The maximum likelihood (ML) estimation procedure of Matthews and Crowther (1995: A maximum likelihood estimation procedure when modelling in terms of constraints. South African Statistical Journal, 29, 29-51) is utilized to fit a continuous distribution to a grouped data set. This grouped data set may be a single frequency distribution or various frequency distributions that arise from a cross classification of several factors in a multifactor design. It will also be shown how to fit a bivariate normal distribution to a two-way contingency table where the two underlying continuous variables are jointly normally distributed. This thesis is organized in three different parts, each playing a vital role in the explanation of analysing grouped data with the ML estimation of Matthews and Crowther. In Part I the ML estimation procedure of Matthews and Crowther is formulated. This procedure plays an integral role and is implemented in all three parts of the thesis. In Part I the exponential distribution is fitted to a grouped data set to explain the technique. Two different formulations of the constraints are employed in the ML estimation procedure and provide identical results. The justification of the method is further motivated by a simulation study. Similar to the exponential distribution, the estimation of the normal distribution is also explained in detail. Part I is summarized in Chapter 5 where a general method is outlined to fit continuous distributions to a grouped data set. Distributions such as the Weibull, the log-logistic and the Pareto distributions can be fitted very effectively by formulating the vector of constraints in terms of a linear model. In Part II it is explained how to model a grouped response variable in a multifactor design. This multifactor design arise from a cross classification of the various factors or independent variables to be analysed. The cross classification of the factors results in a total of T cells, each containing a frequency distribution. Distribution fitting is done simultaneously to each of the T cells of the multifactor design. Distribution fitting is also done under the additional constraints that the parameters of the underlying continuous distributions satisfy a certain structure or design. The effect of the factors on the grouped response variable may be evaluated from this fitted design. Applications of a single-factor and a two-factor model are considered to demonstrate the versatility of the technique. A two-way contingency table where the two variables have an underlying bivariate normal distribution is considered in Part III. The estimation of the bivariate normal distribution reveals the complete underlying continuous structure between the two variables. The ML estimate of the correlation coefficient ρ is used to great effect to describe the relationship between the two variables. Apart from an application a simulation study is also provided to support the method proposed. / Thesis (PhD (Mathematical Statistics))--University of Pretoria, 2007. / Statistics / unrestricted

Multiple testing literature contains ample research on controlling false discoveries for hypotheses classified according to one criterion, which we refer to as `one-way classified hypotheses'. However, one often encounters the scenario of `two-way classified hypotheses' where hypotheses can be partitioned into two sets of groups via two different criteria. Associated multiple testing procedures that incorporate such structural information are potentially more effective than their one-way classified or non-classified counterparts. To the best of our knowledge, very little research has been pursued in this direction. This dissertation proposes two types of multiple testing procedures for two-way classified hypotheses. In the first part, we propose a general methodology for controlling the false discovery rate (FDR) using the Benjamini-Hochberg (BH) procedure based on weighted p-values. The weights can be appropriately chosen to reflect one- or two-way classified structure of hypotheses, producing novel multiple testing procedures for two-way classified hypotheses. Newer results for one-way classified hypotheses have been obtained in this process. Our proposed procedures control the false discovery rate (FDR) non-asymptotically in their oracle forms under positive regression dependence on subset of null p-values (PRDS) and in their data-adaptive forms for independent p-values. Simulation studies demonstrate that our proposed procedures can be considerably more powerful than some contemporary methods in many instances and that our data-adaptive procedures can non-asymptotically control the FDR under certain dependent scenarios. The proposed two-way adaptive procedure is applied to a data set from microbial abundance study, for which it makes more discoveries than an existing method. In the second part, we propose a Local false discovery rate (Lfdr) based multiple testing procedure for two-way classified hypotheses. The procedure has been developed in its oracle form under a model based framework that isolates the effects due to two-way grouping from the significance of an individual hypothesis. Simulation studies show that our proposed procedure successfully controls the average proportion of false discoveries, and is more powerful than existing methods. / Statistics

It is widely believed that the median is “usually” between the mean and the mode for skewed unimodal distributions. However, this inequality is not always true, especially with grouped data. Unavailability of complete raw data further necessitates the importance of evaluating this characteristic in grouped data. There is a gap in the current statistical literature on assessing mean–median–mode inequality for grouped data. The study aims to evaluate the relationship between the mean, median, and mode with unimodal grouped data; derive conditions for their inequalities; and present their application.

Because of the rapid advancements being made in the field of knowledge, educators, as well as those in other fields, must periodically take inventory. Current practices, policies, and methods must be carefully scrutinized to determine if they are the most effective. The group or class method of instruction is one such area. The current philosophy of education held in many parts of the United States today places a great deal of importance upon the individual child. Numerous programs have been inaugurated to give the individual child as much attention as possible and still be able to have a class large enough to be practical financially. This task becomes increasingly difficult when the range of abilities within each classroom is so great. It isn't uncommon in the upper elementary and secondary classes to find a spread of from six to nine years difference in ability or achievement within one classroom. Not only do we have the problem of range within the classroom, but with the increasing school population of today, classes have grown to a prohibitive size. Add these and other problems that stem from the pressures of present day society together, and even with the best possible teacher, we get only average results.

Grouping students with in the school setting has long been a concern to all those associated with the educative process. Much investigation has been devoted to the position of level of aspiration as influential motive forces for educational, occupational, and social achievement. Many avenues of grouping have been investigated and some very thoroughly. However, research concerning grouping in relation to level of aspiration in the school setting appears to be extremely limited. This study is designed to further investigate this issue. It might be said that this study consists of essentially two phases: (1) the development of instruments for measuring classroom aspiration, social aspiration, and educational aspiration; and (2) the determination of level of aspiration relationships which exist between junior high school boys grouped on the basis of ability and those randomly assigned to a classroom.

The existing estimation techniques for grouped data models can be analyzed as a class of estimators of instrumental variable-Generalized Method of Moments (GMM) type with the matrix of group indicators being the set of instruments. Econometric literature (e.g. Smith, 1997; Newey and Smith, 2004) show that, in some cases of empirical relevance, GMM can have shortcomings in terms of the large sample behaviour of the estimator being different from the finite sample properties. Generalized Empirical Likelihood (GEL) estimators are developed that are not sensitive to the nature and number of instruments and possess improved finite sample properties compared to GMM estimators. In this thesis, with the assumption that the data vector is iid within a group, but inid across groups, we developed GEL estimators for grouped data model having population moment conditions of zero mean of errors in each group. First order asymptotic analysis of the estimators show that they are √N consistent (N being the sample size) and normally distributed. The thesis explores second order bias properties that demonstrate sources of bias and differences between choices of GEL estimators. Specifically, the second order bias depends on the third moments of the group errors and correlation among the group errors and explanatory variables. With symmetric errors and no endogeneity all three estimators Empirical Likelihood (EL), Exponential Tilting (ET) and Continuous Updating Estimator (CUE) yield unbiased estimators. A detailed simulation exercise is performed to test comparative performance of the EL, ET and their bias corrected estimators to the standard 2SLS/GMM estimators. Simulation results reveal that while, with a few strong instruments, we can simply use 2SLS/GMM estimators, in case of many and/or weak instruments, increased degree of endogeneity, or varied signal to noise ratio, bias corrected EL, ET estimators dominate in terms of both least bias and accurate coverage proportions of asymptotic confidence intervals even for a considerably large sample. The thesis includes a case where there are within group dependent data, to assess the consequences of a key assumption being violated, namely the within-group iid assumption. Theoretical analysis and simulation results show that ignoring this feature can result in misleading inference. The proposed estimators are used to estimate the returns to an additional year of schooling in the UK using Labour Force Survey data over 1997-2009. Pooling the 13 years data yields roughly the same estimate of 11.27% return for British-born men aged 25-50 using any of the estimation techniques. In contrast using 2009 LFS data only, for a relatively small sample and many weak instruments, the return to first degree holder men is 13.88% using EL bias corrected estimator, where 2SLS estimator yields an estimate of 6.8%.

The key results of this thesis are (1) an extension of N-mixture models to incorporate the additional layer of obfuscation brought by observing counts from a related auxiliary population (rather than the target population), (2) an extension of N-mixture models to allow for grouped counts, the purpose being two-fold: to extend the applicability of N-mixtures to larger population sizes, and to allow for the use of coarse counts in fitting N-mixture models, (3) a new R package allowing the easy application of the new N-mixture models, (4) a new R package allowing for optimization of multi-parameter functions using arbitrary precision arithmetic, which was a necessary tool for optimization of the likelihood in large population abundance N-mixture models, as well as (5) simulation studies validating the new grouped count models and comparing them to the classic N-mixtures models. / Graduate

For over forty years ability grouping has been of professional concern to educators in the United States (Reisner, 1936). There have been fundamental changes in educational theory and practice during that time, one of which is the recognition that educational practices must adjust to individual differences. The interpretation of "equal opportunity" in education has gradually changed from meaning the same methods, standards, and course content for all children, to meaning the full opportunity for each child to develop his own potential in a school program suited to his individual capacities and needs. The interpersonal relationships and social development of school children have received an increasing amount of attention as we have come to realize the pervasive effect they have on educational objectives and as optimal social adjustment has itself become one of these objectives. As Brumbaugh (1960, p. 99) has pointed out: Mental health and social adjustment are words to conjure with when there is discussion about separate grouping. A half century ago, the fear was that stigma would attach to a child in a special class for those with below average intelligence. It is now replaced by anxiety lest those at the other end of the scale would have feelings of superiority and become egotistical little snobs. There are enough studies of children in such classes to indicate that this does not happen but there is also some evidence that there are concomitant effects which are used to oppose ability grouping on a "social segregation" argument. Taba et al. (1952) as an example of this point of view write: Of special interest for intergroup education is the fact that the static single bases for grouping have almost always fixed homogeneity simultaneously along lines of socioeconomic status, race, and religious background. For example, any type of ability grouping also inadervertently introduces segregation by economic class, race, and neighborhood. Because of their cultural handicaps, children from deviant backgrounds tend to be at the bottom of the heap, as far as school achievement is concerned. Hence, in ability grouping, they are thrown together and separated from other children.... This segregation, of course, prevents learning common culture by association with other children. The stigma attached to the lower ability groups further destroys motivation and self-respect, Thus, a basis is built for both physical and psychological isolation. (pp. 138-1939)

Attitudes are usually defined as feelings for or against something (Remmers and Gage, 1955). They are very important in the lives of people because they help determine future success in an individual's life. Because of them one works to get the things he wants, one votes for or against certain issues, one joins a cause, opposes something, or attempts to influence others.

No description available.