This paper reviews some programming techniques in R that have proved useful, particularly for substantial projects. These include several versions of object-oriented programming, used in a large number of R packages. The review tries to clarify the origins and ideas behind the various versions, each of which is valuable in the appropriate context.

R has also been strongly influenced by the ideas of functional programming and, in particular, by the desire to combine functional with object oriented programming.

To clarify how this particular mix of ideas has turned out in the current R language and supporting software, the paper will first review the basic ideas behind object-oriented and functional programming, and then examine the evolution of R with these ideas providing context.

Functional programming supports well-defined, defensible software giving reproducible results. Object-oriented programming is the mechanism par excellence for managing complexity while keeping things simple for the user. The two paradigms have been valuable in supporting major software for fitting models to data and numerous other statistical applications.

The paradigms have been adopted, and adapted, distinctively in R. Functional programming motivates much of R but R does not enforce the paradigm. Object-oriented programming from a functional perspective differs from that used in non-functional languages, a distinction that needs to be emphasized to avoid confusion.

R initially replicated the S language from Bell Labs, which in turn was strongly influenced by earlier program libraries. At each stage, new ideas have been added, but the previous software continues to show its influence in the design as well. Outlining the evolution will further clarify why we currently have this somewhat unusual combination of ideas.

I describe an approach to compiling common idioms in R code directly to native machine code and illustrate it with several examples. Not only can this yield significant performance gains, but it allows us to use new approaches to computing in R. Importantly, the compilation requires no changes to R itself, but is done entirely via R packages. This allows others to experiment with different compilation strategies and even to define new domain-specific languages within R. We use the Low-Level Virtual Machine (LLVM) compiler toolkit to create the native code and perform sophisticated optimizations on the code. By adopting this widely used software within R, we leverage its ability to generate code for different platforms such as CPUs and GPUs, and will continue to benefit from its ongoing development. This approach potentially allows us to develop high-level R code that is also fast, that can be compiled to work with different data representations and sources, and that could even be run outside of R. The approach aims to both provide a compiler for a limited subset of the R language and also to enable R programmers to write other compilers. This is another approach to help us write high-level descriptions of what we want to compute, not how.

One of the big challenges of developing interactive statistical applications is the management of the data pipeline, which controls transformations from data to plot. The user’s interactions needs to be propagated through these modules and reflected in the output representation at a fast pace. Each individual module may be easy to develop and manage, but the dependency structure can be quite challenging. The MVC (Model/View/Controller) pattern is an attempt to solve the problem by separating the user’s interaction from the representation of the data. In this paper we discuss the paradigm of reactive programming in the framework of the MVC architecture and show its applicability to interactive graphics. Under this paradigm, developers benefit from the separation of user interaction from the graphical representation, which makes it easier for users and developers to extend interactive applications. We show the central role of reactive data objects in an interactive graphics system, implemented as the R package cranvas, which is freely available on GitHub and the main developers include the authors of this paper.

This paper reviews strategies for solving problems encountered when analyzing large genomic data sets and describes the implementation of those strategies in R by packages from the Bioconductor project. We treat the scalable processing, summarization and visualization of big genomic data. The general ideas are well established and include restrictive queries, compression, iteration and parallel computing. We demonstrate the strategies by applying Bioconductor packages to the detection and analysis of genetic variants from a whole genome sequencing experiment.

An essential component of inference based on familiar frequentist notions, such as $p$-values, significance and confidence levels, is the relevant sampling distribution. This feature results in violations of a principle known as the strong likelihood principle (SLP), the focus of this paper. In particular, if outcomes $\mathbf{x}^{\ast }$ and $\mathbf{y}^{\ast }$ from experiments $E_{1}$ and $E_{2}$ (both with unknown parameter $\theta $) have different probability models $f_{1}(\cdot)$, $f_{2}(\cdot)$, then even though $f_{1}(\mathbf{x}^{\ast };\theta )=cf_{2}(\mathbf{y}^{\ast };\theta )$ for all $\theta $, outcomes $\mathbf{x}^{\ast }$ and $\mathbf{y}^{\ast }$ may have different implications for an inference about $\theta $. Although such violations stem from considering outcomes other than the one observed, we argue this does not require us to consider experiments other than the one performed to produce the data. David Cox [Ann. Math. Statist. 29 (1958) 357–372] proposes the Weak Conditionality Principle (WCP) to justify restricting the space of relevant repetitions. The WCP says that once it is known which $E_{i}$ produced the measurement, the assessment should be in terms of the properties of $E_{i}$. The surprising upshot of Allan Birnbaum’s [J. Amer. Statist. Assoc. 57 (1962) 269–306] argument is that the SLP appears to follow from applying the WCP in the case of mixtures, and so uncontroversial a principle as sufficiency (SP). But this would preclude the use of sampling distributions. The goal of this article is to provide a new clarification and critique of Birnbaum’s argument. Although his argument purports that [(WCP and SP) entails SLP], we show how data may violate the SLP while holding both the WCP and SP. Such cases also refute [WCP entails SLP].

Deborah Mayo claims to have refuted Birnbaum’s argument that the Likelihood Principle is a logical consequence of the Sufficiency and Conditionality Principles. However, this claim fails because her interpretation of the Conditionality Principle is different from Birnbaum’s. Birnbaum’s proof cannot be so readily dismissed.

We discuss Birnbaum’s result, its relevance to statistical reasoning, Mayo’s objections and the result in [Electron. J. Statist. 7 (2013) 2645–2655] that the proof of this result doesn’t establish what is commonly believed.

This is an invited contribution to the discussion on Professor Deborah Mayo’s paper, “On the Birnbaum argument for the strong likelihood principle,” to appear in Statistical Science. Mayo clearly demonstrates that statistical methods violating the likelihood principle need not violate either the sufficiency or conditionality principle, thus refuting Birnbaum’s claim. With the constraints of Birnbaum’s theorem lifted, we revisit the foundations of statistical inference, focusing on some new foundational principles, the inferential model framework, and connections with sufficiency and conditioning.

In this discussion we demonstrate that fiducial distributions provide a natural example of an inference paradigm that does not obey Strong Likelihood Principle while still satisfying the Weak Conditionality Principle.

The paper by Mayo claims to provide a new clarification and critique of Birnbaum’s argument for showing that sufficiency and conditionality principles imply the likelihood principle. However, much of the arguments go back to arguments made thirty to forty years ago. Also, the main contention in the paper, that Birnbaum’s arguments are not valid, seems to rest on a misunderstanding.

The Neyman–Fisher controversy considered here originated with the 1935 presentation of Jerzy Neyman’s Statistical Problems in Agricultural Experimentation to the Royal Statistical Society. Neyman asserted that the standard ANOVA F-test for randomized complete block designs is valid, whereas the analogous test for Latin squares is invalid in the sense of detecting differentiation among the treatments, when none existed on average, more often than desired (i.e., having a higher Type I error than advertised). However, Neyman’s expressions for the expected mean residual sum of squares, for both designs, are generally incorrect. Furthermore, Neyman’s belief that the Type I error (when testing the null hypothesis of zero average treatment effects) is higher than desired, whenever the expected mean treatment sum of squares is greater than the expected mean residual sum of squares, is generally incorrect. Simple examples show that, without further assumptions on the potential outcomes, one cannot determine the Type I error of the F-test from expected sums of squares. Ultimately, we believe that the Neyman–Fisher controversy had a deleterious impact on the development of statistics, with a major consequence being that potential outcomes were ignored in favor of linear models and classical statistical procedures that are imprecise without applied contexts.

Empirical Bayes methods use the data from parallel experiments, for instance, observations $X_{k}\sim\mathcal{N}(\Theta_{k},1)$ for $k=1,2,\ldots,N$, to estimate the conditional distributions $\Theta_{k}|X_{k}$. There are two main estimation strategies: modeling on the $\theta$ space, called “$g$-modeling” here, and modeling on the $x$ space, called “$f$-modeling.” The two approaches are described and compared. A series of computational formulas are developed to assess their frequentist accuracy. Several examples, both contrived and genuine, show the strengths and limitations of the two strategies.

In the search for genetic factors that are associated with complex heritable human traits, considerable attention is now being focused on rare variants that individually have small effects. In response, numerous recent papers have proposed testing strategies to assess association between a group of rare variants and a trait, with competing claims about the performance of various tests. The power of a given test in fact depends on the nature of any association and on the rareness of the variants in question. We review such tests within a general framework that covers a wide range of genetic models and types of data. We study the performance of specific tests through exact or asymptotic power formulas and through novel simulation studies of over 10,000 different models. The tests considered are also applied to real sequence data from the 1000 Genomes project and provided by the GAW17. We recommend a testing strategy, but our results show that power to detect association in plausible genetic scenarios is low for studies of medium size unless a high proportion of the chosen variants are causal. Consequently, considerable attention must be given to relevant biological information that can guide the selection of variants for testing.