Interval estimation of the kappa coefficient with binary classification and an equal marginal probability model

We derive a likelihood score method for interval estimation of the intraclass version of the kappa coefficient of agreement with binary classification using a general theory of Bartlett (1953, Biometrika 40, 306-317). By exact evaluation, we investigate statistical properties of the score method, the chi-square goodness-of-fit procedure (Donner and Eliasziw, 1992, Statistics in Medicine 11, 1511-1519; Hale and Fleiss, 1993, Biometrics 49, 523-534), and a crude confidence interval for small and medium sample sizes. Actual coverage percentages of the score and chi-square methods are satisfactorily close to the nominal confidence coefficient, while that of the crude method is quite unsatisfactory. The expected length of the score method is shorter than that of the chi-square procedure when the response rate is very small or very large.     

Testing the Intraclass Version of Kappa Coeffcient of Agreement with Binary Scale and Sample Size Determination

The intraclass version of kappa coefficient has been commonly applied as a measure of agreement for two ratings per subject with binary outcome in reliability studies. We present an efficient statistic for testing the strength of kappa agreement using likelihood scores, and derive asymptotic power and sample size formula. Exact evaluation shows that the score test is generally conservative and more powerful than a method based on a chi‐square goodness‐of‐fit statistic (Donner and Eliasziw , 1992, Statistics in Medicine 11 , 1511–1519). In particular, when the research question is one directional, the one‐sided score test is substantially more powerful and the reduction in sample size is appreciable.     

Estimation and comparison of changes in the presence of informative right censoring: conditional linear model

A general linear regression model for the usual least squares estimated rate of change (slope) on censoring time is described as an approximation to account for informative right censoring in estimating and comparing changes of a continuous variable in two groups. Two noniterative estimators for the group slope means, the linear minimum variance unbiased (LMVUB) estimator and the linear minimum mean squared error (LMMSE) estimator, are proposed under this conditional model. In realistic situations, we illustrate that the LMVUB and LMMSE estimators, derived under a simple linear regression model, are quite competitive compared to the pseudo maximum likelihood estimator (PMLE) derived by modeling the censoring probabilities. Generalizations to polynomial response curves and general linear models are also described.     

Comparison of recent estimators of interclass correlation from familial data

The asymptotic and finite-sample properties of several recent estimators of interclass correlation are compared to more traditional estimators in the case of a variable number of siblings per family. It is shown that Karlin's family-weighted pairwise estimator (Karlin, Cameron, and Williams, 1981, Proceedings of the National Academy of Science 78, 2664-2668) is virtually equivalent to the ensemble estimator (Rosner, Donner, and Hennekens, 1977, Applied Statistics 26, 179-187), thus suggesting an estimator of the former's asymptotic variance. Further, an estimator proposed by Srivastava (1984, Biometrika 71, 177-185) is shown to be identical to the modified sib-mean estimator (Konishi, 1982, Annals of the Institute of Statistical Mathematics 34, 505-515) when the sib-sib correlation is estimated by the method of unweighted group means. Although the estimator due to Srivastava has smaller asymptotic variance than the other two, the gain in efficiency is slight, for familial data, both asymptotically and in finite samples.     

Shrinkage and Pretest Nonparametric Estimation of Regression Parameters from Censored Data with Multiple Observations at Each Level of Covariate

A simple linear regression model is considered where the independent variable assumes only a finite number of values and the response variable is randomly right censored. However, the censoring distribution may depend on the covariate values. A class of noniterative estimators for the slope parameter, namely, the noniterative unrestricted estimator, noniterative restricted estimator and noniterative improved pretest estimator are proposed. The asymptotic bias and mean squared errors of the proposed estimators are derived and compared. The relative dominance picture of the estimators is investigated. A simulation study is also performed to asses the properties of the various estimators for small samples.     

Asymptotically efficient two-step estimators of the hazards ratio for follow-up studies and survival data

Asymptotically efficient estimators of a common hazard rate ratio (for follow-up studies) and the proportional hazards ratio (for survival studies) are obtained by a single iteration of the "Mantel-Haenszel" estimator appropriate for each setting. Estimators of their variance are also developed. The two-step estimator for survival data and its variance estimator are shown by simulation to be minimally biased and the estimator is shown to be efficient relative to the Cox partial likelihood estimator in small samples.     

Accuracy of the endpoint assay for virus titration

The statistics of estimators used with the endpoint assay for virus titration were investigated. For a standard assay with 10 wells/dilution, the graphical estimator traditionally used was found to produce estimates with significant positive bias and a relatively low accuracy. Furthermore, the graphical estimator was found to be inconsistent. A superior estimator based on the maximum likelihood principle was developed. The results are discussed in relation to the choice between the endpoint titration assay and the plaque assay, and an alternative two-stage assay is presented.     

Describing Ordinal Odds Ratios for Stratified r × c Tables

For an r × ctable with ordinal responses, odds ratios are commonly used to describe the relationship between the row and column variables. This article shows two types of ordinal odds ratios where local‐global odds ratios are used to compare several groups on a c‐category ordinal response and a global odds ratio is used to measure the global association between a pair of ordinal responses. When there is a stratification factor, we consider Mantel‐Haenszel (MH) type estimators of these odds ratios to summarize the association from several strata. Like the ordinary MH estimator of the common odds ratio for several 2 × 2 contingency tables, the estimators are used when the association is not expected to vary drastically among the strata. Also, the estimators are consistent under the ordinary asymptotic framework in which the number of strata is fixed and also under sparse asymptotics in which the number of strata grows with the sample size. Compared to the maximum likelihood estimators, simulations find that the MH type estimators perform better especially when each stratum has few observations. This article provides variances and covariances formulae for the local‐global odds ratios estimators and applies the bootstrap method to obtain a standard error for the global odds ratio estimator. At the end, we discuss possible ways of testing the homogeneity assumption.     

A Nonrecursive Estimator for the Cox Model

The Cox regression model is one of the most widely used models to incorporate covariates. The frequently used partial likelihood estimator of the regression parameter has to be computed iteratively. In this paper we propose a noniterative estimator for the regression parameter and show that under certain conditions it dominates another noniterative estimator derived by Kalbfleish and Prentice. The new estimator is demonstrated on lifetime data of rats having been subject to insult with a carcinogen.     

Robust inference on the average treatment effect using the outcome highly adaptive lasso

Many estimators of the average effect of a treatment on an outcome require estimation of the propensity score, the outcome regression, or both. It is often beneficial to utilize flexible techniques, such as semiparametric regression or machine learning, to estimate these quantities. However, optimal estimation of these regressions does not necessarily lead to optimal estimation of the average treatment effect, particularly in settings with strong instrumental variables. A recent proposal addressed these issues via the outcome-adaptive lasso, a penalized regression technique for estimating the propensity score that seeks to minimize the impact of instrumental variables on treatment effect estimators. However, a notable limitation of this approach is that its application is restricted to parametric models. We propose a more flexible alternative that we call the outcome highly adaptive lasso. We discuss the large sample theory for this estimator and propose closed-form confidence intervals based on the proposed estimator. We show via simulation that our method offers benefits over several popular approaches.     

Measuring agreement of multivariate discrete survival times using a modified weighted kappa coefficient

Summary .  Assessing agreement is often of interest in clinical studies to evaluate the similarity of measurements produced by different raters or methods on the same subjects. We present a modified weighted kappa coefficient to measure agreement between bivariate discrete survival times. The proposed kappa coefficient accommodates censoring by redistributing the mass of censored observations within the grid where the unobserved events may potentially happen. A generalized modified weighted kappa is proposed for multivariate discrete survival times. We estimate the modified kappa coefficients nonparametrically through a multivariate survival function estimator. The asymptotic properties of the kappa estimators are established and the performance of the estimators are examined through simulation studies of bivariate and trivariate survival times. We illustrate the application of the modified kappa coefficient in the presence of censored observations with data from a prostate cancer study.     

Bias-reduced estimators of conditional odds ratios in matched case-control studies with unmatched confounding

We study bias-reduced estimators of exponentially transformed parameters in general linear models (GLMs) and show how they can be used to obtain bias-reduced conditional (or unconditional) odds ratios in matched case-control studies. Two options are considered and compared: the explicit approach and the implicit approach. The implicit approach is based on the modified score function where bias-reduced estimates are obtained by using iterative procedures to solve the modified score equations. The explicit approach is shown to be a one-step approximation of this iterative procedure. To apply these approaches for the conditional analysis of matched case-control studies, with potentially unmatched confounding and with several exposures, we utilize the relation between the conditional likelihood and the likelihood of the unconditional logit binomial GLM for matched pairs and Cox partial likelihood for matched sets with appropriately setup data. The properties of the estimators are evaluated by using a large Monte Carlo simulation study and an illustration of a real dataset is shown. Researchers reporting the results on the exponentiated scale should use bias-reduced estimators since otherwise the effects can be under or overestimated, where the magnitude of the bias is especially large in studies with smaller sample sizes.     

Estimating the Antimicrobial Log Reduction: Part 2. Presence/Absence Assays

This is part 2 of a pair of papers on antimicrobial assays conducted to estimate the log reduction (LR), in the density of viable microbes, attributable to the germicide. Two alternative definitions of LR were defined in part 1, one based on the mean of the log-transformed densities; the other is based on the logarithm of the mean of densities. In this paper, we evaluate statistical methods for estimating LR from an antimicrobial assay in which the responses are presence/absence observations at each dilution in a series of dilutions. We provide a model for the presence/absence data, and, for each definition of LR, we derive the maximum likelihood estimator (mle). Using computer simulation methods, we compare the mle to several alternative estimators, including an estimator based on averaging the log-transformed most probable number (mpn) values. Standard error formulas for the estimators are also derived and evaluated using computer simulations. This investigation results in the following recommendations. If the parameter of interest is based on the mean of log-transformed densities, then the results favor use of the log-transformed mpn method. If, however, the parameter of interest is based on the logarithm of the mean of densities, then the results show that the mle should be used.     

Estimation of a common effect parameter from sparse follow-up data

Breslow (1981, Biometrika 68, 73-84) has shown that the Mantel-Haenszel odds ratio is a consistent estimator of a common odds ratio in sparse stratifications. For cohort studies, however, estimation of a common risk ratio or risk difference can be of greater interest. Under a binomial sparse-data model, the Mantel-Haenszel risk ratio and risk difference estimators are consistent in sparse stratifications, while the maximum likelihood and weighted least squares estimators are biased. Under Poisson sparse-data models, the Mantel-Haenszel and maximum likelihood rate ratio estimators have equal asymptotic variances under the null hypothesis and are consistent, while the weighted least squares estimators are again biased; similarly, of the common rate difference estimators the weighted least squares estimators are biased, while the estimator employing "Mantel-Haenszel" weights is consistent in sparse data. Variance estimators that are consistent in both sparse data and large strata can be derived for all the Mantel-Haenszel estimators.     

Penalized likelihood phylogenetic inference: bridging the parsimony-likelihood gap

The increasing diversity and heterogeneity of molecular data for phylogeny estimation has led to development of complex models and model-based estimators. Here, we propose a penalized likelihood (PL) framework in which the levels of complexity in the underlying model can be smoothly controlled. We demonstrate the PL framework for a four-taxon tree case and investigate its properties. The PL framework yields an estimator in which the majority of currently employed estimators such as the maximum-parsimony estimator, homogeneous likelihood estimator, gamma mixture likelihood estimator, etc., become special cases of a single family of PL estimators. Furthermore, using the appropriate penalty function, the complexity of the underlying models can be partitioned into separately controlled classes allowing flexible control of model complexity.     

Software Review of Distribution Fitting Programs: Crystal Ball and BestFit Add-In to @RISK

Simulation software programs continue to evolve and to meet the needs of risk analysts. In the past several years, two spreadsheet add-in programs added the capability of fitting distributions to data to their tool kits using classical statistical (i.e., non-Bayesian) methods. Crystal Ball version 4.0 now contains this capability in its standard program (and in Crystal Ball Pro version 4.0), while the BestFit software program is a component of the @RISK Decision Tools Suite that can also be purchased as a stand-alone program. Both programs will automatically fit distributions using maximum likelihood estimators to continuous data and provide goodness-of-fit statistics based on chi-squared, Kolmogorov-Smirnov, and Anderson-Darling tests. BestFit will also fit discrete distributions, and for all distributions it offers the option of optimizing the fit based on the goodness-of-fit parameters. Analysts should be wary of placing too much emphasis on the goodness-of-fit statistics given their limitations, and the fact that only some of the statistics are appropriately corrected to account for the fact that the distribution parameters are also fit using the data. These programs dramatically simplify efforts to use maximum likelihood estimation to fit distributions. However, the fact that a program is used to fit distributions should not be viewed as validation that the data have been fitted and interpreted correctly. Both programs rely heavily on the analyst's judgment and will allow analysts to fit inappropriate distributions. Currently, both programs could be improved by adding the ability to perform extensive basic exploratory data analysis and to give regression diagnostics that are needed to satisfy critical analysts or reviewers. Given that Bayesian methods are central to risk analysis, adding the capability of fitting distributions by combining data with prior information would greatly increase the utility of these programs.     

Homogeneity/Heterogeneity Hypotheses for Standardized Mortality Ratios Based on Minimum Power‐divergence Estimators

This paper analyzes the power divergence estimators when homogeneity/heterogeneity hypotheses among standardized mortality ratios (SMRs) are taken into account. A Monte Carlo study shows that when the standard mortality rate is not external, that is it is estimated from the sample data, these estimators have a good performance even for small sample sets and in particular the minimum chi‐square estimators have a better behavior compared to the classical maximum likelihood estimators. In order to make decisions under homogeneity/heterogeneity hypotheses of SMRs we propose some test‐statistics which consider the minimum power divergence estimators. Through a numerical example focused on SMRs of melanoma mortality ratios in different regions of the US, a homogeneity/heterogeneity study is illustrated.     

Testing hypotheses in case-control studies--equivalence of Mantel-Haenszel statistics and logit score tests.

The two approaches in common use for the analysis of case-control studies are cross-classification by confounding variables, and modeling the logarithm of the odds ratio as a function of exposure and confounding variables. We show here that score statistics derived from the likelihood function in the latter approach are identical to the Mantel-Haenszel test statistics appropriate for the former approach. This identity holds in the most general situation considered, testing for marginal homogeneity in mK tables. This equivalence is demonstrated by a permutational argument which leads to a general likelihood expression in which the exposure variable may be a vector of discrete and/or continuous variables and in which more than two comparison groups may be considered. This likelihood can be used in analyzing studies in which there are multiple controls for each case or in which several disease categories are being compared. The possibility of including continuous variables makes this likelihood useful in situations that cannot be treated using the Mantel-Haenszel cross-classification approach.     

Estimation of regression parameters and the hazard function in transformed linear survival models

An estimator of the regression parameters in a semiparametric transformed linear survival model is examined. This estimator consists of a single Newton-like update of the solution to a rank-based estimating equation from an initial consistent estimator. An automated penalized likelihood algorithm is proposed for estimating the optimal weight function for the estimating equations and the error hazard function that is needed in the variance estimator. In simulations, the estimated optimal weights are found to give reasonably efficient estimators of the regression parameters, and the variance estimators are found to perform well. The methodology is applied to an analysis of prognostic factors in non-Hodgkin's lymphoma.     

Efficient inference for random-coefficient growth curve models with unbalanced data

Growth and dose-response curve studies often result in incomplete or unbalanced data. Random-effects models together with a variety of computer-intensive iterative techniques have been suggested for the analysis of such data. This paper is concerned with a noniterative method for estimating and comparing location parameters in random-coefficient growth curve models. Consistent and asymptotically efficient estimators of the location parameters are obtained using estimated generalized least squares. Two criteria for testing multivariate general linear hypotheses are introduced and their asymptotic properties are investigated. The results are applied to clinical data obtained on the blood ultrafiltration performance of hemodialyzers used in the treatment of patients with end-stage renal disease.