Geometric Standard Deviation - Reply to Bohidar

Bohidar (1) comments that a formula I proposed (2) for geometric standard deviation (GSD) is “absolutely incorrect”. This comment is based on erroneous interpretation of my proposal and to avoid further confusion it may be helpful to point this out. The issue addressed in ref(2) was how to summarize the extent of statistical variation in a sample of data drawn from the lognormal distribution, for which the underlying variation is multiplicative. This situation was contrasted with the normal, or Gaussian, distribution for which the underlying variation is additive. Normally distributed data are conveniently summarized by a mean plus or minus a standard deviation (SD) (or standard error, SE). For lognormal data the geometric mean was available as replacement for the ordinary mean, but no direct analogue existed for SD (or SE).     

The In Vitro and In Vivo Performance of Aqueous Bases Enteric Coats of Neutralised Hydroxypropyl Methyl Cellulose Phthalate

Prednisolone tablets, enteric coated with neutralised hydroxypropyl methylcellulose phthalate (HPMCP) were compared with Deltacortril tablets (Pfizer) by compendial in vitro testing and a pharmacokinetic study in 12 volunteers. Despite satisfactory compliance for both products with the specifications for enteric products of the European Pharmacopoeia and the United States Pharmacopoeia a significant difference in lag time before prednisolone was detected in plasma was observed between the products and only the Deltracortril tablet was concluded to exhibit true enteric properties

The failure of the neutralised HPMCP coating probably results from incomplete gastric conversion to its acidic form due to the majority of subjects having gastric pH values in excess of those stipulated in the compendial in vitro tests. Alternative in vitro testing procedures are proposed     

Advances in Dissolution Technology: Design, Pros and Cons

To date dissolution tests are considered to be the most reliable predictors of bioavailability of drugs. Dissolution tests are critical and difficult to carry out properly. A review of different dissolution apparatii currently in use or employed in the past is present along with the advantages and disadvantages offered by each. Criteria as to design, operation, sensitivity, etc. of an in vitro dissolution system are outlined which would assist in fabrication of a more efficient and reliable apparatus. If one is to obtain meaningful results, care and attention must be given to those aspects that have been identified as crucial. To date no universal dissolutiuon test has yet been devised that in every instance gives the same rank order for in vitro dissolution and in vivo availability for different formulation and batches.     

“Monitoring Punch Forces and Punch Movements as an AID to Developing Robust Tablet Formulations”

The benefits of commencing the pharmaceutical development program for a new drug entity as soon as practical are obvious. However, the ability to do so is controlled by many factors including limited availability of drug substance and very little information on pharmaceutically relevant properties. It is this data which the formulator must begin to collect as soon as any material becomes available. Establishing the physico-chemical characteristics (often referred to as 'preformulation') covers a wide range of attributes. This paper only considers those which can be investigated by use of instrumented compaction equipment

The first experiments should include establishing the intrinsic compactibility of the compound by preparation of simple compacts of drug substance under controlled conditions and with essentially no additives except die wall lubrication. Irrespective of whether compacts are formed or not, the raw data collected from the instrumentation may be used to give an initial reading of this important property of the material

On completion of these tests on drug substance, the development process then follows a pattern of making simple tablets and testing them for dissolution, strength, content uniformity, etc. Instrumentation can improve the efficiency of this process in several ways by providing information on predominant compaction mechanisms, a predictive capability of strength and maybe even dissolution. It will also create a data base of 'normal' results and a rapid, easy screen for variability

If the work is being carried out on a sophisticated test instrument or simulator, the formulations may be subjected to high speed tabletting cycles in order to obtain a measure of the strain rate sensitivity. Several reports have now shown the effect of strain rate on properties such as tensile strength of tablets and their disintegration time

As more drug substance becomes available a set of experiments designed to identify a commercializable formulation is carried out. This is also an appropriate point to transfer the processing to a rotary tablet press (if not already done), but the conditions must be controllable and precisely known. From these experiments a primary and at least one back-up formulation are identified, including processing conditions as well as composition

The first clinical material, using the preferred formula and manufacturing instructions, can now be made. During the ensuing development phases the emphasis on utilisation of the instrumentation changes, but many of the measurements already referred to, are still useful. Much of the data resulting from the overall program described, can provide an impressive addition to the “Formulation Development” section of any submission!     

The control of particle size distribution for fabricated alumina nanoparticles using a thermophoretic separator

Control of the particle size distribution of fabricated alumina nanoparticles from general alumina powder with a large geometric standard deviation (GSD) was studied. A thermophoretic separator was used to control the GSD of the nanoparticles, and unevaporated and primary particles were separated to yield a small GSD. The fabricated nanoparticles were characterized by field emission scanning electron microscopy (FESEM) and a scanning mobility particle sizer (SMPS). We confirmed that the GSD of the nanoparticles was controlled by the thermophoretic separator. A temperature difference between 79 K and 151 K was applied to the thermophoretic separator for control of the nanoparticle GSD. The GSD of the fabricated alumina nanoparticles was improved from 1.74 to 1.44.     

A Goodness-of-Fit Test for the Two Parameter vs. Three Parameter Weibull; Confidence Bounds for ThreshoId

An investigation is described herein of modifications of the two-parameter Weibullgoodness-of-fit test of Mann, Scheuer and Fertig (1973). It is assumed that the sole alternative of interest is any three-parameter Weibull distribution. The power of candidate test statistics is investigated, therefore, only under various three-parameter Weibull alternative hypotheses.

It is found that for a fixed selection of gaps (differences of adjacent order statistics) used in the numerator and in the denominator of the approximately F dist ributed test statistic, nothing is gained by weighting the gaps in order to minimize the variances and thus to maximize the numbers of degrees of freedom.

A test statistic which is a modified version of that of Mann, Scheuer and Fertig is shown to have higher power under three-parameter Weibull alternatives, and a simple method for approximating critical values of the test statistic is described. The test statistic is shown to be a monotone function of an unknown threshold (location) parameter for the three-parameter Weibull model. Hence, the methodology described for testing for a zero threshold parameter can be used to obtiain a confidence interval for this parameter. Methods for combining life-test data for application to progressively censored samples are also described.     

Determination of thoron equilibrium factor from simultaneous long-term thoron and its progeny measurements

With 3-month simultaneous measurements of thoron and its progeny concentrations in the lowest floors of 109 homes, the geometric mean (GM) of thoron equilibrium factor was determined to be 0.019 with a geometric standard deviation (GSD) of 3.63. Combined with the analysis from results obtained from a previous study in 138 homes, the GM of thoron equilibrium factor was determined to be 0.022 with a GSD of 3.02. The results indicate that the F value of 0.02 for thoron recommended by UNSCEAR is a reasonable value for those Canadian homes tested.     

Detection of Discordant Content Uniformity Observations and Compendial Compliance

A discordant observation is a data point whose value is drastically different from that of the rest of the members in the data set. In the context of content uniformity experiments, however, a discordant observation arises in two ways: (i) when the value of an observation is markedly distant from that of the other data points even though it is within the required compendial range, and (ii) when the value of an observation is outside the permissible compendial range. Several statistical tests for detecting one or more discordant observations are presented. Since discordancy distorts the symmetricity of the data, several tests of symmetricity are provided. Tests for detection of group discordancy induced by discordant samples are also included. The compendial requirements are explained in statistical terms. The impact of discordant observations on compendial compliance requirements is assessed. The statistical basis of the construction of compendial limits as well as the assumptions implicit in the construction is elaborated. The results of the statistical analysis of three content uniformity studies are appropriately interpreted.     

On the probability of a model

The posterior probabilities ofK given models when improper priors are used depend on the proportionality constants assigned to the prior densities corresponding to each of the models. It is shown that this assignment can be done using natural geometric priors in multiple regression problems if the normal distribution of the residual errors is truncated. This truncation is a realistic modification of the regression models, and since it will be made far away from the mean, it has no other effect beyond the determination of the proportionality constants, provided that the sample size is not too large. In the caseK=2, the posterior odds ratio is related to the usualF statistic in “classical” statistics. Assuming zero-one losses the optimal selection of a regression model is achieved by maximizing the posterior probability of a submodel. It is shown that the geometric criterion obtained in this way is asymptotically equivalent to Schwarz’s asymptotic Bayesian criterion, sometimes called the BIC criterion. An example of polynomial regression is used to provide numerical comparisons between the new geometric criterion, the BIC criterion and the Akaike information criterion. Villegas and Swartz were partially supported by grants from the Natural Sciences and Engineering Research Council of Canada.     

Construction of upper confidence limit of coefficient of variation for content uniformity

Relative standard deviation (coefficient of variation) plays an important role in meeting the current compendial requirements for content uniformity. Since the sample RSD value would vary from sample to sample in a population (batch), the scientist would need not only the sample estimate of the RSD but also its 95% two-sided upper confidence limit, for making the proper statistical inference as well as for arriving at the appropriate pharmaceutics decisions. The primary purpose of this paper is to depict the five available methods for determining the confidence limit and to discuss their relative merits and similarities in the context of a content uniformity study associated with Product-C. Suitable tables are furnished to facilitate rapid access to the desired RSD confidence limit.     

Containment aerosol characterization during nuclear power plant severe accident

ABSTRACT

The aerosol size distribution and concentration during a postulated severe accident scenario were simulated using a numerical code NAUA to characterize the aerosols dispersed in the containment under various conditions. The model-predicted aerosol concentration and particle size distribution were in general agreement with the measured data reported in previous studies. A large set of sensitivity tests were conducted to examine the effects of input parameters. The main particle removal mechanism in containment was gravitational sedimentation when wall temperature was assumed to be the same as the air temperature, whereas diffusiophoresis became a more important particle removal mechanism when the difference between the wall temperature and air temperature was significant. The operation of a containment filtered venting system (CFVS) resulted in reduced particle concentration, mass median diameter (MMD), and geometric standard deviation (GSD) because aged larger particles were removed by the CFVS. The particle concentration and GSD of the containment aerosol decreased with increasing emitted particle size owing to the effect of gravitational sedimentation that removes large particles selectively. Non-sphericity of particles reduced the particle removal rate due to gravitational sedimentation, leading to higher particle concentration, MMD, and GSD.     

A EWMA Control Chart for Exponential Distributed Quality Based on Moving Average Statistics

We propose an exponentially weighted moving average (EWMA) control chart for monitoring exponential distributed quality characteristics. The proposed control chart first transforms the sample data to approximate normal variables, then calculates the moving average (MA) statistic for each subgroup, and finally constructs the EWMA statistic based on the current and the previous MA statistics. The upper and the lower control limits are derived using the mean and the variance of EWMA statistics. The in‐control and the out‐of‐control average run lengths are derived and tabularized according to process shift parameters and smoothing constants. It is shown that the proposed control chart outperforms the MA control chart for all shift parameters. Copyright © 2015 John Wiley & Sons, Ltd.     

A General Framework for Constructing Control Charts

A general framework for the construction of control charts is presented. The method is based on using the density of the sample subgroup statistic as a measure of how unusual newly observed subgroups are. This methodology includes, as special cases, many common control chart techniques. The method is also easily applied to multivariate and multimodal situations. A non‐parametric control chart is implemented by estimating the density of the sample subgroup statistic using a kernel estimator of the bootstrap distribution of the observed subgroup statistics. Several examples of the method are presented in the parametric and non‐parametric situations. The potential performance of the non‐parametric version of the method is demonstrated through an empirical study of average run length properties. Copyright © 2005 John Wiley & Sons, Ltd.     

Detection of Discordant Content Uniformity Observations and Compendial Compliance

Abstract

A discordant observation is a data point whose value is drastically different from that of the rest of the members in the data set. In the context of content uniformity experiments, however, a discordant observation arises in two ways: (i) when the value of an observation is markedly distant from that of the other data points even though it is within the required compendial range, and (ii) when the value of an observation is outside the permissible compendial range. Several statistical tests for detecting one or more discordant observations are presented. Since discordancy distorts the symmetricity of the data, several tests of symmetricity are provided. Tests for detection of group discordancy induced by discordant samples are also included. The compendial requirements are explained in statistical terms. The impact of discordant observations on compendial compliance requirements is assessed. The statistical basis of the construction of compendial limits as well as the assumptions implicit in the construction is elaborated. The results of the statistical analysis of three content uniformity studies are appropriately interpreted.     

Comparisons of Shewhart-type rank based control charts for monitoring location parameters of univariate processes

The nonparametric (distribution-free) control charts are robust alternatives to the conventional parametric control charts when the form of underlying process distribution is unknown or complicated. In this paper, we consider two new nonparametric control charts based on the Hogg–Fisher–Randle (HFR) statistic and the Savage rank statistic. These are popular statistics for testing location shifts, especially in right-skewed densities. Nevertheless, the control charts based on these statistics are not studied in quality control literature. In the current context, we study phase-II Shewhart-type charts based on the HFR and Savage statistics. We compare these charts with the Wilcoxon rank-sum chart in terms of false alarm rate, out-of-control average run-length and other run length properties. Implementation procedures and some illustrations of these charts are also provided. Numerical results based on Monte Carlo analysis show that the new charts are superior to the Wilcoxon rank-sum chart for a class of non-normal distributions in detecting location shift. New charts also provide better control over false alarm when reference sample size is small.     

Generalized spectral decomposition: a theory and practice to spectral reconstruction

The generalized spectral decomposition (GSD) theorem is introduced, and the generalized fundamental stimulus and metameric black are analyzed to show how they convey the valuable features in terms of color information. The suggestion would be considered as the generalization of Cohen and Kappauf's matrix R theory and its later application in parameric correction by Fairman. The GSD theorem provides a modular model whose arguments can be elaborately set up for high-performance spectral recovery. It is also shown that the suggested methods for spectral decomposition and/or spectral reconstruction proposed by different researchers could be considered as special cases of GSD.     

Uncertainty analysis of the urinary excretion of plutonium

A quantitative estimate of the uncertainty of the urinary excretion of plutonium predicted by available biokinetic models is provided. Urinary excretion is primarily considered here because the monitoring of internal contamination of plutonium mainly relies on measurements of activity in urine samples. A previous paper has identified the most significant transfer rates for urinary plutonium excretion following an acute intake. That analysis is used here as a screening method to reduce the number of model parameters to be considered. A log-normal distribution was assumed for the probability distribution of the model parameters. The spread of the values, represented by the geometric standard deviation (GSD), is explicitly calculated, as few indications of the range of variation of systemic transfer rates are available. Different values for the GSD were considered. Assuming a certain GSD for all the systemic rate constants, random values of the rates were generated (by means of a Monte Carlo simulation with a Latin hypercube sampling scheme) and the resulting predictions of urine bioassay measurements were calculated. The comparison of the mean and variance of the predictions with the available data from several studies performed on different subjects provides information about the GSD of model parameters that represents the intersubject variation of transfer parameters.     

Partial sums of lagged cross-products of AR residuals and a test for white noise

Partial sums of lagged cross-products of AR residuals are defined. By studying the sample paths of these statistics, changes in residual dependence can be detected that might be missed by statistics using only the total sum of cross-products. Also, a test statistic for white noise is proposed. It is shown that the limiting distribution of the test statistic converges weakly to a vector Brownian motion with independent elements under the null hypothesis of no residual autocorrelation. An indication of the circumstances under which the asymptotic results apply in finite-sample situations is obtained through a simulation study. Some considerations are given to the empirical size and power of the test statistic vis-à-vis the Ljung–Box (Biometrika 65:297–303, 1978) portmanteau statistic, and a diagnostic test statistic proposed by Peña and Rodriguez (J. Am. Stat. Assoc. 97:601–610, 2002). An empirical example illustrates the importance of examining partial sums of time series residuals when inadequacies in model fit are anticipated due to a change in autocorrelation structure.     

Comparison of pure and “Latinized” centroidal Voronoi tessellation against various other statistical sampling methods

A recently developed centroidal Voronoi tessellation (CVT) sampling method is investigated here to assess its suitability for use in statistical sampling applications. CVT efficiently generates a highly uniform distribution of sample points over arbitrarily shaped M-dimensional parameter spaces. On several 2-D test problems CVT has recently been found to provide exceedingly effective and efficient point distributions for response surface generation. Additionally, for statistical function integration and estimation of response statistics associated with uniformly distributed random-variable inputs (uncorrelated), CVT has been found in initial investigations to provide superior points sets when compared against latin-hypercube and simple-random Monte Carlo methods and Halton and Hammersley quasi-random sequence methods. In this paper, the performance of all these sampling methods and a new variant (“Latinized” CVT) are further compared for non-uniform input distributions. Specifically, given uncorrelated normal inputs in a 2-D test problem, statistical sampling efficiencies are compared for resolving various statistics of response: mean, variance, and exceedence probabilities.     

Design and implementation of CUSUM exceedance control charts for unknown location

Nonparametric control charts provide a robust alternative in practice when the form of the underlying distribution is unknown. Nonparametric CUSUM (NPCUSUM) charts blend the advantages of a CUSUM with that of a nonparametric chart in detecting small to moderate shifts. In this paper, we examine efficient design and implementation of Phase II NPCUSUM charts based on exceedance (EX) statistics, called the NPCUSUM-EX chart. We investigate the choice of the order statistic from the reference (Phase I) sample that defines the exceedance statistic. We see that choices other than the median, such as the 75th percentile, can yield improved performance of the chart in certain situations. Furthermore, observing certain shortcomings of the average run-length, we use the median run-length as the performance metric. The NPCUSUM-EX chart is compared with the NPCUSUM-Rank chart based on the popular Wilcoxon rank-sum statistic. We also study the choice of the reference value, k, of the CUSUM charts. An illustration with real data is provided.