Nonparametric Two-Sample Tests for General Clinical Data

Starting from the discussion of a practical example a unifying concept for the derivation of meaningfully interpretable nonparametric tests for the two-sample case is developed which may well be adapted for other designs, too. This methodology covers other well-known procedures, e.g. Fisher's exact test, the Wilcoxon-Mann-Whitney and Gehan's tests, and may furthermore be extended to all situations sharing the same fundamental structural property of the sample space, namely its strict order induced by the substantial problem under study. The resulting test procedure is discussed for a randomization argument, exact and approximate, as well as for the general specific test problem. A numerical example is provided.     

Unconditional inferences on the difference of two proportions in cross-sectional studies

In order to analyse a 2 x 2 table it is usual to perform inferences (hypothesis test or interval of confidence) on the difference d = p2 - p1 between two independent proportions. To this end it has been customary to adopt the Fisher conditional method, but nowadays the unconditional method of Barnard is increasingly adopted. However, all the present unconditional inferences are based on a double-binomial model. This article performs these inferences - exact and asymptotic - under a multinomial model, which is the appropriate one when the data proceed from a cross-sectional survey. At http://www.ugr.es/-bioest/SG_ASO.EXE there is a program for performing the said unconditional tests that may be copied.     

Simple Nonparametric Inference for Monotonic Processes

A scoring system for possibly censored monotonic processes is presented which allows to analyze such processes by means of standard parametric or nonparametric tests. In either case the analyses are invariant to monotonic transformations of time. Such analyses are of relevance whenever changes in degree of function are observed rather than the single and 100% loss of function of survival analysis. An example of gradually increasing function after particular reconstructive surgery is supplied.     

Nonparametric Tests for Slope in Linear Regression Problems

In this paper some new, exactly distribution-free tests are offered for the hypothesis about the slope parameter in one-sample, two-sample, and several-sample simple linear regression problems. Asymptotic relative efficiencies of these test procedures are also studied.     

Nonparametric scanning tests of homogeneity for hierarchical models with continuous covariates

In many applications of hierarchical models, there is often interest in evaluating the inherent heterogeneity in view of observed data. When the underlying hypothesis involves parameters resting on the boundary of their support space such as variances and mixture proportions, it is a usual practice to entertain testing procedures that rely on common heterogeneity assumptions. Such procedures, albeit omnibus for general alternatives, may entail a substantial loss of power for specific alternatives such as heterogeneity varying with covariates. We introduce a novel and flexible approach that uses covariate information to improve the power to detect heterogeneity, without imposing unnecessary restrictions. With continuous covariates, the approach does not impose a regression model relating heterogeneity parameters to covariates or rely on arbitrary discretizations. Instead, a scanning approach requiring continuous dichotomizations of the covariates is proposed. Empirical processes resulting from these dichotomizations are then used to construct the test statistics, with limiting null distributions shown to be functionals of tight random processes. We illustrate our proposals and results on a popular class of two-component mixture models, followed by simulation studies and applications to two real datasets in cancer and caries research.     

A semiparametric approach for the nonparametric transformation survival model with multiple covariates

The nonparametric transformation model makes no parametric assumptions on the forms of the transformation function and the error distribution. This model is appealing in its flexibility for modeling censored survival data. Current approaches for estimation of the regression parameters involve maximizing discontinuous objective functions, which are numerically infeasible to implement with multiple covariates. Based on the partial rank (PR) estimator (Khan and Tamer, 2004), we propose a smoothed PR estimator which maximizes a smooth approximation of the PR objective function. The estimator is shown to be asymptotically equivalent to the PR estimator but is much easier to compute when there are multiple covariates. We further propose using the weighted bootstrap, which is more stable than the usual sandwich technique with smoothing parameters, for estimating the standard error. The estimator is evaluated via simulation studies and illustrated with the Veterans Administration lung cancer data set.     

Nonparametric Tests for Interval‐Censored Failure Time Data

This paper discusses two sample nonparametric comparison of survival functions when only interval‐censored failure time data are available. The problem considered often occurs in, for example, biological and medical studies such as medical follow‐up studies and clinical trials. For the problem, we present and study several nonparametric test procedures that include methods based on both absolute and squared survival differences as well as simple survival differences. The presented tests provide alternatives to existing methods, most of which are rank‐based tests and not sensitive to nonproportional or nonmonotone alternatives. Simulation studies are performed to evaluate and compare the proposed methods with existing methods and suggest that the proposed tests work well for nonmonotone alternatives as well as monotone alternatives. An illustrative example is presented.     

Nonparametric variable importance assessment using machine learning techniques

In a regression setting, it is often of interest to quantify the importance of various features in predicting the response. Commonly, the variable importance measure used is determined by the regression technique employed. For this reason, practitioners often only resort to one of a few regression techniques for which a variable importance measure is naturally defined. Unfortunately, these regression techniques are often suboptimal for predicting the response. Additionally, because the variable importance measures native to different regression techniques generally have a different interpretation, comparisons across techniques can be difficult. In this work, we study a variable importance measure that can be used with any regression technique, and whose interpretation is agnostic to the technique used. This measure is a property of the true data‐generating mechanism. Specifically, we discuss a generalization of the analysis of variance variable importance measure and discuss how it facilitates the use of machine learning techniques to flexibly estimate the variable importance of a single feature or group of features. The importance of each feature or group of features in the data can then be described individually, using this measure. We describe how to construct an efficient estimator of this measure as well as a valid confidence interval. Through simulations, we show that our proposal has good practical operating characteristics, and we illustrate its use with data from a study of risk factors for cardiovascular disease in South Africa.     

A Nonparametric Test for Differences in the Dispersion of Dependent Samples

A new statistic Δ to test the hypothesis of a difference in the dispersion of two dependent samples of ordinal data quality is proposed. It draws on the idea of rank assignment originally forwarded by Siegel and Tukey (1960). No exact probability levels can be given for this statistic for the time being, but it is shown that the statistic is linearly related to the so-called Hotelling-Pabst statistic D, and that one can use exact tables of the latter as a substitute in the statistical decision process with small samples. For larger samples, an approximation of Δ to the standard normal distribution is given. The problem of tied observations is not sufficiently solved yet. A conservative procedure of rank assignment is proposed as long as the exact distribution of Δ in the presence of ties is unknown.     

Nonparametric approaches for estimating regional lake nutrient thresholds

Nonparametric approaches including a classification and regression tree (CART), a nonparametric changepoint analysis (nCPA) and a Bayesian hierarchical modeling (BHM) method were developed to determine ecoregional nutrient response thresholds. A CART analysis revealed that hierarchical structure was important for predicting Chl a concentrations from total nitrogen (TN) and total phosphorus (TP). The nCPA and BHM methods confirmed the CART results for each node in the tree, and the 90% confidence interval for each threshold was calculated to quantify uncertainty. The CART, nCPA, and BHM methods suggested that the nutrient criteria differed significantly within certain nutrient ecoregions and that numerical nutrient criteria of 0.0150–0.222 mg/L TP and 0.300–1.766 mg/L TN may control Chl a concentrations in the various lake ecoregions. The results of this analysis suggest that the integration of CART, nCPA and BHM might be useful for determining nutrient thresholds.     

Simplified formulas for three-way chi-square testing

In order to test against contingency in a 3-way table, simplified X²-formulas are given for (1) the general r xcxl-table, (2) for the rxcx2-table, (3) for the 2x2xl-table, and (4) for the 2×2×2-table. The tests are valid only if a sample of N individuals is observed as to 3 discrete or discretisized (dichotomized) variables which are, under H₀, assumed to be totally independent.     

Likelihood methods for incomplete longitudinal binary responses with incomplete categorical covariates

We consider longitudinal studies in which the outcome observed over time is binary and the covariates of interest are categorical. With no missing responses or covariates, one specifies a multinomial model for the responses given the covariates and uses maximum likelihood to estimate the parameters. Unfortunately, incomplete data in the responses and covariates are a common occurrence in longitudinal studies. Here we assume the missing data are missing at random (Rubin, 1976, Biometrika 63, 581-592). Since all of the missing data (responses and covariates) are categorical, a useful technique for obtaining maximum likelihood parameter estimates is the EM algorithm by the method of weights proposed in Ibrahim (1990, Journal of the American Statistical Association 85, 765-769). In using the EM algorithm with missing responses and covariates, one specifies the joint distribution of the responses and covariates. Here we consider the parameters of the covariate distribution as a nuisance. In data sets where the percentage of missing data is high, the estimates of the nuisance parameters can lead to highly unstable estimates of the parameters of interest. We propose a conditional model for the covariate distribution that has several modeling advantages for the EM algorithm and provides a reduction in the number of nuisance parameters, thus providing more stable estimates in finite samples.     

On the multivariate probit model for exchangeable binary data with covariates

This paper considers the use of a multivariate binomial probit model for the analysis of correlated exchangeable binary data. The model can naturally accommodate both cluster and individual level covariates, while keeping a fairly flexible intracluster association structure. We discuss Bayesian estimation when a sample of independent clusters of varying sizes are available, and show how Gibbs sampling may be used to derive the posterior densities of parameters. The methodology is illustrated with two examples: the first involves epidemiological data from a study of familial disease aggregation; the second uses teratological data from a developmental toxicity application.     

Nonparametric testing for DNA copy number induced differential mRNA gene expression

Summary .  The central dogma of molecular biology relates DNA with mRNA. Array CGH measures DNA copy number and gene expression microarrays measure the amount of mRNA. Methods that integrate data from these two platforms may uncover meaningful biological relationships that further our understanding of cancer. We develop nonparametric tests for the detection of copy number induced differential gene expression. The tests incorporate the uncertainty of the calling of genomic aberrations. The test is preceded by a "tuning algorithm" that discards certain genes to improve the overall power of the false discovery rate selection procedure. Moreover, the test statistics are "shrunken" to borrow information across neighboring genes that share the same array CGH signature. For each gene we also estimate its effect, its amount of differential expression due to copy number changes, and calculate the coefficient of determination. The method is illustrated on breast cancer data, in which it confirms previously reported findings, now with a more profound statistical underpinning.     

A study on emergence of alternating reciprocity in a 2 x 2 game with 2-length memory strategy

It is recognized that bilateral cooperation (C), a reward-state in other words, emergently comes up in a 2 × 2 prisoner's dilemma game, if you assume a strategy set with a memory concept. Also observed is a mixture state of cooperation (C) and defect (D), saint- and temptation-state in other words, to obtain a higher payoff than R (R reciprocity) in a hero or leader game that is a chicken-type dilemma game; this phenomenon is called alternating reciprocity (AR) or ST reciprocity. Observing a holistic 2 × 2 game world including trivial games and various dilemma games, where 2-length memory and infinite interactions are assumed, the paper reports on the specific mechanism of AR. It is observed there are three different phases relating to AR, which can be explained by the stress of the dilemma.     

Correction for Covariate Measurement Error in Nonparametric Longitudinal Regression

Summary We introduce a correction for covariate measurement error in nonparametric regression applied to longitudinal binary data arising from a study on human sleep. The data have been surveyed to investigate the association of some hormonal levels and the probability of being asleep. The hormonal effect is modeled flexibly while we account for the error‐prone measurement of its concentration in the blood and the longitudinal character of the data. We present a fully Bayesian treatment utilizing Markov chain Monte Carlo inference techniques, and also introduce block updating to improve sampling and computational performance in the binary case. Our model is partly inspired by the relevance vector machine with radial basis functions, where usually very few basis functions are automatically selected for fitting the data. In the proposed approach, we implement such data‐driven complexity regulation by adopting the idea of Bayesian model averaging. Besides the general theory and the detailed sampling scheme, we also provide a simulation study for the Gaussian and the binary cases by comparing our method to the naive analysis ignoring measurement error. The results demonstrate a clear gain when using the proposed correction method, particularly for the Gaussian case with medium and large measurement error variances, even if the covariate model is misspecified.