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We consider a multivariate situation with means µ₁,...,µ_p and covariance matrix Σ. We wish to derive sequential procedures for testing the hypothesis: H₀: (µ̲ - µ̲ₒ)Σ⁻¹(µ̲ - μ̲₀)’= λ₀²( usually zero) against the alternative: μ̲₀ H₁: (µ̲ - µ̲ₒ)Σ⁻¹µ̲ - μ̲₀)’=λ₁² both for the case where Σ is known (the sequential X²-test) and where Σ is unknown and must be estimated from the sample (the sequential T²-test). These sequential procedures should guarantee that the probability of accepting H₁ when H₀ is true is equal to a and the probability of accepting H₀ when H₁ is true is equal to β. For the case where Σ is known, λ₀² = 0 and λ₁² = λ², the test procedure is as follows: for a sample of n observations form the probability ratio: P_1n/P_0n = e^-nλ²₀F₁(p/2;nλ²X_n²/4) where p denotes the number of variables, _nx[with horizontal bars above and below the x] denotes the vector of the sample means based on n observations, X²_n = n(_nx[with horizontal bars above and below the x] - μ̲₀) Σ⁻¹(_nx[with horizontal bars above and below the x] - μ̲₀)’ and ₀F₁ (c;x) is a type of generalized hypergeometric function. a. If P_1n/P_0n ≤ β/(1-α), accept H₀; b. If P_1n/P_0n ≥ (1- β)/α, accept H₁; c. If β/(1-α) < P_1n/P_0n < (1-β)/α, continue sampling. For the case where Σ is unknown, the procedure is exactly the same except that the probability ratio is now: P_1n/P_0n = e⁻^-nλ²/2 ₁F₁[n/2,p/2;nλ²T_n²/2(n-1+T_n²)] where T_n² = n(_nx[with horizontal bars above and below the x] - μ̲₀)S<sub>n</n>⁻¹(_nx[with horizontal bars above and below the x] - μ̲₀)' , S_ndenotes the sample covariance matrix based on n observations and ₁F₁(a,c;x) is a confluent hypergeometric function. Procedures are also given for the case λ²₀ ≠ 0. Similar procedures are given to test the hypothesis: H₀ = (μ̲₁ - μ̲₂- δ ̲)Σ⁻¹(μ̲₁ - μ̲₂ - δ̲)’ = λ₀² (usually zero) against the alternative: H₁ = (μ̲₁ - μ̲₂- δ ̲)Σ⁻¹(μ̲₁ - μ̲₂ - δ̲)’ = λ₁² It is shown that these sequential procedures all exist in the sense that the risks of accepting H₀ when H₁ is true and of accepting H₁ when H₀ is true are approximately α and β respectively and that these sequential procedures terminate with probability unity. Some of these situation have been generalized to give simultaneous tests and the means and covariance matrix of a sample. No expressions yet exist for the OC or ASN functions although some conjectured values have been determined for the latter and suggest, in comparison with their corresponding fixed-sample tests, substantial reductions in the sample sizes required when either H₀ or H₁ is true. The general problem of tolerances is discussed and then some of these procedures are demonstrated with a numerical example drawn from the field of ballistic missiles. The determination of P_1n/P_0n is quite laborious for both the sequential X² - and T²-tests since it requires the evaluation of a hypergeometric function each time an observation is made. It would be better for each value of n, given p, α, β and λ² under H₁, to compute the values of X²_n or T²_n which would correspond to the boundaries of the tests indicated by β/(1-α) and (1-β)/α. Tables to facilitate both the sequential X²- and T²-tests are given for p = 2,3,...,9; λ² = 0.5, l.0, 2.0; α = β = 0.05 for n ranging from the minimum value necessary to reach a decision to 30, 45 and 60 for λ² = 0.5, 1.0, 2.0 respectively. These tables were prepared on the IBM 650 computer using the Newton-Raphson iterative procedure. Finally, a discussion is given for the hypergeometric function ₀F₁ (c;x) and a table given of this function for c = .5(.5)5.0 and x = .1(.1)1(1) 10(10)100(50)1000. / Doctor of Philosophy

The purpose of this study is to make an empirical evaluation of Multivariate Sequential Procedures for Testing Means as proposed by J. E. Jackson. This was done by simulated sampling from a multivariate normal population with known means, both when the variance-covariance matrix is assumed known (Sequential χ²-Test), and when it is assumed unknown (Sequential T² -Test). The results indicate the test procedure, either the Sequential χ²-Test or the Sequential T² -Test, lead to satisfactory decisions. The tests are conservative but the α and β errors are of an appropriate magnitude. The method (Bhate's Conjecture) used to approximate the Average Sample Number appears to give good estimates of the size of sample needed to obtain a decision. / Master of Science

Exact distributions of statistics for the tests of hypotheses in multivariate analysis of variance and for the test of independence are compared z with the asymptotic -XL distribution for -2 log à s. â Critica1" sample sizes have been recorded which indicate the magnitude of a sample needed so that the approximate technique may produce satisfactory results for testing at the .05 and .01 significance levels. / Master of Science

This dissertation is concerned with the application of a method for decomposing multivariate data, three-mode principal component analysis, to a three-way table with one observation per cell. It is based on the class of multiplicative models for three-way tables (s x t x u) whose general form has expectation E(y_ijk) = μ + α_i + β_j + γ_k + (αβ)_ij + (αγ)_ik + (βγ)_jk + s Σ _p=1 t Σ _q=1 u Σ _r=1 c_pqr g_ip h_jq e_kr. The application is related to tests for nonadditivity in the two-way analysis of variance with noreplication. Three-factor interaction can be assessed for three-way cross classified tables with only one observation per treatment combination. This is done by partitioning the three-factor interaction sum of squares into a portion related to the interaction and a portion associated with random error. In particular, the estimated interaction matrix is decomposed by three-mode principal component analysis to separate significant interaction from random error. Three test procedures are presented for assessing interaction: randomization tests, Monte Carlo methods, and likelihood ratio tests. Examples illustrating the use of these approaches are presented. In addition to the above testing approaches, a graphical procedure, joint plots, is investigated for diagnosing the type of model to fit to three-way arrays of data. The plot is a multiway analogue of the biplot graphical analysis for two-way matrices. Each observation is represented by a linear combination of inner products of markers which are obtained from three-mode principal component analysis. The relationship between various models and the geometrical configurations of the plots on Euclidean spaces of such markers allows one to diagnose the type of model which fits the data. An example is given to illustrate the simplicity of the technique and the usefulness of this graphical approach in diagnosing models. / Ph. D.

Thesis (PhD)--Stellenbosch University, 1997. / One copy microfiche. / ENGLISH ABSTRACT: Discriminani analysis and logistic regression are techniques that can be used to classify entities of unknown origin into one of a number of groups. However, the underlying models and assumptions for application of the two techniques differ. In this study, the two techniques are compared with respect to classification of entities. Firstly, the two techniques were compared in situations where no data dependent variable selection took place. Several underlying distributions were studied: the normal distribution, the double exponential distribution and the lognormal distribution. The number of variables, sample sizes from the different groups and the correlation structure between the variables were varied to' obtain a large number of different configurations. .The cases of two and three groups were studied. The most important conclusions are: "for normal and double' exponential data linear discriminant analysis outperforms logistic regression, especially in cases where the ratio of the number of variables to the total sample size is large. For lognormal data, logistic regression should be preferred, except in cases where the ratio of the number of variables to the total sample size is large. " Variable selection is frequently the first step in statistical analyses. A large number of potenti8.Ily important variables are observed, and an optimal subset has to be selected for use in further analyses. Despite the fact that variable selection is often used, the influence of a selection step on further analyses of the same data, is often completely ignored. An important aim of this study was to develop new selection techniques for use in discriminant analysis and logistic regression. New estimators of the postselection error rate were also developed. A new selection technique, cross model validation (CMV) that can be applied both in discriminant analysis and logistic regression, was developed. ."This technique combines the selection of variables and the estimation of the post-selection error rate. It provides a method to determine the optimal model dimension, to select the variables for the final model and to estimate the post-selection error rate of the discriminant rule. An extensive Monte Carlo simulation study comparing the CMV technique to existing procedures in the literature, was undertaken. In general, this technique outperformed the other methods, especially with respect to the accuracy of estimating the post-selection error rate. Finally, pre-test type variable selection was considered. A pre-test estimation procedure was adapted for use as selection technique in linear discriminant analysis. In a simulation study, this technique was compared to CMV, and was found to perform well, especially with respect to correct selection. However, this technique is only valid for uncorrelated normal variables, and its applicability is therefore limited. A numerically intensive approach was used throughout the study, since the problems that were investigated are not amenable to an analytical approach. / AFRIKAANSE OPSOMMING: Lineere diskriminantanaliseen logistiese regressie is tegnieke wat gebruik kan word vir die Idassifikasie van items van onbekende oorsprong in een van 'n aantal groepe. Die agterliggende modelle en aannames vir die gebruik van die twee tegnieke is egter verskillend. In die studie is die twee tegnieke vergelyk ten opsigte van k1assifikasievan items. Eerstens is die twee tegnieke vergelyk in 'n apset waar daar geen data-afhanklike seleksie van veranderlikes plaasvind me. Verskeie onderliggende verdelings is bestudeer: die normaalverdeling, die dubbeleksponensiaal-verdeling,en die lognormaal verdeling. Die aantal veranderlikes, steekproefgroottes uit die onderskeie groepe en die korrelasiestruktuur tussen die veranderlikes is gevarieer om 'n groot aantal konfigurasies te verkry. Die geval van twee en drie groepe is bestudeer. Die belangrikste gevolgtrekkings wat op grond van die studie gemaak kan word is: vir normaal en dubbeleksponensiaal data vaar lineere diskriminantanalise beter as logistiese regressie, veral in gevalle waar die. verhouding van die aantal veranderlikes tot die totale steekproefgrootte groot is. In die geval van data uit 'n lognormaalverdeling, hehoort logistiese regressie die metode van keuse te wees, tensy die verhouding van die aantal veranderlikes tot die totale steekproefgrootte groot is. Veranderlike seleksie is dikwels die eerste stap in statistiese ontledings. 'n Groot aantal potensieel belangrike veranderlikes word waargeneem, en 'n subversamelingwat optimaal is, word gekies om in die verdere ontledings te gebruik. Ten spyte van die feit dat veranderlike seleksie dikwels gebruik word, word die invloed wat 'n seleksie-stap op verdere ontledings van dieselfde data. het, dikwels heeltemal geYgnoreer.'n Belangrike doelwit van die studie was om nuwe seleksietegniekete ontwikkel wat gebruik kan word in diskriminantanalise en logistiese regressie. Verder is ook aandag gegee aan ontwikkeling van beramers van die foutkoers van 'n diskriminantfunksie wat met geselekteerde veranderlikes gevorm word. 'n Nuwe seleksietegniek, kruis-model validasie (KMV) wat gebruik kan word vir die seleksie van veranderlikes in beide diskriminantanalise en logistiese regressie is ontwikkel. Hierdie tegniek hanteer die seleksie van veranderlikes en die beraming van die na-seleksie foutkoers in een stap, en verskaf 'n metode om die optimale modeldimensiete bepaal, die veranderlikes wat in die model bevat moet word te kies, en ook die na-seleksie foutkoers van die diskriminantfunksie te beraam. 'n Uitgebreide simulasiestudie waarin die voorgestelde KMV-tegniek met ander prosedures in die Iiteratuur. vergelyk is, is vir beide diskriminantanaliseen logistiese regressie ondemeem. In die algemeen het hierdie tegniek beter gevaar as die ander metodes wat beskou is, veral ten opsigte van die akkuraatheid waarmee die na-seleksie foutkoers beraam word. Ten slotte is daar ook aandag gegee aan voor-toets tipeseleksie. 'n Tegniek is ontwikkel wat gebruik maak van 'nvoor-toets berarningsmetode om veranderlikes vir insluiting in 'n lineere diskriminantfunksie te selekteer. Die tegniek ISin 'n simulasiestudie met die KMV-tegniek vergelyk, en vaar baie goed, veral t.o.v. korrekte seleksie. Hierdie tegniek is egter slegs geldig vir ongekorreleerde normaalveranderlikes, wat die gebruik darvan beperk. 'n Numeries intensiewe benadering is deurgaans in die studie gebruik. Dit is genoodsaak deur die feit dat die probleme wat ondersoek is, nie deur middel van 'n analitiese benadering hanteer kan word nie.

published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy

This dissertation investigated the use of a variety of multivariate statistical procedures to answer questions regarding the value of a number of medical tests and procedures in the diagnosis of space-occupying liver disease. Also investigated were some aspects of test ordering behavior by physicians. A basic methodology was developed to deal with archival data. A number of methodological problems were addressed. Discriminant function analysis was used to determine which procedures and tests served to provide the best classification of disease entities. Although the results were not spectacular, some variables, including a physical examination variable and a number of laboratory procedures were identified as being important. A more detailed analysis of the role of the laboratory variables was afforded by the use of stepwise logistic regression. In these analyses pairs of disease classifications were compared. Two of the more specific laboratory tests, total bilirubin and alkaline phosphate, entered into the equations to provide a fit to the data. Logistic regression analyses employing patient variables mirrored the results obtained with the discriminant function analyses. Liver-spleen scan indicants were also employed as predictor variables in a series of logistic regression analyses. In general, for a range of comparisons, those indicants cited in the literature as being valuable in discriminating between disease entities entered into the equations. Log-Linear models were used to investigate test ordering behavior. In general, test ordering was independent of department. The sole exception being that of the Gynecology-oncology department which relies heavily on Ultrasound. Log-Linear analyses investigating the use of a number of procedures showed differential use of procedures consistent with what is usually suggested in the medical literature for the combination of different imaging and more specialized procedures. Finally, a set of analyses investigated the ordering of a number of procedures relative to specific disease classifications. This set of analyses suffers, as do a number of the other analyses, from insufficient numbers of cases. However, some indications of differential performance of tests for different disease classifications were evident. Suggestions for further study concentrated on the development of experimental procedures given the results of this study.