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<p>Craniofacial variability index (CVI), estimated by the standard deviation of z-scores obtained from craniofacial measurements of one or more individuals, is considered in the medical literature as a useful and relatively simple quantitative measure of the degree of dysmorphogenesis in the head and face.</p> <p>CVI obtained from patients diagnosed with syndromes such as Smith-Lemli-Opitz Syndrome (SLOLS) is often compared with CVI for healthy individuals. Moreover, CVI is commonly used to compare degree of dysmorphogenesis among individuals and groups characterized by the presence or absence of certain syndromes and abnormalities. However, the type of comparison provided is often subjective with no statistical comparison of values provided to account for sample to sample variability.</p> <p>We performed a simulation study to compare the performance of tests of variance under the presence of within and between individual dependencies. We compare empirical level and power obtained from 10,000 simulations to assess performance. We considered four variance comparison tests: F-test, Levene’s Test, Fligner-Killeen test and permutation F test. We also provided a detailed analysis of a real data set to illustrate our results. Overall, results from our simulation indicate that the performances of the F and the permutation F tests are better than the other methods. However, for all the tests considered, power for detecting small differences in variance is very low when the sample size is small. An interesting finding in our simulation is that the performance of the tests was actually enhanced by the presence of within individual or group dependence, where the power of the tests increased with increased correlation. On the other hand, between dependence and within and between individual dependence have the effect of lowering the power in comparison with the scenarios of independence and within dependence. It is also observed that the higher the correlation the lower the power. It should be noted that in the case of group comparison the simulation scenario of within and between dependence shows opposite results to those observed in the individual comparison. A higher correlation is associated to both a higher level and power.</p> <p>On another note, the analysis of the Smith-Lemli-Opitz Syndrome (SLOS) dataset has shown that comparison of pattern profile between individuals is a useful tool to identify influential z-scores of craniofacial measurements that affect CVI and the subsequent results of hypothesis testing of equal variances using the classical F test versus Levene-median and Fligner-Killeen tests.</p> / Master of Science (MSc)

The purpose of this study is to examine variances in the case of distributions obtained by truncating a given distribution at various points. In particular, the truncated distributions are restricted to nested increasing intervals, and the question is posed whether the variances of these distributions are monotonically increasing. The answer to this question is relevant to the use of conditional information for purposes of estimation and prediction. Several tables are presented in the thesis which provide evidence of the property of monotonic variance for nested increasing intervals of truncation in the case of univariate distributions., The Monte Carlo procedure is used to determine a table of standard deviations for the standard normal distribution with the same points of truncation reported by Clark(2). Clark's table is given intact, and it is used in comparison with the new table reported here as a check on the Monte Carlo procedure used in the present study. Distributions other than the standard normal distribution are examined as well, namely, a Pearson U-shaped distribution and a bimodal distribution consisting of a mixture or two Pearson distributions. Graphs of the U-shaped and bimodal distributions are given. A section is given in which dispersion for a bivariate case is examined in terms of the bivariate normal distribution. An interesting trend among the covariance matrices is observed in the data reported in that section. A separate computer program for each type of distribution was written and used to calculate the variances of the truncated distributions. FORTRAN programs and flow charts are presented in the Appendix. Explanation of the tables and procedures used to calculate the entries in the body of each table are given in each section as well as some discussion of the results presented. / Master of Science

Sucrose, raffinose, and stachyose make up the majority of the carbohydrates in soybean seeds. While sucrose is a desirable component of soybean seeds, raffinose and stachyose are considered to be anutritive factors and eliminating or reducing them appears to be a beneficial endeavour. The major objective of this study was to determine the genetic mechanism controlling accumulation of soluble saccharides in soybean seeds. An experimental soybean line, V99-5089, with high sucrose (14.6%) content coupled with low amounts of raffinose (0.5%) and stachyose (0.4%) was the center of this study. Three populations were studied and segregation patterns were observed in F2:3 populations. All three sugars were extracted by an aqueous procedure and quantified by high pressure liquid chromatography (HPLC) using a NH2 column and refractive index (RI) detector. Segregation of seeds from F2:3 plants indicated a single, partially recessive gene reduced stachyose content of soybean seeds from about 4% to less than 1%. Estimates of genetic variability indicate the presence of sufficient additive variation in addition to the putative major gene to warrant selection. Raffinose and stachyose were positively correlated to each other and each was negatively correlated with sucrose while there was not a significant correlation between total sugar content and the amount of any of the individual sugars. Agronomic traits evaluated do not appear to be adversely effected by the reduction of stachyose content. Additionally, a negative relationship was observed between inorganic phosphorus and stachyose content of soybean seeds but a relationship was not observed between stachyose and phytate phosphorus or between inorganic phosphorus and phytate phosphorus. / Master of Science

The problem considered in this paper is that of estimating several missing values and analyzing the resulting augmented data in a balanced incomplete block design. The estimates are obtained by Yates' procedure of minimizing the error sum of squares. Explicit formulae are obtained for all cases involving not more than two missing values and for several particular configurations of the missing values within the design. A general solution is obtained which involves the inversion of a symmetric n-square matrix, where n is the number of missing values. An exact analysis of data augmented by missing value estimates is given which eliminates a positive bias in the treatment sum of squares. It is possible to treat a balanced incomplete block design as a randomized block design with missing values. Estimates of the missing entries and a randomized block analysis can then be obtained according to the methods of Glenn and Kramer. An example of this procedure is given, and the results are compared with the results obtained by the usual balanced incomplete block analysis. An example is given illustrating the techniques of missing value estimation and subsequent exact analysis for the balanced incomplete block design. / Master of Science

Various models of the genetic architecture of quantitative traits have been considered to provide the basis for increased genetic progress. The finite polygenic model (FPM), which contains a finite number of unlinked polygenic loci, is proposed as an improvement to the infinitesimal model (IM) for estimating both additive and dominance variance for a wide range of genetic models. Analysis under an additive five-loci FPM by either a deterministic Maximum Likelihood (DML) or a Markov chain Monte Carlo (MCMC) Bayesian method (BGS) produced accurate estimates of narrow-sense heritability (0.48 to 0.50 with true values of h2 = 0.50) for phenotypic data from a five-generation, 6300-member pedigree simulated without selection under either an IM, FPMs containing five or forty loci with equal homozygote difference, or a FPM with eighteen loci of diminishing homozygote difference. However, reducing the analysis to a three- or four-loci FPM resulted in some biased estimates of heritability (0.53 to 0.55 across all genetic models for the 3-loci BGS analysis and 0.47 to 0.48 for the 40-loci FPM and the infinitesimal model for both the 3- and 4-loci DML analyses). The practice of cutting marriage and inbreeding loops utilized by the DML method expectedly produced overestimates of additive genetic variance (55.4 to 66.6 with a true value of sigma squared sub a = 50.0 across all four genetic models) for the same pedigree structure under selection, while the BGS method was mostly unaffected by selection, except for slight overestimates of additive variance (55.0 and 58.8) when analyzing the 40-loci FPM and the infinitesimal model, the two models with the largest numbers of loci. Changes to the BGS method to accommodate estimation of dominance variance by sampling genotypes at individual loci are explored. Analyzing the additive data sets with the BGS method, assuming a five-loci FPM including both additive and dominance effects, resulted in accurate estimates of additive genetic variance (50.8 to 52.2 for true sigma squared sub a = 50.0) and no significant dominance variance (3.7 to 3.9) being detected where none existed. The FPM has the potential to produce accurate estimates of dominance variance for large, complex pedigrees containing inbreeding, whereas the IM suffers severe limitations under inbreeding. Inclusion of dominance effects into the genetic evaluations of livestock, with the potential increase in accuracy of additive breeding values and added ability to exploit specific combining abilities, is the ultimate goal. / Master of Science

M.S.

The purpose of this study is to consider two different types of estimators for reliability using the extreme value distribution as the life-testing model. First the unbiased minimum variance estimator is derived. Then the Bayes' estimators for the uniform, exponential, and inverted gamma prior distributions are obtained, and these results are extended to a whole class of exponential failure models. Each of the Bayes' estimators is compared with the unbiased minimum variance estimator in a Monte Carlo simulation where it is shown that the Bayes' estimator has smaller squared error loss in each case. The problem of obtaining estimators with respect to an exponential type loss function is also considered. The difficulties in such an approach are demonstrated. / Master of Science

The object of this paper was to study lower bounds ·for the variance of uniformly minimum variance unbiased estimators. The lower bounds of Cramer and Rao, Bhattacharyya, Hammersley, Chapman and Robbins, and Kiefer were derived and discussed. Each was compared with the other, showing their relative merits and shortcomings. Of the lower bounds considered all are greater than or equal to the Cramer-Rao lower bound. The Kiefer lower bound is as good as any of the others, or better. We were able to show that the Cramer-Rao lower bound is exactly the first Bhattacharyya lower bound. The Hammersley and the Chapman and Robbins lower bounds are identical when they both have the same parameter space, i.e., when Ω = (a,b). The use of the various lower bounds is illustrated in examples throughout the paper. / M.S.