Isotonic Regression Estimator Research Articles

Meta.shrinkage: An R Package for Meta-Analyses for Simultaneously Estimating Individual Means

Meta-analysis is an indispensable tool for synthesizing statistical results obtained from individual studies. Recently, non-Bayesian estimators for individual means were proposed by applying three methods: the James–Stein (JS) shrinkage estimator, isotonic regression estimator, and pretest (PT) estimator. In order to make these methods available to users, we develop a new R package meta.shrinkage. Our package can compute seven estimators (named JS, JS+, RML, RJS, RJS+, PT, and GPT). We introduce this R package along with the usage of the R functions and the “average-min-max” steps for the pool-adjacent violators algorithm. We conduct Monte Carlo simulations to validate the proposed R package to ensure that the package can work properly in a variety of scenarios. We also analyze a data example to show the ability of the R package.

A Meta-Analysis for Simultaneously Estimating Individual Means with Shrinkage, Isotonic Regression and Pretests

Meta-analyses combine the estimators of individual means to estimate the common mean of a population. However, the common mean could be undefined or uninformative in some scenarios where individual means are “ordered” or “sparse”. Hence, assessments of individual means become relevant, rather than the common mean. In this article, we propose simultaneous estimation of individual means using the James–Stein shrinkage estimators, which improve upon individual studies’ estimators. We also propose isotonic regression estimators for ordered means, and pretest estimators for sparse means. We provide theoretical explanations and simulation results demonstrating the superiority of the proposed estimators over the individual studies’ estimators. The proposed methods are illustrated by two datasets: one comes from gastric cancer patients and the other from COVID-19 patients.

Targeted design for adaptive clinical trials via semiparametric model

Precision medicine approach that assigns treatment according to an individual's personal (including molecular) profile is revolutionizing health care. Existing statistical methods for clinical trial design typically assume a known model to estimate characteristics of treatment outcomes, which may yield biased results if the true model deviates far from the assumed one. This article aims to achieve model robustness in a phase II multi-stage adaptive clinical trial design. We propose and study a semiparametric regression mixture model in which the mixing proportions are specified according to the subjects' profiles, and each sub-group distribution is only assumed to be unimodal for robustness. The regression parameters and the error density functions are estimated by semiparametric maximum likelihood and isotonic regression estimators. The asymptotic properties of the estimates are studied. Simulation studies are conducted to evaluate the performance of the method after a real data analysis.

Nonparametric inference for panel count data with competing risks

In survival and reliability studies, panel count data arise when we investigate a recurrent event process and each study subject is observed only at discrete time points. If recurrent events of several types are possible, we obtain panel count data with competing risks. Such data arise frequently from transversal studies on recurrent events in demography, epidemiology and reliability experiments where the individuals cannot be observed continuously. In the present paper, we propose an isotonic regression estimator for the cause specific mean function of the underlying recurrent event process of a competing risks panel count data. Further, a nonparametric test is proposed to compare the cause specific mean functions of the panel count competing risks data. Asymptotic properties of the proposed estimator and test statistic are studied. A simulation study is conducted to assess the finite sample behaviour of the proposed estimator and test statistic. Finally, the procedures developed are applied to a real data arising from skin cancer chemo prevention trial.

Causal Isotonic Regression.

In observational studies, potential confounders may distort the causal relationship between an exposure and an outcome. However, under some conditions, a causal dose-response curve can be recovered using the G-computation formula. Most classical methods for estimating such curves when the exposure is continuous rely on restrictive parametric assumptions, which carry significant risk of model misspecification. Nonparametric estimation in this context is challenging because in a nonparametric model these curves cannot be estimated at regular rates. Many available nonparametric estimators are sensitive to the selection of certain tuning parameters, and performing valid inference with such estimators can be difficult. In this work, we propose a nonparametric estimator of a causal dose-response curve known to be monotone. We show that our proposed estimation procedure generalizes the classical least-squares isotonic regression estimator of a monotone regression function. Specifically, it does not involve tuning parameters, and is invariant to strictly monotone transformations of the exposure variable. We describe theoretical properties of our proposed estimator, including its irregular limit distribution and the potential for doubly-robust inference. Furthermore, we illustrate its performance via numerical studies, and use it to assess the relationship between BMI and immune response in HIV vaccine trials.

A study on the least squares estimator of multivariate isotonic regression function

AbstractConsider the problem of pointwise estimation of f in a multivariate isotonic regression model Z=f(X1,…,Xd)+ϵ, where Z is the response variable, f is an unknown nonparametric regression function, which is isotonic with respect to each component, and ϵ is the error term. In this article, we investigate the behavior of the least squares estimator of f. We generalize the greatest convex minorant characterization of isotonic regression estimator for the multivariate case and use it to establish the asymptotic distribution of properly normalized version of the estimator. Moreover, we test whether the multivariate isotonic regression function at a fixed point is larger (or smaller) than a specified value or not based on this estimator, and the consistency of the test is established. The practicability of the estimator and the test are shown on simulated and real data as well.

Performance Measures in Dose‐Finding Experiments

SummaryIn the first phase of pharmaceutical development, and assuming that the probability of positive response increases with dose, the main statistical goal is to estimate a percentile of the dose–response function for a given target value . We compare the Maximum Likelihood and centred isotonic regression estimators of the target dose and we discuss several performance criteria to assess inferential precision, the amount of toxicity exposure and the trade‐off between them for a set of some exemplary adaptive designs. We compare these designs using graphical tools. Several scenarios are considered using simulation, including the use of several start‐up rules, the change of slope of the dose‐toxicity function at the target dose and also different theoretical models, as logistic, normal or skew‐normal distribution functions.

The limiting behavior of isotonic and convex regression estimators when the model is misspecified

We study the asymptotic behavior of the least squares estimators when the model is possibly misspecified. We consider the setting where we wish to estimate an unknown function $f_{*}:(0,1)^{d}\rightarrow \mathbb{R}$ from observations $(X,Y),(X_{1},Y_{1}),\cdots ,(X_{n},Y_{n})$; our estimator $\hat{g}_{n}$ is the minimizer of $\sum _{i=1}^{n}(Y_{i}-g(X_{i}))^{2}/n$ over $g\in \mathcal{G}$ for some set of functions $\mathcal{G}$. We provide sufficient conditions on the metric entropy of $\mathcal{G}$, under which $\hat{g}_{n}$ converges to $g_{*}$ as $n\rightarrow \infty $, where $g_{*}$ is the minimizer of $\|g-f_{*}\|\triangleq \mathbb{E}(g(X)-f_{*}(X))^{2}$ over $g\in \mathcal{G}$. As corollaries of our theorem, we establish $\|\hat{g}_{n}-g_{*}\|\rightarrow 0$ as $n\rightarrow \infty $ when $\mathcal{G}$ is the set of monotone functions or the set of convex functions. We also make a connection between the convergence rate of $\|\hat{g}_{n}-g_{*}\|$ and the metric entropy of $\mathcal{G}$. As special cases of our finding, we compute the convergence rate of $\|\hat{g}_{n}-g_{*}\|^{2}$ when $\mathcal{G}$ is the set of bounded monotone functions or the set of bounded convex functions.

The bias of isotonic regression.

We study the bias of the isotonic regression estimator. While there is extensive work characterizing the mean squared error of the isotonic regression estimator, relatively little is known about the bias. In this paper, we provide a sharp characterization, proving that the bias scales as O(n -β/3) up to log factors, where 1 ≤ β ≤ 2 is the exponent corresponding to Hölder smoothness of the underlying mean. Importantly, this result only requires a strictly monotone mean and that the noise distribution has subexponential tails, without relying on symmetric noise or other restrictive assumptions.

The isotonic regression approach for an instrumental variable estimation of the potential outcome distributions for compliers

This paper discusses an instrumental variable estimation of the potential outcome distributions for compliers. The existing nonparametric estimators have a limitation in that they give non-proper cumulative distribution functions that violate the non-decreasing property. Using the least squares representation of the standard nonparametric estimators, a simple isotonic regression approach has been developed. A nonparametric bootstrap method is proposed as an appropriate method used to estimate the variances of the isotonic regression estimators. A simulation study demonstrates that the isotonic regression estimators provide more proper and efficient cumulative distribution functions, with much smaller standard errors than those of the standard nonparametric estimators when the proportion of compliers is small. The methods are illustrated with a study to estimate the distributional causal effect of a veteran status on future earnings.

New types of shrinkage estimators of Poisson means under the normalized squared error loss

ABSTRACTIn estimating p( ⩾ 2) independent Poisson means, Clevenson and Zidek (1975) have proposed a class of estimators that shrink the unbiased estimator to the origin and dominate the unbiased one under the normalized squared error loss. This class of estimators was subsequently enlarged in several directions. This article deals with the problem and proposes new classes of dominating estimators using prior information pertinently. Dominance is shown by partitioning the sample space into disjoint subsets and averaging the loss difference over each subset. Estimation of several Poisson mean vectors is also discussed. Further, simultaneous estimation of Poisson means under order restriction is treated and estimators which dominate the isotonic regression estimator are proposed for some types of order restrictions.

The asymptotic distribution of the isotonic regression estimator over a general countable pre-ordered set

We study the isotonic regression estimator over a general countable pre-ordered set. We obtain the limiting distribution of the estimator and study its properties. It is proved that, under some general assumptions, the limiting distribution of the isotonized estimator is given by the concatenation of the separate isotonic regressions of the certain subvectors of an unrestrecred estimator’s asymptotic distribution. Also, we show that the isotonization preserves the rate of convergence of the underlying estimator. We apply these results to the problems of estimation of a bimonotone regression function and estimation of a bimonotone probability mass function.

Regression with Ordered Predictors via Ordinal Smoothing Splines

Many applied studies collect one or more ordered categorical predictors, which do not fit neatly within classic regression frameworks. In most cases, ordinal predictors are treated as either nominal (unordered) variables or metric (continuous) variables in regression models, which is theoretically and/or computationally undesirable. In this paper, we discuss the benefit of taking a smoothing spline approach to the modeling of ordinal predictors. The purpose of this paper is to provide theoretical insight into the ordinal smoothing spline, as well as examples revealing the potential of the ordinal smoothing spline for various types of applied research. Specifically, we (i) derive the analytical form of the ordinal smoothing spline reproducing kernel, (ii) propose an ordinal smoothing spline isotonic regression estimator, (iii) prove an asymptotic equivalence between the ordinal and linear smoothing spline reproducing kernel functions, (iv) develop large sample approximations for the ordinal smoothing spline, and (v) demonstrate the use of ordinal smoothing splines for isotonic regression and semiparametric regression with multiple predictors. Our results reveal that the ordinal smoothing spline offers a flexible approach for incorporating ordered predictors in regression models, and has the benefit of being invariant to any monotonic transformation of the predictor scores.

Determination of the median effective dose (ED50 ) of spinal chloroprocaine in labour analgesia.

The primary goal of this study was to determine the median effective dose (ED50 ) of spinal chloroprocaine for labour analgesia. Thirty-eight parturients requesting neuraxial analgesia were enrolled. Doses of 1% chloroprocaine were determined by the technique of up-down sequential allocation, with an initial dose of 20 mg and steps of 2 mg. The chloroprocaine spinal dose was given as the spinal component of a combined spinal-epidural, which was then supplemented with an epidural dose of 7.5 μg sufentanil in 7 ml saline. Effective analgesia was defined as a score ≤ 10 mm within 15 min on a 100-mm visual analogue pain scale. Using the isotonic regression estimator method, the ED50 of chloroprocaine for the spinal component of a combined spinal-epidural for labour was calculated to be median (95%CI) 12.0 (9.3-17.0) mg.

Successive comparisons between ordered normal means using isotonic regression estimators

ABSTRACTIn this study we discuss multiple comparison procedures for checking differences among a sequence of normal means with ordered restriction. Lee and Spurrier (1995) proposed a multiple comparison procedure which tests the difference between two adjacent means using the difference of sample means. In this study we propose a multiple comparison procedure modifying Lee and Spurrier's (1995) procedure using isotonic regression estimators instead of sample means. We determine the critical value for pairwise comparisons for a specified significance level. Furthermore, we formulate the power of the test. Finally, we give some numerical examples regarding critical values and power of the test intended to compare our procedure with Lee and Spurrier's (1995) procedure.

Trimmed Mean Isotonic Regression

AbstractThe trimmed mean is well‐known in literature for being more robust and for having better efficiency than the sample mean when data is generated from heavy‐tailed distributions. In this article, the trimmed mean in the isotonic regression setup is proposed, and the asymptotic as well as the robustness properties of the estimator are studied. The usefulness of the proposed estimator is illustrated using different real and simulated data. Further, the performance of the estimator is compared with that of the mean and the median isotonic regression estimators.

Classical Laplace estimation for [formula omitted]-consistent estimators: Improved convergence rates and rate-adaptive inference

We propose a classical Laplace estimator alternative for a large class of n3-consistent estimators, including isotonic regression, monotone hazard, and maximum score estimators. The proposed alternative provides a unified method of smoothing; easier computation is a byproduct in the maximum score case. Depending on input parameter choice and smoothness, the convergence rate of our estimator varies between n3 and (almost) n and its limit distribution varies from Chernoff to normal. We provide a bias reduction method and an inference procedure which automatically adapts to the correct convergence rate and limit distribution.

On the Properties of Estimates of Monotonic Mean Vectors for Multivariate Normal Distributions

Problems concerning estimation of parameters and determination the statistic, when it is known a priori that some of these parameters are subject to certain order restrictions, are of considerable interest. In the present paper, we consider the estimators of the monotonic mean vectors for two dimensional normal distributions and compare those with the unrestricted maximum likelihood estimators under two different cases. One case is that covariance matrices are known, the other one is that covariance matrices are completely unknown and unequal. We show that when the covariance matrices are known, under the squared error loss function which is similar to the mahalanobis distance, the obtained multivariate isotonic regression estimators, motivated by estimators given in Robertson et al. (1988), which are the estimators given by Sasabuchi et al. (1983) and Sasabuchi et al. (1992), have the smaller risk than the unrestricted maximum likelihood estimators uniformly, but when the covariance matrices are unknown and unequal, the estimators have the smaller risk than the unrestricted maximum likelihood estimators only over some special sets which are defined on the covariance matrices. To illustrate the results two numerical examples are presented.

Dose-Finding Trial Designs for Combination Therapies in Oncology

Dose-Finding Trial Designs for Combination Therapies in Oncology

Modifying estimators of ordered positive parameters under the Stein loss

This paper treats the problem of estimating positive parameters restricted to a polyhedral convex cone which includes typical order restrictions, such as simple order, tree order and umbrella order restrictions. In this paper, two methods are used to show the improvement of order-preserving estimators over crude non-order-preserving estimators without any assumption on underlying distributions. One is to use Fenchel’s duality theorem, and then the superiority of the isotonic regression estimator is established under the general restriction to polyhedral convex cones. The use of the Abel identity is the other method, and we can derive a class of improved estimators which includes order-statistics-based estimators in the typical order restrictions. When the underlying distributions are scale families, the unbiased estimators and their order-restricted estimators are shown to be minimax. The minimaxity of the restrictedly generalized Bayes estimator against the prior over the restricted space is also demonstrated in the two dimensional case. Finally, some examples and multivariate extensions are given.