Permuting incomplete paired data: a novel exact and asymptotic correct randomization test

Various statistical tests have been developed for testing the equality of means in matched pairs with missing values. However, most existing methods are commonly based on certain distributional assumptions such as normality, 0-symmetry or homoscedasticity of the data. The aim of this paper is to develop a statistical test that is robust against deviations from such assumptions and also leads to valid inference in case of heteroscedasticity or skewed distributions. This is achieved by applying a clever randomization approach to handle missing data. The resulting test procedure is not only shown to be asymptotically correct but is also finitely exact if the distribution of the data is invariant with respect to the considered randomization group. Its small sample performance is further studied in an extensive simulation study and compared to existing methods. Finally, an illustrative data example is analysed.     

Model‐free causal inference of binary experimental data

For binary experimental data, we discuss randomization‐based inferential procedures that do not need to invoke any modeling assumptions. In addition to the classical method of moments, we also introduce model‐free likelihood and Bayesian methods based solely on the physical randomization without any hypothetical super population assumptions about the potential outcomes. These estimators have some properties superior to moment‐based ones such as only giving estimates in regions of feasible support. Due to the lack of identification of the causal model, we also propose a sensitivity analysis approach that allows for the characterization of the impact of the association between the potential outcomes on statistical inference.     

Randomization-Based Inference within Principal Strata

In randomized studies, treatment comparisons conditional on intermediate post-randomization outcomes using standard analytic methods do not have a causal interpretation. An alternate approach entails treatment comparisons within principal strata defined by the potential outcomes for the intermediate outcome that would be observed under each treatment assignment. In this paper, we develop methods for randomization-based inference within principal strata. The proposed methods are compared with existing large-sample methods as well as traditional intent-to-treat approaches. This research is motivated by HIV prevention studies where few infections are expected and inference is desired within the always-infected principal stratum, i.e., all individuals who would become infected regardless of randomization assignment.     

Nonparametric predictive inference for stock returns

In finance, inferences about future asset returns are typically quantified with the use of parametric distributions and single-valued probabilities. It is attractive to use less restrictive inferential methods, including nonparametric methods which do not require distributional assumptions about variables, and imprecise probability methods which generalize the classical concept of probability to set-valued quantities. Main attractions include the flexibility of the inferences to adapt to the available data and that the level of imprecision in inferences can reflect the amount of data on which these are based. This paper introduces nonparametric predictive inference (NPI) for stock returns. NPI is a statistical approach based on few assumptions, with inferences strongly based on data and with uncertainty quantified via lower and upper probabilities. NPI is presented for inference about future stock returns, as a measure for risk and uncertainty, and for pairwise comparison of two stocks based on their future aggregate returns. The proposed NPI methods are illustrated using historical stock market data.     

A Semiparametric Estimation Approach for Linear Mixed Models

Maximum likelihood approach is the most frequently employed approach for the inference of linear mixed models. However, it relies on the normal distributional assumption of the random effects and the within-subject errors, and it is lack of robustness against outliers. This article proposes a semiparametric estimation approach for linear mixed models. This approach is based on the first two marginal moments of the response variable, and does not require any parametric distributional assumptions of random effects or error terms. The consistency and asymptotically normality of the estimator are derived under fairly general conditions. In addition, we show that the proposed estimator has a bounded influence function and a redescending property so it is robust to outliers. The methodology is illustrated through an application to the famed Framingham cholesterol data. The finite sample behavior and the robustness properties of the proposed estimator are evaluated through extensive simulation studies.     

Robust-efficient fitting of mixed linear models: Methodology and theory

In many areas of application mixed linear models serve as a popular tool for analyzing highly complex data sets. For inference about fixed effects and variance components, likelihood-based methods such as (restricted) maximum likelihood estimators, (RE)ML, are commonly pursued. However, it is well-known that these fully efficient estimators are extremely sensitive to small deviations from hypothesized normality of random components as well as to other violations of distributional assumptions. In this article, we propose a new class of robust-efficient estimators for inference in mixed linear models. The new three-step estimation procedure provides truncated generalized least squares and variance components' estimators with hard-rejection weights adaptively computed from the data. More specifically, our data re-weighting mechanism first detects and removes within-subject outliers, then identifies and discards between-subject outliers, and finally it employs maximum likelihood procedures on the “clean” data. Theoretical efficiency and robustness properties of this approach are established.     

On the use of quasi-likelihood for estimation in hidden-Markov random fields

This paper illustrates the use of quasi-likelihood methods of inference for hidden Markov random fields. These are simple to use and can be employed under circumstances where only the model form and its covariance structure are specified. In particular they can be used to derive the same estimating equations as the E-M algorithm or change of measure methods, which make full distributional assumptions.     

A note on fourth moment-based direction dependence measures when regression errors are non normal

Measures of direction dependence enable researchers to determine the directionality of linear effects in bivariate data. Existing fourth moment-based approaches assume that regression errors are at least mesokurtic. Direction dependence measures based on the co-kurtosis of variables are proposed that relax this assumption. Simulations suggest that co-kurtosis-based measures perform equally well as existing kurtosis-based methods when distributional assumptions of the latter are fulfilled. However, kurtosis-based approaches are sensitive to platy- or leptokurtic errors, while co-kurtosis-based measures protect Type I error and power rates. Data requirements necessary for causal inference and recommendations for selecting proper direction dependence measures are discussed.     

Moment-based estimation of nonlinear regression models with boundary outcomes and endogeneity,with applications to nonnegative and fractional responses

In this article, we suggest simple moment-based estimators to deal with unobserved heterogeneity in a special class of nonlinear regression models that includes as main particular cases exponential models for nonnegative responses and logit and complementary loglog models for fractional responses. The proposed estimators: (i) treat observed and omitted covariates in a similar manner; (ii) can deal with boundary outcomes; (iii) accommodate endogenous explanatory variables without requiring knowledge on the reduced form model, although such information may be easily incorporated in the estimation process; (iv) do not require distributional assumptions on the unobservables, a conditional mean assumption being enough for consistent estimation of the structural parameters; and (v) under the additional assumption that the dependence between observables and unobservables is restricted to the conditional mean, produce consistent estimators of partial effects conditional only on observables.     

A hierarchical modelling approach to analysing longitudinal data with drop-out and non-compliance, with application to an equivalence trial in paediatric acquired immune deficiency syndrome

Longitudinal clinical trials with long follow-up periods almost invariably suffer from a loss to follow-up and non-compliance with the assigned therapy. An example is protocol 128 of the AIDS Clinical Trials Group, a 5-year equivalency trial comparing reduced dose zidovudine with the standard dose for treatment of paediatric acquired immune deficiency syndrome patients. This study compared responses to treatment by using both clinical and cognitive outcomes. The cognitive outcomes are of particular interest because the effects of human immunodeficiency virus infection of the central nervous system can be more acute in children than in adults. We formulate and apply a Bayesian hierarchical model to estimate both the intent-to-treat effect and the average causal effect of reducing the prescribed dose of zidovudine by 50%. The intent-to-treat effect quantifies the causal effect of assigning the lower dose, whereas the average causal effect represents the causal effect of actually taking the lower dose. We adopt a potential outcomes framework where, for each individual, we assume the existence of a different potential outcomes process at each level of time spent on treatment. The joint distribution of the potential outcomes and the time spent on assigned treatment is formulated using a hierarchical model: the potential outcomes distribution is given at the first level, and dependence between the outcomes and time on treatment is specified at the second level by linking the time on treatment to subject-specific effects that characterize the potential outcomes processes. Several distributional and structural assumptions are used to identify the model from observed data, and these are described in detail. A detailed analysis of AIDS Clinical Trials Group protocol 128 is given; inference about both the intent-to-treat effect and average causal effect indicate a high probability of dose equivalence with respect to cognitive functioning.     

Bias in Small-Sample Inference With Count-Data Models

Abstract

Both Poisson and negative binomial regression can provide quasi-likelihood estimates for coefficients in exponential-mean models that are consistent in the presence of distributional misspecification. It has generally been recommended, however, that inference be carried out using asymptotically robust estimators for the parameter covariance matrix. As with linear models, such robust inference tends to lead to over-rejection of null hypotheses in small samples. Alternative methods for estimating coefficient estimator variances are considered. No one approach seems to remove all test bias, but the results do suggest that the use of the jackknife with Poisson regression tends to be least biased for inference.     

Estimating efficacy in a proposed randomized trial with initial and later non-compliance

Summary.  A controversial topic in obstetrics is the effect of walking on the probability of Caesarean section among women in labour. A major reason for the controversy is the presence of non-compliance that complicates the estimation of efficacy, the effect of treatment received on outcome. The intent-to-treat method does not estimate efficacy, and estimates of efficacy that are based directly on treatment received may be biased because they are not protected by randomization. However, when non-compliance occurs immediately after randomization, the use of a potential outcomes model with reasonable assumptions has made it possible to estimate efficacy and still to retain the benefits of randomization to avoid selection bias. In this obstetrics application, non-compliance occurs initially and later in one arm. Consequently some parameters cannot be uniquely estimated without making strong assumptions. This difficulty is circumvented by a new study design involving an additional randomization group and a novel potential outcomes model (principal stratification).     

Estimating standard error of intra-class correlation coefficients up to three level unbalanced nested clinical trials

ABSTRACT

Very often researchers plan a balanced design for cluster randomization clinical trials in conducting medical research, but unavoidable circumstances lead to unbalanced data. By adopting three or more levels of nested designs, they usually ignore the higher level of nesting and consider only two levels, this situation leads to underestimation of variance at higher levels. While calculating the sample size for three-level nested designs, in order to achieve desired power, intra-class correlation coefficients (ICCs) at individual level as well as higher levels need to be considered and must be provided along with respective standard errors. In the present paper, the standard errors of analysis of variance (ANOVA) estimates of ICCs for three-level unbalanced nested design are derived. To conquer the strong appeal of distributional assumptions, balanced design, equality of variances between clusters and large sample, general expressions for standard errors of ICCs which can be deployed in unbalanced cluster randomization trials are postulated. The expressions are evaluated on real data as well as highly unbalanced simulated data.     

An alternative parameterization of Bayesian logistic hierarchical models for mixed treatment comparisons

Mixed treatment comparison (MTC) models rely on estimates of relative effectiveness from randomized clinical trials so as to respect randomization across treatment arms. This approach could potentially be simplified by an alternative parameterization of the way effectiveness is modeled. We introduce a treatment‐based parameterization of the MTC model that estimates outcomes on both the study and treatment levels. We compare the proposed model to the commonly used MTC models using a simulation study as well as three randomized clinical trial datasets from published systematic reviews comparing (i) treatments on bleeding after cirrhosis, (ii) the impact of antihypertensive drugs in diabetes mellitus, and (iii) smoking cessation strategies. The simulation results suggest similar or sometimes better performance of the treatment‐based MTC model. Moreover, from the real data analyses, little differences were observed on the inference extracted from both models. Overall, our proposed MTC approach performed as good, or better, than the commonly applied indirect and MTC models and is simpler, fast, and easier to implement in standard statistical software. Copyright © 2015 John Wiley & Sons, Ltd.     

A Bayesian growth and yield model for slash pine plantations

We formulate a traditional growth and yield model as a Bayes model. We attempt to introduce as few new assumptions as possible. Zellner's Bayesian method of moments procedure is used, because the published model did not include any distributional assumptions. We generate predictive posterior samples for a number of stand variables using the Gibbs sampler. The means of the samples compare favorably with the predictions from the published model. In addition, our model delivers distributions of outcomes, from which it is easy to establish measures of uncertainty, such as highest posterior density regions.     

Randomization,balance, and the validity and efficiency of design-adaptive allocation methods

Few topics have stirred as much discussion and controversy as randomization. A reading of the literature suggests that clinical trialists generally feel randomization is necessary for valid inference, while biostatisticians using model-based inference often appear to prefer nearly optimal designs irrespective of any induced randomness. Dissection of the methods of treatment assignment shows that there are five basic approaches; pure randomizers, true randomizers, quasi-randomizers, permutation testers, and conventional modelers. Four of these have coherent design and analysis strategies, even though they are not mutually consistent, but the fifth and most prevalent approach (quasi-randomization) has little to recommend it. Design-adaptive allocation is defined, it is shown to provide valid inference, and a simulation indicates its efficiency advantage. In small studies, or large studies with many important prognostic covariates or analytic subgroups, design-adaptive allocation is an extremely attractive method of treatment assignment.     

A Permutation Approach to Split-Plot Experiments

Split-plot design may be refer to a common experimental setting where a particular type of restricted randomization has occurred during a planned experiment. The aim of this article is to suggest a new method to perform inference on split-plot experiments by combination-based permutation tests. This novel nonparametric approach has been studied and validated using a Monte Carlo simulation study where we compared it with the parametric and nonparametric procedures proposed in the literature. Results suggest that in each experimental situation where normality is hard to justify and especially when errors have heavy-tailed distribution, the proposed nonparametric procedure can be considered as a valid solution.     

A Bayesian growth and yield model for slash pine plantations

We formulate a traditional growth and yield model as a Bayes model. We attempt to introduce as few new assumptions as possible. Zellner's Bayesian method of moments procedure is used, because the published model did not include any distributional assumptions. We generate predictive posterior samples for a number of stand variables using the Gibbs sampler. The means of the samples compare favorably with the predictions from the published model. In addition, our model delivers distributions of outcomes, from which it is easy to establish measures of uncertainty, such as highest posterior density regions.     

Correlated Errors in the Parameters Estimation of the ARFIMA Model: A Simulated Study

Kurtosis, usually as measured by the standardised fourth central moment, has been examined on a number of occasions by observing the effect of contaminating the distribution, that is, mixing in another distribution. However, superficial treatment can lead, and indeed has led, to misunderstandings. This paper considers, firstly for a symmetric distribution contaminated at two points symmetrically placed around its centre and then for a mixture of two continuous symmetric distributions, the behaviour of three measures of kurtosis. This is done in general and not just as the mixing proportion tends to zero as in the influence function approach. It is seen that when both scale and kurtosis change, the latter is not necessarily intuitive. It is also illustrated that parameter interpretation in terms of distributional properties such as shape can be misleading without the use of the appropriate distributional partial ordering     

Randomization as a basis for inference in noninferiority trials

Noninferiority testing in clinical trials is commonly understood in a Neyman-Pearson framework, and has been discussed in a Bayesian framework as well. In this paper, we discuss noninferiority testing in a Fisherian framework, in which the only assumption necessary for inference is the assumption of randomization of treatments to study subjects. Randomization plays an important role in not only the design but also the analysis of clinical trials, no matter the underlying inferential field. The ability to utilize permutation tests depends on assumptions around exchangeability, and we discuss the possible uses of permutation tests in active control noninferiority analyses. The other practical implications of this paper are admittedly minor but lead to better understanding of the historical and philosophical development of active control noninferiority testing. The conclusion may also frame discussion of other complicated issues in noninferiority testing, such as the role of an intention to treat analysis.