Generalized PValues and Confidence Intervals: A Novel Approach for Analyzing Lognormally Distributed Exposure Data
The problem of assessing occupational exposure using the mean of a lognormal distribution is addressed. The novel concepts of generalized pvalues and generalized confidence intervals are applied for testing hypotheses and computing confidence intervals for a lognormal mean. The proposed methods perform well, they are applicable to small sample sizes, and they are easy to implement. Power studies and sample size calculation are also discussed. Computational details and a source for the computer program are given. The procedures are also extended to compare two lognormal means and to make inference about a lognormal variance. In fact, our approach based on generalized pvalues and generalized confidence intervals is easily adapted to deal with any parametric function involving one or two lognormal distributions. Several examples involving industrial exposure data are used to illustrate the methods. An added advantage of the generalized variables approach is the ease of computation and implementation. In fact, the procedures can be easily coded in a programming language for implementation. Furthermore, extensive numerical computations by the authors show that the results based on the generalized pvalue approach are essentially equivalent to those based on the Land's method. We want to draw the attention of the industrial hygiene community to this accurate and unified methodology to deal with any parameter associated with the lognormal distribution.
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 "The computation of confidence limits for some rather complicated parameters comes up in many industrial hygiene applications, and the concept of a generalized confidence interval has proved very fruitful to address such problems. In a series of articles, Krishnamoorthy and Mathew (2002, 2009) and Krishnamoorthy et al., (2006 Krishnamoorthy et al., ( , 2007) have successfully applied the generalized confidence interval idea for the analysis of industrial hygiene data. In particular, Krishnamoorthy and Mathew (2009) have developed an accurate upper confidence limit for the symmetricrange accuracy , using the generalized confidence interval approach, in the context of normally distributed sample measurements. "
[Show abstract] [Hide abstract] ABSTRACT: The symmetricrange accuracy of a sampler is defined as the fractional range, symmetric about the true concentration, that includes a specified proportion of sampler measurements. In this article, we give an explicit expression for assuming that the sampler measurements follow a oneway random model so as to capture different components of variability, for example, variabilities among and within different laboratories or variabilities among and within exposed workers. We derive an upper confidence limit for based on the concept of a ‘generalized confidence interval’. A convenient approximation is also provided for computing the upper confidence limit. Both balanced and unbalanced data situations are investigated. Monte Carlo evaluation indicates that the proposed upper confidence limit is satisfactory even for small samples. The statistical procedures are illustrated using an example.0Comments 0Citations 
 "A confidence interval for g can be obtained along the lines of the method for the percentiles given in earlier section and using the generalized variable approach. For more details on the generalized variable approach in the present context, see the articles by Krishnamoorthy et al. (2006 Krishnamoorthy et al. ( , 2011) and Krishnamoorthy and Mathew (2009b). Specifically, an approximate 'generalized pivotal quantity (GPQ)' for g, which can be constructed following the lines of Krishnamoorthy et al. (2010), as follows. "
[Show abstract] [Hide abstract] ABSTRACT: The problem of assessing occupational exposure using the mean or an upper percentile of a lognormal distribution is addressed. Inferential methods for constructing an upper confidence limit for an upper percentile of a lognormal distribution and for finding confidence intervals for a lognormal mean based on samples with multiple detection limits are proposed. The proposed methods are based on the maximum likelihood estimates. They perform well with respect to coverage probabilities as well as power and are applicable to small sample sizes. The proposed approaches are also applicable for finding confidence limits for the percentiles of a gamma distribution. Computational details and a source for the computer programs are given. An advantage of the proposed approach is the ease of computation and implementation. Illustrative examples with real data sets and a simulated data set are given.0Comments 2Citations 
 "Generalized procedures have been successfully applied to several problems of practical importance. The areas of applications include comparison of means, testing and estimation of functions of parameters of normal and related distributions (Weerahandi,2345, Krishnamoorthy and Mathew [6], Johnson and Weerahandi [7], Gamage, Mathew and Weerahandi [8]); testing fixed effects and variance components in repeated measures and mixed effects ANOVA models (Zhou and Mathew [9], Gamage and Weerahandi [10], Chiang [11], Krishnamoorthy and Mathew [6], Weerahandi [5], Mathew and Webb [12], Arendacka [13]); interlaboratory testing (Iyer, Wang and Mathew [14]); bioequivalence (McNally, Iyer and Mathew [15]); growth curve modeling (Weerahandi and Berger [16], Lin and Lee [17]); reliability and system engineering (Roy and Mathew [18], Tian and Cappelleri [19], Mathew, Kurian and Sebastian [20]); process control (Burdick, Borror and Montgomery [21], Mathew, Kurian and Sebastian [22]); environmental health (Krishnamoorthy, Mathew and Ramachandran [23]) and many others. The simulation studies in Johnson and Weerahandi [7], Weerahandi [4,5] Zhou and Mathew [9], Gamage and Weerahandi [10], among others have demonstrated the success of the generalized procedure in many problems where the classical approach fails to yield adequate confidence intervals. "
[Show abstract] [Hide abstract] ABSTRACT: Generalized confidence intervals provide confidence intervals for complicated parametric functions in many common practical problems. They do not have exact frequentist coverage in general, but often provide coverage close to the nominal value and have the correct asymptotic coverage. However, in many applications generalized confidence intervals do not have satisfactory finite sample performance. We derive expansions of coverage probabilities of onesided generalized confidence intervals and use the expansions to explain the nonuniform performance of the generalized intervals. We then show how to use these expansions to obtain improved coverage by suitable calibration. The benefits of the proposed modification are illustrated via several examples.0Comments 3Citations
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