The least mean squares (LS) regression method produced the best linear unbiased estimates under the normal error distribution. However, many researchers have noted that the optimal condition is rarely met in real data analyses. To remedy the impact of potential data contamination, several advantages of the least median squares (LMS) regression are illustrated using a user-friendly software program, "Program for RObust reGRESSion" (PROGRESS). A public data base (the Longitudinal Study...

Topics: ERIC Archive, Computer Software, Estimation (Mathematics), Least Squares Statistics, Prediction,...

An algorithm is presented for the best least-squares fitting correlation matrix approximating a given missing value or improper correlation matrix. The proposed algorithm is based on a solution for C. I. Mosier's oblique Procrustes rotation problem offered by J. M. F. ten Berge and K. Nevels (1977). It is shown that the minimization problem belongs to a certain class of convex programs in optimization theory. A necessary and sufficient condition for a solution to yield the unique global minimum...

Topics: ERIC Archive, Algorithms, Computer Software, Correlation, Estimation (Mathematics), Least Squares...

Tests of mean equality proposed by Alexander and Govern (1994) and Tsakok (1978) were compared to the well-known procedures of Brown and Forsythe (1974), James (1951), and Welch (1951) for their ability to limit the number of Type I errors in one-way designs where the underlying distributions were nonnormal, variances were nonhomogeneous, and group sizes were unequal. These tests were compared when the usual method of least squares was applied to estimate group means and variances and when...

Topics: ERIC Archive, Comparative Analysis, Estimation (Mathematics), Foreign Countries, Least Squares...

This paper explains in user-friendly terms why multivariate statistics are so important in educational research. The basic logic of canonical correlation analysis is presented as a simple or bivariate Pearson "r" procedure. It is noted that all statistical tests implicitly involve the calculation of least squares weights, and that all parametric tests can be conducted using canonical analysis, since canonical analysis subsumes parametric methods as special cases. Canonical analysis is...

Topics: ERIC Archive, Educational Research, Heuristics, Least Squares Statistics, Multiple Regression...

This paper examines the differences between multilevel modeling and weighted ordinary least squares (OLS) regression for analyzing data from the National Educational Longitudinal Study 1988 (NELS:88). The final sample consisted of 718 students in 298 schools. Eighteen variables from the NELS:88 dataset were used, with the dependent variable being the science item response theory estimated number right standardized t-score. Results from the analyses yield no single criterion for choosing one...

Topics: ERIC Archive, Least Squares Statistics, National Surveys, Regression (Statistics), Research Design,...

Gross Domestic Product of any country often influence economic decisions by policy makers, market participants and econometricians on policy recommendations, evaluation and forecasting. However these decisions are often based on preliminary data announcements by statistical agencies. It is therefore important to ensure that the preliminary GDP announcements are efficient and can be relied on. This paper focuses on South Africa's preliminary announcements of quarterly GDP estimates by examining...

Topics: ERIC Archive, Foreign Countries, Economic Factors, Computation, Data, Least Squares Statistics,...

Least squares methods are sophisticated mathematical curve fitting procedures used in all classical parametric methods. The linear least squares approximation is most often associated with finding the "line of best fit" or the regression line. Since all statistical analyses are correlational and all classical parametric methods are least square procedures, it becomes imperative to understand just what the least squares procedure is and how it works. This paper illustrates the least...

Topics: ERIC Archive, Goodness of Fit, Least Squares Statistics, Matrices, Regression (Statistics)

This digest introduces hierarchical data structure, describes how hierarchical models work, and presents three approaches to analyzing hierarchical data. Hierarchical, or nested data, present several problems for analysis. People or creatures that exist within hierarchies tend to be more similar to each other than people randomly sampled from the entire population; for this reason, observations based on these individuals are not fully independent. Hierarchical linear modeling can address...

Topics: ERIC Archive, Least Squares Statistics, Models, Statistical Analysis, Osborne, Jason W.

Langmuir's model is studied for the situation where epsilon is independently and identically normally distributed. The "Y/x" versus "Y" plot had a 90% mid-range that did not contain the true curve in a vast portion of the range of "x". The "1/Y" versus "1/chi" plot had undefined expected values, and this problem worsens as sample size increases. The use of non-linear least squares is recommended. In non-linear regression, it is demonstrated that...

Topics: ERIC Archive, Equations (Mathematics), Estimation (Mathematics), Least Squares Statistics,...

Homoscedasticity is an important assumption of linear regression. This paper explains what it is and why it is important to the researcher. Graphical and mathematical methods for testing the homoscedasticity assumption are demonstrated. Sources of homoscedasticity and types of homoscedasticity are discussed, and methods for correction are demonstrated. Graphs are used to illustrate different patterns that may be caused by heteroscedasticity. An extensive example for using Weighted Least Squares...

Topics: ERIC Archive, Graphs, Least Squares Statistics, Regression (Statistics), Thompson, Russel L.

Homogeneity analysis, or multiple correspondence analysis, is usually applied to k separate variables. In this paper, it is applied to sets of variables by using sums within sets. The resulting technique is referred to as OVERALS. It uses the notion of optimal scaling, with transformations that can be multiple or single. The single transformations consist of three types: (1) nominal; (2) ordinal; and (3) numerical. The corresponding OVERALS computer program minimizes a least squares loss...

Topics: ERIC Archive, Algorithms, Computer Software, Least Squares Statistics, Linear Programing,...

The information that is gained through various analyses of the residual scores yielded by the least squares regression model is explored. In fact, the most widely used methods for detecting data that do not fit this model are based on an analysis of residual scores. First, graphical methods of residual analysis are discussed, followed by a review of several quantitative approaches. Only the more widely used approaches are discussed. Example data sets are analyzed through the use of the...

Topics: ERIC Archive, Graphs, Identification, Least Squares Statistics, Regression (Statistics), Research...

The nature of the criterion (dependent) variable may play a useful role in structuring a list of classification/prediction problems. Such criteria are continuous in nature, binary dichotomous, or multichotomous. In this paper, discussion is limited to the continuous normally distributed criterion scenarios. For both cases, it is assumed that the predictor variables are continuous multivariate normal. For the binary variable case, the multivariate normal assumption is conditioned on the binary...

Topics: ERIC Archive, Classification, Correlation, Error of Measurement, Estimation (Mathematics), Least...

This study investigated the effects of maternal participation in the Special Supplemental Nutrition Program for Women, Infants, and Children (WIC) on the birth weight, motor and social skills, and temperament of a national sample of children born between 1990 and 1996 to women participating in the National Longitudinal Survey of Youth (NLSY). Sibling fixed effect models were used to account for persistent differences in difficult to measure characteristics of mothers that affect participation...

Topics: ERIC Archive, Child Development, Infants, Least Squares Statistics, Mothers, Nutrition, Siblings,...

Many aspects of the geometry of linear statistical models and least squares estimation are well known. Discussions of the geometry may be found in many sources. Some aspects of the geometry relating to the partitioning of variation that can be explained using a little-known theorem of Pappus and have not been discussed previously are the topic of this report. I discuss, using the theorem, how geometric explanation helps us understand issues relating to contributions of independent variables to...

Topics: ERIC Archive, Least Squares Statistics, Geometry, Geometric Concepts, Statistical Analysis,...

This paper describes structural equation modeling (SEM) in comparison with another overarching analysis within the general linear model (GLM) analytic family: canonical correlation analysis. The uninitiated reader can gain an understanding of SEM's basic tenets and applications. Latent constructs discovered via a measurement model are explored and the structural models that "connect" latent constructs are described. In addition to reviewing SEM constructs, the paper uses the analysis...

Topics: ERIC Archive, Correlation, Equations (Mathematics), Heuristics, Least Squares Statistics,...

It has been increasingly realized that (1) multivariate methods are essential in most quantitative studies (Fish, 1988; Thompson, 1992), and (2) all conventional parametric analytic methods are correlational and invoke least squares weights (e.g., the beta weights in regression) (Knapp, 1978; Thompson, 1991). The present paper reviews one very popular multivariate analytic method that explicitly invokes weighting to optimize one criterion: the analytic method that researchers have come to call...

Topics: ERIC Archive, Correlation, Least Squares Statistics, Measurement Techniques, Multivariate Analysis,...

Three simplifying conditions are given for obtaining least squares (LS) estimates for a nonlinear submodel of a linear model. If these are satisfied, and if the subset of nonlinear parameters may be LS fit to the corresponding LS estimates of the linear model, then one attains the desired LS estimates for the entire submodel. Two illustrative analyses employing this method are given, each involving an Eckart-Young (LS) decomposition of a matrix of linear LS estimates. In each case the factors...

Topics: ERIC Archive, Analysis of Variance, Least Squares Statistics, Mathematical Models, Mathematics,...

Criteria for prediction of multinomial responses are examined in terms of estimation bias. Logarithmic penalty and least squares are quite similar in behavior but quite different from maximum probability. The differences ultimately reflect deficiencies in the behavior of the criterion of maximum probability.

Topics: ERIC Archive, Probability, Prediction, Classification, Computation, Statistical Bias, Least Squares...

This report provides empirical results of attempts to achieve consistency of estimates between two National Center for Education Statistics (NCES) surveys. These surveys are the 1991- 92 Private School Survey (PSS) and the Private School Component of the 1990-91 Schools and Staffing Survey (SASS). Consistency was sought in the numbers of schools, teachers, and students from these two sources. Comparisons are made among statistical and computational procedures that might serve to bring about the...

Topics: ERIC Archive, Classification, Elementary Secondary Education, Estimation (Mathematics), Least...

Maximum likelihood and least-squares estimates of parameters from the logistic regression model are derived from an iteratively reweighted linear regression algorithm. Empirical Bayes estimates are derived using an m-group regression model to regress the within-group estimates toward common values. The m-group regression model assumes that the parameter vectors from "m" groups are independent, and identically distributed, observations from a multivariate normal "prior"...

Topics: ERIC Archive, Bayesian Statistics, Estimation (Mathematics), Least Squares Statistics, Maximum...

A high-breakdown estimator is a robust statistic that can withstand a large amount of contaminated data. In linear regression, high-breakdown estimators can detect outliers and distinguish between good and bad leverage points. This paper summarizes the case for high-breakdown regression and emphasizes the least quartile difference estimator (LQD) proposed by C. Croux, P. J. Rousseeuw, and O. Hossjer (1994). This regression method examines the absolute differences between every pair of residuals...

Topics: ERIC Archive, Computer Software, Estimation (Mathematics), Least Squares Statistics, Regression...

The conceptualization of analysis of covariance (ANCOVA), as an analysis of variance (ANOVA) on the residual scores that are obtained when the dependent variable is regressed on the covariate, is mathematically incorrect. If residuals are obtained from the pooled within-groups regression coefficient, ANOVA on the residuals results in an inflated alpha-level. If the regression coefficient for the total sample combined into one group is used, ANOVA on the residuals yields an inappropriately...

Topics: ERIC Archive, Analysis of Covariance, Analysis of Variance, Least Squares Statistics, Mathematical...

Although investigation of school security measures and their relationships to various outcomes including school crime rates (Gottfredson, 2001), perpetuation of social inequality (Ferguson, 2001; Nolan, 2011; Welch & Payne, 2010), and the impact on childhood experiences has seen significant growth within the last 20 years (Newman, 2004; Kupchik, 2010), few studies have sought to explore the impacts of these measures on suspension rates. Using data from the Educational Longitudinal Study...

Topics: ERIC Archive, School Security, Suspension, Discipline, Race, Parent Participation, Longitudinal...

The Welch-James (WJ) and Improved General Approximation (IGA) tests for the within-subjects main and interaction effects in a split-plot repeated measurement design were investigated when least squares estimates and robust estimates based on trimmed means were used. Variables manipulated in the Monte Carlo study included the degree of multivariate normality, degree of departure from the assumption of multisample sphericity, total sample size, degree of sample size imbalance, and number of...

Topics: ERIC Archive, Foreign Countries, Least Squares Statistics, Monte Carlo Methods, Research Design,...

Ordinary least-squares regression treats the variables asymmetrically, designating a dependent variable and one or more independent variables. When it is not obvious how to make this distinction, a researcher may prefer to use orthogonal regression, which treats the variables symmetrically. However, the usual procedure for orthogonal regression is not equivariant. A simple modification is proposed to overcome this serious defect. Illustrative computations involving 15 observations on 5...

Topics: ERIC Archive, Equations (Mathematics), Estimation (Mathematics), Least Squares Statistics,...

Whenever one uses ordinary least squares regression, one is making an implicit assumption that all of the independent variables have been measured without error. Such an assumption is obviously unrealistic for most social data. One approach for estimating such regression models is to measure implied coefficients between latent variables for which one had multiple manifest indicators. The problem with this approach is that overidentified models yield multiple estimates of the associations among...

Topics: ERIC Archive, Computer Programs, Factor Analysis, Least Squares Statistics, Mathematical Models,...

Least squares fitting process as a method of data reduction is presented. The general strategy is to consider fitting (linear) models as partitioning data into a fit and residuals. The fit can be parsimoniously represented by a summary of the data. A fit is considered adequate if the residuals are small enough so that manipulating their signs and locations does not affect the summary more than a pre-specified amount. The effect of the residuals on the summary is shown to be (approximately)...

Topics: ERIC Archive, Goodness of Fit, Least Squares Statistics, Mathematical Models, Measurement...

The purpose of this paper is to assist researchers, practitioners, and graduate students in identifying and addressing key questions related to the task of choosing among the analytic techniques designed to analyze a dichotomized dependent variable with a set of independent variables. The discussion is limited to (1) the analysis of data by the analytic procedures of ordinary least squares regression, discriminant analysis, or logistic regression; (2) the use of the Statistical Package for the...

Topics: ERIC Archive, Discriminant Analysis, Least Squares Statistics, Regression (Statistics), Research...

This study developed a robust linear regression technique based on the idea of weighted least squares. In this technique, a subsample of the full data of interest is drawn, based on a measure of distance, and an initial set of regression coefficients is calculated. The rest of the data points are then taken into the subsample, one after another, and a weighted least squares procedure is performed each time a new data point is brought in, until all data points are included. The weighted average...

Topics: ERIC Archive, Estimation (Mathematics), Least Squares Statistics, Regression (Statistics),...

This study examined the effect of type of correlation matrix on the robustness of LISREL maximum likelihood and unweighted least squares structural parameter estimates for models with categorical manifest variables. Two types of correlation matrices were analyzed; one containing Pearson product-moment correlations and one containing tetrachoric, polyserial, and product-moment correlations as appropriate. Using continuous variables generated according to the equations defining the population...

Topics: ERIC Archive, Computer Software, Correlation, Estimation (Mathematics), Goodness of Fit, Hypothesis...

Trust is one crucial prerequisite for the welfare state. However, very few empirical studies exist that help us understand the mechanisms through which trust affects the welfare state. Influencing public support for developing friendly public policies might be one of these mechanisms. In this study, we use unique micro data from 34 countries to investigate the relationship between trust and support for public education expenditures. We use the Life in Transition Survey (LiTS) conducted by the...

Topics: ERIC Archive, Foreign Countries, Trust (Psychology), Public Policy, Surveys, Expenditures, Public...

It is argued that analysis of variance (ANOVA) and related methods should be taught using a general linear model (GLM) approach, rather than a classical ordinary sums of squares approach. The GLM approach emphasizes the linkages among conventional parametric methods, emphasizing that all classical parametric methods are least squares procedures that implicitly or explicitly use weights, focus on latent synthetic variables, and yield effect sizes analogous to "r" squared (are...

Topics: ERIC Archive, Analysis of Variance, Correlation, Effect Size, Higher Education, Introductory...

This paper reviews recent work in factor analysis of categorical variables. Emphasis is on the generalized least squares solution. A section on maximum likelihood solution focuses on extensions of the classical model, especially the normal case. Many of the recent developments have taken place within this context, and it provides a unified framework of exposition against which other models may be introduced in contrast. Section 2 provides a brief review of factor analysis of measured variables,...

Topics: ERIC Archive, Correlation, Estimation (Mathematics), Factor Analysis, Factor Structure, Latent...

This article looked at non-experimental data via an ordinary least squares (OLS) model and compared its results to ridge regression models in terms of cross-validation predictor weighting precision when using fixed and random predictor cases and small and large p/n ratio models. A majority of the time with two random predictor cases, ridge regression accuracy was superior to OLS in estimating beta weights. Thus, ridge regression was very useful under this condition. However, when the fixed...

Topics: ERIC Archive, Regression (Statistics), Prediction, Least Squares Statistics, Computation,...

To eliminate maturation as a factor in the pretest-posttest design, pretest scores can be converted to anticipate posttest scores using grade equivalent scores from standardized tests. This conversion, known as historical regression, assumes that without specific intervention, growth will continue at the rate (grade equivalents per year of schooling) obtained at the time of pretest. Data were taken from reports of 213 Title I compensatory education programs in New York State to examine the...

Topics: ERIC Archive, Academic Achievement, Achievement Gains, Elementary Education, Grade Equivalent...

This study estimates the extent that state financial support for higher education raises college attainment. Despite its manifest importance for policy, this is the first study to estimate this effect directly. Many studies have estimated the effect of college price on attendance, but state support for higher education and college price do not have a one-to-one correspondence. Moreover, state support for higher education can affect enrollment through college quality, not just price. A two-stage...

Topics: ERIC Archive, Educational Finance, Financial Support, Higher Education, State Aid, Educational...

Instrumental variables (IV) methods allow for consistent estimation of causal effects, but suffer from poor finite-sample properties and data availability constraints. IV estimates also tend to have relatively large standard errors, often inhibiting the interpretability of differences between IV and non-IV point estimates. Lastly, instrumental variables' idiosyncratic nature reduces their availability in data sets alongside outcome and other variables of interest. Most prior work on two-sample...

Topics: ERIC Archive, Least Squares Statistics, Labor Supply, Measurement Techniques, Error of Measurement,...

The proposed paper studies the bias in the two-stage least squares, or 2SLS, estimator that is caused by the compliance-effect covariance (hereafter, the compliance-effect bias). It starts by deriving the formula for the bias in an infinite sample (i.e., in the absence of finite sample bias) under different circumstances. Specifically, it considers the following cases: (1) A single site study with one mediator; (2) A multiple site study with one mediator; and (3) A multiple site study with...

Topics: ERIC Archive, Least Squares Statistics, Bias, Compliance (Psychology), Context Effect, Educational...

The papers were presented at the Social Statistics Section, the Government Statistics Section, and the Section on Survey Research Methods. The following papers are included in the Social Statistics Section and Government Statistics Section, "Overcoming the Bureaucratic Paradigm: Memorial Session in Honor of Roger Herriot": "1995 Roger Herriot Award Presentation" (Daniel Kasprzyk, Fritz Scheuren, and Dan Levine); "Space/Time Variations in Survey Estimates" (Leslie...

Topics: ERIC Archive, Elementary Secondary Education, Least Squares Statistics, Longitudinal Studies,...

The passage of the Chicago (Illinois) School Reform Act introduced a model of schools called the production model. This model defines the structures of the school by its inputs, throughputs (or production process), and outputs. The production model produces quantitative reports describing the fiscal condition and the quality of output, thus creating a fiscal scorecard. Survey responses from 10,170 teachers from 331 Chicago schools were the data for an analysis using partial least squares path...

Topics: ERIC Archive, Economic Factors, Educational Change, Elementary Secondary Education, Institutional...

Background: The importance of information technology to current business practices has long drawn the attention of practitioners and academicians. Aim: This paper aims to broaden understanding about service innovation as a critical organizational capability through which information technology adoption influences the competitive advantage of a firm. In the context of financial firms, this study examines how information technology is adopted and managed to enhance service innovation practices...

Topics: ERIC Archive, Information Technology, Business, Finance Occupations, Foreign Countries, Technology...

In this paper, we used simulations to compare the performance of classical and Bayesian estimations in logistic regression models using small samples. In the performed simulations, conditions were varied, including the type of relationship between independent and dependent variable values (i.e., unrelated and related values), the type of variable (i.e., binary and continuous), and different Binomial distribution values and symmetry (i.e., symmetry and positive asymmetry). Iteratively...

Topics: ERIC Archive, Regression (Statistics), Models, Simulation, Least Squares Statistics, Computation,...

This report provides empirical results of attempts to achieve consistency of estimates between two National Center for Education Statistics (NCES) surveys, the 1993-94 Private School Survey (PSS) and the Schools and Staffing Survey (SASS). Comparisons are made among statistical and computational procedures that may achieve the desired consistency between estimates. An overview describes the problem of consistency and why it matters. Sections 2 through 4 of this report explore the adjustment...

Topics: ERIC Archive, Classification, Elementary Secondary Education, Estimation (Mathematics), Least...

In research, data sets often occur in which the variance of the distribution of the dependent variable at given levels of the predictors is a function of the values of the predictors. In this situation, the use of weighted least-squares (WLS) or techniques is required. Weights suitable for use in a WLS regression analysis must be estimated. A variety of techniques have been proposed for the empirical selection of weights with the ultimate objective being a better "fit." The outcomes...

Topics: ERIC Archive, Error of Measurement, Estimation (Mathematics), Goodness of Fit, Least Squares...

The question of least-squares weights versus equal weights has been a subject of great interest to researchers for over 60 years. Several researchers have compared the efficiency of equal weights and that of least-squares weights under different conditions. Recently, S. V. Paunonen and R. C. Gardner stressed that the necessary and sufficient condition for equal-weights aggregation is that the predictors satisfy the requirements of psychometric parallelism. In this study, the effect of...

Topics: ERIC Archive, Correlation, Error of Measurement, Least Squares Statistics, Predictor Variables,...

The economic benefit that communities derive from in-migration of retired persons has been well recognized in rural development literature. This paper examines the impact of Georgia county attributes on net migration by persons 55 years old and older from 1975 to 1980. Data were obtained from the 1982 County-City Data Book, the U.S. Census Migration Estimates for States and Counties, the Georgia Department of Industry and Trade, and the Georgia Atlas. An empirical model was used to test the...

Topics: ERIC Archive, Community Characteristics, Influences, Least Squares Statistics, Middle Aged Adults,...

This study evaluated the comparative stability and agreement of three approaches to calculating school effects given both student-level and school-level data. The approaches were hierarchical linear modeling (HLM), ordinary least squares (OLS), and weighted least squares (WLS). Analyses were conducted using data from the 1998 Maryland School Performance Assessment Program for 23,461 third graders and 21,226 fifth graders. A two-level model was used for computing HLM school effects with four...

Topics: ERIC Archive, Elementary Education, Elementary School Students, Least Squares Statistics,...

Instrumental variable estimators hold the promise of enabling researchers to estimate the effects of educational treatments that are not (or cannot be) randomly assigned but that may be affected by randomly assigned interventions. Examples of the use of instrumental variables in such cases are increasingly common in educational and social science research. The most commonly used instrumental variables estimator is two-stage least squares (2SLS). Many of the properties of the 2SLS estimator are...

Topics: ERIC Archive, Social Science Research, Least Squares Statistics, Computation, Correlation,...

This paper tests the degree of overlap between operational definitions of transformational and transactional leadership, the nature of the relationships between the constructs of transformational and transactional leadership, and specified outcomes in an empirically derived data set by the application of two forms of analysis. Based on Bass's (1985) model, canonical analysis and partial least-squares analysis are applied to derive two path models. The data set was obtained from 1991 Canadian...

Topics: ERIC Archive, Educational Improvement, Elementary Secondary Education, Foreign Countries,...