Reliable Inference in Categorical Regression Analysis for Non‐randomly Coarsened Observations

In most surveys, one is confronted with missing or, more generally, coarse data. Traditional methods dealing with these data require strong, untestable and often doubtful assumptions, for example, coarsening at random. But due to the resulting, potentially severe bias, there is a growing interest in approaches that only include tenable knowledge about the coarsening process, leading to imprecise but reliable results. In this spirit, we study regression analysis with a coarse categorical‐dependent variable and precisely observed categorical covariates. Our (profile) likelihood‐based approach can incorporate weak knowledge about the coarsening process and thus offers a synthesis of traditional methods and cautious strategies refraining from any coarsening assumptions. This also allows a discussion of the uncertainty about the coarsening process, besides sampling uncertainty and model uncertainty. Our procedure is illustrated with data of the panel study ‘Labour market and social security' conducted by the Institute for Employment Research, whose questionnaire design produces coarse data.     

Identification and estimation of statistical functionals using incomplete data

Incomplete data, due to missing observations or interval measurement of variables, usually cause parameters of interest in applications to be unidentified except under untestable and often controversial assumptions. However, it is often possible to identify sharp bounds on parameters without making untestable assumptions about the process through which data become incomplete. The bounds contain all logically possible values of the parameters and can be estimated consistently by replacing the population distribution of the data with the empirical distribution. This is straightforward in some circumstances but computationally burdensome in others. This paper describes the general problem and presents an empirical illustration.     

Sensitivity Analysis of Continuous Incomplete Longitudinal Outcomes

Even though models for incomplete longitudinal data are in common use, they are surrounded with problems, largely due to the untestable nature of the assumptions one has to make regarding the missingness mechanism. Two extreme views on how to deal with this problem are (1) to avoid incomplete data altogether and (2) to construct ever more complicated joint models for the measurement and missingness processes. In this paper, it is argued that a more versatile approach is to embed the treatment of incomplete data within a sensitivity analysis. Several such sensitivity analysis routes are presented and applied to a case study, the milk protein trial analyzed before by Diggle and Kenward (1994) . Apart from the use of local influence methods, some emphasis is put on pattern-mixture modeling. In the latter case, it is shown how multiple-imputation ideas can be used to define a practically feasible modeling strategy.     

How to Choose a Working Model for Measuring the Statistical Evidence About a Regression Parameter

Consider using a likelihood ratio to measure the strength of statistical evidence for one hypothesis over another. Recent work has shown that when the model is correctly specified, the likelihood ratio is seldom misleading. But when the model is not, misleading evidence may be observed quite frequently. Here we consider how to choose a working regression model so that the statistical evidence is correctly represented as often as it would be under the true model. We argue that the criteria for choosing a working model should be how often it correctly represents the statistical evidence about the object of interest (regression coefficient in the true model). We see that misleading evidence about the object of interest is more likely to be observed when the working model is chosen according to other criteria (e.g., parsimony or predictive accuracy).     

An alternative quasi likelihood approach,Bayesian analysis and data-based inference for model specification

This paper studies an alternative quasi likelihood approach under possible model misspecification. We derive a filtered likelihood from a given quasi likelihood (QL), called a limited information quasi likelihood (LI-QL), that contains relevant but limited information on the data generation process. Our LI-QL approach, in one hand, extends robustness of the QL approach to inference problems for which the existing approach does not apply. Our study in this paper, on the other hand, builds a bridge between the classical and Bayesian approaches for statistical inference under possible model misspecification. We can establish a large sample correspondence between the classical QL approach and our LI-QL based Bayesian approach. An interesting finding is that the asymptotic distribution of an LI-QL based posterior and that of the corresponding quasi maximum likelihood estimator share the same “sandwich”-type second moment. Based on the LI-QL we can develop inference methods that are useful for practical applications under possible model misspecification. In particular, we can develop the Bayesian counterparts of classical QL methods that carry all the nice features of the latter studied in  White (1982). In addition, we can develop a Bayesian method for analyzing model specification based on an LI-QL.     

On the analysis of multivariate growth curves

Growth curve data arise when repeated measurements are observed on a number of individuals with an ordered dimension for occasions. Such data appear frequently in almost all fields in which statistical models are used, for instance in medicine, agriculture and engineering. In medicine, for example, more than one variable is often measured on each occasion. However, analyses are usually based on exploration of repeated measurements of only one variable. The consequence is that the information contained in the between-variables correlation structure will be discarded.  In this study we propose a multivariate model based on the random coefficient regression model for the analysis of growth curve data. Closed-form expressions for the model parameters are derived under the maximum likelihood (ML) and the restricted maximum likelihood (REML) framework. It is shown that in certain situations estimated variances of growth curve parameters are greater for REML. Also a method is proposed for testing general linear hypotheses. One numerical example is provided to illustrate the methods discussed. Received: 22 February 1999     

Scoring functions for ordered classifications in statistical analysis

The problem of scoring ordered classifications prior to the further statistical analysis is discussed. A review of some methods of scoring is provided. This includes linear transformations of integer scores, where previous applications to two way classifications are introduced. Also reviewed are scores based on canonical correlations, maximum likelihood scores under assumed logistic distributions for variables, ridits, and conditional mean scoring functions. The latter are shown to satisfy a reasonable set of postulates, and demonstrates that some earlier attempts to do this were incomplete. Examples of the conditional mean scoring function under different distributional assumptions are given. Methods based on compounded functions of proportions for categorical data are applied to many of the scores reviewed and introduced. Appropriate algorithms for these methods are introduced and exemplified. Through the medium of a range of existing data sets the sensitivity of their results to differing scoring systems applied to two way classifications is examined. It is seen that apart from data arising from highly skewed distributions little is to be lost by using simple integer scores.     

Measuring the impact of nonignorability in panel data with non‐monotone nonresponse

The analysis of panel data with non‐monotone nonresponse often relies on the critical and untestable assumption of ignorable missingness. It is important to assess the consequences of departures from the ignorability assumption. Non‐monotone nonresponse, however, can often make such sensitivity analysis infeasible because the likelihood functions for alternative models involve high‐dimensional and difficult‐to‐evaluate integrals with respect to missing outcomes. We develop an extension of the local sensitivity method that overcomes computational difficulty and completely avoids fitting alternative models and evaluating these high‐dimensional integrals. The proposed method is applicable to a wide range of panel outcomes. We apply the method to a Smoking Trend dataset where we relax the standard ignorability assumption and evaluate how smoking‐trend estimates in different groups of US young adults are affected by alternative assumptions about the missing‐data mechanism. The main finding is that the standard estimate in the black male group is sensitive to nonignorable missingness but those in other groups are reasonably robust. Copyright © 2010 John Wiley & Sons, Ltd.     

Relative Efficiency of the Fuzzy p-Value Approach to Hypothesis Testing

In missing data problems, it is often the case that there is a natural test statistic for testing a statistical hypothesis had all the data been observed. A fuzzy  p -value approach to hypothesis testing has recently been proposed which is implemented by imputing the missing values in the "complete data" test statistic by values simulated from the conditional null distribution given the observed data. We argue that imputing data in this way will inevitably lead to loss in power. For the case of scalar parameter, we show that the asymptotic efficiency of the score test based on the imputed "complete data" relative to the score test based on the observed data is given by the ratio of the observed data information to the complete data information. Three examples involving probit regression, normal random effects model, and unidentified paired data are used for illustration. For testing linkage disequilibrium based on pooled genotype data, simulation results show that the imputed Neyman Pearson and Fisher exact tests are less powerful than a Wald-type test based on the observed data maximum likelihood estimator. In conclusion, we caution against the routine use of the fuzzy  p -value approach in latent variable or missing data problems and suggest some viable alternatives.     

Bounds for best response functions in binary games

This paper studies the identification of best response functions in binary games without making strong parametric assumptions about the payoffs. The best response function gives the utility maximizing response to a decision of the other players. This is analogous to the response function in the treatment-response literature, taking the decision of the other players as the treatment, except that the best response function has additional structure implied by the associated utility maximization problem. Further, the relationship between the data and the best response function is not the same as the relationship between the data and the response function in the treatment-response literature. We focus especially on the case of a complete information entry game with two firms. We also discuss the case of an entry game with many firms, non-entry games, and incomplete information. Our analysis of the entry game is based on the observation of realized entry decisions, which we then link to the best response functions under various assumptions including those concerning the level of rationality of the firms, including the assumption of Nash equilibrium play, the symmetry of the payoffs between firms, and whether mixed strategies are admitted.     

A Foundational Justification for a Weighted Likelihood Approach to Inference

Two types of probability are discussed, one of which is additive whilst the other is non-additive. Popular theories that attempt to justify the importance of the additivity of probability are then critically reviewed. By making assumptions the two types of probability put forward are utilised to justify a method of inference which involves betting preferences being revised in light of the data. This method of inference can be viewed as a justification for a weighted likelihood approach to inference where the plausibility of different values of a parameter θ based on the data is measured by the quantity q (θ) = l ( , θ) w (θ), where l ( , θ) is the likelihood function and w (θ) is a weight function. Even though, unlike Bayesian inference, the method has the disadvantageous property that the measure q (θ) is generally non-additive, it is argued that the method has other properties which may be considered very desirable and which have the potential to imply that when everything is taken into account, the method is a serious alternative to the Bayesian approach in many situations. The methodology that is developed is applied to both a toy example and a real example.     

Three Sides of Smoothing: Categorical Data Smoothing, Nonparametric Regression, and Density Estimation

The past forty years have seen a great deal of research into the construction and properties of nonparametric estimates of smooth functions. This research has focused primarily on two sides of the smoothing problem: nonparametric regression and density estimation. Theoretical results for these two situations are similar, and multivariate density estimation was an early justification for the Nadaraya-Watson kernel regression estimator.
A third, less well-explored, strand of applications of smoothing is to the estimation of probabilities in categorical data. In this paper the position of categorical data smoothing as a bridge between nonparametric regression and density estimation is explored. Nonparametric regression provides a paradigm for the construction of effective categorical smoothing estimates, and use of an appropriate likelihood function yields cell probability estimates with many desirable properties. Such estimates can be used to construct regression estimates when one or more of the categorical variables are viewed as response variables. They also lead naturally to the construction of well-behaved density estimates using local or penalized likelihood estimation, which can then be used in a regression context. Several real data sets are used to illustrate these points.     

ML– and semiparametric estimation in logistic models with incomplete covariate data

ML–estimation of regression parameters with incomplete covariate information usually requires a distributional assumption regarding the concerned covariates that implies a source of misspecification. Semiparametric procedures avoid such assumptions at the expense of efficiency. In this paper a simulation study with small sample size is carried out to get an idea of the performance of the ML–estimator under misspecification and to compare it with the semiparametric procedures when the former is based on a correct assumption. The results show that there is only a little gain by correct parametric assumptions, which does not justify the possibly large bias when the assumptions are not met. Additionally, a simple modification of the complete case estimator appears to be nearly semiparametric efficient.     

Maximum likelihood estimation of monotone and concave production frontiers

In this paper we bring together the previously separate parametric and nonparametric approaches to production frontier estimation by developing composed error models for maximum likelihood estimation from nonparametrically specified classes of frontiers. This approach avoids the untestable restrictions of parametric functional forms and also provides a statistical foundation for nonparametric frontier estimation. We first examine the single output setting and then extend our formulation to the multiple output setting. The key step in developing the estimation problems is to identify operational constraint sets to ensure estimation from the desired class of frontiers. We also suggest algorithms for solving the resulting constrained likelihood function optimization problems.The refereeing process of this paper was handled through R. Robert Russell. Helpful comments from Bob Russell and two anonymous referees are gratefully acknowedged. We are, of course, solely responsible for any remaining errors or omissions.     

Nonlinear regression models with single‐index heteroscedasticity

We consider nonlinear heteroscedastic single‐index models where the mean function is a parametric nonlinear model and the variance function depends on a single‐index structure. We develop an efficient estimation method for the parameters in the mean function by using the weighted least squares estimation, and we propose a “delete‐one‐component” estimator for the single‐index in the variance function based on absolute residuals. Asymptotic results of estimators are also investigated. The estimation methods for the error distribution based on the classical empirical distribution function and an empirical likelihood method are discussed. The empirical likelihood method allows for incorporation of the assumptions on the error distribution into the estimation. Simulations illustrate the results, and a real chemical data set is analyzed to demonstrate the performance of the proposed estimators.     

Convenient links for the estimation of hedonic price indexes: the case of unique,infrequently traded assets

Hedonic methods are a prominent approach in the construction of quality‐adjusted price indexes. This paper shows that the process of computing such indexes is substantially simplified if arithmetic (geometric) price indexes are computed based on exponential (log‐linear) hedonic functions estimated by the Poisson pseudo‐maximum likelihood (ordinary least squares) method. A Monte Carlo simulation study based on housing data illustrates the convenience of the links identified and the very attractive properties of the Poisson estimator in the hedonic framework.     

Intensity estimation for inhomogeneous Gibbs point process with covariates‐dependent chemical activity

Recent development of intensity estimation for inhomogeneous spatial point processes with covariates suggests that kerneling in the covariate space is a competitive intensity estimation method for inhomogeneous Poisson processes. It is not known whether this advantageous performance is still valid when the points interact. In the simplest common case, this happens, for example, when the objects presented as points have a spatial dimension. In this paper, kerneling in the covariate space is extended to Gibbs processes with covariates‐dependent chemical activity and inhibitive interactions, and the performance of the approach is studied through extensive simulation experiments. It is demonstrated that under mild assumptions on the dependence of the intensity on covariates, this approach can provide better results than the classical nonparametric method based on local smoothing in the spatial domain. In comparison with the parametric pseudo‐likelihood estimation, the nonparametric approach can be more accurate particularly when the dependence on covariates is weak or if there is uncertainty about the model or about the range of interactions. An important supplementary task is the dimension reduction of the covariate space. It is shown that the techniques based on the inverse regression, previously applied to Cox processes, are useful even when the interactions are present. © 2014 The Authors. Statistica Neerlandica © 2014 VVS.     

INFORMATION THEORETIC APPROACHES IN ECONOMICS

Economics has seen a recent rise in interest in information theory as an alternative framework to the conventional notion of equilibrium as a fixed state, such as Walrasian market‐clearing general equilibrium. The information theoretic approach is predicated on the notion of statistical equilibrium (SE) that takes a distribution over all possible states as an equilibrium, and therefore predicts the endogenous fluctuations of the system along with its central tendency simultaneously. For this reason, SE approaches can explain the observed data without relying on arbitrary assumptions about random noise and provide useful insights for many interesting economic problems that conventional methods have not been able to satisfactorily deal with. In this paper, we review the key elements of information theory focusing on the notions and applications of entropy and SE in economics, particularly paying attention to how entropy concepts open up a new frontline of economic research.     

A simulation estimation analysis of the external debt crises of developing countries

In this paper we develop models of the incidence and extent of external financing crises of developing countries, which lead to multiperiod multinomial discrete choice and discrete/continuous econometric specifications with flexible correlation structures in the unobservables. We show that estimation of these models based on simulation methods has attractive statistical properties and is computationally tractable. Three such simulation estimation methods are exposited, analysed theoretically, and used in practice: a method of smoothly simulated maximum likelihood (SSML) based on a smooth recursive conditioning simulator (SRC), a method of simulated scores (MSS) based on a Gibbs sampling simulator (GSS), and an MSS estimator based on the SRC simulator. The data set used in this study comprises 93 developing countries observed through the 1970–88 period and contains information on external financing responses that was not available to investigators in the past. Moreover, previous studies of external debt problems had to rely on restrictive correlation structures in the unobservables to overcome otherwise intractable computational difficulties. The findings show that being able for the first time to allow for flexible correlation patterns in the unobservables through estimation by simulation has a substantial impact on the parameter estimates obtained from such models. This suggests that past empirical results in this literature require a substantial re-evaluation.