Conditional Inference Procedures Research Articles

Comparison of tree-based methods used in survival data

Abstract Survival trees and forests are popular non-parametric alternatives to parametric and semi-parametric survival models. Conditional inference trees (Ctree) form a non-parametric class of regression trees embedding tree-structured regression models into a well-defined theory of conditional inference procedures. The Ctree is applicable in a varietyof regression-related issues, involving nominal, ordinal, numeric, censored, as well as multivariate response variables and arbitrary measurement scales of covariates. Conditional inference forests (Cforest) consitute a survival forest method which combines a large number of Ctrees. The Cforest provides a unified and flexible framework for ensemble learning in the presence of censoring. The random survival forests (RSF) methodology extends the random forests method enabling the approximation of rich classes of functions while maintaining generalisation errors low. In the present study, the Ctree, Cforest and RSF methods are discussed in detail and the performances of the survival forest methods, namely the Cforest and RSF have been compared with a simulation study. The results of the simulation demonstrate that the RSF method with a log-rank score distinction criteria outperforms the Cforest and the RSF with log-rank distinction criteria.

Conditional estimation and inference to address observed covariate imbalance in randomized clinical trials.

Baseline covariate imbalance (between treatment groups) is a common problem in randomized clinical trials which often raises questions about the validity of trial results. Answering these questions requires careful consideration of the statistical implications of covariate imbalance. The possibil ity of having covariate imbalance contributes to the marginal variance of an unadjusted treatment difference estimator, which can be reduced by making appropriate adjustments. Actual observed imbalance introduces a conditional bias into an unadjusted estimator, which may increase the conditional size of an unadjusted test. This article provides conditional estimation and inference procedures to address the conditional bias due to observed imbalance. Interestingly, it is possible to use the same adjusted treatment difference estimator to address the marginal variance issue and the conditional bias issue associated with covariate imbalance. Its marginal variance estimator tends to be conservative for conditional inference, and we propose a conditionally appropriate variance estimator. We also provide an estimator of the conditional bias in an unadjusted treatment difference estimator, together with a conditional variance estimator. The proposed methodology is illustrated with real data from a stroke trial and evaluated in simulation experiments based on the same trial. The simulation results show that covariate imbalance can result in a substantial conditional bias and that the proposed methods deal with the conditional bias quite effectively. We recommend that the proposed methodology be used routinely to address the observed covariate imbalance in randomized clinical trials.

Predictive Factors of Surgical Outcome in Frontal Lobe Epilepsy Explored with Stereoelectroencephalography.

Resective surgery established treatment for pharmacoresistant frontal lobe epilepsy (FLE), but seizure outcome and prognostic indicators are poorly characterized and vary between studies. To study long-term seizure outcome and identify prognostic factors. We retrospectively analyzed 42 FLE patients having undergone surgical resection, mostly preceded by invasive recordings with stereoelectroencephalography (SEEG). Postsurgical outcome up to 10-yr follow-up and prognostic indicators were analyzed using Kaplan-Meier analysis and multivariate and conditional inference procedures. At the time of last follow-up, 57.1% of patients were seizure-free. The estimated chance of seizure freedom was 67% (95% confidence interval [CI]: 54-83) at 6 mo, 59% (95% CI: 46-76) at 1 yr, 53% (95% CI: 40-71) at 2 yr, and 46% (95% CI: 32-66) at 5 yr. Most relapses (83%) occurred within the first 12 mo. Multivariate analysis showed that completeness of resection of the epileptogenic zone (EZ) as defined by SEEG was the main predictor of seizure outcome. According to conditional inference trees, in patients with complete resection of the EZ, focal cortical dysplasia as etiology and focal EZ were positive prognostic indicators. No difference in outcome was found in patients with positive vs negative magnetic resonance imaging. Surgical resection in drug-resistant FLE can be a successful therapeutic approach, even in the absence of neuroradiologically visible lesions. SEEG may be highly useful in both nonlesional and lesional FLE cases, because complete resection of the EZ as defined by SEEG is associated with better prognosis.

Decision tree modeling using R.

In machine learning field, decision tree learner is powerful and easy to interpret. It employs recursive binary partitioning algorithm that splits the sample in partitioning variable with the strongest association with the response variable. The process continues until some stopping criteria are met. In the example I focus on conditional inference tree, which incorporates tree-structured regression models into conditional inference procedures. While growing a single tree is subject to small changes in the training data, random forests procedure is introduced to address this problem. The sources of diversity for random forests come from the random sampling and restricted set of input variables to be selected. Finally, I introduce R functions to perform model based recursive partitioning. This method incorporates recursive partitioning into conventional parametric model building.

Long runs under a conditional limit distribution

This paper presents a sharp approximation of the density of long runs of a random walk conditioned on its end value or by an average of a function of its summands as their number tends to infinity. In the large deviation range of the conditioning event it extends the Gibbs conditional principle in the sense that it provides a description of the distribution of the random walk on long subsequences. An approximation of the density of the runs is also obtained when the conditioning event states that the end value of the random walk belongs to a thin or a thick set with a nonempty interior. The approximations hold either in probability under the conditional distribution of the random walk, or in total variation norm between measures. An application of the approximation scheme to the evaluation of rare event probabilities through importance sampling is provided. When the conditioning event is in the range of the central limit theorem, it provides a tool for statistical inference in the sense that it produces an effective way to implement the Rao-Blackwell theorem for the improvement of estimators; it also leads to conditional inference procedures in models with nuisance parameters. An algorithm for the simulation of such long runs is presented, together with an algorithm determining the maximal length for which the approximation is valid up to a prescribed accuracy.

Generalized Maximally Selected Statistics

Maximally selected statistics for the estimation of simple cutpoint models are embedded into a generalized conceptual framework based on conditional inference procedures. This powerful framework contains most of the published procedures in this area as special cases, such as maximally selected chi(2) and rank statistics, but also allows for direct construction of new test procedures for less standard test problems. As an application, a novel maximally selected rank statistic is derived from this framework for a censored response partitioned with respect to two ordered categorical covariates and potential interactions. This new test is employed to search for a high-risk group of rectal cancer patients treated with a neo-adjuvant chemoradiotherapy. Moreover, a new efficient algorithm for the evaluation of the asymptotic distribution for a large class of maximally selected statistics is given enabling the fast evaluation of a large number of cutpoints.

Regression Analysis of Panel Count Data with Dependent Observation Times

Panel count data often occur in long-term studies that concern occurrence rate of a recurrent event. Methods have been proposed for regression analysis of panel count data, but most of the existing research focuses on situations where observation times are independent of longitudinal response variables and therefore rely on conditional inference procedures given the observation times. This article considers a different situation where the independence assumption may not hold. That is, the observation times and the response variable may be correlated. For inference, estimating equation approaches are proposed for estimation of regression parameters and both large and finite sample properties of the proposed estimates are established. An illustrative example from a cancer study is provided.

Unbiased Recursive Partitioning: A Conditional Inference Framework

Recursive binary partitioning is a popular tool for regression analysis. Two fundamental problems of exhaustive search procedures usually applied to fit such models have been known for a long time: overfitting and a selection bias towards covariates with many possible splits or missing values. While pruning procedures are able to solve the overfitting problem, the variable selection bias still seriously affects the interpretability of tree-structured regression models. For some special cases unbiased procedures have been suggested, however lacking a common theoretical foundation. We propose a unified framework for recursive partitioning which embeds tree-structured regression models into a well defined theory of conditional inference procedures. Stopping criteria based on multiple test procedures are implemented and it is shown that the predictive performance of the resulting trees is as good as the performance of established exhaustive search procedures. It turns out that the partitions and therefore the models induced by both approaches are structurally different, confirming the need for an unbiased variable selection. Moreover, it is shown that the prediction accuracy of trees with early stopping is equivalent to the prediction accuracy of pruned trees with unbiased variable selection. The methodology presented here is applicable to all kinds of regression problems, including nominal, ordinal, numeric, censored as well as multivariate response variables and arbitrary measurement scales of the covariates. Data from studies on glaucoma classification, node positive breast cancer survival and mammography experience are re-analyzed.

Semiparametric Regression Analysis of Longitudinal Data With Informative Observation Times

Statistical analysis of longitudinal data has been discussed by many authors, and a number of methods have been proposed. Most of the research have focused on situations where observation times are independent of or carry no information about the response variable and therefore rely on conditional inference procedures given the observation times. This article considers a different situation, where the independence assumption may not hold; that is, the observation times may carry information about the response variable. For inference, estimating equation approaches are proposed, and both large-sample and final-sample properties of the proposed methods are established. The methodology is applied to a bladder cancer study that motivated this investigation.

CONDITIONAL INFERENCE PROCEDURES FOR THE PARETO AND POWER FUNCTION DISTRIBUTIONS BASED ON TYPE-II RIGHT CENSORED SAMPLES

In this paper we consider conditional inference procedures for the Pareto and power function distributions. We develop procedures for obtaining confidence intervals for the location and scale parameters as well as upper and lower n probability tolerance intervals for a proportion g, given a Type-II right censored sample from the corresponding distribution. The intervals are exact, and are obtained by conditioning on the observed values of the ancillary statistics. Since, for each distribution, the procedures assume that a shape parameter x is known, a sensitivity analysis is also carried out to see how the procedures are affected by changes in x.

Approximate Inference for the Factor Loading of a Simple Factor Analysis Model

We study a factor analysis model with two normally distributed observations and one factor. Two approximate conditional inference procedures for the factor loading are developed. The first proposal is a very simple procedure but it is not very accurate. The second proposal gives extremely accurate results even for very small sample size. Moreover, the calculations require only the signed log‐likelihood ratio statistic and a measure of the standardized maximum likelihood departure. Simulations are used to study the accuracy of the proposed procedures.

Conditional inference procedures for the Laplace distribution when the observed samples are progressively censored

In this paper, we develop procedures for obtaining confidence intervals for the parameters of a Laplace distribution as well as upper and lower γ probability tolerance intervals for a proportion β, given a progressively Type-II right censored sample from the Laplace distribution. The intervals are obtained by conditioning on the observed values of the ancillary statistics. The intervals are exact, do not require numerical integration, and generalize the work of Childs and Balakrishnan (1996) who considered the conventional Type-II right censored case.

Conditional inference procedures for the Laplace distribution based on Type-II right censored samples

In this paper we develop procedures for obtaining confidence intervals for the parameters of a Laplace distribution as well as upper and lower γ probability tolerance intervals for a proportion β, given a Type-II right censored sample from the Laplace distribution. The intervals are obtained by conditioning on the observed values of the ancillary statistics. The intervals are exact, and will generalize the work of Kappenman (1975, 1977) who considered the full sample case.

Effect of the Sampling Network on Estimates of the Covariance Function of Stationary Fields

AbstractThe covariance function of the stochastic component of various soil properties plays an essential role in the estimation and interpolation of spatial phenomena using kriging techniques and in the solution of transport problems in heterogeneous media using stochastic continuum transport modeling. In this study the effect of the sampling network on the estimates of covariance functions of two‐dimensional, isotropic, second‐order stationary processes, characterized by different underlying correlation scales, was analyzed. Estimates of the covariance function based on a standard systematic sampling network were compared with estimates of the covariance functions based on a modified sampling network. Results of these comparisons showed that the modified sampling network was better suited than the systematic network to resolving the statistical properties of the underlying stochastic process at small separation distances. This may result in an improvement in structure identification. Analyses of the effect of the sampling network used for estimating the covariance function on the results of conditional inference procedures such as kriging showed that the kriging estimates were essentially insensitive to the geometric configuration of the sampling network. Uncertainties about these estimates were reduced when using the modified sampling network, however. In the case of unconditional inferences, such as with stochastic analyses of transport problems in heterogeneous media, predictions of the head variance provided by using estimates of the parameters of the covariance function of the log saturated hydraulic conductivity based on the modified sampling network fit the theoretical head variance better than those based on the systematic sampling network. The effect of the sampling network used for estimating the covariance function on the predicted head variance was significant, particularly when the underlying correlation scale was relatively small, when the porous medium texture became finer, and when the hydraulic conductivity variations were treated as being three‐dimensional.

Correction: Some Classes of Conditional Inference Procedures

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Some classes of conditional inference procedures

Data classified into several groups usually depend on structural and accessory parameters, the structural parameter θ being a constant for all the data, while the accessory parameters ϕ i vary between groups.For the structural parameter, inferences that are free of the accessory parameters can be made when the ϕ i admit sufficient statistics or surrogates Si, by conditioning on the Si or by constructing statistics independent of the Si.When the Si are functions of the structural parameter θ, θ is said to be endomorphic. Conditional likelihood methods are not unequivocal for endomorphic parameters; instead, inference is based on statistics independent of the Si, and so free of the ϕ i, derived from conditional distribution functions.With endomorphic parameters, since the surrogates and the statistics independent of them are functions of θ, tests of independence of these sets of statistics provide a means of making inferences about θ.

Some classes of conditional inference procedures

Data classified into several groups usually depend on structural and accessory parameters, the structural parameter θ being a constant for all the data, while the accessory parameters ϕ i vary between groups. For the structural parameter, inferences that are free of the accessory parameters can be made when the ϕ i admit sufficient statistics or surrogates Si , by conditioning on the Si or by constructing statistics independent of the Si . When the Si are functions of the structural parameter θ, θ is said to be endomorphic. Conditional likelihood methods are not unequivocal for endomorphic parameters; instead, inference is based on statistics independent of the Si , and so free of the ϕ i , derived from conditional distribution functions. With endomorphic parameters, since the surrogates and the statistics independent of them are functions of θ, tests of independence of these sets of statistics provide a means of making inferences about θ.

Inferences on the Parameters and Current System Reliability for a Time Truncated Weibull Process

Conditional inference procedures are discussed for the shape parameter and for the current system reliability for a time truncated Weibull process. These tests are shown to be uniformly most powerful unbiased. Approximate confidence limits for the scale parameter are also developed.

Inferences on the Parameters and Current System Reliability for a Time Truncated Weibull Process

Conditional inference procedures are discussed for the shape parameter and for the current system reliability for a time truncated Weibull process. These tests are shown to be uniformly most powerful unbiased. Approximate confidence limits for the scale parameter are also developed.

Specifically objective stochastic latency mechanisms

A two parameter additive model for psychological response latencies f vi ( t) = ( θ v + ϵ i ) exp(−( θ v + ϵ i ) t) is suggested. Conditional inference procedures are derived for the psychometric latency model. If the model is considered as a theory for speed test construction the unrestricted comparability of measurements obtained under “speed” and “power” conditions is an attractive feature of the model. An extension of the parametric structure gives the “linear exponential model” f vi ( t) = ( θ v + Σ ι m a iι η ι ) exp(−( θ v + Σ ι m a iι η ι ) t), where the time constants of the latencies are conceived as linear combinations of “basic parameters” corresponding to the experimental conditions of the latency situations. The linear exponential model includes as special cases some well known series latency mechanisms. Some generalizations of the distributional assumptions are discussed, especially the Weibull distributions. The estimation of parameters by conditional inference procedures and the evaluation of the fit of the model by conditional likelihood ratio tests is illustrated by means of the numerical data from a free recall experiment.