Information from various public and private data sources of extremely large sample sizes are now increasingly available for research purposes. Statistical methods are needed for using information from such big data sources while analyzing data from individual studies that may collect more detailed information required for addressing specific hypotheses of interest. In this article, we consider the problem of building regression models based on individual-level data from an “internal” study while using summary-level information, such as information on parameters for reduced models, from an “external” big data source. We identify a set of very general constraints that link internal and external models. These constraints are used to develop a framework for semiparametric maximum likelihood inference that allows the distribution of covariates to be estimated using either the internal sample or an external reference sample. We develop extensions for handling complex stratified sampling designs, such as case-co...

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Constrained Maximum Likelihood Estimation for Model Calibration Using Summary-Level Information From External Big Data Sources

SUMMARY Not until recently has much attention been given to deriving maximum likelihood methods for estimating the intercept and slope parameters from a binormal ROC curve that assesses the accuracy of a continuous diagnostic test. We propose two new methods for estimating these parameters. The first method uses the profile likelihood and a simple algorithm to produce fully efficient estimates. The second method is based on a pseudo-maximum likelihood that can easily accommodate adjusting for covariates that could affect the accuracy of the continuous test.

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Semi-parametric estimation of the binormal ROC curve for a continuous diagnostic test.

Summary. In many situations information from a sample of individuals can be supplemented by population level information on the relationship between a dependent variable and explan atory variables. Inclusion of the population level information can reduce bias and increase the efficiency of the parameter estimates. Population level information can be incorporated via con straints on functions of the model parameters. In general the constraints are non-linear, making the task of maximum likelihood estimation more difficult. We develop an alternative approach exploiting the notion of an empirical likelihood. It is shown that, within the framework of gener alized linear models, the population level information corresponds to linear constraints, which are comparatively easy to handle. We provide a two-step algorithm that produces parameter estimates by using only unconstrained estimation. We also provide computable expressions for the standard errors. We give an application to demographic hazard modelling by combining panel survey data with birth registration data to estimate annual birth probabilities by parity.

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Generalized linear models incorporating population level information: an empirical‐likelihood‐based approach

Receiver operating characteristic (ROC) curve is widely applied in measuring discriminatory ability of diagnostic or prognostic tests. This makes the ROC analysis one of the most active research areas in medical statistics. Many parametric and semiparametric estimation methods have been proposed for estimating the ROC curve and its functionals. In this paper, we propose the Bayesian bootstrap (BB), a fully nonparametric estimation method, for the ROC curve and its functionals, such as the area under the curve (AUC). The BB method offers a bandwidth-free smoothing approach to the empirical estimate, and gives credible bounds. The accuracy of the estimate of the ROC curve in the simulation studies is examined by the integrated absolute error. In comparison with other existing curve estimation methods, the BB method performs well in terms of accuracy, robustness and simplicity. We also propose a procedure based on the BB approach to test the binormality assumption.

https://www4.stat.ncsu.edu/~ghosal/papers/ROCBB.pdf

Bayesian bootstrap estimation of ROC curve

Preface 1. Econometric information recovery Part I. Traditional Parametric and Semiparametric Probability Models: Estimation and Inference: 2. Formulation and analysis of parametric and semiparametric linear models 3. Method of moments, GMM, and estimating equations Part II. Formulation and Solution of Stochastic Inverse Problems: 4. A stochastic-empirical likelihood inverse problem: formulation and estimation 5. A stochastic-empirical likelihood inverse problem: inference 6. Kullback-Leibler information and the maximum empirical exponential likelihood Part III. A Family of Minimum Discrepancy Estimators: 7. The Cressie-Read family of divergence measures and likelihood functions 8. Cressie-Read-MEL-type estimators in practice: evidence of estimation and inference sampling performance Part IV. Binary Discrete Choice MPD-EML Econometric Models: 9. Family of distribution functions for the binary response-choice model 10. Estimation and inference for the binary response model based on the MPD family of distributions Part V. Optimal Convex Divergence: 11. Choosing the optimal divergence under quadratic loss 12. Epilogue.

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An Information Theoretic Approach to Econometrics

Imbens & Lancaster (1994) pointed out that census reports can be interpreted as providing nearly exact knowledge of moments of the marginal distribution of economic variables. In this paper we show that empirical likelihood can effectively incorporate auxiliary information like this as long as it can be summarised as unbiased estimating equations. By combining empirical and parametric likelihoods, we show that the combined likelihood can produce valid inferences for the underlying parameters. A Wilks' type theorem is proved for the combined likelihood ratio statistic. Simulation results demonstrate that the performance of the combined likelihood ratio confidence intervals is better than conventional confidence intervals that use a normal approximation.

Miscellanea. Combining parametric and empirical likelihoods

SUMMARY Estimation of a receiver operating characteristic, ROC, curve is usually based either on a fully parametric model such as a normal model or on a fully nonparametric model. In this paper, we explore a semiparametric approach by assuming a density ratio model for disease and disease-free densities. This model has a natural connection with the logistic regression model. The proposed semiparametric approach is more robust than a fully parametric approach and is more efficient than a fully nonparametric approach. Two real examples demonstrate that the ROC curve estimated by our semiparametric method is much smoother than that estimated by the nonparametric method.

Jing Qin

Papers

Miscellanea. Combining parametric and empirical likelihoods

Using logistic regression procedures for estimating receiver operating characteristic curves