Comments on goodness-of-fitmeasures in binary choice models by frank windmeijer

Why study Pseudo-R²,s for limited dependent variable models? After all, even in the much clearer ordinary least squares case, R² is a poor guide to model selection, at least when used by itself, because it never decreases and typically increases whenever an independent variable is added. There are even cases where R² will tend to one when there is no relationship among the (nonstationary) variables whatsoever (Granger and Newbold, 1974). Surely applied researchers would not want to bother with such a statistic in the limited dependent variable case, particularly when the intuitive explainedvariation- to- total- variation interpretation is no longer available.     

Relative efficiency of three estimators in a polynomial regression with measurement errors

In a polynomial regression with measurement errors in the covariate, the latter being supposed to be normally distributed, one has (at least) three ways to estimate the unknown regression parameters: one can apply ordinary least squares (OLS) to the model without regard to the measurement error or one can correct for the measurement error, either by correcting the estimating equation (ALS) or by correcting the mean and variance functions of the dependent variable, which is done by conditioning on the observable, error ridden, counter part of the covariate (SLS). While OLS is biased, the other two estimators are consistent. Their asymptotic covariance matrices and thus their relative efficiencies can be compared to each other, in particular for the case of a small measurement error variance. In this case, it appears that ALS and SLS become almost equally efficient, even when they differ noticeably from OLS.     

Power and sample size when multiple endpoints are considered

A common approach to analysing clinical trials with multiple outcomes is to control the probability for the trial as a whole of making at least one incorrect positive finding under any configuration of true and false null hypotheses. Popular approaches are to use Bonferroni corrections or structured approaches such as, for example, closed-test procedures. As is well known, such strategies, which control the family-wise error rate, typically reduce the type I error for some or all the tests of the various null hypotheses to below the nominal level. In consequence, there is generally a loss of power for individual tests. What is less well appreciated, perhaps, is that depending on approach and circumstances, the test-wise loss of power does not necessarily lead to a family wise loss of power. In fact, it may be possible to increase the overall power of a trial by carrying out tests on multiple outcomes without increasing the probability of making at least one type I error when all null hypotheses are true. We examine two types of problems to illustrate this. Unstructured testing problems arise typically (but not exclusively) when many outcomes are being measured. We consider the case of more than two hypotheses when a Bonferroni approach is being applied while for illustration we assume compound symmetry to hold for the correlation of all variables. Using the device of a latent variable it is easy to show that power is not reduced as the number of variables tested increases, provided that the common correlation coefficient is not too high (say less than 0.75). Afterwards, we will consider structured testing problems. Here, multiplicity problems arising from the comparison of more than two treatments, as opposed to more than one measurement, are typical. We conduct a numerical study and conclude again that power is not reduced as the number of tested variables increases.     

THE USE OF AGGREGATE DATA TO ESTIMATE GOMPERTZ‐TYPE OLD‐AGE MORTALITY IN HETEROGENEOUS POPULATIONS

We consider two related aspects of the study of old‐age mortality. One is the estimation of a parameterized hazard function from grouped data, and the other is its possible deceleration at extreme old age owing to heterogeneity described by a mixture of distinct sub‐populations. The first is treated by half of a logistic transform, which is known to be free of discretization bias at older ages, and also preserves the increasing slope of the log hazard in the Gompertz case. It is assumed that data are available in the form published by official statistical agencies, that is, as aggregated frequencies in discrete time. Local polynomial modelling and weighted least squares are applied to cause‐of‐death mortality counts. The second, related, problem is to discover what conditions are necessary for population mortality to exhibit deceleration for a mixture of Gompertz sub‐populations. The general problem remains open but, in the case of three groups, we demonstrate that heterogeneity may be such that it is possible for a population to show decelerating mortality and then return to a Gompertz‐like increase at a later age. This implies that there are situations, depending on the extent of heterogeneity, in which there is at least one age interval in which the hazard function decreases before increasing again.     

A note on some conditions for optimality of certain ridge regression estimates

It is noted that the results obtained by Alam and Hawkes show that the ridge estimates suggested by Hoerl, Kennard and Baldwin, and by Lawless and Wang, give guaranteed lower mean square error than least squares for design matrices having none or at least two very small eigenvalues, but not necessarily for those having only one very small eigenvalue.     

A generalization of the alias matrix

The investigation of aliases or biases is important for the interpretation of the results from factorial experiments. For two-level fractional factorials this can be facilitated through their group structure. For more general arrays the alias matrix can be used. This tool is traditionally based on the assumption that the error structure is that associated with ordinary least squares. For situations where that is not the case, we provide in this article a generalization of the alias matrix applicable under the generalized least squares assumptions. We also show that for the special case of split plot error structure, the generalized alias matrix simplifies to the ordinary alias matrix.     

OPTIMAL QUADRATIC UNBIASED ESTIMATION FOR MODELS WITH LINEAR TOEPLITZ COVARIANCE STRUCTURE

This paper deals with the problem of quadratic unbiased estimation for models with linear Toeplitz covariance structure. These serial covariance models are very useful to modelize time or spatial correlations by means of linear models. Optimality and local optimality is examined in different ways. For the nested Toeplitz models, it is shown that there does not exist a Uniformly Minimum Variance Quadratic Unbiased Estimator for at least one linear combination of covariance parameters. Moreover, empirical unbiased estimators are identified as Locally Minimum Variance Quadratic Unbiased Estimators for a particular choice on covariance parameters corresponding to the case where the covariance matrix of the observed random vector is proportional to the identity matrix. The complete Toeplitz-circulant model is also studied. For this model, the existence of a Uniformly Minimum Variance Quadratic Unbiased Estimator for each covariance parameter is proved.     

Matrix–geometric analysis of the discrete time Gi/G/1 system

In this paper, we show that the discrete GI/G/1 system can be analysed as a QBD process with infinite blocks. Most importantly, we show that Matrix–geometric method can be used for analyzing this general queue system including establishing its stability criterion and for obtaining the explicit stationary probability and the waiting time distributions. This also settles the unwritten myth that Matrix–geometric method is limited to cases with at least one Markov based characterizing parameter, i.e. either interarrival or service times, in the case of queueing systems.     

Studentized robust statistics for,main effects in a two-factor manova

Multiresponse experiments in two-faoior manova are considered. StalibLical procedures of the test and estimation, based on studentized robust statistics. for location parameters in the models arc piupused. Large sample properties of their procedures as the cell sizes tend to infinity are investigated. Although Fisher's consistency is assumed in the theory ol ili-estimators, it is not needed. in this paper. For the univariate case, it is found that the asymptotic relative efficiencies (ARE's) of the proposed procedures relative to classical procedures agrees with the classical A/Sisresults of Huber's one sample Mestimator relative to the sample mean. By simulation studies, it can be seen that the proposed estimators are more efficient than the least squares estimators except for the case where the underlying distribution is normal     

Condition-based maintenance policy for linear consecutive-k-out-of-n: F system

In the article, a condition-based maintenance policy is proposed for a linear consecutive-k-out-of-n: F system. The failure times of components are assumed to be independent and identically distributed. It is assumed that the component states in the system can be known at any time and the system failure can be detected immediately. The preventive maintenance action is based on the number of working components in minimal cut sets of the system. If there is at least one minimal cut set consisting of only one working component, the system is maintained preventively after a certain time interval. The proposed policy is compared with corrective maintenance and age-based maintenance policies. As an extended case, it is assumed that the component states can only be known by inspection, but the system failure can be detected immediately. In this case, the system is inspected periodically and is also maintained preventively based on the system state at inspection. Numerical examples are studied to evaluate the performance of the proposed policy and investigate the effects of cost parameters on the expected cost rate.     

A Classroom Approach to the Construction of an Approximate Confidence Interval of a Poisson Mean Using One Observation

Even elementary statistical problems may give rise to a deeper and broader discussion of issues in probability and statistics. The construction of an approximate confidence interval for a Poisson mean turns out to be such a case. The simple standard two-sided Wald confidence interval by normal approximation is discussed and compared with the score interval. The discussion is partly in the form of an imaginary dialog between a teacher and a student, where the latter is supposed to have studied mathematical statistics for at least one semester.     

Estimating the Number of Block Boundaries from Diagonal Blockwise Matrices Without Penalization

In computational biology, numerous recent studies have been dedicated to the analysis of the chromatin structure within the cell by two‐dimensional segmentation methods. Motivated by this application, we consider the problem of retrieving the diagonal blocks in a matrix of observations. The theoretical properties of the least squares estimators of both the boundaries and the number of blocks are investigated. More precisely, the contribution of the paper is to establish the consistency of these estimators. A surprising consequence of our results is that, contrary to the one‐dimensional case, a penalty is not needed for retrieving the true number of diagonal blocks. Finally, the results are illustrated on synthetic data.     

Stein-rule estimation in models with a lagged-dependemt variable

There is an extensive literature on the Stein-rule estimation of the parameters in a regression model. An important result in this literature is that the Stein-rule does not dominate the least squares estimator in the lower mean squared error sense when there are two or less regressors in the model. However, we note that not much is known about the Stein-rule estimation in dynamic models. This paper is a modest attempt in this direction and it shows that Stein-rule estimation of the lagged coefficient does not dominate the least square indicating that the result of regresion model goes through in the dynamic case.     

Balanced 2m factorial designs of resolution v which allow search and estimation of one extra unknown effect, 4 ≤ m ≤ 8

In this paper, we obtain balanced resolution V plans for 2^m factorial experiments (4 ≤ m ≤ 8), which have an additional feature. Instead of assuming that the three factor and higher order effects are all zero, we assume that there is at most one nonnegligible effect among them; however, we do not know which particular effect is nonnegligible. The problem is to search which effect is non-negligible and to estimate it, along with estimating the main effects and two factor interactions etc., as in an ordinary resolution V design. For every value of N (the number of treatments) within a certain practical range, we present a design using which the search and estimation can be carried out. (Of course, as in all statistical problems, the probability of correct search will depend upon the size of “error” or “noise” present in the observations. However, the designs obtained are such that, at least in the noiseless case, this probability equals 1.) It is found that many of these designs are identical with optimal balanced resolution V designs obtained earlier in the work of Srivastava and Chopra.     

A pragmatic adaptive enrichment design for selecting the right target population for cancer immunotherapies

One of the challenges in the design of confirmatory trials is to deal with uncertainties regarding the optimal target population for a novel drug. Adaptive enrichment designs (AED) which allow for a data-driven selection of one or more prespecified biomarker subpopulations at an interim analysis have been proposed in this setting but practical case studies of AEDs are still relatively rare. We present the design of an AED with a binary endpoint in the highly dynamic setting of cancer immunotherapy. The trial was initiated as a conventional trial in early triple-negative breast cancer but amended to an AED based on emerging data external to the trial suggesting that PD-L1 status could be a predictive biomarker. Operating characteristics are discussed including the concept of a minimal detectable difference, that is, the smallest observed treatment effect that would lead to a statistically significant result in at least one of the target populations at the interim or the final analysis, respectively, in the setting of AED.     

Multiple comparisons with ‘best’ for multivariate normal populations

Comparisons of multivariate normal populations are made using a mul-tivariate approach (instead of reducing the problem to a univariate one). A rather negative finding is that, for comparisons with the ‘best’ of each variate, repeated univariate comparisons appear to be almost as efficient as multivariate comparisons, at least for the bivariate case and, under certain circumstances, for higher dimensional cases. Investigations are done on comparisons with the ‘MAX-best’ population (that one having the largest maximum of the marginal means), the ‘MIN-best’ (having the largest minimum) and the ‘O-best’ (being closest to largest in all marginal means). Detailed results are given for the bivariate normal with extensions indicated for the multivariate.     

A LOTTO SYSTEMS PROBLEM

A Lotto Systems bet allows the player to nominate n numbers from which (s)he believes the winning six numbers will be drawn, and to bet on all combinations of six of these n numbers. Assume that the winning six numbers come from the nominated n. How many combinations must be entered to guarantee that one combination will include at least five of the winning six numbers? The problem is generalized in this paper, and the method of simulated annealing is used to find solutions for various situations. The case where two supplementary numbers are drawn after the initial six winning numbers is also considered.     

On the efficiency of nonparametric variance estimation in sequential dose-finding

Dose-finding in clinical studies is typically formulated as a quantile estimation problem, for which a correct specification of the variance function of the outcomes is important. This is especially true for sequential study where the variance assumption directly involves in the generation of the design points and hence sensitivity analysis may not be performed after the data are collected. In this light, there is a strong reason for avoiding parametric assumptions on the variance function, although this may incur efficiency loss. In this paper, we investigate how much information one may retrieve by making additional parametric assumptions on the variance in the context of a sequential least squares recursion. By asymptotic comparison, we demonstrate that assuming homoscedasticity achieves only a modest efficiency gain when compared to nonparametric variance estimation: when homoscedasticity in truth holds, the latter is at worst 88% as efficient as the former in the limiting case, and often achieves well over 90% efficiency for most practical situations. Extensive simulation studies concur with this observation under a wide range of scenarios.     

Outlyingness: Which variables contribute most?

Outlier detection is an inevitable step to most statistical data analyses. However, the mere detection of an outlying case does not always answer all scientific questions associated with that data point. Outlier detection techniques, classical and robust alike, will typically flag the entire case as outlying, or attribute a specific case weight to the entire case. In practice, particularly in high dimensional data, the outlier will most likely not be outlying along all of its variables, but just along a subset of them. If so, the scientific question why the case has been flagged as an outlier becomes of interest. In this article, a fast and efficient method is proposed to detect variables that contribute most to an outlier’s outlyingness. Thereby, it helps the analyst understand in which way an outlier lies out. The approach pursued in this work is to estimate the univariate direction of maximal outlyingness. It is shown that the problem of estimating that direction can be rewritten as the normed solution of a classical least squares regression problem. Identifying the subset of variables contributing most to outlyingness, can thus be achieved by estimating the associated least squares problem in a sparse manner. From a practical perspective, sparse partial least squares (SPLS) regression, preferably by the fast sparse NIPALS (SNIPLS) algorithm, is suggested to tackle that problem. The performed method is demonstrated to perform well both on simulated data and real life examples.