were extracted. The data synthesis was conducted using summary statistics and a narrative format. RESULTS: Our review identified 44 proportionate intervention trials, comprising 28 trial results, 13 protocols and three secondary analyses. These were mostly described as stepped care (n=37) and mainly focussed on mental health research (n=30). The other studies were aimed at finding an optimal adaptive treatment strategy (n=7) in a variety of therapeutic areas. Further terminology used included adaptive intervention, staged intervention, sequentially multiple assignment trial or a two-phase design. The median number of decision stages in the interventions was two and only one study explicitly evaluated the effect of the individual stages. CONCLUSIONS: Trials of proportionate staged interventions are being used predominantly within the mental health field. However, few studies consider the different stages of the interventions, either at the design or the analysis phase, and how they may interact with one another. There is a need for further guidance on the design, analyses and reporting across trials of proportionate interventions. TRIAL REGISTRATION: Prospero, CRD42016033781. Registered on 2 February 2016.

Appropriate statistical methods for analysing partially nested randomised controlled trials with continuous outcomes: a simulation study.

BACKGROUND: In individually randomised trials we might expect interventions delivered in groups or by care providers to result in clustering of outcomes for participants treated in the same group or by the same care provider. In partially nested randomised controlled trials (pnRCTs) this clustering only occurs in one trial arm, commonly the intervention arm. It is important to measure and account for between-cluster variability in trial design and analysis. We compare analysis approaches for pnRCTs with continuous outcomes, investigating the impact on statistical inference of cluster sizes, coding of the non-clustered arm, intracluster correlation coefficient (ICCs), and differential variance between intervention and control arm, and provide recommendations for analysis. METHODS: We performed a simulation study assessing the performance of six analysis approaches for a two-arm pnRCT with a continuous outcome. These include: linear regression model; fully clustered mixed-effects model with singleton clusters in control arm; fully clustered mixed-effects model with one large cluster in control arm; fully clustered mixed-effects model with pseudo clusters in control arm; partially nested homoscedastic mixed effects model, and partially nested heteroscedastic mixed effects model. We varied the cluster size, number of clusters, ICC, and individual variance between the two trial arms. RESULTS: All models provided unbiased intervention effect estimates. In the partially nested mixed-effects models, methods for classifying the non-clustered control arm had negligible impact. Failure to account for even small ICCs resulted in inflated Type I error rates and over-coverage of confidence intervals. Fully clustered mixed effects models provided poor control of the Type I error rates and biased ICC estimates. The heteroscedastic partially nested mixed-effects model maintained relatively good control of Type I error rates, unbiased ICC estimation, and did not noticeably reduce power even with homoscedastic individual variances across arms. CONCLUSIONS: In general, we recommend the use of a heteroscedastic partially nested mixed-effects model, which models the clustering in only one arm, for continuous outcomes similar to those generated under the scenarios of our simulations study. However, with few clusters (3-6), small cluster sizes (5-10), and small ICC (≤0.05) this model underestimates Type I error rates and there is no optimal model.

What is an ROC curve?

The paper by Body et al is concerned with the evaluation of decision aids, which can be used to identify potential acute coronary syndromes (ACS) in the ED. The authors previously developed the Manchester Acute Coronary Syndromes model (MACS) decision aid, which uses several clinical variables and two biomarkers to 'rule in' and 'rule out' ACS. However, one of the two biomarkers (heart-type fatty acid bindingprotein, H-FABP) is not widely used so a revised decision aid has been developed (Troponin-only Manchester Acute Coronary Syndromes, T-MACS), which include a single biomarker hs-cTnT. In this issue, the authors show how they derive a revised decision aid and describe its performance in a number of independent diagnostic cohort studies. Decision aids (as well as other types of 'diagnostic tests') are often evaluated in terms of diagnostic testing parameters such as the area under the receiver operating characteristic (ROC) curve, sensitivity and specificity. In this article, we explain how the ROC analysis is conducted and why it is an essential step towards developing a test with the desirable levels of sensitivity and specificity.

Evaluation of biases present in the cohort multiple randomised controlled trial design: a simulation study.

BACKGROUND: The cohort multiple randomised controlled trial (cmRCT) design provides an opportunity to incorporate the benefits of randomisation within clinical practice; thus reducing costs, integrating electronic healthcare records, and improving external validity. This study aims to address a key concern of the cmRCT design: refusal to treatment is only present in the intervention arm, and this may lead to bias and reduce statistical power. METHODS: We used simulation studies to assess the effect of this refusal, both random and related to event risk, on bias of the effect estimator and statistical power. A series of simulations were undertaken that represent a cmRCT trial with time-to-event endpoint. Intention-to-treat (ITT), per protocol (PP), and instrumental variable (IV) analysis methods, two stage predictor substitution and two stage residual inclusion, were compared for various refusal scenarios. RESULTS: We found the IV methods provide a less biased estimator for the causal effect when refusal is present in the intervention arm, with the two stage residual inclusion method performing best with regards to minimum bias and sufficient power. We demonstrate that sample sizes should be adapted based on expected and actual refusal rates in order to be sufficiently powered for IV analysis. CONCLUSION: We recommend running both an IV and ITT analyses in an individually randomised cmRCT as it is expected that the effect size of interest, or the effect we would observe in clinical practice, would lie somewhere between that estimated with ITT and IV analyses. The optimum (in terms of bias and power) instrumental variable method was the two stage residual inclusion method. We recommend using adaptive power calculations, updating them as refusal rates are collected in the trial recruitment phase in order to be sufficiently powered for IV analysis.

Cohort Multiple Randomised Controlled Trials (cmRCT) design: efficient but biased? A simulation study to evaluate the feasibility of the Cluster cmRCT design.

BACKGROUND: The Cohort Multiple Randomised Controlled Trial (cmRCT) is a newly proposed pragmatic trial design; recently several cmRCT have been initiated. This study tests the unresolved question of whether differential refusal in the intervention arm leads to bias or loss of statistical power and how to deal with this. METHODS: We conduct simulations evaluating a hypothetical cluster cmRCT in patients at risk of cardiovascular disease (CVD). To deal with refusal, we compare the analysis methods intention to treat (ITT), per protocol (PP) and two instrumental variable (IV) methods: two stage predictor substitution (2SPS) and two stage residual inclusion (2SRI) with respect to their bias and power. We vary the correlation between treatment refusal probability and the probability of experiencing the outcome to create different scenarios. RESULTS: We found ITT to be biased in all scenarios, PP the most biased when correlation is strong and 2SRI the least biased on average. Trials suffer a drop in power unless the refusal rate is factored into the power calculation. CONCLUSIONS: The ITT effect in routine practice is likely to lie somewhere between the ITT and IV estimates from the trial which differ significantly depending on refusal rates. More research is needed on how refusal rates of experimental interventions correlate with refusal rates in routine practice to help answer the question of which analysis more relevant. We also recommend updating the required sample size during the trial as more information about the refusal rate is gained.

Collider Bias Is Only a Partial Explanation for the Obesity Paradox.

BACKGROUND: "Obesity paradox" refers to an association between obesity and reduced mortality (contrary to an expected increased mortality). A common explanation is collider stratification bias: unmeasured confounding induced by selection bias. Here, we test this supposition through a realistic generative model. METHODS: We quantify the collider stratification bias in a selected population using counterfactual causal analysis. We illustrate the bias for a range of scenarios, describing associations between exposure (obesity), outcome (mortality), mediator (in this example, diabetes) and an unmeasured confounder. RESULTS: Collider stratification leads to biased estimation of the causal effect of exposure on outcome. However, the bias is small relative to the causal relationships between the variables. CONCLUSIONS: Collider bias can be a partial explanation of the obesity paradox, but unlikely to be the main explanation for a reverse direction of an association to a true causal relationship. Alternative explanations of the obesity paradox should be explored. See Video Abstract at http://links.lww.com/EDE/B51.