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.     

Semiparametric Transformation Rate Model for Recurrent Event Data

In this article, we propose a class of semiparametric transformation rate models for recurrent event data subject to right censoring and potentially stopped by a terminating event (e.g., death). These transformation models include both additive rates model and proportional rates model as special cases. Respecting the property that no recurrent events can occur after the terminating event, we model the conditional recurrent event rate given survival. Weighted estimating equations are constructed to estimate the regression coefficients and baseline rate function. In particular, the baseline rate function is approximated by wavelet function. Asymptotic properties of the proposed estimators are derived and a data-dependent criterion is proposed for selecting the most suitable transformation. Simulation studies show that the proposed estimators perform well for practical sample sizes. The proposed methods are used in two real-data examples: a randomized trial of rhDNase and a community trial of vitamin A.     

Estimating Incidence Rates from Population‐Based Case‐Control Studies in the Presence of Nonrespondents

In population‐based case‐control studies, it is of great public‐health importance to estimate the disease incidence rates associated with different levels of risk factors. This estimation is complicated by the fact that in such studies the selection probabilities for the cases and controls are unequal. A further complication arises when the subjects who are selected into the study do not participate (i.e. become nonrespondents) and nonrespondents differ systematically from respondents. In this paper, we show how to account for unequal selection probabilities as well as differential nonresponses in the incidence estimation. We use two logistic models, one relating the disease incidence rate to the risk factors, and one modelling the predictors that affect the nonresponse probability. After estimating the regression parameters in the nonresponse model, we estimate the regression parameters in the disease incidence model by a weighted estimating function that weights a respondent's contribution to the likelihood score function by the inverse of the product of his/her selection probability and his/her model‐predicted response probability. The resulting estimators of the regression parameters and the corresponding estimators of the incidence rates are shown to be consistent and asymptotically normal with easily estimated variances. Simulation results demonstrate that the asymptotic approximations are adequate for practical use and that failure to adjust for nonresponses could result in severe biases. An illustration with data from a cardiovascular study that motivated this work is presented.     

Modified Gaussian estimation for correlated binary data

In this paper, we develop a Gaussian estimation (GE) procedure to estimate the parameters of a regression model for correlated (longitudinal) binary response data using a working correlation matrix. A two‐step iterative procedure is proposed for estimating the regression parameters by the GE method and the correlation parameters by the method of moments. Consistency properties of the estimators are discussed. A simulation study was conducted to compare 11 estimators of the regression parameters, namely, four versions of the GE, five versions of the generalized estimating equations (GEEs), and two versions of the weighted GEE. Simulations show that (i) the Gaussian estimates have the smallest mean square error and best coverage probability if the working correlation structure is correctly specified and (ii) when the working correlation structure is correctly specified, the GE and the GEE with exchangeable correlation structure perform best as opposed to when the correlation structure is misspecified.     

Survival estimation using splines

A nonparametric maximum likelihood procedure is given for estimating the survivor function from right-censored data. It approximates the hazard rate by a simple function such as a spline, with different approximations yielding different estimators. A special case is that proposed by Nelson (1969, Journal of Quality Technology 1, 27-52) and Altshuler (1970, Mathematical Biosciences 6, 1-11). The estimators are uniformly consistent and have the same asymptotic weak convergence properties as the Kaplan-Meier (1958, Journal of the American Statistical Association 53, 457-481) estimator. However, in small and in heavily censored samples, the simplest spline estimators have uniformly smaller mean squared error than do the Kaplan-Meier and Nelson-Altshuler estimators. The procedure is extended to estimate the baseline hazard rate and regression coefficients in the Cox (1972, Journal of the Royal Statistical Society, Series B 34, 187-220) proportional hazards model and is illustrated using experimental carcinogenesis data.     

Overweight in adolescents: Implications for health expenditures

We consider a compelling research question raised by the growing prevalence of overweight among adolescents: do overweight adolescents incur greater health care expenditures than adolescents of normal weight? To address this question, we use data from the Medical Expenditure Panel Survey (MEPS) and estimate a two-part, generalized linear model (GLM) of health spending. Considering separate models by gender, we find that overweight females incur $790 more in annual expenditures than those of normal weight but we find no expenditure differences by bodyweight for males. We find that mental health spending is associated with part of the disparity in expenditures for adolescent females but establishing causality between mental health problems and weight-related health expenditure differences is challenging.     

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.     

Multivariate survival analysis for case-control family data

Multivariate survival data arise from case-control family studies in which the ages at disease onset for family members may be correlated. In this paper, we consider a multivariate survival model with the marginal hazard function following the proportional hazards model. We use a frailty-based approach in the spirit of Glidden and Self (1999) to account for the correlation of ages at onset among family members. Specifically, we first estimate the baseline hazard function nonparametrically by the innovation theorem, and then obtain maximum pseudolikelihood estimators for the regression and correlation parameters plugging in the baseline hazard function estimator. We establish a connection with a previously proposed generalized estimating equation-based approach. Simulation studies and an analysis of case-control family data of breast cancer illustrate the methodology's practical utility.     

Estimation of the Density of G Given Observations of Y=G+E

We study the problem of estimating the density of a random variable G, given observations of a random variable Y = G + E. The random variable E is independent of G and its probability distribution function is considered as known. We build a family of estimators of the density of G using characteristic functions. We then derive a family of estimators of the density of Y based on the model for Y. The estimators are shown to be asymptotically unbiased and consistent. Simulations show that these estimators are better, as measured by integrated squared error, than the standard kernel estimators. Finally, we give an example of the use of this method for the detection of major genes in animal populations.     

Adjustment for Missingness Using Auxiliary Information in Semiparametric Regression

Summary .  In this article, we study the estimation of mean response and regression coefficient in semiparametric regression problems when response variable is subject to nonrandom missingness. When the missingness is independent of the response conditional on high-dimensional auxiliary information, the parametric approach may misspecify the relationship between covariates and response while the nonparametric approach is infeasible because of the curse of dimensionality. To overcome this, we study a model-based approach to condense the auxiliary information and estimate the parameters of interest nonparametrically on the condensed covariate space. Our estimators possess the double robustness property, i.e., they are consistent whenever the model for the response given auxiliary covariates or the model for the missingness given auxiliary covariate is correct. We conduct a number of simulations to compare the numerical performance between our estimators and other existing estimators in the current missing data literature, including the propensity score approach and the inverse probability weighted estimating equation. A set of real data is used to illustrate our approach.     

Median regression models for longitudinal data with dropouts

Summary .  Recently, median regression models have received increasing attention. When continuous responses follow a distribution that is quite different from a normal distribution, usual mean regression models may fail to produce efficient estimators whereas median regression models may perform satisfactorily. In this article, we discuss using median regression models to deal with longitudinal data with dropouts. Weighted estimating equations are proposed to estimate the median regression parameters for incomplete longitudinal data, where the weights are determined by modeling the dropout process. Consistency and the asymptotic distribution of the resultant estimators are established. The proposed method is used to analyze a longitudinal data set arising from a controlled trial of HIV disease ( Volberding et al., 1990 , The New England Journal of Medicine 322, 941–949). Simulation studies are conducted to assess the performance of the proposed method under various situations. An extension to estimation of the association parameters is outlined.     

Estimating mutation rate and generation time from longitudinal samples of DNA sequences

We present in this paper a simple method for estimating the mutation rate per site per year which also yields an estimate of the length of a generation when mutation rate per site per generation is known. The estimator, which takes advantage of DNA polymorphisms in longitudinal samples, is unbiased under a number of population models, including population structure and variable population size over time. We apply the new method to a longitudinal sample of DNA sequences of the env gene of human immunodeficiency virus type 1 (HIV-1) from a single patient and obtain 1.62 x 10(-2) as the mutation rate per site per year for HIV-1. Using an independent data set to estimate the mutation rate per generation, we obtain 1.8 days as the length of a generation of HIV-1, which agrees well with recent estimates based on viral load data. Our estimate of generation time differs considerably from a recent estimate by Rodrigo et al. when the same mutation rate per site per generation is used. Some factors that may contribute to the difference among different estimators are discussed.     

Applications of multiple imputation to the analysis of censored regression data.

The first part of the article reviews the Data Augmentation algorithm and presents two approximations to the Data Augmentation algorithm for the analysis of missing-data problems: the Poor Man's Data Augmentation algorithm and the Asymptotic Data Augmentation algorithm. These two algorithms are then implemented in the context of censored regression data to obtain semiparametric methodology. The performances of the censored regression algorithms are examined in a simulation study. It is found, up to the precision of the study, that the bias of both the Poor Man's and Asymptotic Data Augmentation estimators, as well as the Buckley-James estimator, does not appear to differ from zero. However, with regard to mean squared error, over a wide range of settings examined in this simulation study, the two Data Augmentation estimators have a smaller mean squared error than does the Buckley-James estimator. In addition, associated with the two Data Augmentation estimators is a natural device for estimating the standard error of the estimated regression parameters. It is shown how this device can be used to estimate the standard error of either Data Augmentation estimate of any parameter (e.g., the correlation coefficient) associated with the model. In the simulation study, the estimated standard error of the Asymptotic Data Augmentation estimate of the regression parameter is found to be congruent with the Monte Carlo standard deviation of the corresponding parameter estimate. The algorithms are illustrated using the updated Stanford heart transplant data set.     

Estimating marginal and incremental effects on health outcomes using flexible link and variance function models

We propose an extension to the estimating equations in generalized linear models to estimate parameters in the link function and variance structure simultaneously with regression coefficients. Rather than focusing on the regression coefficients, the purpose of these models is inference about the mean of the outcome as a function of a set of covariates, and various functionals of the mean function used to measure the effects of the covariates. A commonly used functional in econometrics, referred to as the marginal effect, is the partial derivative of the mean function with respect to any covariate, averaged over the empirical distribution of covariates in the model. We define an analogous parameter for discrete covariates. The proposed estimation method not only helps to identify an appropriate link function and to suggest an underlying distribution for a specific application but also serves as a robust estimator when no specific distribution for the outcome measure can be identified. Using Monte Carlo simulations, we show that the resulting parameter estimators are consistent. The method is illustrated with an analysis of inpatient expenditure data from a study of hospitalists.     

Joint regression analysis for discrete longitudinal data

Summary We introduce an approximation to the Gaussian copula likelihood of Song, Li, and Yuan (2009, Biometrics 65, 60–68) used to estimate regression parameters from correlated discrete or mixed bivariate or trivariate outcomes. Our approximation allows estimation of parameters from response vectors of length much larger than three, and is asymptotically equivalent to the Gaussian copula likelihood. We estimate regression parameters from the toenail infection data of De Backer et al. (1996, British Journal of Dermatology 134, 16–17), which consist of binary response vectors of length seven or less from 294 subjects. Although maximizing the Gaussian copula likelihood yields estimators that are asymptotically more efficient than generalized estimating equation (GEE) estimators, our simulation study illustrates that for finite samples, GEE estimators can actually be as much as 20% more efficient.     

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.     

Generalization of the Mantel-Haenszel estimating function for sparse clustered binary data

We extend the Mantel-Haenszel estimating function to estimate both the intra-cluster pairwise correlation and the main effects for sparse clustered binary data. We propose both a composite likelihood approach and an estimating function approach for the analysis of such data. The proposed estimators are consistent and asymptotically normally distributed. Simulation results demonstrate that the two approaches are comparable in terms of bias and efficiency; however, the estimating equation approach is computationally simpler. Analysis of the Georgia High Blood Pressure survey is used for illustration.     

Alternative estimation procedures for Pr(X less than Y) in categorized data

Consider two independent random variables X and Y. The functional R = Pr(X less than Y) [or gamma = Pr(X less than Y) - Pr(Y less than X)] is of practical importance in many situations, including clinical trials, genetics, and reliability. In this paper several approaches to estimation of gamma when X and Y are presented in discretized (categorical) form are analyzed and compared. Asymptotic formulas for the variances of the estimators are derived; use of the bootstrap to estimate variances is also discussed. Computer simulations indicate that the choice of the best estimator depends on the value of gamma, the underlying distribution, and the sparseness of the data. It is shown that the bootstrap provides a robust estimate of variance. Several examples are treated.     

Elementary methods of cohort analysis with several exposure levels.

This paper proposes an extension of the Mantel-Haenszel rate ratio for the dichotomous exposure to the multiple exposure levels. This extension is based on the unbiased estimating function approach and yields closed-form Mantel-Haenszel rate ratio estimators. Dually consistent variance and covariance estimators of the estimating functions are given and a quasi-score-based confidence interval for individual common rate ratio is provided. A similar extension to the common rate difference case is also given.     

Mutational and structural-based analyses of the osmolyte effect on protein stability

It is known that several naturally occurring substances known as osmolytes increase the conformational stability of proteins. Bolen and co-worker proposed the osmophobic theory, which asserts the osmolyte effect occurs because of an unfavorable interaction of osmolytes mainly with the protein backbone, based on the results on the transfer Gibbs energy of amino acids (Deltag) [Bolen and Baskakov (2001) J. Mol. Biol. 310, 955-963]. In this paper, we report the effect of sarcosine on the conformational stability (DeltaG) of RNase Sa (96 residues and one disulfide bond) and four mutant proteins. The thermal denaturation curves for RNase Sa in sarcosine fitted a two-state model on nonlinear least-squares analysis. All the RNase Sa proteins were stabilized by sarcosine. For example, the increase in stability of the wild-type protein in 4 M sarcosine due to the osmolyte effect (Delta(o)DeltaG) is 3.2 kcal/mol. Mutational analysis of the osmolyte effect indicated that the changed Delta(o)DeltaG values upon mutation (Delta(m)Delta(o)DeltaG), as estimated from the Deltag values, are similar to the experimental values. Structural-based analysis of the osmolyte effect was also performed using model denatured structures: (a) a fully extended model (single chain) with no disulfide bond, (b) two-part, unfolded models (two chains) with a disulfide bond constructed through molecular dynamic (MD) simulation, and (c) a two-part, folded model (two chains). The two-part, unfolded models were expected to be more suitable as denatured structures. The Delta(o)DeltaG values calculated using the two-part, unfolded models were more consistent with experimental values than those calculated using the fully extended and two-part, folded models. This suggests that MD simulation is useful for testing denatured structures. These results indicate that the osmophobic theory can explain the osmolyte effect on protein stability.