Limitation of permutation-based differential correlation analysis.

The comparison of biological systems, through the analysis of molecular changes under different conditions, has played a crucial role in the progress of modern biological science. Specifically, differential correlation analysis (DCA) has been employed to determine whether relationships between genomic features differ across conditions or outcomes. Because ascertaining the null distribution of test statistics to capture variations in correlation is challenging, several DCA methods utilize permutation which can loosen parametric (e.g., normality) assumptions. However, permutation is often problematic for DCA due to violating the assumption that samples are exchangeable under the null. Here, we examine the limitations of permutation-based DCA and investigate instances where the permutation-based DCA exhibits poor performance. Experimental results show that the permutation-based DCA often fails to control the type I error under the null hypothesis of equal correlation structures.

On multi-type Cannings models and multi-type exchangeable coalescents.

A multi-type neutral Cannings population model with migration and fixed subpopulation sizes is analyzed. Under appropriate conditions, as all subpopulation sizes tend to infinity, the ancestral process, properly time-scaled, converges to a multi-type coalescent sharing the exchangeability and consistency property. The proof gains from coalescent theory for single-type Cannings models and from decompositions of transition probabilities into parts concerning reproduction and migration respectively. The following section deals with a different but closely related multi-type Cannings model with mutation and fixed total population size but stochastically varying subpopulation sizes. The latter model is analyzed forward and backward in time with an emphasis on its behavior as the total population size tends to infinity. Forward in time, multi-type limiting branching processes arise for large population size. Its backward structure and related open problems are briefly discussed.

Discussion on "LEAP: the latent exchangeability prior for borrowing information from historical data" by Ethan M. Alt, Xiuya Chang, Xun Jiang, Qing Liu, May Mo, H. Amy Xia, and Joseph G. Ibrahim.

In the following discussion, we describe the various assumptions of exchangeability that have been made in the context of Bayesian borrowing and related models. In this context, we are able to highlight the difficulty of dynamic Bayesian borrowing under the assumption of individuals in the historical data being exchangeable with the current data and thus the strengths and novel features of the latent exchangeability prior. As borrowing methods are popular within clinical trials to augment the control arm, some potential challenges are identified with the application of the approach in this setting.

Rejoinder to the discussion on "LEAP: the latent exchangeability prior for borrowing information from historical data".

The discussions of our paper provide insights into the practical considerations of the latent exchangeability prior while also highlighting further extensions. In this rejoinder, we briefly summarize the discussions and provide comments.

Discussion on "LEAP: the latent exchangeability prior for borrowing information from historical data" by Ethan M. Alt, Xiuya Chang, Xun Jiang, Qing Liu, May Mo, H. Amy Xia, and Joseph G. Ibrahim.

This discussion provides commentary on the paper by Ethan M. Alt, Xiuya Chang, Xun Jiang, Qing Liu, May Mo, H. Amy Xia, and Joseph G. Ibrahim entitled "LEAP: the latent exchangeability prior for borrowing information from historical data". The authors propose a novel method to bridge the incorporation of supplemental information into a study while also identifying potentially exchangeable subgroups to better facilitate information sharing. In this discussion, we highlight the potential relationship with other Bayesian model averaging approaches, such as multisource exchangeability modeling, and provide a brief numeric case study to illustrate how the concepts behind latent exchangeability prior may also improve the performance of existing methods. The results provided by Alt et al. are exciting, and we believe that the method represents a meaningful approach to more efficient information sharing.

Learning about treatment effects in a new target population under transportability assumptions for relative effect measures.

Investigators often believe that relative effect measures conditional on covariates, such as risk ratios and mean ratios, are "transportable" across populations. Here, we examine the identification of causal effects in a target population using an assumption that conditional relative effect measures are transportable from a trial to the target population. We show that transportability for relative effect measures is largely incompatible with transportability for difference effect measures, unless the treatment has no effect on average or one is willing to make even stronger transportability assumptions that imply the transportability of both relative and difference effect measures. We then describe how marginal (population-averaged) causal estimands in a target population can be identified under the assumption of transportability of relative effect measures, when we are interested in the effectiveness of a new experimental treatment in a target population where the only treatment in use is the control treatment evaluated in the trial. We extend these results to consider cases where the control treatment evaluated in the trial is only one of the treatments in use in the target population, under an additional partial exchangeability assumption in the target population (i.e., an assumption of no unmeasured confounding in the target population with respect to potential outcomes under the control treatment in the trial). We also develop identification results that allow for the covariates needed for transportability of relative effect measures to be only a small subset of the covariates needed to control confounding in the target population. Last, we propose estimators that can be easily implemented in standard statistical software and illustrate their use using data from a comprehensive cohort study of stable ischemic heart disease.

Generalized cumulative shrinkage process priors with applications to sparse Bayesian factor analysis.

The paper discusses shrinkage priors which impose increasing shrinkage in a sequence of parameters. We review the cumulative shrinkage process (CUSP) prior of Legramanti et al. (Legramanti et al. 2020 Biometrika 107, 745-752. (doi:10.1093/biomet/asaa008)), which is a spike-and-slab shrinkage prior where the spike probability is stochastically increasing and constructed from the stick-breaking representation of a Dirichlet process prior. As a first contribution, this CUSP prior is extended by involving arbitrary stick-breaking representations arising from beta distributions. As a second contribution, we prove that exchangeable spike-and-slab priors, which are popular and widely used in sparse Bayesian factor analysis, can be represented as a finite generalized CUSP prior, which is easily obtained from the decreasing order statistics of the slab probabilities. Hence, exchangeable spike-and-slab shrinkage priors imply increasing shrinkage as the column index in the loading matrix increases, without imposing explicit order constraints on the slab probabilities. An application to sparse Bayesian factor analysis illustrates the usefulness of the findings of this paper. A new exchangeable spike-and-slab shrinkage prior based on the triple gamma prior of Cadonna et al. (Cadonna et al. 2020 Econometrics 8, 20. (doi:10.3390/econometrics8020020)) is introduced and shown to be helpful for estimating the unknown number of factors in a simulation study. This article is part of the theme issue 'Bayesian inference: challenges, perspectives, and prospects'.

Exchangeability of Measures of Association Before and After Exposure Status Is Flipped: Its Relationship With Confounding in the Counterfactual Model.

BACKGROUND: The counterfactual definition of confounding is often explained in the context of exchangeability between the exposed and unexposed groups. One recent approach is to examine whether the measures of association (eg, associational risk difference) are exchangeable when exposure status is flipped in the population of interest. We discuss the meaning and utility of this approach, showing their relationships with the concept of confounding in the counterfactual framework. METHODS: Three hypothetical cohort studies are used, in which the target population is the total population. After providing an overview of the notions of confounding in distribution and in measure, we discuss the approach from the perspective of exchangeability of measures of association (eg, factual associational risk difference vs counterfactual associational risk difference). RESULTS: In general, if the measures of association are non-exchangeable when exposure status is flipped, confounding in distribution is always present, although confounding in measure may or may not be present. Even if the measures of association are exchangeable when exposure status is flipped, there could be confounding both in distribution and in measure. When we use risk difference or risk ratio as a measure of interest and the exposure prevalence in the population is 0.5, testing the exchangeability of measures of association is equivalent to testing the absence of confounding in the corresponding measures. CONCLUSION: The approach based on exchangeability of measures of association essentially does not provide a definition of confounding in the counterfactual framework. Subtly differing notions of confounding should be distinguished carefully.

On latent idealized models in symbolic datasets: unveiling signals in noisy sequencing data.

Data taking values on discrete sample spaces are the embodiment of modern biological research. "Omics" experiments based on high-throughput sequencing produce millions of symbolic outcomes in the form of reads (i.e., DNA sequences of a few dozens to a few hundred nucleotides). Unfortunately, these intrinsically non-numerical datasets often deviate dramatically from natural assumptions a practitioner might make, and the possible sources of this deviation are usually poorly characterized. This contrasts with numerical datasets where Gaussian-type errors are often well-justified. To overcome this hurdle, we introduce the notion of latent weight, which measures the largest expected fraction of samples from a probabilistic source that conform to a model in a class of idealized models. We examine various properties of latent weights, which we specialize to the class of exchangeable probability distributions. As proof of concept, we analyze DNA methylation data from the 22 human autosome pairs. Contrary to what is usually assumed in the literature, we provide strong evidence that highly specific methylation patterns are overrepresented at some genomic locations when latent weights are taken into account.

Composite Likelihoods with Bounded Weights in Extrapolation of Data.

Among many efforts to facilitate timely access to safe and effective medicines to children, increased attention has been given to extrapolation. Loosely, it is the leveraging of conclusions or available data from adults or older age groups to draw conclusions for the target pediatric population when it can be assumed that the course of the disease and the expected response to a medicinal product would be sufficiently similar in the pediatric and the reference population. Extrapolation then can be characterized as a statistical mapping of information from the reference (adults or older age groups) to the target pediatric population. The translation, or loosely mapping of information, can be through a composite likelihood approach where the likelihood of the reference population is weighted by exponentiation and that this exponent is related to the value of the mapped information in the target population. The weight is bounded above and below recognizing the fact that similarity (of the disease and the expected response) is still valid despite variability of response between the cohorts. Maximum likelihood approaches are then used for estimation of parameters, and asymptotic theory is used to derive distributions of estimates for use in inference. Hence, the estimation of effects in the target population borrows information from the reference population. In addition, this manuscript also talks about how this method is related to the Bayesian statistical paradigm.

Are Nonsynonymous Transversions Generally More Deleterious than Nonsynonymous Transitions?

It has been suggested that, due to the structure of the genetic code, nonsynonymous transitions are less likely than transversions to cause radical changes in amino acid physicochemical properties so are on average less deleterious. This view was supported by some but not all mutagenesis experiments. Because laboratory measures of fitness effects have limited sensitivities and relative frequencies of different mutations in mutagenesis studies may not match those in nature, we here revisit this issue using comparative genomics. We extend the standard codon model of sequence evolution by adding the parameter η that quantifies the ratio of the fixation probability of transitional nonsynonymous mutations to that of transversional nonsynonymous mutations. We then estimate η from the concatenated alignment of all protein-coding DNA sequences of two closely related genomes. Surprisingly, η ranges from 0.13 to 2.0 across 90 species pairs sampled from the tree of life, with 51 incidences of η < 1 and 30 incidences of η >1 that are statistically significant. Hence, whether nonsynonymous transversions are overall more deleterious than nonsynonymous transitions is species-dependent. Because the corresponding groups of amino acid replacements differ between nonsynonymous transitions and transversions, η is influenced by the relative exchangeabilities of amino acid pairs. Indeed, an extensive search reveals that the large variation in η is primarily explainable by the recently reported among-species disparity in amino acid exchangeabilities. These findings demonstrate that genome-wide nucleotide substitution patterns in coding sequences have species-specific features and are more variable among evolutionary lineages than are currently thought.

Dynamic borrowing in the presence of treatment effect heterogeneity.

A number of statistical approaches have been proposed for incorporating supplemental information in randomized clinical trials. Existing methods often compare the marginal treatment effects to evaluate the degree of consistency between sources. Dissimilar marginal treatment effects would either lead to increased bias or down-weighting of the supplemental data. This represents a limitation in the presence of treatment effect heterogeneity, in which case the marginal treatment effect may differ between the sources solely due to differences between the study populations. We introduce the concept of covariate-adjusted exchangeability, in which differences in the marginal treatment effect can be explained by differences in the distributions of the effect modifiers. The potential outcomes framework is used to conceptualize covariate-adjusted and marginal exchangeability. We utilize a linear model and the existing multisource exchangeability models framework to facilitate borrowing when marginal treatment effects are dissimilar but covariate-adjusted exchangeability holds. We investigate the operating characteristics of our method using simulations. We also illustrate our method using data from two clinical trials of very low nicotine content cigarettes. Our method has the ability to incorporate supplemental information in a wider variety of situations than when only marginal exchangeability is considered.

A group-sequential randomized trial design utilizing supplemental trial data.

Definitive clinical trials are resource intensive, often requiring a large number of participants over several years. One approach to improve the efficiency of clinical trials is to incorporate historical information into the primary trial analysis. This approach has tremendous potential in the areas of pediatric or rare disease trials, where achieving reasonable power is difficult. In this article, we introduce a novel Bayesian group-sequential trial design based on Multisource Exchangeability Models, which allows for dynamic borrowing of historical information at the interim analyses. Our approach achieves synergy between group sequential and adaptive borrowing methodology to attain improved power and reduced sample size. We explore the frequentist operating characteristics of our design through simulation and compare our method to a traditional group-sequential design. Our method achieves earlier stopping of the primary study while increasing power under the alternative hypothesis but has a potential for type I error inflation under some null scenarios. We discuss the issues of decision boundary determination, power and sample size calculations, and the issue of information accrual. We present our method for a continuous and binary outcome, as well as in a linear regression setting.

Ensuring exchangeability in data-based priors for a Bayesian analysis of clinical trials.

In many orphan diseases and pediatric indications, the randomized controlled trials may be infeasible because of their size, duration, and cost. Leveraging information on the control through a prior can potentially reduce sample size. However, unless an objective prior is used to impose complete ignorance for the parameter being estimated, it results in biased estimates and inflated type-I error. Hence, it is essential to assess both the confirmatory and supplementary knowledge available during the construction of the prior to avoid "cherry-picking" advantageous information. For this purpose, propensity score methods are employed to minimize selection bias by weighting supplemental control subjects according to their similarity in terms of pretreatment characteristics to the subjects in the current trial. The latter can be operationalized through a proposed measure of overlap in propensity-score distributions. In this paper, we consider single experimental arm in the current trial and the control arm is completely borrowed from the supplemental data. The simulation experiments show that the proposed method reduces prior and data conflict and improves the precision of the of the average treatment effect.

Hypothesis testing in multivariate normal models with block circular covariance structures.

In this article, we address the problem of simultaneous testing hypothesis about mean and covariance matrix for repeated measures data when both the mean vector and covariance matrix are patterned. In particular, tests about the mean vector under block circular and doubly exchangeable covariance structures have been considered. The null distributions are established for the corresponding likelihood ratio test statistics, and expressions for the exact or near-exact probability density and cumulative distribution functions are obtained. The application of the results is illustrated by both a simulation study and a real-life data example.

Borrowing from supplemental sources to estimate causal effects from a primary data source.

The increasing multiplicity of data sources offers exciting possibilities in estimating the effects of a treatment, intervention, or exposure, particularly if observational and experimental sources could be used simultaneously. Borrowing between sources can potentially result in more efficient estimators, but it must be done in a principled manner to mitigate increased bias and Type I error. Furthermore, when the effect of treatment is confounded, as in observational sources or in clinical trials with noncompliance, causal effect estimators are needed to simultaneously adjust for confounding and to estimate effects across data sources. We consider the problem of estimating causal effects from a primary source and borrowing from any number of supplemental sources. We propose using regression-based estimators that borrow based on assuming exchangeability of the regression coefficients and parameters between data sources. Borrowing is accomplished with multisource exchangeability models and Bayesian model averaging. We show via simulation that a Bayesian linear model and Bayesian additive regression trees both have desirable properties and borrow under appropriate circumstances. We apply the estimators to recently completed trials of very low nicotine content cigarettes investigating their impact on smoking behavior.

Calibrated dynamic borrowing using capping priors.

Multisource exchangeability models (MEMs), a BayeTsian approach for dynamically integrating information from multiple clinical trials, are a promising approach for gaining efficiency in randomized controlled trials. When the supplementary trials are considerably larger than the primary trial, care must be taken when integrating supplementary data to avoid overwhelming the primary trial. In this paper, we propose "capping priors," which controls the extent of dynamic borrowing by placing an a priori cap on the effective supplemental sample size. We demonstrate the behavior of this technique via simulation, and apply our method to four randomized trials of very low nicotine content cigarettes.

Exchangeable Markov multi-state survival processes.

We consider exchangeable Markov multi-state survival processes, which are temporal processes taking values over a state-space S , with at least one absorbing failure state b ∈ S that satisfy the natural invariance properties of exchangeability and consistency under subsampling. The set of processes contains many well-known examples from health and epidemiology including survival, illness-death, competing risk, and comorbidity processes. Here, an extension leads to recurrent event processes. We characterize exchangeable Markov multi-state survival processes in both discrete and continuous time. Statistical considerations impose natural constraints on the space of models appropriate for applied work. In particular, we describe constraints arising from the notion of composable systems. We end with an application to irregularly sampled and potentially censored multi-state survival data, developing a Markov chain Monte Carlo algorithm for inference.

Bayesian hierarchical meta-analytic methods for modeling surrogate relationships that vary across treatment classes using aggregate data.

Surrogate endpoints play an important role in drug development when they can be used to measure treatment effect early compared to the final clinical outcome and to predict clinical benefit or harm. Such endpoints are assessed for their predictive value of clinical benefit by investigating the surrogate relationship between treatment effects on the surrogate and final outcomes using meta-analytic methods. When surrogate relationships vary across treatment classes, such validation may fail due to limited data within each treatment class. In this paper, two alternative Bayesian meta-analytic methods are introduced which allow for borrowing of information from other treatment classes when exploring the surrogacy in a particular class. The first approach extends a standard model for the evaluation of surrogate endpoints to a hierarchical meta-analysis model assuming full exchangeability of surrogate relationships across all the treatment classes, thus facilitating borrowing of information across the classes. The second method is able to relax this assumption by allowing for partial exchangeability of surrogate relationships across treatment classes to avoid excessive borrowing of information from distinctly different classes. We carried out a simulation study to assess the proposed methods in nine data scenarios and compared them with subgroup analysis using the standard model within each treatment class. We also applied the methods to an illustrative example in colorectal cancer which led to obtaining the parameters describing the surrogate relationships with higher precision.

You Can't Drive a Car With Only Three Wheels.

Authors aiming to estimate causal effects from observational data frequently discuss 3 fundamental identifiability assumptions for causal inference: exchangeability, consistency, and positivity. However, too often, studies fail to acknowledge the importance of measurement bias in causal inference. In the presence of measurement bias, the aforementioned identifiability conditions are not sufficient to estimate a causal effect. The most fundamental requirement for estimating a causal effect is knowing who is truly exposed and unexposed. In this issue of the Journal, Caniglia et al. (Am J Epidemiol. 2019;000(00):000-000) present a thorough discussion of methodological challenges when estimating causal effects in the context of research on distance to obstetrical care. Their article highlights empirical strategies for examining nonexchangeability due to unmeasured confounding and selection bias and potential violations of the consistency assumption. In addition to the important considerations outlined by Caniglia et al., authors interested in estimating causal effects from observational data should also consider implementing quantitative strategies to examine the impact of misclassification. The objective of this commentary is to emphasize that you can't drive a car with only three wheels, and you also cannot estimate a causal effect in the presence of exposure misclassification bias.