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Journal of algebraic statistics 10(1), 30-50 (2019). doi:10.18409/jas.v10i1.74 special issue: "Special Volume in honor of memory of S.E.Fienberg", Published by Istanbul

Symmetry 13(12), 2297 (2021). doi:10.3390/sym13122297 special issue: "Special Issue "Advances in Quasi-Symmetry Models" / Special Issue Editor: Dr. Kouji Tahata, Guest Editor", Published by MDPI, Basel

The asymptotic posterior normality (APN) of the latent variable vector in an item response theory (IRT) model is a crucial argument in IRT modeling approaches. In case of a single latent trait and under general assumptions, Chang and Stout (Psychometrika, 58(1):37–52, 1993) proved the APN for a broad class of latent trait models for binary items. Under the same setup, they also showed the consistency of the latent trait’s maximum likelihood estimator (MLE). Since then, several modeling approaches have been developed that consider multivariate latent traits and assume their APN, a conjecture which has not been proved so far. We fill this theoretical gap by extending the results of Chang and Stout for multivariate latent traits. Further, we discuss the existence and consistency of MLEs, maximum a-posteriori and expected a-posteriori estimators for the latent traits under the same broad class of latent trait models.

The paper introduces a new class of row-column (RC) association models for contingency tables by allowing the user to select both the scale on which interactions are measured as in Kateri and Papaioannou (1994) and the type of logit (local, global, continuation) suitable for the row and column classification variables as in Bartolucci and Forcina (2002). These choices determine the matrix of interactions which is subjected to rank constraints. Examples are provided where the new models fit substantially better than traditional ones; an intuitive explanation for this behavior is outlined and supported by numerical investigations. An extension of the optimality property in Kateri and Papaioannou (1994) is derived, leading to a general representation theorem and reconstruction formulas for the joint probabilities. These results are the key to show that, given marginal logits, the generalized interactions introduced in this paper determine uniquely the bivariate distribution; in addition, for each pair of logit types, we establish which kind of positive association is implied by our extended interactions being non negative. Quick model selection within this wide class can be performed by an efficient algorithm for computing maximum likelihood estimates which allows for additional linear constraints both on marginal logits and interactions.

Summary This paper extends the ordinary quasi-symmetry (QS) model for square contingency tables with commensurable classification variables. The proposed generalised QS model is defined in terms of odds ratios that apply to ordinal variables. In particular, we present QS models based on global, cumulative and continuation odds ratios and discuss their properties. Finally, the conditional generalised QS model is introduced for local and global odds ratios. These models are illustrated through the analysis of two data sets.

Step-stress models form an essential part of accelerated life testing procedures. Under a step-stress model, the test units are exposed to stress levels that increase at intermediate time points of the experiment. The goal is to develop statistical inference for, e.g., the mean lifetime under each stress level, targeting to the extrapolation under normal operating conditions. This is achieved through an appropriate link function that connects the stress level to the associated mean lifetime. The assumptions made about the time points of stress level change, the termination point of the experiment, the underlying lifetime distributions, the type of censoring (if present), and the way of monitoring lead to alternative models. Step-stress models can be designed for single or multiple samples. We discuss recent developments in designing and analyzing step-stress models based on hazard rates. The inference approach adopted is mainly the maximum likelihood, but Bayesian approaches are briefly discussed.

This article proposes new approaches for optimal planning of step-stress accelerated life testing models. The experiment considered is time constrained with the tested items not monitored continuously but inspected at particular time points instead. The inspection points are primarily the points of stress level change and the experiment’s termination point, but the inclusion of additional intermediate inspection points is possible. The underlying lifetimes in each stress level follow a general-scale family of distributions having, among others, the exponential and the Weibull as special cases. For this model, the optimal allocation of the inspection points is studied in terms of the classical A-, C-, D- and E-optimality criteria, as well as in the context of minimizing the probability of nonexistence of the maximum likelihood estimators of the model’s parameters. For the determination of the inspection intervals’ length, a deterministic and a hazard rate-based approach are introduced. Simulation study results indicate that these new approaches outperform the standard ones of equal spacing and equal probability. All the considered designs are comparatively evaluated and discussed on the basis of simulation studies.

Electronic journal of statistics : EJS 15(1), 830-879 (2021). doi:10.1214/21-EJS1797, Published by Cornell University Library, Ithaca, NY

Capacity degradation of lithium-ion batteries under long-term cyclic aging is modeled via a flexible sigmoidal-type regression set-up, where the regression parameters can be interpreted. Different approaches known from the literature are discussed and compared with the new proposal. Statistical procedures, such as parameter estimation, confidence and prediction intervals are presented and applied to real data. The long-term capacity degradation model may be applied in second-life scenarios of batteries. Using some prior information or training data on the complete degradation path, the model can be fitted satisfactorily even if only short-term degradation data is available. The training data may arise from a single battery.

Summary We consider simple ordinal model-based probability effect measures for comparing distributions of two groups, adjusted for explanatory variables. An “ordinal superiority” measure summarizes the probability that an observation from one distribution falls above an independent observation from the other distribution, adjusted for explanatory variables in a model. The measure applies directly to normal linear models and to a normal latent variable model for ordinal response variables. It equals Φ(β/2) for the corresponding ordinal model that applies a probit link function to cumulative multinomial probabilities, for standard normal cdf Φ and effect β that is the coefficient of the group indicator variable. For the more general latent variable model for ordinal responses that corresponds to a linear model with other possible error distributions and corresponding link functions for cumulative multinomial probabilities, the ordinal superiority measure equals exp(β)/[1+exp(β)] with the log–log link and equals approximately exp(β/2)/[1+exp(β/2)] with the logit link, where β is the group effect. Another ordinal superiority measure generalizes the difference of proportions from binary to ordinal responses. We also present related measures directly for ordinal models for the observed response that need not assume corresponding latent response models. We present confidence intervals for the measures and illustrate with an example.

The odds ratio is the predominant measure of association in 2 × 2 contingency tables, which, for inferential purposes, is usually considered on the log-scale. Under an information theoretic set-up, it is connected to the Kullback–Leibler divergence. Considering a generalized family of divergences, the ϕ divergence, alternative association measures are derived for 2 × 2 contingency tables. Their properties are studied and asymptotic inference is developed. For some members of this family, the estimated association measures remain finite in the presence of a sampling zero while for a subset of these members the estimators of these measures have finite variance as well. Special attention is given to the power divergence, which is a parametric family. The role of its parameter λ , in terms of the asymptotic confidence intervals’ coverage probability and average relative length, is further discussed. In special probability table structures, for which the performance of the asymptotic confidence intervals for the classical log odds ratio is poor, the measure corresponding to λ = 1 / 3 is suggested as an alternative.

In lifetime analysis of lithium-ion batteries, the failure of a battery is commonly defined as the time point at which the battery holds only a certain percentage of its initial storage capacity. Usually, lifetime experiments are carried out to record degradation up to a certain level, and thus do not offer an appropriate data base for fitting lifetime distributions to different end-of-life (EoL) criteria by means of residual capacity. However, for a second use of batteries, e.g. in a home energy storage system, lifetime modeling for long-term usage is of specific interest. Based on real data from long-term cyclic aging experiments with lithium-ion batteries, the fitting of three of the most popular lifetime distributions, namely Weibull, lognormal and inverse Gaussian distributions, is examined with respect to some particular EoL criteria. The impact of chosen lifetime distributions on confidence intervals (CIs) for the mean lifetime and certain quantiles is studied. Moreover, since data of capacity decline are commonly interval-censored, comparisons of resulting CIs under full and interval-censored information are made by means of simulations.

A step-stress accelerated life testing model is constructed that deals with type-I censored experiments for which a continuous monitoring of the tested items is infeasible and only their inspection at particular time points is possible, producing thus grouped data. A general scale family of distributions is considered for the underlying lifetimes, which allows for flexible modeling by permitting different lifetime distributions for different stress levels. The maximum likelihood estimators of its parameters and their density functions are derived explicitly only when the inspection points coincide with the points of stress-level change. In case of additional inspection points, the estimates are obtained numerically. Asymptotic, exact (whenever possible) and bootstrap confidence intervals (CIs) are considered. For the bootstrap CIs a smoothing-modification is introduced, accounting for the categorical nature of the data.

A multi-sample model for general step-stress experiments based on sequential order statistics is proposed and analyzed, where the numbers of observations under each stress level are pre-specified. Contrary to common step-stress approaches, an arbitrary absolutely continuous lifetime distribution can be chosen, and, due to the experimental design of repeated Type-II censoring, the existence of maximum likelihood estimators for the parameters of interest is always guaranteed. Estimation and testing of the model parameters associated with the stress levels is examined, and inference under a log-linear link function is considered. Simulation studies illustrate some of the statistical procedures.

In step-stress modeling with the cumulative exposure model it is well known that, for underlying exponential distributions, explicit expressions can be obtained for maximum likelihood estimators of parameters as well as for their conditional density functions or conditional moment generation functions, given their existence. Applying a failure rate based approach instead, similar results can be also obtained for underlying lifetime distributions out of a general scale family of distributions, which allows for a flexible modeling. Exemplarily, respective results are presented for Type-I and Type-II censored experiments.

We present an overview of some important and/or interesting contributions to the latent variable literature for the analysis of multivariate categorical responses, beginning with Lazarsfeld's introduction of latent class models. There is by now an enormous literature on latent variable models for categorical responses, especially in the context of including random effects in generalized linear mixed models, so this is necessarily a highly selective overview. Due to space considerations, we summarize the main ideas, suppressing details. As part of our presentation, we raise a couple of questions that may suggest future research work.

When the goal of a study is to compare two groups on an ordinal categorical scale, a large number of inferential methods are available. Most methods are designed to detect a location effect, such as by focusing a single-degree-of-freedom test on an effect parameter. Often, rather than merely summarizing by a P-value to describe the evidence against a null hypothesis, it is of interest to consider whether a stronger conclusion can be made. For example, can we conclude that the population distributions are stochastically ordered? For parameter space regions described by order restrictions, frequentist methods are not well designed for significance testing. For example, a frequentist P-value for testing identical distributions against an alternative of stochastically ordered distributions can be very small even when the sample distributions give clear evidence that the distributions do not have the ordering property. The Bayesian approach seems better equipped to handle such questions. We discuss this in the context of stochastic ordering and other types of ordinal odds ratio structure, for the two-group comparison and for more general contexts. For Dirichlet priors, simple simulations provide posterior probabilities of particular ordinal odds ratio structures. Copyright © 2013 John Wiley & Sons, Ltd.

We consider simple ordinal model-based probability effect measures for comparing distributions of two groups, adjusted for explanatory variables. An "ordinal superiority" measure summarizes the probability that an observation from one distribution falls above an independent observation from the other distribution, adjusted for explanatory variables in a model. The measure applies directly to normal linear models and to a normal latent variable model for ordinal response variables. It equals Φ(β/2) for the corresponding ordinal model that applies a probit link function to cumulative multinomial probabilities, for standard normal cdf Φ and effect β that is the coefficient of the group indicator variable. For the more general latent variable model for ordinal responses that corresponds to a linear model with other possible error distributions and corresponding link functions for cumulative multinomial probabilities, the ordinal superiority measure equals exp(β)/[1+exp(β)] with the log-log link and equals approximately exp(β/2)/[1+exp(β/2)] with the logit link, where β is the group effect. Another ordinal superiority measure generalizes the difference of proportions from binary to ordinal responses. We also present related measures directly for ordinal models for the observed response that need not assume corresponding latent response models. We present confidence intervals for the measures and illustrate with an example.

In simple step-stress experiments under Type-II censoring with the cumulative exposure model and exponentially distributed lifetimes, maximum likelihood estimates (MLE) of the expected lifetimes may not exist due to the absence of failure times either before or after the stress change point. For this reason, when planning a step-stress experiment, the change point could be chosen so as to minimize the probability of non-existence of the MLE. These non-existence probabilities are examined and compared in the one- as well as the two-sample situations. Moreover, the optimal allocations of the change points are discussed and the effects of the use of non-optimal choices for the change points are assessed.

In the model of sequential order statistics, prior distributions are considered for the model parameters, which, for example, describe increasing load put on remaining components. Gamma priors are examined as well as priors out of a class of extended truncated Erlang distributions (ETED), which is introduced along with some properties. The choice of independent priors in both set-ups leads to respective independent, conjugate posterior distributions for the model parameters of sequential order statistics. Since, in practical applications, the model parameters will often be increasingly ordered, a multivariate prior is applied being the joint distribution of common ETED-order statistics. Whatever baseline distribution of the sequential order statistics is chosen, the joint posterior distribution turns out to be a Weinman multivariate exponential distribution. Posterior moments are given explicitly, and HPD credible sets for the model parameters are stated. Journal of Multivariate Analysis

For general step-stress experiments with arbitrary baseline distributions, wherein the stress levels change immediately after having observed pre-specified numbers of observations under each stress level, a sequential order statistics model is proposed and associated inferential issues are discussed. Maximum likelihood estimators (MLEs) of the mean lifetimes at different stress levels are derived, and some useful properties of the MLEs are established. Joint MLEs are also derived when an additional location parameter is introduced into the model, and estimation under order restriction of the parameters at different stress levels is finally discussed. Annals of the Institute of Statistical Mathematics

The step-stress model is a special case of accelerated life testing that allows for testing of units under different levels of stress with changes occurring at various intermediate stages of the experiment. Interest then lies on inference for the mean lifetime at each stress level. All step-stress models discussed so far in the literature are based on a single experiment. For the situation when data have been collected from different experiments wherein all the test units had been exposed to the same levels of stress but with possibly different points of change of stress, we introduce a model that combines the different experiments and facilitates a meta-analysis for the estimation of the mean lifetimes. We then discuss in detail the likelihood inference for the case of simple step-stress experiments under exponentially distributed lifetimes with Type-II censoring.

Square contingency tables with the same row and column classification occur frequently in a wide range of statistical applications, e.g. whenever the members of a matched pair are classified on the same scale, which is usually ordinal. Such tables are analysed by choosing an appropriate loglinear model. We focus on the models of symmetry, triangular, diagonal and ordinal quasi symmetry. The fit of a specific model is tested by the chi-squared test or the likelihood-ratio test, where p-values are calculated from the asymptotic chi-square distribution of the test statistic or, if this seems unjustified, from the exact conditional distribution. Since the calculation of exact p-values is often not feasible, we propose alternatives based on algebraic statistics combined with MCMC methods.

In the one- and multi-sample cases, in the context of life-testing reliability experiments, we introduce minimal repair processes under a simple step-stress test, based on exponential distributions and an associated cumulative exposure model, and then develop likelihood inference for such a model.

Association models constitute an attractive alternative to the usual log-linear models for modeling the dependence between classification variables. They impose special structure on the underlying association by assigning scores on the levels of each classification variable, which can be fixed or parametric. Under the general row-column (RC) association model, both row and column scores are unknown parameters without any restriction concerning their ordinality. However, when the classification variables are ordinal, order restrictions on the scores arise naturally. Under such restrictions, we adopt an alternative parameterization and draw inferences about the equality of adjacent scores using the Bayesian approach. To achieve that, we have constructed a reversible jump Markov chain Monte Carlo algorithm for moving across models of different dimension and estimate accurately the posterior model probabilities which can be used either for model comparison or for model averaging. The proposed methodology is evaluated through a simulation study and illustrated using actual datasets.

In this paper, we present a general model for predicting the fatigue behavior for any stress level and amplitude using the exponential model. Based on the W\"ohler field for fixed stress level, a compatibility functional equation enables us to derive the general model with eight parameters. The problem of parameter estimation is then discussed and some methods are described. Some examples are finally presented to illustrate the derived model and the proposed methods of estimation. Communications in Statistics - Simulation and Computation

In the context of multiple step-stress models, which is a special type of accelerated life-testing model, interest lies on the expected lifetimes of the experimental units under different stress levels. Although the expected lifetime is shortened as the stress level increases, this information has not been incorporated so far into the associated inferential procedures. For this reason, we develop here the order restricted maximum likelihood estimation (MLE) for multiple step-stress models with exponentially distributed lifetimes under Type-I, and Type-II censored sampling situations. Moreover, the existence of the unrestricted MLE for a certain stress level is conditional on observing failures at that particular stress level. Under the order restriction, MLE exist even for stress levels without observed failures, provided that these stress levels are internal. We also discuss hypothesis testing problems under order restrictions. IEEE TRANSACTIONS ON RELIABILITY

The estimation of parameters of a Weibull distribution, requiring numerical methods, has been discussed by numerous authors. In this paper, an alternative approach based on a very simple and easy-to-apply graphical method is proposed, which also readily shows the existence and uniqueness of the maximum likelihood estimates. Furthermore, for censored data from a large sample size, a closed-form estimator for the shape parameter is also obtained.

The likelihood ratio is used for measuring the strength of statistical evidence. The probability of observing strong misleading evidence along with that of observing weak evidence evaluate the performance of this measure. When the corresponding likelihood function is expressed in terms of a parametric statistical model that fails, the likelihood ratio retains its evidential value if the likelihood function is robust [Royall, R., Tsou, T.S., 2003. Interpreting statistical evidence by using imperfect models: robust adjusted likelihood functions. J. Roy. Statist. Soc. Ser. B 65, 391 404]. In this paper, we extend the theory of Royall and Tsou [2003. Interpreting statistical evidence by using imperfect models: robust adjusted likelihood functions. J. Roy. Statist. Soc., Ser. B 65, 391 404] to the case when the assumed working model is a characteristic model for two-way contingency tables (the model of independence, association and correlation models).We observe that association and correlation models are not equivalent in terms of statistical evidence. The association models are bounded by the maximum of the bump function while the correlation models are not. Journal of Statistical Planning and Inference