Secondary science and mathematics teachers and gender equity: Attitudes and attempted interventions

Although the literature documents the potential impact of teachers on the achievement of young women in science and math, few studies investigate teacher attitudes and the use of teacher interventions. A survey containing both fixed-response and open-ended questions was administered to 56 teachers from eight high schools. Open-ended questions were content analyzed, and log-linear modeling and chi-square tests of independence were used to analyze relationships between categorical variables representing demographic characteristics, teacher attitudes, and teacher use of interventions. Results provide evidence that teachers, regardless of demographic characteristics, are concerned about gender inequity in their classrooms, although they are generally not familiar with the wide range of possible causes (including their own actions). Reported interventions are consistent with those recommended in the literature, although teachers are not familiar with the effectiveness of their interventions, and numerous teachers feel that attempting interventions is a form of reverse discrimination. © 1996 John Wiley & Sons, Inc.     

A Two-Stage Approach to Synthesizing Covariance Matrices in Meta-Analytic Structural Equation Modeling

A great obstacle for wider use of structural equation modeling (SEM) has been the difficulty in handling categorical variables. Two data sets with known structure between 2 related binary outcomes and 4 independent binary variables were generated. Four SEM strategies and resulting apparent validity were tested: robust maximum likelihood (ML), tetrachoric correlation matrix input followed by SEM ML analysis, SEM ML estimation for the sum of squares and cross-products (SSCP) matrix input obtained by the log-linear model that treated all variables as dependent, and asymptotic distribution-free (ADF) SEM estimation. SEM based on the SSCP matrix obtained by the log-linear model and SEM using robust ML estimation correctly identified the structural relation between the variables. SEM using ADF added an extra parameter. SEM based on tetrachoric correlation input did not specify the data generating process correctly. Apparent validity was similar for all models presented. Data transformation used in log-linear modeling can serve as an input for SEM.     

Loglinear analysis of cross-classified ordinal data: applications in developmental research

This article provides an introduction to loglinear analysis of cross-classification tables, including tables with nominal and ordinal variables. Loglinear models offer several advantages over the more commonly used chi-square test of independence, including the ability to analyze 3-, 4-, and higher-way interactions, the ability to determine whether the association between variables is linear or nonlinear, and the ability to interpret scale scores assigned to categories of an ordinal variable. After a review of the advantages of loglinear modeling, the chi-square test of independence is compared with the loglinear model of independence. This comparison serves to introduce the notation and terminology of loglinear modeling. The overall strategy of loglinear modeling is introduced next; then special loglinear models for ordinal data are reviewed. Each model discussed in the article is applied to data from the developmental literature.     

Students' approaches to exploring relationships between categorical variables

In the context of an afterschool program in which students explore relatively large authentic datasets, we investigated how 11- to 14-year old students worked with categorical variables. During the program, students learned to use the Common Online Data Analysis Platform (CODAP), a statistical analysis platform specifically designed for middle and high school students, to create and interpret graphs. Following the program, we conducted individual clinical interviews, during which students used CODAP to answer questions about relationships between variables. Here, we describe how students engaged in exploratory data analysis that involved looking at relationships between two categorical variables. Students worked from data in table form and created “contingency graphs,” a variant of contingency tables, which they used to analyze and draw insights from the data. Our research identified four strategies that students used to examine the data in order to explore patterns, make comparisons, and answer questions with the data.     

Quintilian: Wisdom and Warning

The implementation of online texts, videos, homework, and tests has changed the process of instruction in introductory college mathematics courses. With this change, more of the students’ learning takes place outside of the traditional college classroom and in places such as tutoring centers and dorm rooms. A combination of chi-square tests—for independence with unordered categorical data—and Mann-Whitney two-sample rank-sum tests—for continuous data and ordered categorical data—were used to analyze student outcomes generated from College Algebra and Applied Calculus courses with class sizes ranging from 37 to 129, with common syllabi, homework, quizzes, and tests. These tests showed that medium classes (30–55 students) had little to no benefit over large classes (110–130 students) in student learning and student achievement, with large classes having small to medium positive-effect sizes over medium classes in the area of student satisfaction. The only area in which the small classes had a small positive effect was in the area of student engagement.     

Modeling of Nonrecursive Structural Equation Models With Categorical Indicators

Nonrecursive structural equation models generally take the form of feedback loops, involving 2 latent variables that are connected by 2 unidirectional paths, 1 starting with each variable and terminating in the other variable. Nonrecursive models belong to a larger class of path models that require the use of instrumental variables (IVs) to achieve model identification. Prior research has focused on SEM parameter estimation with IVs when indicators were continuous and normally distributed. Much less is known about how estimators function in the presence of categorical indicators, which are commonly used in the social sciences, such as with cognitive and affective instruments. In this study, there was specific interest in comparing the 2-stage least squares (2SLS) estimator and its categorical variant to other recommended estimators. This study compares the performance of several estimation approaches for fitting structural equation models with categorical indicator variables when IVs are necessary to obtain proper model estimates. Across conditions, 1 extension of the nonlinear 2SLS (N2SLS) approach, the nonlinear 3-stage least squares (N3SLS), which accounts for correlated errors among regressors within each model (as does the N2SLS), as well as correlations of errors across models, which N2SLS does not, appears to work the best among methods compared.     

Measurement Invariance in International Surveys: Categorical Indicators and Fit Measure Performance

In spite of the challenges inherent in making dozens of comparisons across heterogeneous populations, a relatively recent interest in scale-score equivalence for non-achievement measures in an international context has emerged. Until recently, operational procedures for establishing measurement invariance using multiple-groups analyses were typically supported with research that was limited in scope to few groups and relatively small sample sizes. Recent research that examined situations more representative of international surveys recommended some revisions to typically used fit measures. The current study extends this research and evaluates the performance of several fit measures when data are assumed to have an ordered categorical, rather than the typically assumed continuous, scale. Using a simulation study based on empirical results, findings indicated that classic measures and associated criteria were either unsuitable in a large-group and varied sample-size context or should be adjusted, particularly when observed variables were not normally distributed. We provide specific recommendations for revising currently used criteria for evaluating overall and relative fit based on the chi-square test, root mean-squared error of approximation, and comparative fit index (CFI).     

Vocabulary Burden

Abstract

The purpose of the study was to assess high school students’ knowledge in the area of consumer economics and to identify school variables related to that knowledge. Six hundred fifty-three seniors from five Pennsylvania school districts were administered the “Consumer Economics Content Area Measure” of the Adult Performance Level Program (APL). Multiple regression analysis and chi-square procedures were used to analyze the data. Of the school variables examined, only the Scholastic Aptitude Test (SAT) scores in mathematics and reading showed a moderate correlation with the APL scores.     

对数线性模型的统计信息功能

通过一个四维列联表算例研究了对数线性模型在统计分析中的信息功能．结果表明：对数线性模型可通过拟合优度统计检验发现多维分类变量中存在的统计独立关系;可通过参数估计分别针对列联表的每一个组格回答各个单个变量以及2个、3个、4个变量的各种交互对形成该组格概率的贡献;无法提供变量间统计相依程度强弱的信息．     

A General 3-Step Maximum Likelihood Approach to Estimate the Effects of Multiple Latent Categorical Variables on a Distal Outcome

The 3-step approach has been recently advocated over the simultaneous 1-step approach to model a distal outcome predicted by a latent categorical variable. We generalize the 3-step approach to situations where the distal outcome is predicted by multiple and possibly associated latent categorical variables. Although the simultaneous 1-step approach has been criticized, simulation studies have found that the performance of the two approaches is similar in most situations (Bakk & Vermunt, 2016). This is consistent with our findings for a 2-LV extension when all model assumptions are satisfied. Results also indicate that under various degrees of violation of the normality and conditional independence assumption for the distal outcome and indicators, both approaches are subject to bias but the 3-step approach is less sensitive. The differences in estimates using the two approaches are illustrated in an analysis of the effects of various childhood socioeconomic circumstances on body mass index at age 50.     

Fit of Different Models for Multitrait-Multimethod Experiments

In the past, several models have been developed for the estimation of the reliability and validity of measurement instruments from multitrait-multimethod (MTMM) experiments. Suggestions have been made for additive, multiplicative and correlated uniqueness models, whereas recently Coenders and Saris (2000) suggested a procedure to test these models against one another. In this article, the different models suggested for the analysis of MTMM matrixes have been compared for their fit to 87 data sets collected in the United States (Andrews, 1984; Rodgers, Andrews, & Herzog, 1992), Austria (Koltringer, 1995), and the Netherlands (Scherpenzeel & Saris, 1997). As most variables are categorical, the analysis has been carried out on the basis of polychoric-polyserial correlation coefficients and of Pearson correlations. The fit of the models based on polychoric correlations is much worse than the fit of models based on product moment correlations, but in both cases a model that assumes additive method effects fits most data sets better than the other models, including the so-called multiplicative models.     

Examining Chi-Square Test Statistics Under Conditions of Large Model Size and Ordinal Data

This study examined the effect of model size on the chi-square test statistics obtained from ordinal factor analysis models. The performance of six robust chi-square test statistics were compared across various conditions, including number of observed variables (p), number of factors, sample size, model (mis)specification, number of categories, and threshold distribution. Results showed that the unweighted least squares (ULS) robust chi-square statistics generally outperform the diagonally weighted least squares (DWLS) robust chi-square statistics. The ULSM estimator performed the best overall. However, when fitting ordinal factor analysis models with a large number of observed variables and small sample size, the ULSM-based chi-square tests may yield empirical variances that are noticeably larger than the theoretical values and inflated Type I error rates. On the other hand, when the number of observed variables is very large, the mean- and variance-corrected chi-square test statistics (e.g., based on ULSMV and WLSMV) could produce empirical variances conspicuously smaller than the theoretical values and Type I error rates lower than the nominal level, and demonstrate lower power rates to reject misspecified models. Recommendations for applied researchers and future empirical studies involving large models are provided.     

Measuring the Impact of Judge Severity on Examination Scores

In order to obtain objective measurement for examinations that are graded by judges, an extension of the Rasch model designed to analyze examinations with more than two facets (items/examinees) is used. This extended Rasch model calibrates the elements of each facet of the examination (i.e., examinee performances, items, and judges) on a common log-linear scale. A network for assigning judges to examinations is used to link all facets. Real examination data from the "clinical assessment" part of a certification examination are used to illustrate the application. A range of item difficulties and judge severities were found. Comparison of examinee raw scores with objective linear measures corrected for variations in judge severity shows that judge severity can have a substantial impact on a raw score. Correcting for judge severity improves the fairness of examinee measures and of the subsequent pass-fail decisions because the uncorrected raw scores favor examinee performances graded by lenient judges.     

Is Parceling Really Necessary? A Comparison of Results From Item Parceling and Categorical Variable Methodology

This study examined the efficacy of 4 different parceling methods for modeling categorical data with 2, 3, and 4 categories and with normal, moderately nonnormal, and severely nonnormal distributions. The parceling methods investigated were isolated parceling in which items were parceled with other items sharing the same source of variance, and distributed parceling in which items were parceled with items influenced by different factors. These parceling strategies were crossed with strategies in which items were either parceled with similarly distributed or differently distributed items, to create 4 different parceling methods. Overall, parceling together items influenced by different factors and with different distributions resulted in better model fit, but high levels of parameter estimate bias. Across all parceling methods, parameter estimate bias ranged from 20% to over 130%. Parceling strategies were contrasted with use of the WLSMV estimator for categorical, unparceled data. Results based on this estimator are encouraging, although some bias was found when high levels of nonnormality were present. Values of the chi-square and root mean squared error of approximation based on WLSMV also resulted in Type II error rates for misspecified models when data were severely nonnormally distributed.     

Structural Models for Binary Repeated Measures: Linking Modern Longitudinal Structural Equation Models to Conventional Categorical Data Analysis for Matched Pairs

The current widespread availability of software packages with estimation features for testing structural equation models with binary indicators makes it possible to investigate many hypotheses about differences in proportions over time that are typically only tested with conventional categorical data analyses for matched pairs or repeated measures, such as McNemar’s chi-square. The connection between these conventional tests and simple longitudinal structural equation models is described. The equivalence of several conventional analyses and structural equation models reveals some foundational concepts underlying common longitudinal modeling strategies and brings to light a number of possible modeling extensions that will allow investigators to pursue more complex research questions involving multiple repeated proportion contrasts, mixed between-subjects × within-subjects interactions, and comparisons of estimated membership proportions using latent class factors with multiple indicators. Several models are illustrated, and the implications for using structural equation models for comparing binary repeated measures or matched pairs are discussed.     

Selection of models in contingency tables: A reexamination

Qualitative data often need analysis using log-linear models, as pointed out by Liu and Sanders. In this article, references to the state-of-the-art literature and software packages for analyzing such data are presented. We also illustrate the two types of log-linear models and present procedures for investigating both types.     

Models and Strategies for Factor Mixture Analysis: An Example Concerning the Structure Underlying Psychological Disorders

The factor mixture model (FMM) uses a hybrid of both categorical and continuous latent variables. The FMM is a good model for the underlying structure of psychopathology because the use of both categorical and continuous latent variables allows the structure to be simultaneously categorical and dimensional. This is useful because both diagnostic class membership and the range of severity within and across diagnostic classes can be modeled concurrently. Although the conceptualization of the FMM has been explained in the literature, the use of the FMM is still not prevalent. One reason is that there is little research about how such models should be applied in practice and, once a well-fitting model is obtained, how it should be interpreted. In this article, the FMM is explored by studying a real data example on conduct disorder. By exploring this example, this article aims to explain the different formulations of the FMM, the various steps in building a FMM, and how to decide between an FMM and alternative models.     

Log-Linear Analysis: An Introduction for Researchers in Counseling

This article serves as an introduction to log-linear analysis, a technique used in the analysis of qualitative data arranged in multidimensional contingency tables. The steps involved in log-linear analysis are presented in the context of an investigation of a personality characteristic of a group of counselor trainees. The references cited and supplemental readings allow interested readers to pursue the topic further.     

Replication of a path analysis model of secondary physics enrollments: 20 years later

The percentage of enrollment in physics (PEP) at the secondary level in the United States has been approximately 20% for the past few decades. In an attempt to isolate variables related to physics enrollment. Bryant (1977) conducted a path analysis model of secondary physics enrollment in New York State. This present study is a replication and extension of the study conducted by Bryant. Current data were used to verify the Bryant model and construct models with data available since Bryant's work. Data gathered by the New York State Education Department for the 1990–1991 school year were used. The source of this data included surveys completed by teachers and administrators on student characteristics and school facilities. A data analysis similar to Bryant's was conducted to determine whether the relationships between a set of predictor variables and physics enrollment had changed in the past 20 years. The findings indicate that there has been no change in the set of variables related to physics enrollment. © 1996 John Wiley & Sons, Inc.