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The STEP 1 randomized trial evaluated the effect of taking semaglutide vs placebo on body weight over a 68 week duration. As with any study evaluating an intervention delivered over a sustained period, non-adherence was observed. This was addressed in the original trial analysis within the Estimand Framework by viewing non-adherence as an intercurrent event. The primary analysis applied a treatment policy strategy which viewed it as an aspect of the treatment regimen, and thus made no adjustment for its presence. A supplementary analysis used a hypothetical strategy, targeting an estimand that would have been realised had all participants adhered, under the assumption that no post-baseline variables confounded adherence and change in body weight. In this paper we propose an alternative Instrumental Variable method to adjust for non-adherence which does not rely on the same `unconfoundedness' assumption and is less vulnerable to positivity violations (e.g., it can give valid results even under conditions where non-adherence is guaranteed). Unlike many previous Instrumental Variable approaches, it makes full use of the repeatedly measured outcome data, and allows for a time-varying effect of treatment adherence on a participant's weight. We show that it provides a natural vehicle for defining two distinct hypothetical estimands: the treatment effect if all participants would have adhered to semaglutide, and the treatment effect if all participants would have adhered to both semaglutide and placebo. When applied to the STEP 1 study, they both suggest a sustained, slowly decaying weight loss effect of semaglutide treatment.

Instrumental Variables provide a way of addressing bias due to unmeasured confounding when estimating treatment effects using observational data. As instrument prescription preference of individual healthcare providers has been proposed. Because prescription preference is hard to measure and often unobserved, a surrogate measure constructed from available data is often required for the analysis. Different construction methods for this surrogate measure are possible, such as simple rule-based methods which make use of the observed treatment patterns, or more complex model-based methods that employ formal statistical models to explain the treatment behaviour whilst considering measured confounders. The choice of construction method relies on aspects like data availability within provider, missing data in measured confounders, and possible changes in prescription preference over time. In this paper we conduct a comprehensive simulation study to evaluate different construction methods for surrogates of prescription preference under different data conditions, including: different provider sizes, missing covariate data, and change in preference. We also propose a novel model-based construction method to address between provider differences and change in prescription preference. All presented construction methods are exemplified in a case study of the relative glucose lowering effect of two type 2 diabetes treatments in observational data. Our study shows that preference-based Instrumental Variable methods can be a useful tool for causal inference from observational health data. The choice of construction method should be driven by the data condition at hand. Our proposed method is capable of estimating the causal treatment effect without bias in case of sufficient prescription data per provider, changing prescription preference over time and non-ignorable missingness in measured confounders., Comment: 44 pages, 11 figures

Observational studies can play a useful role in assessing the comparative effectiveness of competing treatments. In a clinical trial the randomization of participants to treatment and control groups generally results in well-balanced groups with respect to possible confounders, which makes the analysis straightforward. However, when analysing observational data, the potential for unmeasured confounding makes comparing treatment effects much more challenging. Causal inference methods such as Instrumental Variable and Prior Even Rate Ratio approaches make it possible to circumvent the need to adjust for confounding factors that have not been measured in the data or measured with error. Direct confounder adjustment via multivariable regression and Propensity score matching also have considerable utility. Each method relies on a different set of assumptions and leverages different aspects of the data. In this paper, we describe the assumptions of each method and assess the impact of violating these assumptions in a simulation study. We propose the prior outcome augmented Instrumental Variable method that leverages data from before and after treatment initiation, and is robust to the violation of key assumptions. Finally, we propose the use of a heterogeneity statistic to decide if two or more estimates are statistically similar, taking into account their correlation. We illustrate our causal framework to assess the risk of genital infection in patients prescribed Sodium-glucose co-transporter-2 inhibitors versus Dipeptidyl peptidase-4 inhibitors as second-line treatment for Type 2 Diabets using observational data from the Clinical Practice Research Datalink., Comment: 35 pages, 10 figures

Causal inference methods are gaining increasing prominence in pharmaceutical drug development in light of the recently published addendum on estimands and sensitivity analysis in clinical trials to the E9 guideline of the International Council for Harmonisation. The E9 addendum emphasises the need to account for post-randomization or `intercurrent' events that can potentially influence the interpretation of a treatment effect estimate at a trial's conclusion. Instrumental Variables (IV) methods have been used extensively in economics, epidemiology and academic clinical studies for `causal inference', but less so in the pharmaceutical industry setting until now. In this tutorial paper we review the basic tools for causal inference, including graphical diagrams and potential outcomes, as well as several conceptual frameworks that an IV analysis can sit within. We discuss in detail how to map these approaches to the Treatment Policy, Principal Stratum and Hypothetical `estimand strategies' introduced in the E9 addendum, and provide details of their implementation using standard regression models. Specific attention is given to discussing the assumptions each estimation strategy relies on in order to be consistent, the extent to which they can be empirically tested and sensitivity analyses in which specific assumptions can be relaxed. We finish by applying the methods described to simulated data closely matching two recent pharmaceutical trials to further motivate and clarify the ideas, Comment: 29 pages, 9 Figures, 1 Table

Many partial identification problems can be characterized by the optimal value of a function over a set where both the function and set need to be estimated by empirical data. Despite some progress for convex problems, statistical inference in this general setting remains to be developed. To address this, we derive an asymptotically valid confidence interval for the optimal value through an appropriate relaxation of the estimated set. We then apply this general result to the problem of selection bias in population-based cohort studies. We show that existing sensitivity analyses, which are often conservative and difficult to implement, can be formulated in our framework and made significantly more informative via auxiliary information on the population. We conduct a simulation study to evaluate the finite sample performance of our inference procedure and conclude with a substantive motivating example on the causal effect of education on income in the highly-selected UK Biobank cohort. We demonstrate that our method can produce informative bounds using plausible population-level auxiliary constraints. We implement this method in the R package selectioninterval., Comment: 23 pages, 6 figures

Mendelian randomization (MR) is a method of exploiting genetic variation to unbiasedly estimate a causal effect in presence of unmeasured confounding. MR is being widely used in epidemiology and other related areas of population science. In this paper, we study statistical inference in the increasingly popular two-sample summary-data MR design. We show a linear model for the observed associations approximately holds in a wide variety of settings when all the genetic variants satisfy the exclusion restriction assumption, or in genetic terms, when there is no pleiotropy. In this scenario, we derive a maximum profile likelihood estimator with provable consistency and asymptotic normality. However, through analyzing real datasets, we find strong evidence of both systematic and idiosyncratic pleiotropy in MR, echoing the omnigenic model of complex traits that is recently proposed in genetics. We model the systematic pleiotropy by a random effects model, where no genetic variant satisfies the exclusion restriction condition exactly. In this case we propose a consistent and asymptotically normal estimator by adjusting the profile score. We then tackle the idiosyncratic pleiotropy by robustifying the adjusted profile score. We demonstrate the robustness and efficiency of the proposed methods using several simulated and real datasets., Comment: 59 pages, 5 figures, 6 tables

Instrumental variable analysis is a widely used method to estimate causal effects in the presence of unmeasured confounding. When the instruments, exposure and outcome are not measured in the same sample, Angrist and Krueger (1992) suggested to use two-sample instrumental variable (TSIV) estimators that use sample moments from an instrument-exposure sample and an instrument-outcome sample. However, this method is biased if the two samples are from heterogeneous populations so that the distributions of the instruments are different. In linear structural equation models, we derive a new class of TSIV estimators that are robust to heterogeneous samples under the key assumption that the structural relations in the two samples are the same. The widely used two-sample two-stage least squares estimator belongs to this class. It is generally not asymptotically efficient, although we find that it performs similarly to the optimal TSIV estimator in most practical situations. We then attempt to relax the linearity assumption. We find that, unlike one-sample analyses, the TSIV estimator is not robust to misspecified exposure model. Additionally, to nonparametrically identify the magnitude of the causal effect, the noise in the exposure must have the same distributions in the two samples. However, this assumption is in general untestable because the exposure is not observed in one sample. Nonetheless, we may still identify the sign of the causal effect in the absence of homogeneity of the noise., Comment: 21 pages, 1 figure, 6 tables

Response-adaptive randomisation (RAR) can considerably improve the chances of a successful treatment outcome for patients in a clinical trial by skewing the allocation probability towards better performing treatments as data accumulates. There is considerable interest in using RAR designs in drug development for rare diseases, where traditional designs are not feasible or ethically objectionable. In this paper we discuss and address a major criticism of RAR: the undesirable type I error inflation due to unknown time trends in the trial. Time trends can appear because of changes in the characteristics of recruited patients - so-called "patient drift". Patient drift is a realistic concern for clinical trials in rare diseases because these typically recruit patients over a very long period of time. We compute by simulations how large the type I error inflation is as a function of the time trend magnitude in order to determine in which contexts a potentially costly correction is actually necessary. We then assess the ability of different correction methods to preserve type I error in this context and their performance in terms of other operating characteristics, including patient benefit and power. We make recommendations of which correction methods are most suitable in the rare disease context for several RAR rules, differentiating between the two-armed and the multi-armed case. We further propose a RAR design for multi-armed clinical trials, which is computationally cheap and robust to several time trends considered.

Mendelian randomization is the use of genetic variants to make causal inferences from observational data. The field is currently undergoing a revolution fuelled by increasing numbers of genetic variants demonstrated to be associated with exposures in genome-wide association studies, and the public availability of summarized data on genetic associations with exposures and outcomes from large consortia. A Mendelian randomization analysis with many genetic variants can be performed relatively simply using summarized data. However, a causal interpretation is only assured if each genetic variant satisfies the assumptions of an instrumental variable. To provide some protection against failure of these assumptions, robust methods for instrumental variable analysis have been proposed. Here, we develop three extensions to instrumental variable methods using: i) robust regression, ii) the penalization of weights from candidate instruments with heterogeneous causal estimates, and iii) L1 penalization. Results from a wide variety of robust methods, including the recently-proposed MR-Egger and median-based methods, are compared in an extensive simulation study. We demonstrate that two methods, robust regression in an inverse-variance weighted method and a simple median of the causal estimates from the individual variants, have considerably improved Type 1 error rates compared with conventional methods in a wide variety of scenarios when up to 30% of the genetic variants are invalid instruments. While the MR-Egger method gives unbiased estimates when its assumptions are satisfied, these estimates are less efficient than those from other methods and are highly sensitive to violations of the assumptions. Methods that make different assumptions should be used routinely to assess the robustness of findings from applied Mendelian randomization investigations with multiple genetic variants.

Mendelian randomization is the use of genetic variants as instrumental variables to assess whether a risk factor is a cause of a disease outcome. Increasingly, Mendelian randomization investigations are conducted on the basis of summarized data, rather than individual-level data. These summarized data comprise the coefficients and standard errors from univariate regression models of the risk factor on each genetic variant, and of the outcome on each genetic variant. A causal estimate can be derived from these associations for each individual genetic variant, and a combined estimate can be obtained by inverse-variance weighted meta-analysis of these causal estimates. Various proposals have been made for how to calculate this inverse-variance weighted estimate. In this paper, we show that the inverse-variance weighted method as originally proposed (equivalent to a two-stage least squares or allele score analysis using individual-level data) can lead to over-rejection of the null, particularly when there is heterogeneity between the causal estimates from different genetic variants. Random-effects models should be routinely employed to allow for this possible heterogeneity. Additionally, over-rejection of the null is observed when associations with the risk factor and the outcome are obtained in overlapping participants. The use of weights including second-order terms from the delta method is recommended in this case.

The funnel plot is a graphical visualisation of summary data estimates from a meta-analysis, and is a useful tool for detecting departures from the standard modelling assumptions. Although perhaps not widely appreciated, a simple extension of the funnel plot can help to facilitate an intuitive interpretation of the mathematics underlying a meta-analysis at a more fundamental level, by equating it to determining the centre of mass of a physical system. We used this analogy, with some success, to explain the concepts of weighing evidence and of biased evidence to a young audience at the Cambridge Science Festival, without recourse to precise definitions or statistical formulae. In this paper we aim to formalise this analogy at a more technical level using the estimating equation framework: firstly, to help elucidate some of the basic statistical models employed in a meta-analysis and secondly, to forge new connections between bias adjustment in the evidence synthesis and causal inference literatures.

Multi-armed bandit problems (MABPs) are a special type of optimal control problem well suited to model resource allocation under uncertainty in a wide variety of contexts. Since the first publication of the optimal solution of the classic MABP by a dynamic index rule, the bandit literature quickly diversified and emerged as an active research topic. Across this literature, the use of bandit models to optimally design clinical trials became a typical motivating application, yet little of the resulting theory has ever been used in the actual design and analysis of clinical trials. To this end, we review two MABP decision-theoretic approaches to the optimal allocation of treatments in a clinical trial: the infinite-horizon Bayesian Bernoulli MABP and the finite-horizon variant. These models possess distinct theoretical properties and lead to separate allocation rules in a clinical trial design context. We evaluate their performance compared to other allocation rules, including fixed randomization. Our results indicate that bandit approaches offer significant advantages, in terms of assigning more patients to better treatments, and severe limitations, in terms of their resulting statistical power. We propose a novel bandit-based patient allocation rule that overcomes the issue of low power, thus removing a potential barrier for their use in practice., Comment: Published at http://dx.doi.org/10.1214/14-STS504 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Inference for causal effects can benefit from the availability of an instrumental variable (IV) which, by definition, is associated with the given exposure, but not with the outcome of interest other than through a causal exposure effect. Estimation methods for instrumental variables are now well established for continuous outcomes, but much less so for dichotomous outcomes. In this article we review IV estimation of so-called conditional causal odds ratios which express the effect of an arbitrary exposure on a dichotomous outcome conditional on the exposure level, instrumental variable and measured covariates. In addition, we propose IV estimators of so-called marginal causal odds ratios which express the effect of an arbitrary exposure on a dichotomous outcome at the population level, and are therefore of greater public health relevance. We explore interconnections between the different estimators and support the results with extensive simulation studies and three applications., Comment: Published in at http://dx.doi.org/10.1214/11-STS360 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)