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Time-series data are widespread and have inspired numerous research works in machine learning and data analysis fields for the classification and clustering of temporal data. While there are several clustering methods for univariate time series and a few for multivariate series, most methods are based on distance and/or dissimilarity measures that do not fully utilize the time-dependency information inherent to time-series data. To highlight the main dynamic structure of a set of multivariate time series, this study extends the use of standard variance–covariance matrices in principal component analysis to cross-autocorrelation matrices at time lags k=1,2,…. This results in “principal component time series”. Simulations and a sign language dataset are used to demonstrate the effectiveness of the proposed method and its benefits in exploring the main structural features of multiple time series.

In this paper, we discuss the theory behind calibration curve experiments and their application to a zinc (Zn) bioavailability study with broiler chickens. Seven replicates of 16 male commercial broiler chicks were fed starter diets for 14 days. Six diets had different levels of a potential Zn source and one was a positive control with standard industry levels of Zn for comparison. Four commonly used methods of calculating bioavailability means and confidence intervals (CI) from a calibration curve (standard curve) experiment to estimate the bioavailability of a new zinc source in broiler chickens were compared. The methods compared were the following: 1) the Counter-Intuitive Method uses a multiple-range test to compare unknown test and standard samples; 2) the Intuitive Method uses standard linear regression and inverts the equation to predict Zn bioavailability for each replicate of test samples; 3) the Abductive Method uses Graybill's Equation, based on theory and observation, to estimate CI's; and 4) the Sophistic Method uses reverse regression, and calculates Zn bioavailability values directly from the equation. The Counter-Intuitive Method only gives information about which standards the test samples are, or are not, significantly different from respectively (average available Zn not predicted). The Intuitive Method ignores error about the standard curve and theoretically cannot estimate the CI directly (

One of the common issues in large dataset analyses is to detect and construct homogeneous groups of objects in those datasets. This is typically done by some form of clustering technique. In this study, we present a divisive hierarchical clustering method for two monothetic characteristics of histogram data. Unlike classical data points, a histogram has internal variation of itself as well as location information. However, to find the optimal bipartition, existing divisive monothetic clustering methods for histogram data consider only location information as a monothetic characteristic and they cannot distinguish histograms with the same location but different internal variations. Thus, a divisive clustering method considering both location and internal variation of histograms is proposed in this study. The method has an advantage in interpreting clustering outcomes by providing binary questions for each split. The proposed clustering method is verified through a simulation study and applied to a large U.S. house property value dataset.

With contemporary data collection capacity, data sets containing large numbers of different multivariate time series relating to a common entity (e.g., fMRI, financial stocks) are becoming more prevalent. One pervasive question is whether or not there are patterns or groups of series within the larger data set (e.g., disease patterns in brain scans, mining stocks may be internally similar but themselves may be distinct from banking stocks). There is a relatively large body of literature centered on clustering methods for univariate and multivariate time series, though most do not utilize the time dependencies inherent to time series. This paper develops an exploratory data methodology which in addition to the time dependencies, utilizes the dependency information between S series themselves as well as the dependency information between p variables within the series simultaneously while still retaining the distinctiveness of the two types of variables. This is achieved by combining the principles of both canonical correlation analysis and principal component analysis for time series to obtain a new type of covariance/correlation matrix for a principal component analysis to produce a so-called “principal component time series”. The results are illustrated on two data sets.

Since Aitchison's founding research work, compositional data analysis has attracted growing attention in recent decades. As a powerful technique for exploratory analysis, principal component analysis (PCA) has been extended to compositional data. Despite extensive efforts in PCA on compositional data parts as variables, this paper contributes to modeling PCA for compositional data vectors. Based on algebraic operators in Simplex space, the PCA process is deduced and transformed into calculating some inner products. Properties of principal components are also investigated. Two real-data examples illustrate the merits of the proposed PCA for compositional data vectors.

How cuticular proteins (CPs) interact with chitin and with each other in the cuticle remains unresolved. We employed LC-MS/MS to identify CPs from 5–6 day-old adults of Anopheles gambiae released after serial extraction with PBS, EDTA, 2-8M urea, and SDS as well as those that remained unextracted. Results were compared to published data on time of transcript abundance, localization of proteins within structures and within the cuticle, as well as properties of individual proteins, length, pI, percent histidine, tyrosine, glutamine, and number of AAP[A/V/L] repeats. Thirteen proteins were solubilized completely, all were CPRs, most belonging to the RR-1 group. Eleven CPs were identified in both soluble fractions and the final pellet, including 5 from other CP families. Forty-three were only detected from the final pellet. These included CPRs and members of the CPAP1, CPF, CPFL, CPLCA, CPLCG, CPLCP, and TWDL families, as well as several low complexity CPs, not assigned to families and named CPLX. For a given protein, many histidines or tyrosines or glutamines appear to be potential participants in cross-linking since we could not identify any peptide bearing these residues that was consistently absent. We failed to recover peptides from the amino-terminus of any CP. Whether this implicates that location in sclerotization or some modification that prevents detection is not known. Soluble CPRs had lower isoelectric points than those that remained in the final pellet; most members of other CP families had isoelectric points of 8 or higher. Obviously, techniques beyond analysis of differential solubility will be needed to learn how CPs interact with each other and with chitin.

The focus of this paper is to propose an approach to construct histogram values for the principal components of interval-valued observations. Le-Rademacher and Billard (J Comput Graph Stat 21:413---432, 2012) show that for a principal component analysis on interval-valued observations, the resulting observations in principal component space are polytopes formed by the convex hulls of linearly transformed vertices of the observed hyper-rectangles. In this paper, we propose an algorithm to translate these polytopes into histogram-valued data to provide numerical values for the principal components to be used as input in further analysis. Other existing methods of principal component analysis for interval-valued data construct the principal components, themselves, as intervals which implicitly assume that all values within an observation are uniformly distributed along the principal components axes. However, this assumption is only true in special cases where the variables in the dataset are mutually uncorrelated. Representation of the principal components as histogram values proposed herein more accurately reflects the variation in the internal structure of the observations in a principal component space. As a consequence, subsequent analyses using histogram-valued principal components as input result in improved accuracy.

Nowadays, most government agencies and local authorities regularly and routinely collect a large amount of data from censuses and surveys and officially publish them for public purposes. The most frequently used form for the publication is as statistical tables and it is usually not possible to access the raw data for those tables due to privacy issues. Under these situations, we have to analyze data using only those aggregated tables. These tables typically have formats summarized by ordinal or nominal items. Tables for quantitative variables have histogram-valued formats and those for qualitative variables are represented by multimodal-valued types. Both are classes of the so-called symbolic data. In this study, we propose dissimilarity measures and a divisive clustering algorithm for symbolic multimodal-valued data. In order to split a partition efficiently at each stage, the algorithm extends the monothetic method for binary data. The proposed method is verified by simulation studies and applied to a work-related nonfatal injury and illness dataset.

Clustering is an explanatory procedure which helps to understand data with complex structure and multivariate relationships, and is a very useful method to extract knowledge and information especially from large datasets. When such datasets are aggregated into categories (as driven by scientific questions underlying the analysis), the resulting observations will perforce be expressed as so-called symbolic data (though symbolic data can occur ''naturally'' in any sized datasets). The focus of this work is to provide a divisive polythetic algorithm to establish clusters for p-dimensional histogram-valued data. In addition, two cluster validity indexes for use in establishing the optimal number of clusters are also developed. Finally, the proposed procedure is applied to a large forestry cover type dataset.

One feature of contemporary datasets is that instead of the single point value in the p-dimensional space ℜp seen in classical data, the data may take interval values thus producing hypercubes in ℜp. This paper studies the vertices principal components methodology for interval-valued data; and provides enhancements to allow for so-called ‘trivial’ intervals, and generalized weight functions. It also introduces the concept of vertex contributions to the underlying principal components, a concept not possible for classical data, but one which provides a visualization method that further aids in the interpretation of the methodology. The method is illustrated in a dataset using measurements of facial characteristics obtained from a study of face recognition patterns for surveillance purposes. A comparison with analyses in which classical surrogates replace the intervals, shows how the symbolic analysis gives more informative conclusions. A second example illustrates how the method can be applied even when the number of parameters exceeds the number of observations, as well as how uncertainty data can be accommodated. © 2011 Wiley Periodicals, Inc. Statistical Analysis and Data Mining 4: 229–246 2011 © 2011 Wiley Periodicals, Inc.

Increasingly, datasets are so large they must be summarized in some fashion so that the resulting summary dataset is of a more manageable size, while still retaining as much knowledge inherent to the entire dataset as possible. One consequence of this situation is that the data may no longer be formatted as single values such as is the case for classical data, but rather may be represented by lists, intervals, distributions, and the like. These summarized data are examples of symbolic data. This article looks at the concept of symbolic data in general, and then attempts to review the methods currently available to analyze such data. It quickly becomes clear that the range of methodologies available draws analogies with developments before 1900 that formed a foundation for the inferential statistics of the 1900s, methods largely limited to small (by comparison) datasets and classical data formats. The scarcity of available methodologies for symbolic data also becomes clear and so draws attention to an enor...

Census data are used in a variety of ways. In the United States, their constitutionally mandated use is apportioning congressional seats in the House of Representatives based on population. After the 1910 Census, Congress fixed the number of seats permanently at 435. This had the inevitable consequence that any subsequent shifts in population would result in some states gaining seats at the expense of other states. It was now a zero-sum game, and, since politicians typically are loath to give up power, the stage was set for potential battles in the future. It is

Quantitative plant-disease epidemiology gained its primary motivation with the publication of VanderPlank's (1963) mathematical treatise, since which a range of predominantly deterministic models has been proposed by plant pathologists to describe the temporal dynamic of plant diseases. In this article, a number of results are reviewed. In particular, a summary is made of the primary models developed in the literature together with diseases for which they were proposed; estimation of the infection rate is considered for the generalized logistic model; and a bivariate stochastic model for stem and leaf infection is developed and used to obtain measures of plant disease stress. L'epidemiologie quantitative des maladies vegetales a acquis sa motivation premiere avec la publication du traite mathematique de VanderPlank (1963), d'apres lequel une variete de modeles majoritairement deterministes ont ete proposes par des pathologistes des plantes afin de decrire la dynamique temporelle des maladies vegetales. Dans cet article, un certain nombre de resultats sont passes en revue. En particulier, nous fournissons un sommaire des modeles primaires developpes dans la litterature, et des maladies pour lesquels ils sont proposes; nous considerons l'estimation du taux d'infection pour le modele logistique generalise; nous developpons un modele stochastique bivarie pour l'infection de la tige et de la feuille, afin d'obtenir des mesures du stress de la maladie des plantes.

(1997). A Voyage of Discovery. Journal of the American Statistical Association: Vol. 92, No. 437, pp. 1-12.

This article addresses the issue of empirical Bayes estimation procedures for the transition probability matrix of a single stationary Markov chain. The distribution of the frequency count matrix is obtained by relying heavily on the Whittle distributions and on the natural conjugate priors. Thence empirical Bayes estimates are obtained, and the limiting distribution is derived

Mallows' L2 distance allows for decomposition of total inertia into within and between inertia according to Huygens theorem. It can be decomposed into three terms: the location term, the spread term and the shape term; a simple and straightforward proof of this theorem is presented. These characteristics are very helpful in the interpretation of the results for some distance-based methods, such as clustering by k-means and classical multidimensional scaling. For histogram-type data, Mallows' L2 distance is preferable because its calculation is simple, even when the number and length of the histograms' subintervals differ. An illustration of its use on population pyramids for 14 East European countries in the period 1995–2015 is presented. The results provide an insight into the information that this distance can extract from a complex dataset.

International audience; Contemporary computers collect databases that can be too large for classical methods to handle. The present work takes data whose observations are distribution functions (rather than the single numerical point value of classical data) and presents a computational statistical approach of a new methodology to group the distributions into classes. The clustering method links the searched partition to the decomposition of mixture densities, through the notions of a function of distributions and of multi-dimensional copulas. The new clustering technique is illustrated by ascertaining distinct temperature and humidity regions for a global climate dataset and shows that the results compare favorably with those obtained from the standard EM algorithm method.

Cardiologists are interested in determining whether the type of hospital pathway followed by a patient is predictive of survival. The study objective was to determine whether accounting for hospital pathways in the selection of prognostic factors of one-year survival after acute myocardial infarction (AMI) provided a more informative analysis than that obtained by the use of a standard regression tree analysis (CART method). Information on AMI was collected for 1095 hospitalized patients over an 18-month period. The construction of pathways followed by patients produced symbolic-valued observations requiring a symbolic regression tree analysis. This analysis was compared with the standard CART analysis using patients as statistical units described by standard data selected TIMI score as the primary predictor variable. For the 1011 (84, resp.) patients with a lower (higher) TIMI score, the pathway variable did not appear as a diagnostic variable until the third (second) stage of the tree construction. For an ecological analysis, again TIMI score was the first predictor variable. However, in a symbolic regression tree analysis using hospital pathways as statistical units, the type of pathway followed was the key predictor variable, showing in particular that pathways involving early admission to cardiology units produced high one-year survival rates.

In a survey of 640 human subjects, a subgroup of 356 persons without recent exposure to antibiotics demonstrated that those with a high prevalence of Hg resistance in their intestinal floras were significantly more likely to also have resistance to two or more antibiotics. This observation led us to consider the possibility that mercury released from amalgam ("silver") dental restorations might be a selective agent for both mercury- and antibiotic-resistant bacteria in the oral and intestinal floras of primates. Resistances to mercury and to several antibiotics were examined in the oral and intestinal floras of six adult monkeys prior to the installation of amalgam fillings, during the time they were in place, and after replacement of the amalgam fillings with glass ionomer fillings (in four of the monkeys). The monkeys were fed an antibiotic-free diet, and fecal mercury concentrations were monitored. There was a statistically significant increase in the incidence of mercury-resistant bacteria during the 5 weeks following installation of the amalgam fillings and during the 5 weeks immediately following their replacement with glass ionomer fillings. These peaks in incidence of mercury-resistant bacteria correlated with peaks of Hg elimination (as high as 1 mM in the feces) immediately following amalgam placement and immediately after replacement of the amalgam fillings. Representative mercury-resistant isolates of three selected bacterial families (oral streptococci, members of the family Enterobacteriaceae, and enterococci) were also resistant to one or more antibiotics, including ampicillin, tetracycline, streptomycin, kanamycin, and chloramphenicol. While such mercury- and antibiotic-resistant isolates among the staphylococci, the enterococci, and members of the family Enterobacteriaceae have been described, this is the first report of mercury resistance in the oral streptococci. Many of the enterobacterial strains were able to transfer mercury and antibiotic resistances together to laboratory bacterial recipients, suggesting that the loci for these resistances are genetically linked. Our findings indicate that mercury released from amalgam fillings can cause an enrichment of mercury resistance plasmids in the normal bacterial floras of primates. Many of these plasmids also carry antibiotic resistance, implicating the exposure to mercury from dental amalgams in an increased incidence of multiple antibiotic resistance plasmids in the normal floras of nonmedicated subjects.

SUMMARY Using results of Anderson & Goodman (1957), we develop empirical Bayes estimates for the transition probabilities of a finite stationary Markov chain. Some asymptotic properties of these estimators are also derived.

Contemporary computers bring us very large datasets, datasets which can be too large for those same computers to analyse properly. One approach is to aggregate these data (by some suitably scientific criteria) to provide more manageably-sized datasets. These aggregated data will perforce be symbolic data consisting of lists, intervals, histograms, etc. Now an observation is a p-dimensional hypercube or Cartesian product of p distributions in Rp, instead of the p-dimensional point in in Rp of classical data. Other data can be naturally symbolic. We give a brief overview of interval-valued data and show briefly that it is important to use symbolic analysis methodology since, e.g., analyses based on classical surrogates ignore some of the information in the dataset.

While symbolic data exist in their own right, contemporary datasets can be too large to analyse using traditional statistical methodologies. Aggregation of these large datasets into sets of more managable size perforce produce datasets whose entries are symbolic data. This paper studies the derivation of basic description statistics, in particular, histograms and mean and variances plus joint histograms for interval-valued datasets when logical dependency rules are present. Algorithms for calculating these histograms are also provided.

The International Biometric Society is an international society for the advancement of biological science through the development of quantitative theories and the application, development and dissemination of effective mathematical and statistical techniques. We consider some of the nonscientific and scientific issues being addressed by researchers in and across Regions and Groups of our Biometric world. These run the gamut of theoretical to applied mathematics and statistics with the applications spanning many fields and, not surprisingly, forming a rich source of new theoretical developments.

“Do (workers elsewhere) Immediately embrace the statistical method as an integral thread in the fabric of their own enterprises and decision making ventures, or have we become Isolated and irrelevant?” Set against the backdrop of ASA's first century of writers, we attempt to cast a critical eye at the role of statistics and the role of the statistician, both in Its practice and in the training and preparation of the practitioner for the next century. This includes addressing the role of the Association and the profession.

One of the concerns of a plant scientist is the need to control disease and its spread through crops, plants, and pasture. This leads to the study of the progress of disease as it spreads through the plant and, in particular, to plant stress measured by the area under the disease progress curve. A nice review of this and related concepts has been provided elsewhere, but that work restricts attention to deterministic models. In this work, we develop a stochastic model for disease spread. Hence, plant stress is a stochastic variable. We show that this stress follows a mixture distribution related to sums of known exponential random variables, and we derive the corresponding mean and variance of stress experienced by the plant until prespecified proportions become infected. Through numerical illustrations, we discuss the significance of the model parameters. We also derive limiting formulas as the number of leaves/size of plants becomes large.

Partial table of contents: ENVIRONMENTAL HAZARDS Spatial Pattern Analyses to Detect Rare Disease Clusters (L. Waller, et al.) Prediction Models for Personal Ozone Exposure Assessment (D. Wypij & L.-J Liu) FORESTRY, FISHERIES, GENETICS Estimating Pine Seedling Response to Ozone and Acidic Rain (J. Rawlings & S. Spruill) Survival Analysis for Size Regulation of Atlantic Halibut (S. Smith, et al.) HABITAT AND ANIMAL STUDIES Spatial Association Learning in Hummingbirds (J. Graham & A. Petkau) Time-Series Analyses of Beaver Body Temperatures (P. Reynolds) HEALTH CARE AND PUBLIC HEALTH POLICY Analysis of Attitudes Towards Workplace Smoking Restrictions (S. Bull) CLINICAL TRIALS Early Lung Cancer Detection Studies (B. Flehinger & M. Kimmel) Quality Control for Bone Mineral Density Scans (S. Wong & N. Lane) EPIDEMIOLOGY, TOXICOLOGY Patterns of Lung Cancer Risk in Ex-Smokers (B. Gillespie, et al.) Drug Interactions Between Morphine and Marijuana (C. Gennings, et al.) Appendix References Indexes.

It is an honor and a privilege to thank Regional Whynott Chairman of the Regional Municipality of Hamilton-Wentworth for his warm welcome to Canada and to Hamilton and also our thanks through him for the support and hospitality of the Tourism and Conventions Office of the Department of Economic Development. It is also a privilege to thank Dean Bob McNutt for his greetings on behalf of McMaster University for the occasion of our Seventeenth International Biometric Conference. May I take this opportunity to thank Judith O'Fallon, President of the Host Region, for the invitation to hold our IBC in this Eastern North American Region of the IBS. I wish on behalf of the International Society to thank the Chair of the Local Organizing Committee Peter MacDonald who together with his committee has worked long and hard for the success of this week. To this we add our thanks to Byron Morgan and his Program Committee who have organized an outstanding scientific program for us to savor and enjoy. We shall be thinking, dreaming, and breathing statistics this week. To ensure we retain a balanced perspective on life, I refer you to the famous American Robert Frost's well known poem "The Road Not Taken." With apologies to Robert Frost, rather than focussing on roads not taken, we shall briefly consider instead roads we have taken, or "The Roads Travelled." Past Presidential Addresses have largely taken the form of a state of the profession or state of the society review type format, though some have focussed on some specific statistical issue. One of these latter is that of Professor Cochran who in 1955 addressed the Society under the title "The 1954 trial of the poliomyelitis vaccine in the United States." A cross-reading of the registration of participants in 1955 and today 1994 reveals that, except for David Finney and P. V. Sukhatme, none of us heard that presentation. Perhaps Professor Finney and Suhkatme can verify for us, but my reading of this address is that it was surely a compelling presentation dealing as it did with a disease that was seemingly running rampant and which almost surely was capturing everyone's attention. As Cochran (1955) stated

Recently, Billard, Lacayo and Langberg (1979) considered the classical simple epidemic process from the viewpoint of interinfection times. However, this concept can be expanded to include arbitrary generalized simple epidemic processes as well as non-exponential interinfection times. This work finds the distribution of the number of infectives for these generalizations as well as the mean and variance.

SUMMARY Based on intermittent observations from several independent regression models with autocorrelated errors, we derive tests of homogeneity, and the maximum likelihood estimate of the common regression parameter. The estimate for the regression parameter is based on aggregated data in the form of sample means at each time point. The limit distributions of the test statistics, and the estimate are considered. Basawa, Billard & Srinivasan (1984) discussed testing the homogeneity of several independent autoregressive processes. They assumed that equally-spaced time series data were available from all the realizations. In many practical situations, however, observa- tions may be missing at some time points, and consequently the sample may consist of intermittent observations from each series. Also, in some cases, only the pooled data in the form of sample means of observations at successive time points may be recorded, these sample means being based on an unequal number of sample sizes at different time points. In the present paper, we first indicate how the tests of homogeneity of Basawa et al. (1984) can be applied to the present situation for testing the equality of regression parameters for several independent regression models with autocorrelated errors. Assum- ing homogeneity, we then consider the estimation of the common regression parameter based on the pooled data of sample means. Section 2 gives a formulation of the basic model and the tests of homogeneity are considered in ? 3. Finally, in ? 4 we study the estimation problem, and discuss the limit distribution. It is possible to extend the results

SUMMARY Recently Billard (1977) introduced a truncated partial sequential procedure for testing a null hypothesis about a normal mean with known variance against a two-sided alternative hypothesis. That procedure had a disadvantage that a large number of observations was necessary if the null hypothesis was to be accepted. A new procedure is introduced which reduces the expected sample size for all mean values with considerable reductions for values near the null mean value. Theoretical operating characteristic and average sample number functions are derived.

A simple epidemic process in which the number of individuals who can become infected at any point in time is itself a random variable is described. The discrete asymptotic behaviour of such a process is discussed. In particular, the associated marginal distribution of the limiting process is considered.

SUMMARY This paper considers the situation in which it is desired to compare a null hypothesis concerning the unknown mean of a normal distribution whose variance is known against a twosided alternative hypothesis. A procedure is introduced in which an initial number of observations is taken. Then a truncated sequential test procedure of the type described by Armitage (1957) is implemented. An advantage of the resultant procedure is that the average sample number function under the null hypothesis has been reduced from that obtained by the Armitage procedure.

SUMMARY Limit distributions both under the null and a sequence of alternative hypotheses are derived for the likelihood ratio and related statistics for testing the homogeneity of several autoregressive processes. These results are extended to mixed autoregressive and moving average processes. It is often required to test whether two or more sets of data can be considered as coming from a common time series model. If the hypothesis of homogeneity is accepted one can obviously achieve parsimony in model specification and also obtain better estimates of the model parameters by pooling the data sets. On the other hand, if there is a significant difference between the sets one may be interested in alternative models which may vary between sets but are still close to each other for large samples. When dealing with regression models the homogeneity tests usually lead to analysis of variance techniques. In this note we consider the comparison of autoregressive and also of mixed autoregressive-moving average models. A unified method of deriving limit distributions of the likelihood ratio and related test statistics is suggested in ? 2. The method is illustrated in ? 3 for autoregressive processes augmented by regression variables, and for mixed autoregressive-moving average processes.

Classical epidemic models have invariably proved to be mathematically intractable. By considering the distribution of the number of infectives in a simple epidemic process as a convolution of exponential waiting times, the solution to the classical model is obtained easily giving more insight into the underlying structure. The idea can be extended to other simple epidemic models.

Ever since Lotka (1925) and Volterra (1926), (1931) first considered mathematical formulations for prey-predator processes, the resultant equations have resisted attempts to solve them. However, over the intervening 50 years, standard techniques have allowed a few isolated results to be obtained for some simplified versions of the original process, but the classical equations for the stochastic model have remained unsolved. We give here solutions to the classical process for the case in which interactions occur over a sufficiently short period of time that no births occur. The technique used is one recently developed by Severo (1969a), (1969b), (1971). The approach can be easily generalised to allow solution for the case in which births do occur, as well as for the simplified versions of the original process.

The recurrent epidemic model has hitherto proved mathematically intractable. It is shown here that by imposing the mild condition that the source of new susceptibles is initially of a constant size M for M sufficiently large that there is zero probability of more than M new susceptibles being added to the population in the course of the epidemic, then state probabilities for the stochastic recurrent epidemic model can be found.

An optimum partial sequential procedure for testing a null hypothesis concerning the binomial parameter with a two-sided alternative hypothesis is described. Formulas for its operating characteristic and average sample number functions are derived. By approximating an Armitage procedure by a special case of this partial procedure, approximate values can be obtained for his operating characteristic and average sample number functions.