N. J. Le Roux

Discriminating between the end products of the earlier Middle Stone Age sub-stages at Klasies River using biplot methodology

The difference between the culture-stratigraphic entities, the MSA I and MSA II, in the Klasies River sequence is explored by statistical analysis of the end products. Technological analysis of the cores, end products and waste products suggested that the MSA I and MSA II represent distinct technological traditions aimed at producing different end products. To quantify the difference between the end products, points and blades, extensive univariate and multivariate statistical analyses of continuous variables have been undertaken. Biplot methodology is adopted for enhancing the statistical analysis. Canonical variate analysis biplots are constructed and alpha-bags added for visual displays of the overlap and separation among the different groups. It is demonstrated that the platform thickness relative to length is useful in discriminating between the end products of the MSA I and MSA II. Statistical analyses support a clear distinction between the MSA I and MSA II.

A synthesis of canonical variate analysis, generalised canonical correlation and Procrustes analysis

Canonical variate analysis (CVA) is concerned with the analysis of J classes of samples, all described by the same variables. Generalised canonical correlation analysis (GCCA) is concerned with the analysis of K sets of variables, all describing the same samples. A generalised procrustes analysis context is used for data partitioned into J classes of samples and K sets of variables to explore the links between GCCA and CVA. Biplot methodology is used to exploit the visualisation properties of these techniques. This methodology is illustrated by an example of 1425 samples described by three sets of variables (K = 3), the initial analysis of which suggests a grouping of the samples into four classes (J = 4), followed by subsequent more detailed analyses.

Variable selection and error rate estimation in discriminant analysis

We investigate the problems of variable selection and error rate estimation in discriminant analysis. A cross model validation based method that simultaneously addresses both these issues, is proposed. The performance of this method is compared to that of some procedures in the literature by means of a Monte Carlo simulation study. The actual error rate of the classification rule based on the selected variables and the probability of correct model selection are used to evaluate the selection performance of the procedures, and unconditional mean squared error is used as criterion to judge estimation accuracy. We find that the new proposal largely improves on existing procedures.

Extending a scatterplot for displaying group structure in multivariate data : a case study

The power of canonical variate analysis (CVA) biplots, when regarded as extensions of ordinary scatterplots to describe variation and group structure in multivariate observations, is demonstrated by presenting a case study from the South African wood pulp industry. It is shown how multidimensional standards specified by users of a product may be added to the biplot in the form of acceptance regions such that the roles of the respective variables that influence the product can be ascertained. The case study considers an alternative to CVA and multivariate analysis of variance (MANOVA) when the application of these procedures becomes questionable as a result of dealing with small sample sizes and heterogeneity of covariance matrices. It is explained how analysis of distance (AOD) analogous to analysis of variance may be performed in such cases. Biplots to accompany AOD are provided. The biplots and AOD illustrated in the case study from the wood pulp industry have the potential to be used widely where a primary product, influenced by several variables, is produced and where this product is of importance to various secondary manufacturers depending on which set of multidimensional specifications are met.

School results and access test results as indicators of first-year performance at university

The goals set by the National Plan for Higher Education, the fact that many schools are still severely disadvantaged as well as far-reaching changes in the school system demand that South African universities urgently reconsider their admission procedures. Redesigning admission procedures calls for a thorough understanding of the interrelationships between school marks, results in existing access tests and first-year university performance. These interrelationships were statistically investigated in the case of the 1999, 2000 and 2001 intake groups, who were compelled to write access tests before being admitted to Stellenbosch University. The results of this investigation confirm an alarming degree of unpreparedness among many prospective students regarding what is expected of them at university. This is aggravated by school marks creating a totally unrealistic expectation of performance in the first year at university. It is emphasised that schools and authorities dealing with admission of prospective students at universities should be cognisant of the findings reported here. Furthermore, the statistical analyses demonstrate several novel techniques for investigating the interrelationship between school marks, access test results and university performance. Keywords : Access test, admission criteria, boxplot, first-year university performance, school marks ORiON Vol. 20 (1) 2004: pp. 73-88

MODELLING HAIL FREQUENCY USING A GENERALIZED ADDITIVEINTERACTIVE TECHNIQUE

ABSTRACT Relationships between hail day frequency (HDF), altitude and latitude in the summer rainfall region of South Africa are investigated non-para- metrically by means of generalized additive models. Two dimensional and three dimensional graphs are produced displaying the qualitative and quantitative character of these interrelationships.

Efficient maximin distance designs for experiments in mixtures

In this paper, different dissimilarity measures are investigated to construct maximin designs for compositional data. Specifically, the effect of different dissimilarity measures on the maximin design criterion for two case studies is presented. Design evaluation criteria are proposed to distinguish between the maximin designs generated. An optimization algorithm is also presented. Divergence is found to be the best dissimilarity measure to use in combination with the maximin design criterion for creating space-filling designs for mixture variables.

Statistical properties of indicators of first-year performance at university

Appraisal of admission procedures is a matter of urgency for South African universities, as well as for schools producing the prospective students. In this article the focus is on how various statistical procedures can be used to assess admission measures. Properties of the statistical distributions related to school results, access test results and first-year university performance are vital for decision-makers in schools preparing the prospective students and for those who wish to refine university admission procedures. These properties are scrutinised for the 1999, 2000 and 2001 intake groups required to write access tests before being admitted to Stellenbosch University. Using kernel density estimates the univariate distributions of all variables concerned are described in detail. Bagplots are proposed for visual displays of important features like location, spread, correlation, skewness, outliers and tails of bivariate distributions composed of university average performance and a school result or access test variable. Evidence is provided that certain access tests (Mathematics, Science and Numeracy Skills) have statistical distributions similar to that of average first-year university performance, but that average school marks could not be trusted to discriminate between potentially successful and unsuccessful university students.

Biplots for displaying multidimensional financial performance data graphically

Invariably financial performance of an enterprise is not judged on a single criterion but is described in terms of various related financial performance indicators. Financial decision makers are thus typically confronted with multidimensional data. This paper demonstrates to financial managers several uses of the biplot for displaying multidimensional data graphically. It is shown that biplots not only serve in understanding multidimensional data better but also might reveal features of such data not perceptible by graphing each variable on its own. Two different types of biplots are discussed: One for describing variation in multidimensional data and the other for separating different groups optimally. The first of these types of biplots is applied to a data set originating in risk management and the second to a data set where, on the basis of various financial ratio indicators, the aim is to separate enterprises in danger of financial failure within a year from those not in such danger. In the latter example a distinction is also made between companies classified as manufacturing enterprises and those classified as trading enterprises.

A comparison of the post selection error rate behaviour of the normal linear and quadratic discriminant rules.

The normal linear discriminant rule (NLDR) and the normal quadratic discriminant rule (NQDR) are popular classifiers when working with normal populations. Several papers in the literature have been devoted to a comparison of these rules with respect to classification performance. An aspect which has, however, not received any attention is the effect of an initial variable selection step on the relative performance of these classification rules. Cross model validation variabie selection has been found to perform well in the linear case, and can be extended to the quadratic case. We report the results of a simulation study comparing the NLDR and the NQDR with respect to the post variable selection classification performance. It is of interest that the NQDR generally benefits from an initial variable selection step. We also comment briefly on the problem of estimating the post selection error rates of the two rules.