Generalized P-values 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 p-values 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 p-values 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 p-value 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.     

Evaluation of statistical methods for left-censored environmental data with nonuniform detection limits

Monte Carlo simulations were used to evaluate statistical methods for estimating 95% upper confidence limits of mean constituent concentrations for left-censored data with nonuniform detection limits. Two primary scenarios were evaluated: data sets with 15 to 50% nondetected samples and data sets with 51 to 80% nondetected samples. Sample size and the percentage of nondetected samples were allowed to vary randomly to generate a variety of left-censored data sets. All statistical methods were evaluated for efficacy by comparing the 95% upper confidence limits for the left-censored data with the 95% upper confidence limits for the noncensored data and by determining percent coverage of the true mean (micro). For data sets with 15 to 50% nondetected samples, the trimmed mean, Winsorization, Aitchison's, and log-probit regression methods were evaluated. The log-probit regression was the only method that yielded sufficient coverage (99-100%) of micro, as well as a high correlation coefficient (r2 = 0.99) and small average percent residuals (-0.1%) between upper confidence limits for censored versus noncensored data sets. For data sets with 51 to 80% nondetected samples, a bounding method was effective (r2 = 0.96 - 0.99, average residual = -5% to -7%, 95-98% coverage of micro), except when applied to distributions with low coefficients of variation (standard deviation/micro < 0.5). Thus, the following recommendations are supported by this research: data sets with 15 to 50% nondetected samples--log-probit regression method and use of Chebyshev theorem to estimate 95% upper confidence limits; data sets with 51 to 80% nondetected samples-bounding method and use of Chebyshev theorem to estimate 95% upper confidence limits.     

A simple method for assessing occupational exposure via the one-way random effects model

A one-way random effects model is postulated for the log-transformed shift-long personal exposure measurements, where the random effect in the model represents an effect due to the worker. Simple closed-form confidence intervals are proposed for the relevant parameters of interest using the method of variance estimates recovery (MOVER). The performance of the confidence bounds is evaluated and compared with those based on the generalized confidence interval approach. Comparison studies indicate that the proposed MOVER confidence bounds are better than the generalized confidence bounds for the overall mean exposure and an upper percentile of the exposure distribution. The proposed methods are illustrated using a few examples involving industrial hygiene data.     

Upper limits for exceedance probabilities under the one-way random effects model

In this article, we propose statistical methods for setting upper limits on (i) the probability that the mean exposure of an individual worker exceeds the occupational exposure limit (OEL) and (ii) the probability that the exposure of a worker exceeds the OEL. The proposed method for (i) is obtained using the generalized variable approach, and the one for (ii) is based on an approximate method for constructing one-sided tolerance limits in the one-way random effects model. Even though tolerance limits can be used to assess the proportion of exposure measurements exceeding the OEL, the upper limits on these probabilities are more informative than tolerance limits. The methods are conceptually as well as computationally simple. Two data sets involving industrial exposure data are used to illustrate the methods.     

Inferences on the mean of zero-inflated lognormal data: the generalized variable approach

This paper considers the problems of confidence interval estimation and hypothesis testing for the mean of zero-inflated lognormal data. We present an unified novel approach based on the concepts of generalized test variables and generalized pivotal quantities. Simulation results demonstrate that the coverage accuracy of proposed confidence intervals and the type I error control of the proposed exact tests are satisfactory.     

A measure of goodness-of-fit for the lognormal model applied to occupational exposures.

The lognormal distribution is often applied to occupational exposures, yet the assumption of lognormality is rarely verified. This lack of rigor in evaluating the appropriateness of the lognormal model has resulted, in part, from the difficulty of applying formal goodness-of-fit tests. When evaluation of model fit has been attempted, occupational hygienists have relied upon probability plotting of exposures rather than upon formal statistical methods. The goal of this work was to develop for the occupational hygienist a simple quantitative evaluation to supplement the probability plot. A measure of goodness-of-fit to the lognormal model based on the ratio of two estimators of the mean of the distribution, the simple or direct estimate of the mean and the maximum likelihood estimate of the mean of a lognormal distribution, is described. This new measure, the ratio metric, is a simple extension of calculations made routinely by many occupational hygienists. Results from using the ratio metric were compared to probability plotting and to two traditional measures of goodness-of-fit, the Lilliefors test and the W test, for two occupational exposure data sets. The results of the ratio and W tests are comparable for a variety of occupational exposure data, but the Lilliefors test is overly conservative and does not detect several cases of gross deviations from lognormality. The ratio metric is an effective alternative to the Lilliefors test and is easier to perform than the W test for the range of data usually encountered by occupational hygienists. Occupational hygienists are encouraged to use the ratio metric in conjunction with the probability plot in evaluating the lognormal assumption.     

A comparison of several methods for analyzing censored data

The purpose of this study was to compare the performance of several methods for statistically analyzing censored datasets [i.e. datasets that contain measurements that are less than the field limit-of-detection (LOD)] when estimating the 95th percentile and the mean of right-skewed occupational exposure data. The methods examined were several variations on the maximum likelihood estimation (MLE) and log-probit regression (LPR) methods, the common substitution methods, several non-parametric (NP) quantile methods for the 95th percentile and the NP Kaplan-Meier (KM) method. Each method was challenged with computer-generated censored datasets for a variety of plausible scenarios where the following factors were allowed to vary randomly within fairly wide ranges: the true geometric standard deviation, the censoring point or LOD and the sample size. This was repeated for both a single-laboratory scenario (i.e. single LOD) and a multiple-laboratory scenario (i.e. three LODs) as well as a single lognormal distribution scenario and a contaminated lognormal distribution scenario. Each method was used to estimate the 95th percentile and mean for the censored datasets (the NP quantile methods estimated only the 95th percentile). For each scenario, the method bias and overall imprecision (as indicated by the root mean square error or rMSE) were calculated for the 95th percentile and mean. No single method was unequivocally superior across all scenarios, although nearly all of the methods excelled in one or more scenarios. Overall, only the MLE- and LPR-based methods performed well across all scenarios, with the robust versions generally showing less bias than the standard versions when challenged with a contaminated lognormal distribution and multiple LODs. All of the MLE- and LPR-based methods were remarkably robust to departures from the lognormal assumption, nearly always having lower rMSE values than the NP methods for the exposure scenarios postulated. In general, the MLE methods tended to have smaller rMSE values than the LPR methods, particularly for the small sample size scenarios. The substitution methods tended to be strongly biased, but in some scenarios had the smaller rMSE values, especially for sample sizes <20. Surprisingly, the various NP methods were not as robust as expected, performing poorly in the contaminated distribution scenarios for both the 95th percentile and the mean. In conclusion, when using the rMSE rather than bias as the preferred comparison metric, the standard MLE method consistently outperformed the so-called robust variations of the MLE-based and LPR-based methods, as well as the various NP methods, for both the 95th percentile and the mean. When estimating the mean, the standard LPR method tended to outperform the robust LPR-based methods. Whenever bias is the main consideration, the robust MLE-based methods should be considered. The KM method, currently hailed by some as the preferred method for estimating the mean when the lognormal distribution assumption is questioned, did not perform well for either the 95th percentile or mean and is not recommended.     

Interval estimates for the ratio and difference of two lognormal means

Health research often gives rise to data that follow lognormal distributions. In two sample situations, researchers are likely to be interested in estimating the difference or ratio of the population means. Several methods have been proposed for providing confidence intervals for these parameters. However, it is not clear which techniques are most appropriate, or how their performance might vary. Additionally, methods for the difference of means have not been adequately explored. We discuss in the present article five methods of analysis. These include two methods based on the log-likelihood ratio statistic and a generalized pivotal approach. Additionally, we provide and discuss the results of a series of computer simulations. Finally, the techniques are applied to a real example.     

Comparison of Means of Two Lognormal Distributions Based on Samples with Multiple Detection Limits

The problem of comparing the means of two lognormal distributions based on samples with multiple detection limits is considered. Tests and confidence intervals for the ratio of the two means, based on pivotal quantities involving the maximum likelihood estimators, are proposed. The merits of the proposed approaches are evaluated by Monte Carlo simulation. Simulation study indicates that the procedures are satisfactory in terms of coverage probabilities of confidence intervals, and powers of tests. The proposed approach can also be applied to find confidence intervals for the difference between the means of the two lognormal distributions. Illustrative examples with a real data set and with a simulated data set are given.     

Computation of profile likelihood-based confidence intervals for reference limits with covariates

Many biochemical quantities depend on age or some other covariate. Reference limits that allow for these dependencies help physicians to interpret the results of biochemical tests. Because reference limits must be estimated, it is important to assess their precision with, for example, confidence intervals. This paper relies on the assumption that data can be modeled by a generalized linear model and presents a method for calculating approximate profile likelihood-based confidence intervals for reference limits. The calculation of confidence intervals is based on a new method that draws on profile likelihood-based confidence intervals in general statistical models. The asset of this new method is that only two constrained optimization problems have to be solved instead of several in the standard method. We motivate our confidence interval calculation method with two applications. The first is for data on immunoglobulin concentration in the context of a generalized linear model with gamma distribution. This model is compared with the often used lognormal model. The second application handles data on serum alpha-fetoprotein and is presented in a linear regression situation. In the latter application the widths of the calculated profile confidence intervals are compared with exact and approximate regression-based intervals and the actual confidence levels are determined by simulation. Copyright (c) 2007 John Wiley & Sons, Ltd.     

Exact expressions for the bias and variance of estimators of the mean of a lognormal distribution.

Exact mathematical expressions are given for the bias and variance of the arithmetic and maximum likelihood estimators of the first moment (mean) of a lognormal distribution. On the basis of these exact expressions, and without the need for simulation, statistics on the bias and variance have been computed for a range of sample sizes and geometric standard deviations. The results reaffirm that an unbiased maximum likelihood estimator exists that has minimum variance. Contrary to some recent recommendations, this is the preferred estimator if the data truly come from a lognormal distribution.     

Unconditional efficient one-sided confidence limits for the odds ratio based on conditional likelihood

We compare various one-sided confidence limits for the odds ratio in a 2 x 2 table. The first group of limits relies on first-order asymptotic approximations and includes limits based on the (signed) likelihood ratio, score and Wald statistics. The second group of limits is based on the conditional tilted hypergeometric distribution, with and without mid-P correction. All these limits have poor unconditional coverage properties and so we apply the general transformation of Buehler (J. Am. Statist. Assoc. 1957; 52:482-493) to obtain limits which are unconditionally exact. The performance of these competing exact limits is assessed across a range of sample sizes and parameter values by looking at their mean size. The results indicate that Buehler limits generated from the conditional likelihood have the best performance, with a slight preference for the mid-P version. This confidence limit has not been proposed before and is recommended for general use, especially when the underlying probabilities are not extreme.     

Exact one-sided confidence limits for the difference between two correlated proportions

We construct exact and optimal one-sided upper and lower confidence bounds for the difference between two probabilities based on matched binary pairs using well-established optimality theory of Buehler. Starting with five different approximate lower and upper limits, we adjust them to have coverage probability exactly equal to the desired nominal level and then compare the resulting exact limits by their mean size. Exact limits based on the signed root likelihood ratio statistic are preferred and recommended for practical use.     

Construction of confidence limits about effect measures: a general approach

It is widely accepted that confidence interval construction has important advantages over significance testing for the presentation of research results, as now facilitated by readily available software. However, for a number of effect measures, procedures are either not available or not satisfactory in samples of small to moderate size. In this paper, we describe a general approach for estimating a difference between effect measures, which can also be used to obtain confidence limits for a risk ratio and a lognormal mean. Numerical evaluation shows that this closed-form procedure outperforms existing methods, including the bootstrap.     

Assessment of occupational exposures to industrial hazardous substances. IV. A proposed method based on one-sided tolerance limits

Occupational exposures to potentially hazardous substances may vary considerably because of interday environmental behavioral fluctuations in the contaminant concentration. Such occupational exposures including those of non-monitored days can be theoretically evaluated by the following three ways: 1) assessment of geometric mean and geometric standard deviation, 2) assessment of arithmetic mean, and 3) assessment of upper limits of daily exposure distribution. In our previous report, an evaluation method on 95% upper limit or arithmetic mean of exposures was proposed. The method is useful, particularly, in case where only one or two days are being monitored, but may provide an approximate estimate because of statistical assumption. A sampling and decision scheme using one-sided tolerance limits (OTL) proposed by Tuggle (1982) can precisely evaluate the upper limits of exposures. However, many cases would be evaluated as "no decision," unless the sample size is extremely large in number. We developed a revised method based on OTL for assessment of occupational exposures. The characteristic features of this method can be summarized as follows: 1. Upper limits of lognormally distributed 8-h exposure concentrations can be evaluated in comparison with an established standard. 2. A third OTL factor was introduced into Tuggle's scheme in which two OTL factors were used. A comparison between the upper limits of exposures and the standard can be made at 50% confidence level with the factor. The factor was calculated using non-central t-distribution. 3. The usefulness of the third OTL factor in the assessment of occupational exposures was confirmed by examining the performance characteristics of the method.(ABSTRACT TRUNCATED AT 250 WORDS)     

The 4-parameter lognormal (SB) model of human exposure

This paper explores the 4-parameter lognormal distribution (or Johnson S(B) distribution) as a model for occupational exposures to airborne contaminants. This model can incorporate extreme values when they are known a priori, or alternatively, they can be estimated from the data. This additional flexibility may be of value in estimating background and/or maximum exposures, as well as improving the fitting process and subsequent estimation of mean exposures. In addition, the model is physically consistent with the definition of concentration and provides a basis for linking stochastic and deterministic exposure modeling approaches. There is some additional computational burden in estimating the mean and variance of exposure relative to the usual 2-parameter lognormal model.     

Tractor controls actuating force limits for Indian operators

In four-wheel tractors, proper design of controls is important for comfortable and safe operation of the tractor. The design involves location and dimensions of controls as well as strength limits for operating these controls. The present study was aimed to quantify human strength for operation of tractor controls and to recommend the maximum control actuating forces for normal operation of tractors based on strength capability of 3,423 Indian male agricultural workers. The 5th percentile values of strength parameters i.e. leg strength sitting (left and right), foot strength sitting (right), torque strength (both hands) sitting, push strength (left hand and right hand) sitting and pull strength (left hand and right hand) sitting of agricultural workers collected using a strength measurement set-up were taken into consideration for the study. It was recommended that the maximum actuating forces for normal operation of frequently operated brake and clutch pedals of tractors should not exceed 260 N and 125 N based on 5th percentile values of right and left leg strength of male agricultural workers, respectively. The maximum actuating force required in steering wheel operation should not exceed 51 N based on 5th percentile value of torque strength (both hands) sitting of workers. The maximum actuating forces required for operating frequently operated levers viz. gear selection, speed selection, hydraulic control and hand throttle of Indian tractors should not exceed 46 N, 46 N, 25 N and 25 N, respectively. It may be concluded that the maximum actuating force limits as given in Bureau of Indian Standards IS 10703 are very high as compared to the findings of the study based on strength data of Indian male operators, which highlight the need to revise the standard.     

Determination of the appropriate quarantine period following smallpox exposure: an objective approach using the incubation period distribution

Determination of the most appropriate quarantine period for those exposed to smallpox is crucial to the construction of an effective preparedness program against a potential bioterrorist attack. This study reanalyzed data on the incubation period distribution of smallpox to allow the optimal quarantine period to be objectively calculated. In total, 131 cases of smallpox were examined; incubation periods were extracted from four different sets of historical data and only cases arising from exposure for a single day were considered. The mean (median and standard deviation (SD)) incubation period was 12.5 (12.0, 2.2) days. Assuming lognormal and gamma distributions for the incubation period, maximum likelihood estimates (and corresponding 95% confidence interval (CI)) of the 95th percentile were 16.4 (95% CI: 15.6, 17.9) and 16.2 (95% CI: 15.5, 17.4) days, respectively. Using a non-parametric method, the 95th percentile point was estimated as 16 (95% CI: 15, 17) days. The upper 95% CIs of the incubation periods at the 90th, 95th and 99th percentiles were shorter than 17, 18 and 23 days, respectively, using both parametric and non-parametric methods. These results suggest that quarantine measures can ensure non-infection among those exposed to smallpox with probabilities higher than 95-99%, if the exposed individuals are quarantined for 18-23 days after the date of contact tracing.     

Response times follow lognormal or gamma distribution in arthritis patients

ObjectiveThis study evaluated the statistical distribution of time to treatment response in patients with rheumatic diseases.Study Design and SettingThe study used a secondary data analysis design. Data from the trial of etanercept and methotrexate with radiographic patient outcomes were used to model the response times for etanercept (ETN), methotrexate (MTX), and combined ETN + MTX in patients with rheumatoid arthritis. The German etanercept registry was used to evaluate the response time distributions in patients with juvenile idiopathic arthritis.ResultsFor MTX, the lognormal distribution was considered to be the best model for the outcome American College of Rheumatology (ACR20), lognormal, generalized gamma, and log-logistic distributions for ACR50, and lognormal and generalized gamma for ACR70. For ETN, the lognormal model was best for ACR20, the generalized gamma for ACR50, and both lognormal and generalized gamma distributions for ACR70. For combined treatment, the best model was the log-logistic distribution for ACR20, generalized gamma for ACR50, and both lognormal and generalized gamma distributions for ACR70. For the German etanercept registry, the lognormal distribution was the best model for all three outcomes of pediatric ACR30, ACR50, and ACR70 without interval censoring.ConclusionStudy designs might be more efficient if the response distributions are taken into consideration during planning.     

Confidence limits based on the first occurrence of an event

Confidence limits for population prevalence based on the first occurrence of an item in a medical database, or for incidence based on time to first occurrence, should be based on the geometric or exponential distributions, respectively. These intervals are presented and compared with the corresponding intervals based on the binomial and Poisson distributions. The lower confidence limits are shown to be the same, but the upper limits are smaller, hence leading to shorter intervals. Applications of these intervals are also presented.