CROSS-DESIGN SYNTHESIS FOR EXTENDING THE APPLICABILITY OF TRIAL EVIDENCE WHEN TREATMENT EFFECT IS HETEROGENEOUS-I. METHODOLOGY

Randomized controlled trials (RCTs) provide reliable evidence for approval of new treatments, informing clinical practice, and coverage decisions. The participants in RCTs are often not a representative sample of the larger at-risk population. Hence it is argued that the average treatment effect from the trial is not generalizable to the larger at-risk population. An essential premise of this argument is that there is significant heterogeneity in the treatment effect (HTE). We present a new method to extrapolate the treatment effect from a trial to a target group that is inadequately represented in the trial, when HTE is present. Our method integrates trial and observational data (cross-design synthesis). The target group is assumed to be well-represented in the observational database. An essential component of the methodology is the estimation of calibration adjustments for unmeasured confounding in the observational sample. The estimate of treatment effect, adjusted for unmeasured confounding, is projected onto the target sample using a weighted G-computation approach. We present simulation studies to demonstrate the methodology for estimating the marginal treatment effect in a target sample that differs from the trial sample to varying degrees. In a companion paper, we demonstrate and validate the methodology in a clinical application

BB: An R Package for Solving a Large System of Nonlinear Equations and for Optimizing a High-Dimensional Nonlinear Objective Function

We discuss <code>R</code> package <b>BB</b>, in particular, its capabilities for solving a nonlinear system of equations. The function <code>BBsolve</code> in <b>BB</b> can be used for this purpose. We demonstrate the utility of these functions for solving: (a) large systems of nonlinear equations, (b) smooth, nonlinear estimating equations in statistical modeling, and (c) non-smooth estimating equations arising in rank-based regression modeling of censored failure time data. The function <code>BBoptim</code> can be used to solve smooth, box-constrained optimization problems. A main strength of <b>BB</b> is that, due to its low memory and storage requirements, it is ideally suited for solving high-dimensional problems with thousands of variables

CROSS-DESIGN SYNTHESIS FOR EXTENDING THE APPLICABILITY OF TRIAL EVIDENCE WHEN TREATMENT EFFECT IS HETEROGENEOUS. PART II. APPLICATION AND EXTERNAL VALIDATION

Randomized controlled trials (RCTs) generally provide the most reliable evidence. When participants in RCTs are selected with respect to characteristics that are potential treatment effect modifiers, the average treatment effect from the trials may not be applicable to a specific target population. We present a new method to project the treatment effect from a RCT to a target group that is inadequately represented in the trial when there is heterogeneity in the treatment effect (HTE). The method integrates RCT and observational data through cross-design synthesis. An essential component is to identify HTE and a calibration factor for unmeasured confounding for the observational study relative to the RCT. The estimate of treatment effect adjusted for unmeasured confounding is projected onto the target sample using G-computation with standardization weights. We call the method Calibrated Risk-Adjusted Modeling (CRAM) and apply it to estimate the effect of angiotensin converting enzyme inhibition to prevent heart failure hospitalization or death. External validation shows that when there is adequate overlap between the RCT and the target sample, risk-based standardization is less biased than CRAM. However, when there is poor overlap between the trial and the target sample, CRAM provides superior estimates of treatment effect

Squared Extrapolation Methods (SQUAREM): A New Class of Simple and Efficient Numerical Schemes for Accelerating the Convergence of the EM Algorithm

We derive a new class of iterative schemes for accelerating the convergence of the EM algorithm, by exploiting the connection between fixed point iterations and extrapolation methods. First, we present a general formulation of one-step iterative schemes, which are obtained by cycling with the extrapolation methods. We, then square the one-step schemes to obtain the new class of methods, which we call SQUAREM. Squaring a one-step iterative scheme is simply applying it twice within each cycle of the extrapolation method. Here we focus on the first order or rank-one extrapolation methods for two reasons, (1) simplicity, and (2) computational efficiency. In particular, we study two first order extrapolation methods, the reduced rank extrapolation (RRE1) and minimal polynomial extrapolation (MPE1). The convergence of the new schemes, both one-step and squared, is non-monotonic with respect to the residual norm. The first order one-step and SQUAREM schemes are linearly convergent, like the EM algorithm but they have a faster rate of convergence. We demonstrate, through five different examples, the effectiveness of the first order SQUAREM schemes, SqRRE1 and SqMPE1, in accelerating the EM algorithm. The SQUAREM schemes are also shown to be vastly superior to their one-step counterparts, RRE1 and MPE1, in terms of computational efficiency. The proposed extrapolation schemes can fail due to the numerical problems of stagnation and near breakdown. We have developed a new hybrid iterative scheme that combines the RRE1 and MPE1 schemes in such a manner that it overcomes both stagnation and near breakdown. The squared first order hybrid scheme, SqHyb1, emerges as the iterative scheme of choice based on our numerical experiments. It combines the fast convergence of the SqMPE1, while avoiding near breakdowns, with the stability of SqRRE1, while avoiding stagnations. The SQUAREM methods can be incorporated very easily into an existing EM algorithm. They only require the basic EM step for their implementation and do not require any other auxiliary quantities such as the complete data log likelihood, and its gradient or hessian. They are an attractive option in problems with a very large number of parameters, and in problems where the statistical model is complex, the EM algorithm is slow and each EM step is computationally demanding

SQUAREM: An R Package for Off-the-Shelf Acceleration of EM, MM and Other EM-Like Monotone Algorithms

We discuss the R package SQUAREM for accelerating iterative algorithms which exhibit slow, monotone convergence. These include the well-known expectation-maximization algorithm, majorize-minimize (MM), and other EM-like algorithms such as expectation conditional maximization, and generalized EM algorithms. We demonstrate the simplicity, generality, and power of SQUAREM through a wide array of applications of EM/MM problems, including binary Poisson mixture, factor analysis, interval censoring, genetics admixture, and logistic regression maximum likelihood estimation (an MM problem). We show that SQUAREM is easy to apply, and can accelerate any fixed-point, smooth, contraction mapping with linear convergence rate. The squared iterative scheme (SQUAREM) algorithm provides significant speed-up of EM-like algorithms. The margin of the advantage for SQUAREM is especially huge for high-dimensional problems or when the EM step is relatively time-consuming to evaluate. SQUAREM can be used off-the-shelf since there is no need for the user to tweak any control parameters to optimize performance. Given its remarkable ease of use, SQUAREM may be considered as a default accelerator for slowly converging EM-like algorithms. All the comparisons of CPU computing time in the paper are made on a quad-core 2.3 GHz Intel Core i7 Mac computer. R package SQUAREM is available from the Comprehensive R Archive Network (CRAN) at https://CRAN.R-project.org/package=SQUAREM/

Fitting Additive Binomial Regression Models with the R Package blm

The R package blm provides functions for fitting a family of additive regression models to binary data. The included models are the binomial linear model, in which all covariates have additive effects, and the linear-expit (lexpit) model, which allows some covariates to have additive effects and other covariates to have logisitc effects. Additive binomial regression is a model of event probability, and the coefficients of linear terms estimate covariate-adjusted risk differences. Thus, in contrast to logistic regression, additive binomial regression puts focus on absolute risk and risk differences. In this paper, we give an overview of the methodology we have developed to fit the binomial linear and lexpit models to binary outcomes from cohort and population-based case-control studies. We illustrate the blm packages methods for additive model estimation, diagnostics, and inference with risk association analyses of a bladder cancer nested case-control study in the NIH-AARP Diet and Health Study

Unifying Optimization Algorithms to Aid Software System Users: optimx for R

R users can often solve optimization tasks easily using the tools in the optim function in the stats package provided by default on R installations. However, there are many other optimization and nonlinear modelling tools in R or in easily installed add-on packages. These present users with a bewildering array of choices. optimx is a wrapper to consolidate many of these choices for the optimization of functions that are mostly smooth with parameters at most bounds-constrained. We attempt to provide some diagnostic information about the function, its scaling and parameter bounds, and the solution characteristics. optimx runs a battery of methods on a given problem, thus facilitating comparative studies of optimization algorithms for the problem at hand. optimx can also be a useful pedagogical tool for demonstrating the strengths and pitfalls of different classes of optimization approaches including Newton, gradient, and derivative-free methods