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This volume is an exposition of a mathematical modeling technique for use in the evaluation and solution of complex educational problems at all levels. It explores in detail the application of simple algebraic techniques to such issues as program reduction, fiscal rollbacks, and computer curriculum planning. Part I ("Introduction to the Modeling Formulation") discusses four stages of modeling development related to acceptance of a modeling formulation for decision-making, contextual assessment, planning for modeling development, and the specific design stage. These chapters use specific real-world problems to illustrate the purpose and application of the model. Part II ("Presentation of a Full Formulation"), also composed of four chapters, provides a detailed description of the multiple alternatives framework as a strategy for purchasing computer equipment. The chapters cover conceptualization, development of the framework, format for execution, and approaches to exploring results. Part III ("Example of a Complete Quantitative Solution") is a single chapter focusing on implementation of the model for determining fiscal rollbacks and program terminations during a crisis. Part IV consists of two chapters focusing on current extensions under research and development and a summary touching on related topics. Three appendixes provide a course outline and syllabus for Multiple Alternatives Analysis instruction and an outline for a field project proposal. (TE)

This report provides an extensive discussion of the use of criterion referenced, mathematical modeling procedures to determine which budget reductions minimize reduction in the quality of educational programs. Part I, "Evaluation of Potential Budgeting Roll-backs under Educational Fiscal Crisis," explains the basic design of multiple alternatives analysis and the context for its use. Chapters include (1) philosophical foundation for fiscal modeling, (2) program budgeting for an allocation/deallocation fiscal strategy, (3) traditional modeling via cost analytical design, (4) operations research and the evaluation of feasible alternatives, and (5) simulation modeling within a criterion-impact design. Included also is a background to the field investigation and outline of the technical report. Part II, "Multiple Alternatives Analysis as a Mathematical Decisioning [sic] Model," provides the technical and mathematical details of the analysis, including both construction and validation. Part III, "Field Application of the Rolbak Model," contains an extensive sampling of the use of these procedures in reducing a budget within a local school district, including chapters on (1) construction of the database, (2) initial T-normal transformations, (3) formulation of the rolbak mathematical model, (4) search for regional feasibility as a benchmark, (5) cyclic optimization of the restricted model, (6) cyclic optimization of the relaxed model, and (7) comparison of the restricted and relaxed "decisioning" models. Part IV summarizes the Multiple Alternatives Model and assesses its future. (TE)

Since research on sex differences in occupational attainment suggests that working men and women attain essentially the same mean level of occupational attainment and do so through quite similar processes, the censoring problem as a potential source of bias in estimating equations for these comparative occupational attainments is investigated. (The censoring problem is defined as a sampling problem in that the sample of working women contains an overrepresentation of successful women, since women who can afford not to work will stay out of the labor force unless they find a job commensurate with their education.) After reviewing and rejecting some alternatives that could correct this problem, a model (developed from an extension of Heckman's set of equations that relate to women's occupational characteristics) is presented for obtaining the structural parameters for the whole female population by accounting for the censoring problem. The censoring problem was found to be minimal and not a reasonable explanation for the apparent similarities between men and women in the process of occupational attainment. It was concluded that before accepting the finding of sexual equality in occupational rewards, the following possibilities should be explored: (1) That the model is misspecified and if certain variables (for example, the status of first job) were included, the process of occupational attainment for the sexes would differ, and (2) that certain dimensions of sexual inequality in occupational rewards are not being tapped by the concept, occupational status. (TA)

Many economists have tried to explain existing wage differentials between men and women. A new approach compares the relative importance of occupational discrimination with that of wage discrimination. This model allows for variation both in occupational distribution and in wages resulting from differences in job qualifications and productivity indicators. It was demonstrated that the usual approach to wage discrimination is a special case of this general model with some restrictive implicit assumptions. A multinomial logit model was used to stimulate the occupational distribution of women that would exist if they faced the same structure of occupational determination as men. Results indicate that there would be more women in managerial and skilled labor jobs and fewer women clerical and service workers. Wages were then estimated as a function of productivity measures for men and women in each occupation so that the components of the wage differential could be calculated. Results indicate that almost the entire differential could be eliminated by ending both forms of discrimination, with occupational discrimination accounting for one-third to one-half of the differential and pure wage discrimination the remainder. (WL)

Neglected issues of simultaneous statistical inference and statistical power in survey research applications of the general linear model are reviewed, and it was found that classical hypothesis testing as it is currently applied, is inadequate for the purposes of social research. The intelligent use of statistical inference demands control over the overall level of Type I error and knowledge of the magnitude of effects one is likely to detect. Techniques are suggested that can be used to routinely incorporate considerations of simultaneous inference and power into statistical analysis of survey data. Several examples of applications of these techniques are presented. (RC)

The Educational Simulation Model (ESM), developed by UNESCO, is designed for use at the national level for long range simulation of educational system fluctuations. The model is a set of equations which describe an educational program and the changes and flows in student enrollment in all the courses of an educational system. Thus the model is restricted to the internal logic of an educational system which is simulated deterministically. Some of the variables employed by the model are population figures for each age and also promotion, repetition, distribution, and leaving rates for each course. The simulation model can be adapted to determine the demand for teachers, the costs, and the cumulative output of any educational system. The model has been employed in several countries, and its implementation on a computer would allow a more objective simulation with more refined results. (WH)

This study investigates how local school districts behave in allocating their budgets among teachers, other professional educators, support personnel, and nonpersonnel inputs. Demands for school inputs are estimated using U.S. Office of Education data on the expenditure and staffing patterns of large (enrollments in excess of 10,000 students) school districts in the 1969-70 school year. In general, results suggest that districts' allocative behavior at the margin is quite different from their average allocative behavior. In particular, relatively small proportions of budget increases (occasioned, perhaps, by reforms in school finance) would be allocated to employing additional teachers. (Author)

A randomization model appropriate for evaluating priority effects in free recall (i.e., whether "new" items are recalled prior to "old" items) is discussed and related to well-known nonparametric significance tests. Since the bases for the measures that have been suggested in the psychological literature may be interpreted either in terms of Wilcoxon's rank sum statistic or through a specific entry in a 2 x 2 contingency table, alternative indices of priority can be adopted directly from classical nonparametric statistics. Finally, the mean and variance formulas for a general correlational statistic are provided that specialize to the moments for the two measures already in common use. (Author)

This report evaluates alternative approaches to and measures of equity in state school finance plans. It also evaluates the Federal Expenditure Disparity Measure (FEDM). Following an introduction, Part II describes alternative conceptions of equity in education and identifies three components of these conceptions: groups treated, alternative treatments, and criteria for equity. Part III presents numerical models that show how alternative combinations of group, treatment, and criterion can yield different conclusions when applied to a hypothetical school district. In Part IV, the value judgments that underly the alternative conceptions of equity are examined. Since few value judgments are widely held, a number of equity conceptions must be considered valid. Also in Part IV, the value judgments inherent in Per Pupil Expenditure Disparity Measures (PPEDMs) are reviewed and analyzed. In Part V, the theory behind the FEDM and the way the Office of Education applies the FEDM to calculate measures from state data are discussed. In Part VI, recommendations concerning equity measurement in federal regulations and particular procedures used by the Office of Education to measure expenditure disparities are presented. A lengthy appendix considers social welfare theory, social choice theory, equity measures calculations, and the tax price of education. (Author/JM)

A multiparameter, programmable model was developed to examine the interactive influence of certain parameters on the probability of deciding that an examinee had attained a specified degree of mastery. It was applied within the simulated context of performance testing of military trainees. These parameters included: (1) the number of assumed mastery states--master, nonmaster, and perhaps intermediate (likely to soon achieve mastery); (2) the prior distribution of scores from similar examinee groups; and (3) the number of test trials or items administered. The results of several simulations showed that the degree of confidence that a decisionmaker can have about the testee's mastery is markedly affected by the values for the three parameters, and the effects of their combination. Using the Bayesian model, test length and costs could be reduced--as long as the prior information was accurate and valid for the particular group of examinees. Results of the simulation also showed that a test may be too short to be of decision-making value. (Author/GDC)

Alternative mathematical models for scoring and decision making with criterion referenced tests are described, especially as they concern appropriate test length and methods of establishing statistically valid cutting scores. Several of these approaches are reviewed and compared on formal-analytic and empirical grounds: (1) Block's approach to setting mastery standards, involving student performance and academic requirements; (2) Crehan's classification, comparing scores of students who have and who have not completed training; (3) the probabilistic models of Macready, Dayton, and Emrick, which assume an equal proportion of masters and nonmasters; (4) the binomial distribution model, which allows for partial acquisition of proficiency; (5) the Bayesian model, which considers prior experience; (6) Rasch's one-parameter logistic model, which yields person-free test calibrations and item-free person measurements; and (7) the regression approach of classical test theory, which enables the estimation of true scores to be made from observed scores. Examples of these approaches are given, as well as their advantages, disadvantages, and ambiguities. (GDC)

A communication model suggests that in almost all cases where dissemination and utilization of knowledge has taken place, one or more of seven general factors can be identified as having been active in aiding the phenomena. The proposed factors are: linkage, structure, capacity, openess, reward, proximity, and synergy. In order to use this model to investigate the ability of the National Science Foundation (NSF) to communicate its mathematics and science curricula to the nation's classrooms, the BACPRO was developed. The BACPRO is a self-report instrument to be completed by principals and teachers. Items are divided into seven subscales relating to the seven factors of the model. A pilot test was conducted among principals of the Chicago City School System followed by a field test among secondary school principals and mathematics teachers in Wisconsin. (BB)

A psychometric model called Saltus, which represents the qualitative aspects of hierarchical development in a form applicable to additive measurement, was applied. Both Piaget's theory of cognitive development and Gagne's theory of learning hierarchies were used to establish the common features of hierarchical development: (1) gappiness--the logical construction of the hierarchy which occurs when there is no state between adjacent stages; and (2) rigidity--learning behavior, exhibited by a fixed sequence of progression through stages. Saltus assumes a theory with gappiness expressed through items or tasks, and estimates the rigidity of data, thus testing the hypothesized gappiness. Four experiments tested the Saltus model in elementary school Australian children and with subjects aged 3, 4, 5, 8, 12, and 21 years. Three Piagetian tasks of rule assessment involving proportionality (balance scale, projection of shadows, and a probability task) gave clear evidence of rigidity in the step from the pre-operational stage to the concrete operational stage. The next step, to the formal operational stage, did not show rigidity, although gappiness was evident. The hypothesized existence of a gap to split the concrete operational stage was not supported. The Gagnean data on a subtraction task showed strong rigidity. (Author/GDC)

The stability of Rasch test item difficulty parameters was investigated under varying conditions. Data were taken from a mathematics achievement test administered to over 2,000 Australian students. The experiments included: (1) relative stability of the Rasch, traditional, and z-item difficulty parameters using different sample sizes and designs; (2) effect of different sample types and sizes on the Rasch item fit estimator; (3) effects, on the item fit parameter and the Rasch item difficulty parameter, of removing some less appropriate items from the test; and (4) an examination of Wright's statement that the standard error of the item difficulty parameter is a good estimator of its variance, and that it has an inverse square root relationship to the sample size. Results showed that Rasch and z-item difficulty parameters were similar. Item fit increased as sample size increased. The removal of poorly fitting items improved fit values for the entire test, but worsened them for the remaining items. The Rasch standard error parameter was an appropriate measure of the true error of estimation as calculated from the square root of the sampling variance of the item difficulty index. (Implications for test calibration are concluded, and detailed item analyses are appended.) (GDC)

Competitive bidding has recently received considerable attention in the literature of economics and operations research. Unfortunately, the classic formulation of the problem developed by Friedman and still expounded (Stark and Mayer) can lead to an incorrect estimate of the expected profit for a bidder when uncertainty about the cost of fulfilling the contract exists. The purpose of this paper is to point out the deficiency in the formulation and to indicate how it may be corrected. (Author)

Estimating the probability that a proposed bid will win is obviously a central problem in competitive bidding theory. Unfortunately, a method for doing this, which originated with Friedman and which continues to be expounded in the literature, is logically incorrect. This paper points out the deficiency in that formulation. (Author/MLF)

The contention is made that group performance data are useful in the construction and interpretation of criterion-referenced tests. The Mastery Learning Test Model, which was developed for analyzing criterion-referenced test data, is described. An estimate of the proportion of students in an instructional group having achieved the referent objectives is usable as a prior probability in interpreting individual responses. Considering instructional group performance enhances estimates of individual performance. Correlational data from a set of test items and a representative population of students are used to estimate the required item parameters. (Author)

Slightly over 180,000 draftees were surveyed 10 months after leaving the military to gain data regarding training, employment, occupation and wages if working, marital status, education, Armed Forces Qualification Test (AFQT) score, age, race, military occupation, and home. Differences in earnings functions among smaller, homogeneous labor markets are examined in terms of urban and regional location and black and white individuals. Aggregate statistics about the structure of earnings, characteristics of earnings functions, regional variations in earnings, and black-white differentials are analyzed; 10 tables supplement the discussion. Conclusions state the estimated effect of schooling and ability differences is low, with an additional year of schooling worth about 5 percent higher income and one decile move in the AFQT score matched by 1 percent increase in earnings for whites, lower for blacks. The estimation of separate earnings functions for labor markets across the country is significant, with the choice of region for employment often equivalent to the marginal earnings of several years of schooling. Racial difference in earnings stems from differences in the earning functions for blacks and whites, not in schooling, ability, or experience levels. Sample and population characteristics are discussed in the appendix. The nonrepresentative character of the sample is emphasized. (LH)

The thrust of this research exploration is aimed at one of the most pressing social problems facing America today: segregation in the public schools. The school system is only one of a complex set of systems, all interrelated, that comprise the entity that we call a city. The geography of the school system--the location of facilities and the location and movement of students to those facilities--is the only observable expression of a complex geo-political decision making system. Many considerations go into the making of essentially spatial decisions that are not primarily concerned with spatial efficiency or with the physical system's day to day operation. It was for this reason that we decided to explore the nature of school location systems in the context of a hypothetical city as a means of minimizing nongeographical influences. We have borrowed the concept of spatial efficiency in school systems. Distance minimization is the comparative measure of the efficiency of alternate arrangements. In order to partially conform to conventional wisdom on urban spatial structure, we chose a polar grid system as a concession to the classical sector and ring theories. The execution of the problem was done using the Linear Programming Algorithm from the IBM Mathematical Programming Systems package on the Pennsylvania State University's IBM/360 computer system. (Author/JM)

The testing of research hypotheses is directly comparable to the dichotomous decision-making of medical diagnosis or jury trials--not ill/ill, or innocent/guilty decisions. There are costs in both kinds of error, type I errors of falsely rejecting a null hypothesis or type II errors of falsely rejecting an alternative hypothesis. It is important to consider the power of a statistical test (that is, the likelihood of avoiding type II errors) as well as the significance level (the likelihood of avoiding type I errors) before an experiment is begun. Techniques for estimating the power of a statistical test for a particular experimental design are presented. (CTM)

This report details the data requirements and procedures used to develop new school bus routes for Six Mile Elementary School in South Carolina. The project examined the current routes of the school and applied computer based techniques to develop new routes given the existing bus stops. Bus routes were developed so that distance and travel time were minimized. Before the routes could be generated, the map of Six Mile had to be modeled as a network, requiring numerous modifications to fit the heuristics used by the computer program. Using Floyd's Algorithm, the shortest path between every pair of bus stops was computed. The bus stops were clustered into routes, considering the number of children allowed on a bus. A model was developed which estimated the travel time based on a description of the route, and was validated by an actual bus ride through the route. The new routes were evaluated by comparing the time and distance of the old and new routes. The appendix contains a complete description of the seven new routes that were developed, including a listing of all stops and turns the buses must make and the roads the buses are to travel on. Contains numerous tables and figures and 16 references. (KS)

This paper aims to describe a Bayesian approach to modeling and estimating cognitive models both in terms of statistical machinery and actual instrument development. Such a method taps the knowledge of experts to provide initial estimates for the probabilistic relationships among the variables in a multivariate latent variable model and refines these estimates using Markov Chain Monte Carlo procedures. This process is described in terms of NetPASS, a complex simulation based assessment in the domain of computer networking. The paper describes a parameterization of the relationships in NetPASS via an ordered polytomous item response model and details the updating of the model with observed data via Bayesian statistical procedures ultimately being provided by Markov Chain Monte Carlo estimation. (Contains 12 tables, 9 figures, and 47 references.) (Author/SLD)