Lognormal Means Research Articles

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.

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.

Efficient estimation of log-normal means with application to pharmacokinetic data

In this paper, the problem of interest is efficient estimation of log-normal means. Several existing estimators are reviewed first, including the sample mean, the maximum likelihood estimator, the uniformly minimum variance unbiased estimator and a conditional minimal mean squared error estimator. A new estimator is then proposed, and we show that it improves over the existing estimators in terms of squared error risk. The improvement is more significant with small sample sizes and large coefficient of variations, which is common in clinical pharmacokinetic (PK) studies. In addition, the new estimator is very easy to implement, and provides us with a simple alternative to summarize PK data, which are usually modelled by log-normal distributions. We also propose a parametric bootstrap confidence interval for log-normal means around the new estimator and illustrate its nice coverage property with a simulation study. Our estimator is compared with the existing ones via theoretical calculations and applications to real PK studies.

A better estimate of lognormal means on pharmacokinetic data

It is well accepted that pharmacokinetic parameters, especially area under the concentration-time curve (AUC) and maximum concentration (Cmax), should be analyzed on the log-scale with the assumption of lognormality. Currently pharmacokinetic data are typically summarized by arithmetic means and geometric means. A deeper look at the lognormal distribution reveals that they are actually estimating different parameters of the distribution. Arithmetic means are naÏve estimates of population means, while geometric means are plug-in estimates of population medians. Suppose population means are the primary interest then we propose another procedure to estimate them. This procedure can provide an alternative way to summarize pharmacokinetic data. In fact, we show that this procedure is a more precise estimate for population means, especially when the standard deviation is large. The procedure is compared with the existing ones via simulations and applied to real pharmacokinetic studies. Some simulation results based on a four-period crossover study are illustrated in the following table, which shows our estimates have much smaller mean squared errors in most cases. (See table) Clinical Pharmacology & Therapeutics (2004) 75, P42–P42; doi: 10.1016/j.clpt.2003.11.160 Regimen MSEOE/MSEAR MSEOE/MSEGM A 0.97 0.79 B 0.96 0.78 C 0.86 0.59 D 0.84 0.55 AR- Arithmetic means; OE-Our Estimates; GM- Geometric means.

Predicting incidence of variant Creutzfeldt-Jakob disease from UK dietary exposure to bovine spongiform encephalopathy for the 1940 to 1969 and post-1969 birth cohorts.

To investigate variant Creutzfeldt-Jakob disease (vCJD) incubation period, transmission barrier, and short-term vCJD predictions for methionine homozygotes in 1940-1969 and post-1969 birth cohorts by use of gender- and age-specific exposure intensities to bovine spongiform encephalopathy (BSE), based on consumption of beef mechanically recovered meat (MRM) and head meat. Simulation (from vCJD infections generated randomly from gender and age-specific dietary exposure intensities to BSE), constrained to equal the 47 and 64 vCJD onsets before 2001 in 1940-1969 and post-1969 birth cohorts, was used to estimate lognormal (and other) incubation mean and standard deviation which fitted the calendar year distribution of observed vCJD onsets; and to explore exponential decay in susceptibility to infection with age above 15 years. For the post-1969 birth cohort, the best-fitting lognormal incubation period mean of 11 years (SD 1.5 years and 195 infections) was associated with 194 vCJD onsets (64 before 2001, 105 in 2001-2005, and 25 in 2006-2010). About one-fifth of simulated vCJD onsets before 2001 arose from infections in 1990-1996; age and gender of simulated and observed vCJD patients agreed closely. For the 1940-1969 birth cohort, well-fitting lognormal means ranged widely, the marginally best fitting being 26 years (SD 16.5 years and 382 infections; 47 vCJD onsets before 2001, 58 in 2001-2005, and 63 in 2006-2010). An age-dependent susceptibility function was required to match the age distribution of vCJD patients in the 1940-1969 birth cohort. About three-fifths of predicted vCJD onsets are expected to be in males, and nearly two-thirds of vCJD onsets in 2001-2005 are expected to be in post-1969 birth cohort according to best-fitting predictions.

Inferences on the means of lognormal distributions using generalized p-values and generalized confidence intervals

The lognormal distribution is widely used to describe the distribution of positive random variables; in particular, it is used to model data relevant to occupational hygiene and to model biological data. A problem of interest in this context is statistical inference concerning the mean of the lognormal distribution. For obtaining confidence intervals and tests for a single lognormal mean, the available small sample procedures are based on a certain conditional distribution, and are computationally very involved. Occupational hygienists have in fact pointed out the difficulties in applying these procedures. In this article, we have first developed exact confidence intervals and tests for a single lognormal mean using the ideas of generalized p-values and generalized confidence intervals. The resulting procedures are easy to compute and are applicable to small samples. We have also developed similar procedures for obtaining confidence intervals and tests for the ratio (or the difference) of two lognormal means. Our work appears to be the first attempt to obtain small sample inference for the latter problem. We have also compared our test to a large sample test. The conclusion is that the large sample test is too conservative or too liberal, even for large samples, whereas the test based on the generalized p-value controls type I error quite satisfactorily. The large sample test can also be biased, i.e., its power can fall below type I error probability. Examples are given in order to illustrate our results. In particular, using an example, it is pointed out that simply comparing the means of the logged data in two samples can produce a different conclusion, as opposed to comparing the means of the original data.

Evaluation of Three Measures of Exposure Concentration: A Case Study of Surface Sediment Concentrations in the Passaic River

Arithmetic and lognormal means historically have been used as estimates of exposure concentration for both human health and ecological risk assessment. Past risk assessment guidance has emphasized the need to include all available informa tion, particularly regarding the spatial and temporal aspects of data and exposure. One estimate of exposure concentration that includes spatial information is the 95% upper confidence limit (UCL) of the area weighted mean. An area weighted mean is calculated using weightings based on the ratio of the area associated with a sample location and the total area of the study area. Currently, U.S. Environmental Protection Agency (USEPA) risk assessment guidance recommends the use of the 95% UCL of the arithmetic mean as the most appropriate measure of exposure concentration for soil and sediment in human health and ecological risk assess ments. The purpose of this study was two fold: (1) to evaluate the use of the 95% UCL of the area weighted mean vs. the 95% UCL of the arithmetic mean for the lognormal distribution and the normal distribution using a large data set from a recent riverine sediment characterization study, and (2) to evaluate the potential associated with the different approaches for reduction in surface sediment exposure concentrations for 2,3,7,8 TCDD and coplanar polychlorinated biphenyls (PCBs) for several subsets of the data. The results of this evaluation indicated that there was no significant reduction in exposure concentrations of the study area surface sediments after any of the selected higher concentration data points had been removed from the various data subsets. In addition, this evaluation indicated that, due to its dependence on the variance of the data, the 95% UCL of the arithmetic mean for a lognormal distribution produced subset exposure concentrations that were higher than the calculated concentrations of the full data set and produced the smallest percent changes in concentration compared to the other two measures. Based on this analysis, the 95% UCL of the area weighted mean was determined to be the most stable and appropriate estimator of exposure concentration for surface sediments.

Confidence intervals for the log-normal mean .

In this paper we conduct a stimulation study to evaluate coverage error, interval width and relative bias of four main methods for the construction of confidence intervals of log-normal means: the naive method; Cox's method; a conservative method; and a parametric bootstrap method. The simulation study finds that the naive method is inappropriate, that Cox's method has the smallest coverage error for moderate and large sample sizes, and that the bootstrap method has the smallest coverage error for small sample sizes. In addition, Cox's method produces the smallest interval width among the three appropriate methods. We also apply the four methods to a real data set to contrast the differences.

Comparison of lognormal population means

Comparisons of two lognormal population means are investigated. For large samples, the conventional test for significance of the population means is the ordinary Student t-test with normal critical value. By Chen and Loh’s result (Ann. Statist. 20 (1992)), however, the transformed t-test based on log-data is asymptotically more powerful than the ordinary t-test. In this paper, a new power transformation (hence a new transformed t-test) is proposed. The new transformed t-test is proven to be asymptotically more powerful than the one based on log-data. Both small sample and large sample properties of the proposed estimate for the power-transformation parameter are studied. A simulation study shows that the advantages of the new test over the t-test based on log-data are overwhelming and evident for sizes of the two samples as small as 20 and 30, or even 10 and 15. The simulation results also show that the new test has greater asymptotic power than Rao’s efficient score test.

Correlation of Laboratory Analyses of Soil Properties and Infiltrometer Seepage from Drainwater Evaporation Ponds

ABSTRACT CONSTRUCTION of drainwater evaporation ponds for disposal of subsurface drainage effluent has raised concern over the extent to which seepage from these ponds affect shallow groundwater. Estimates of seepage rates based on laboratory measurements of soil properties are considered through comparison of results of laboratory and field measurements at 90 sites within a pond. Lognormal means of field measured seepage rates were over six times larger than associated means of hydraulic conductivities measured in the laboratory. The discrepancy can be attributed to potential compaction of extracted cores used in the laboratory, layering of pond soils contained within infiltrometers, and hydraulic gradients greater than unity in pond soils. Of the laboratory measurements considered, field seepage rates were most closely correlated with permeameter-based measurements of hydraulic conductivity and the clay content of the soil. Due to the number of factors involved in correlating results of laboratory and field experiments with respect to soil heterogeneities and sample consolidation, additional work in this area is necessary to validate use of laboratory analyses for estimates of seepage in the field.

Preferential release of oxytocin in response to vasoactive intestinal polypeptide in rats

Vasoactive intestinal polypeptide (VIP) was infused into the aorta of pentobarbitone-anesthetized rats (n = 12) in stepwise increasing doses of 0.001 to 10 μg/rat at rates varying from 0.3 pmol/min/kg to 3000 pmol/min/kg over 3 min. Blood was withdrawn from the vena cava inferior for the measurement of oxytocin (OT) and vasopressin (AVP) by RIA. The loss of blood was compensated for by infusion of isotonic saline (0.9% NaCl with 0.5% human serum albumin). Control rats received this solution only (n = 11). VIP infusions resulted in a dose-dependent increase in plasma OT which was significantly greater than the slight rise observed in the controls. The difference from controls was significant at infusion rates of 3 pmol/min/kg and more. Plasma AVP, on the other hand, did not rise in response to VIP infusions until the infusion rate was increased to 300 and 3000 pmol/min/kg. At these infusion rates, the increments in AVP were much smaller than those of OT, the levels during the highest infusion rates rising to 8.6 ± 2.8 and 27.2 ± 4.8 μU/ml, respectively (log normal means). The preferential release of OT in response to exogenous VIP in rats differs from the response in cats where intracarotid administration of VIP resulted in the release of proportionately more AVP than OT. Immunoreactive VIP is found in the hypothalamo-neurohypophyseal system of rats in close proximity of some of the magnocellular neurons as well as within the nerve terminals. This, together with our data, suggests that endogenous VIP may participate in the release mechanism for OT in rats.

A bimodal lognormal model of the distribution of strength of carbon fibres: effects of electrodeposition of titanium di (dioctyl pyrophosphate) oxyacetate

The tensile strength distribution of Fortafil-3 carbon fibres of circular cross-section has been investigated at different gauge lengths. The unimodal Weibull, unimodal lognormal and bimodal lognormal models were tested. Estimating the model parameters by the method of maximum likelihood, and testing each model by the Kolmogorov-Smirnov goodness-of-fit statistic at prescribed levels of significance, it is found that the data for untreated unsized fibres fit a bimodal lognormal model best. The proportions of the low and high strength populations,p andq, respectively, did not show any well-defined trend with gauge length and had average values close to 0.5. But the lognormal mean for each population showed an increasing trend with decreasing gauge length. It is inferred thatp andq are, respectively, related to the presence of surface flaws and internal defects, both of which probably have the same structural origin. After electrodeposition of titanium di (dioctyl pyrophosphate) oxyacetate (TDPO), the fibre strength was still best approximated by a bimodal lognormal distribution. But the weighting factorp for the weak population was reduced markedly, with a corresponding increase ofq, indicating the healing of surface flaws during electrodeposition of a protective layer of TDPO. Furthermore, in contrast to the observations with untreated fibres, the lognormal means for both the low and high strength populations of the electrocoated fibres were essentially unchanged with gauge length. Changes were also indicated in the number and severity of surface flaws, caused by concurrent electrochemical processes.

A Test for the Scale Parameters of Two Gamma Distributions using the Generalized Likelihood Ratio

Abstract The method of maximum likelihood is used to develop a statistical test for the scale parameters of two gamma distributions with common shape factors. A simple method of determining the power of the test using the non-central chi square distribution is also presented. The results of applying the test to gamma-scale parameters are compared with results obtained by applying the “t” test to normal and log-normal means. The likelihood ratio test for differences in gamma-scale parameters is more powerful than the “t” test applied to log-normal means. The power is nearly equal for the likelihood test of gamma-scale parameters and the “t” test for non-transformed means, although the latter test may not be as robust as is the likelihood ratio test. Since many meteorological variables are known to be gamma distributed, this test should have several applications in meteorology.