A simple alternative to the likelihood ratio test in a logistic discrimination problem

We consider the problem of hypothesis-testing under a logistic model with two dichotomous independent variables. In particular, we consider the case in which the coefficients β₁, and β₂ of these variables are known on an a priori basis to not be of opposite sign. For this situation we show that there exists a simple nonparametric altenative to the likelihood ratio test for testing H₀: β₁ = β₂ = 0 VS.H₁ at least one β₁ = 0. We find the asympotic relative efficiency of this test and show that it exceeds 0.90 under a wide range of conditions. We also given an example.     

A Comparison of Large-Sample Confidence Interval Methods for the Difference of Two Binomial Probabilities

A simulation study was done to compare seven confidence interval methods, based on the normal approximation, for the difference of two binomial probabilities. Cases considered included minimum expected cell sizes ranging from 2 to 15 and smallest group sizes (NMIN) ranging from 6 to 100. Our recommendation is to use a continuity correction of 1/(2 NMIN) combined with the use of (N ? 1) rather than N in the estimate of the standard error. For all of the cases considered with minimum expected cell size of at least 3, this method gave coverage probabilities close to or greater than the nominal 90% and 95%. The Yates method is also acceptable, but it is slightly more conservative. At the other extreme, the usual method (with no continuity correction) does not provide adequate coverage even at the larger sample sizes. For the 99% intervals, our recommended method and the Yates correction performed equally well and are reasonable for minimum expected cell sizes of at least 5. None of the methods performed consistently well for a minimum expected cell size of 2.     

Two-stage hierarchical modeling for analysis of subpopulations in conditional distributions

In this work, we develop modeling and estimation approach for the analysis of cross-sectional clustered data with multimodal conditional distributions where the main interest is in analysis of subpopulations. It is proposed to model such data in a hierarchical model with conditional distributions viewed as finite mixtures of normal components. With a large number of observations in the lowest level clusters, a two-stage estimation approach is used. In the first stage, the normal mixture parameters in each lowest level cluster are estimated using robust methods. Robust alternatives to the maximum likelihood estimation are used to provide stable results even for data with conditional distributions such that their components may not quite meet normality assumptions. Then the lowest level cluster-specific means and standard deviations are modeled in a mixed effects model in the second stage. A small simulation study was conducted to compare performance of finite normal mixture population parameter estimates based on robust and maximum likelihood estimation in stage 1. The proposed modeling approach is illustrated through the analysis of mice tendon fibril diameters data. Analyses results address genotype differences between corresponding components in the mixtures and demonstrate advantages of robust estimation in stage 1.     

A Survey regarding the Reporting of Simulation Studies

Hoaglin and Andrews (1975) proposed standards for computational practice and the reporting of computation-based studies. They observed that “statisticians … often pay too little attention to their own principles of design.” To see if the design and reporting had improved, we surveyed five major statistical journals for 1975, 1978, and 1981 to ascertain whether reported simulation studies involved a specified design, justified the choice of the number of iterations, and specified the random number generator(s) used. Eighteen percent of the 1,198 papers surveyed included results based on simulation. We found that 9% of the papers including a simulation study justified the choice of the number of iterations and 44% at least partially specified the random number generator. Hoaglin and Andrews's observation appears still to be true.     

Two-step jackknife bias reduction for logistic regression mles

Maximum likelihood estimates (MLEs) for logistic regression coefficients are known to be biased in finite samples and consequently may produce misleading inferences. Bias adjusted estimates can be calculated using the first-order asymptotic bias derived from a Taylor series expansion of the log likelihood. Jackknifing can also be used to obtain bias corrected estimates, but the approach is computationally intensive, requiring an additional series of iterations (steps) for each observation in the dataset.Although the one-step jackknife has been shown to be useful in logistic regression diagnostics and i the estimation of classification error rates, it does not effectively reduce bias. The two-step jackknife, however, can reduce computation in moderate-sized samples, provide estimates of dispersion and classification error, and appears to be effective in bias reduction. Another alternative, a two-step closed-form approximation, is found to be similar to the Taylo series method in certain circumstances. Monte Carlo simulations indicate that all the procedures, but particularly the multi-step jackknife, may tend to over-correct in very small samples. Comparison of the various bias correction proceduresin an example from the medical literature illustrates that bias correction can have a considerable impact on inference