Semiparametric Likelihood Research Articles

Constrained Maximum Likelihood Estimation under Logistic Regression Models Based on Case-Control Data

Abstract We study constrained maximum likelihood estimation and tests under the logistic regression model based on case-control data. Our approach is based on the semiparametric profile log likelihood function under a two-sample semiparametric model, which is equivalent to the assumed logistic regression model. We show that the semiparametric likelihood ratio statistic, the Lagrangian multiplier statistic, and the Wald statistic are asymptotically equivalent and that they have an asymptotic chi-squared distribution under the null hypothesis and an asymptotic noncentral chi-squared distribution under local alternatives to the null hypothesis. Moreover, we demonstrate that the three test statistics and their asymptotic distributions may be obtained by fitting the prospective logistic regression model to case-control data. We present some results on simulation and on the analysis of two real data sets.

Nonparametric analysis of doubly truncated data

In many application, statistical data are frequently observed subject to a retrospective sampling criterion resulting in double-truncated data. For double-truncated data, the distribution of lifetime T is observable only when U ≤ T ≤ V, where U and V are the left-truncated and right-truncated time, respectively. Under the assumption of independence between T and (U, V), the nonparametric maximum likelihood estimate (NPMLE) of the distribution function of T (denoted by was developed by Efron and Petrosian (1999). For the special case V = U + d 0, when the distribution of U is parameterized as G(x; θ) and F is left unspecified, a semiparametric MLE estimator of F (denoted by can be obtained using the arguments similar to those of Wang (1989). Further, we can obtain interval estimation for F using the bootstrap method (Qin and Wang, 2001) or the semiparametric likelihood ratio (SLR) method proposed by Li et al. (1997). In this article, a simulation study is conducted to compare the performance between and . ...

Persistence in Inflation: Does Aggregation Cause Long Memory?

This paper examines persistence in Turkish inflation rates using data from consumer and wholesale price indices. The inflationary process in Turkey is believed to be highly inertial, which should lead to strongly persistent inflation series. Persistence of seventy-five inflation series at various aggregation levels are examined by estimating models that allow long memory through fractional differencing. The order of fractional differencing is estimated using several semiparametric and maximum likelihood methods. Persistence of each series is evaluated using the time required for a given percentage of the effect of a shock to dissipate. We find that disaggregate inflation series show no significant persistence. We found that only twelve out of seventy-five series require more than six months for 99 percent of the effect of a shock to dissipate. Thus, the paper finds evidence of spurious long memory due to aggregation.

Merging information for a semiparametric projection density estimation

A semiparametric density estimator is proposed under a m-sample density ratio model, which specifies that the ratio of m − 1 probability density functions with respect to the mth is of a known parametric form without reference to any parametric model. This model arises naturally from retrospective studies and multinomial logistic regression model. A projection density estimator is constructed by smoothing the increments of the maximum semiparametric likelihood estimator of the underlying distribution function, using the combined data from all the samples. We also establish some asymptotic results on the proposed projection density estimator. To cite this article: J.-B. Aubin, S. Leoni-Aubin, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Using entropy distance measures for testing odds ratio parameters under logistic regression models based on case–control data

We propose two retrospective test statistics for testing the vector of odds ratio parameters under the logistic regression model based on case–control data by exploiting the density ratio structure under a two-sample semiparametric model, which is equivalent to the assumed logistic regression model. The proposed test statistics are based on Kullback–Leibler entropy distance and are particularly relevant to the case–control sampling plan. These two test statistics have identical asymptotic chi-squared distributions under the null hypothesis and identical asymptotic noncentral chi-squared distributions under local alternatives to the null hypothesis. Moreover, the proposed test statistics require computation of the maximum semiparametric likelihood estimators of the underlying parameters, but are otherwise easily computed. We present some results on simulation and on the analysis of two real data sets.

Can Risk Aversion Explain Schooling Attainments? Evidence from Italy

Using unique Italian panel data, in which individual differences in behavior toward risk are measured from answers to a lottery question, we investigate if (and to what extent) risk aversion can explain differences in schooling attainments. We formulate the schooling decision process as a reduced-form dynamic discrete choice. The model is estimated with a degree of flexibility virtually compatible with semiparametric likelihood techniques. We analyze how grade transition from one level to the next varies with preference heterogeneity (risk aversion), parental human capital, socioeconomic variables and persistent unobserved (to the econometrician) heterogeneity. We present evidence that schooling attainments decrease with risk aversion, but despite a statistically significant effect, differences in attitudes toward risk account for a modest portion of the probability of entering higher education. Differences in ability(ies) and in parental human capital are much more important. in the most general version of the model, the likelihood function is the joint probability of schooling attainments, and post-schooling wealth and risk aversion.

Can Risk Aversion Explain Schooling Attainments? Evidence from Italy

Using unique Italian panel data, in which individual differences in behavior toward risk are measured from answers to a lottery question, we investigate if (and to what extent) risk aversion can explain differences in schooling attainments. We formulate the schooling decision process as a reduced-form dynamic discrete choice. The model is estimated with a degree of flexibility virtually compatible with semiparametric likelihood techniques. We analyze how grade transition from one level to the next varies with preference heterogeneity (risk aversion), parental human capital, socioeconomic variables and persistent unobserved (to the econometrician) heterogeneity. We present evidence that schooling attainments decrease with risk aversion, but despite a statistically significant effect, differences in attitudes toward risk account for a modest portion of the probability of entering higher education. Differences in ability(ies) and in parental human capital are much more important. in the most general version of the model, the likelihood function is the joint probability of schooling attainments, and post-schooling wealth and risk aversion.

Density estimation under a two-sample semiparametric model

We consider estimating a density function under a two-sample semiparametric model in which the log ratio of two density functions is a quadratic function of data. This two-sample semiparametric model, arising naturally from case–control studies and logistic discriminant analysis, can be regarded as a biased sampling model. Under this model, the difference between the two samples is quantified. A kernel-based density estimator is constructed by smoothing the increments of the maximum semiparametric likelihood estimator of the underlying distribution function. The required computation for our method can be accomplished by using the standard statistical software packages for categorical data analysis. We establish some asymptotic results on the proposed kernel density estimator. In addition, we present some results on a simulation study and on the analysis of two data sets to demonstrate the utility of the proposed density estimator.

Likelihood-based kernel estimation in semiparametric errors-in-covariables models with validation data

We present methods to handle error-in-variables models. Kernel-based likelihood score estimating equation methods are developed for estimating conditional density parameters. In particular, a semiparametric likelihood method is proposed for sufficiently using the information in the data. The asymptotic distribution theory is derived. Small sample simulations and a real data set are used to illustrate the proposed estimation methods.

Analysis of two-sample truncated data using generalized logistic models

Parallel to Cox's [JRSS B34 (1972) 187–230] proportional hazards model, generalized logistic models have been discussed by Anderson [Bull. Int. Statist. Inst. 48 (1979) 35–53] and others. The essential assumption is that the two densities ratio has a known parametric form. A nice property of this model is that it naturally relates to the logistic regression model for categorical data. In astronomic, demographic, epidemiological, and other studies the variable of interest is often truncated by an associated variable. This paper studies generalized logistic models for the two-sample truncated data problem, where the two lifetime densities ratio is assumed to have the form exp { α + φ ( x ; β ) } . Here φ is a known function of x and β , and the baseline density is unspecified. We develop a semiparametric maximum likelihood method for the case where the two samples have a common truncation distribution. It is shown that inferences for β do not depend the nonparametric components. We also derive an iterative algorithm to maximize the semiparametric likelihood for the general case where different truncation distributions are allowed. We further discuss how to check goodness of fit of the generalized logistic model. The developed methods are illustrated and evaluated using both simulated and real data.

A Semiparametric Two‐Component “Compound” Mixture Model and Its Application to Estimating Malaria Attributable Fractions

Malaria remains a major epidemiologic problem in many developing countries. Malaria is defined as the presence of parasites and symptoms (usually fever) due to the parasites. In endemic areas, an individual may have symptoms attributable either to malaria or to other causes. From a clinical viewpoint, it is important to correctly diagnose an individual who has developed symptoms so that the appropriate treatments can be given. From an epidemiologic and economic viewpoint, it is important to determine the proportion of malaria-affected cases in individuals who have symptoms so that policies on intervention program can be developed. Once symptoms have developed in an individual, the diagnosis of malaria can be based on the analysis of the parasite levels in blood samples. However, even a blood test is not conclusive as in endemic areas many healthy individuals can have parasites in their blood slides. Therefore, data from this type of study can be viewed as coming from a mixture distribution, with the components corresponding to malaria and non-malaria cases. A unique feature in this type of data, however, is the fact that a proportion of the non-malaria cases have zero parasite levels. Therefore, one of the component distributions is itself a mixture distribution. In this article, we propose a semiparametric likelihood approach for estimating the proportion of clinical malaria using parasite-level data from a group of individuals with symptoms. Our approach assumes the density ratio for the parasite levels in clinical malaria and nonclinical malaria cases can be modeled using a logistic model. We use empirical likelihood to combine the zero and nonzero data. The maximum semiparametric likelihood estimate is more efficient than existing nonparametric estimates using only the frequencies of zero and nonzero data. On the other hand, it is more robust than a fully parametric maximum likelihood estimate that assumes a parametric model for the nonzero data. Simulation results show that the performance of the proposed method is satisfactory. The proposed method is used to analyze data from a malaria survey carried out in Tanzania.

Marginal likelihood, conditional likelihood and empirical likelihood: Connections and applications

Marginal likelihood and conditional likelihood are often used for eliminating nuisance parameters. For a parametric model, it is well known that the full likelihood can be decomposed into the product of a conditional likelihood and a marginal likelihood. This property is less transparent in a nonparametric or semiparametric likelihood setting. In this paper we show that this nice parametric likelihood property can be carried over to the empirical likelihood world. We discuss applications in case-control studies, genetical linkage analysis, genetical quantitative traits analysis, tuberculosis infection data and unordered-paired data, all of which can be treated as semiparametric finite mixture models. We consider the estimation problem in detail in the simplest case of unordered-paired data where we can only observe the minimum and maximum values of two random variables; the identities of the minimum and maximum values are lost. The profile empirical likelihood approach is used for maximum semiparametric likelihood estimation. We present some large-sample results along with a simulation study.

Testing the Goodness of Fit of Multivariate Multiplicative-intercept Risk Models Based on Case-control Data

The validity of the multivariate multiplicative-intercept risk model with I +1 categories based on casecontrol data is tested. After reparametrization, the assumed risk model is equivalent to an (I +1) -sample semiparametric model in which the I ratios of two unspecified density functions have known parametric forms. By identifying this (I +1) -sample semiparametric model, which is of intrinsic interest in general (I +1) -sample problems, with an (I +1) -sample semiparametric selection bias model, we propose a weighted Kolmogorov-Smirnov-type statistic to test the validity of the multivariate multiplicativeintercept risk model. Established are some asymptotic results associated with the proposed test statistic, also established is an optimal property for the maximum semiparametric likelihood estimator of the parameters in the (I +1) -sample semiparametric selection bias model. In addition, a bootstrap procedure along with some results on analysis of two real data sets is proposed.

A semiparametric hypothesis testing procedure for the ROC curve area under a density ratio model

We propose a semiparametric Wald statistic to test whether a diagnostic test is capable of discriminating between diseased and nondiseased subjects based on the receiver operating characteristic (ROC) curve area under a two-sample semiparametric density ratio model. The proposed Wald test is constructed on the basis of the maximum semiparametric likelihood estimator of the ROC curve area. The proposed test statistic has an asymptotic chi-squared distribution under the null hypothesis and an asymptotic noncentral chi-squared distribution under local alternatives to the null hypothesis. We present some results on a simulation study and on the analysis of two data sets.

Nonparametric and Semiparametric Models for Missing Covariates in Parametric Regression

Robustness of covariate modeling for the missing-covariate problem in parametric regression is studied under the missing-at-random assumption. For a simple missing-covariate pattern, nonparametric covariate model is proposed and is shown to yield a consistent and semiparametrically efficient estimator for the regression parameter. Total robustness is achieved in this situation. For more general missingcovariate patterns, a novel semiparametric modeling approach is proposed for the covariates. In this approach, the covariate distribution is first decomposed into the product of a series of conditional distributions according to the overall missing-data patterns, and the conditional distributions are then represented in the general odds ratio form. The general odds ratios are modeled parametrically, and the other components of the covariate distribution are modeled nonparametrically. Maximum semiparametric likelihood is used to find the parameter estimates. The proposed method yields a consistent estimator for the regression parameter when the odds ratios are modeled correctly. In general, the semiparametric covariate modeling strategy increases the robustness against covariate model misspecification when compared with the parametric modeling strategy proposed by Lipsitz and Ibrahim. The new covariate modeling approach can also be incorporated into the doubly robust procedure of Robins et al. to increase protection against misspecification of the missing-data mechanism. In addition, the proposed modeling strategy avoids the usually intractable integrations involved in the maximization of the incomplete-data likelihood with parametric covariate models. The proposed method can be applied to many regression models to handle incomplete covariates.

Assessing goodness‐of‐fit of categorical regression models based on case‐control data

SummaryDemonstrated equivalence between a categorical regression model based on case‐control data and an I‐sample semiparametric selection bias model leads to a new goodness‐of‐fit test. The proposed test statistic is an extension of an existing Kolmogorov–Smirnov‐type statistic and is the weighted average of the absolute differences between two estimated distribution functions in each response category. The paper establishes an optimal property for the maximum semiparametric likelihood estimator of the parameters in the I‐sample semiparametric selection bias model. It also presents a bootstrap procedure, some simulation results and an analysis of two real datasets.

Bias-corrected maximum semiparametric likelihood estimation under logistic regression models based on case–control data

We consider the corrective approach (Theoretical Statistics, Chapman & Hall, London, 1974, p. 310) and preventive approach (Biometrica 80 (1993) 27) to bias reduction of maximum likelihood estimators under the logistic regression model based on case–control data. The proposed bias-corrected maximum likelihood estimators are based on the semiparametric profile log likelihood function under a two-sample semiparametric model, which is equivalent to the assumed logistic regression model. We show that the prospective and retrospective analyses on the basis of the corrective approach to bias reduction produce identical bias-corrected maximum likelihood estimators of the odds ratio parameter, but this does not hold when using the preventive approach unless the case and control sample sizes are identical. We present some results on simulation and on the analysis of two real data sets.

Semiparametric estimation of tag loss and reporting rates for tag-recovery experiments using exact time-at-liberty data.

We present a semiparametric likelihood approach to estimating reporting rates and tag-loss rates from the tags returned from capture-recapture studies. Such studies are commonly used to estimate critical population parameters. Tag loss rates are estimated using double-tagged animals, while reporting rates are estimated using information from high-reward tags. A likelihood function is constructed based on the conditional distribution of the type of tag returned (low or high reward, single or double tag), given that a tag has been returned. This involves many sparse 5 x 1 tag-return contingency tables, and choosing a good functional form for the tag loss rate is difficult with such data. We model tag-loss rates using monotone-smoothing splines, and use these nonparametric estimates to diagnose the parametric form of the tag-loss rate. The nonparametric methods can also be used directly to model tag-loss rates.

A semiparametric likelihood approach to joint modeling of longitudinal and time-to-event data.

Joint models for a time-to-event (e.g., survival) and a longitudinal response have generated considerable recent interest. The longitudinal data are assumed to follow a mixed effects model, and a proportional hazards model depending on the longitudinal random effects and other covariates is assumed for the survival endpoint. Interest may focus on inference on the longitudinal data process, which is informatively censored, or on the hazard relationship. Several methods for fitting such models have been proposed, most requiring a parametric distributional assumption (normality) on the random effects. A natural concern is sensitivity to violation of this assumption; moreover, a restrictive distributional assumption may obscure key features in the data. We investigate these issues through our proposal of a likelihood-based approach that requires only the assumption that the random effects have a smooth density. Implementation via the EM algorithm is described, and performance and the benefits for uncovering noteworthy features are illustrated by application to data from an HIV clinical trial and by simulation.

Double-Semiparametric Method for Missing Covariates in Cox Regression Models

The problem of nuisance covariate model specification is considered in Cox regression where the maximum semiparametric likelihood method is used to handle the missing covariates. A component of the covariates is modeled nonparametrically to achieve robustness against covariate model misspecification and to reduce the number of possibly intractable integrations involved in the parametric modeling of the covariates. The statistical properties of the proposed method are examined. It is found that in some important situations, the maximum semiparametric likelihood can be applied without making any additional parametric model assumptions on covariates. The proposed method can yield a more efficient estimator than the nonparametric imputation methods and does not require specification of the missingness mechanism when compared with the inverse probability weighting method. A real data example is analyzed to demonstrate use of the proposed method.