Ordinary Least Squares and Quantile Regression: An Inquiry-Based Learning Approach to a Comparison of Regression Methods

We present a short, inquiry-based learning course on concepts and methods underlying ordinary least squares (OLS), least absolute deviation (LAD), and quantile regression (QR). Students investigate squared, absolute, and weighted absolute distance functions (metrics) as location measures. Using differential calculus and properties of convex functions, students discover the sample mean, median, and pth quantile by minimizing sums of distances. Students use these metrics to define loss functions for OLS, LAD, and QR, and explore methods to minimize them. We discuss classroom experiences over two semesters. Classroom activities, Maple worksheets, and R demonstration code are available at a companion website.     

Examining the Reliability of Student Growth Percentiles Using Multidimensional IRT

Student growth percentiles (SGPs, Betebenner, 2009) are used to locate a student's current score in a conditional distribution based on the student's past scores. Currently, following Betebenner (2009), quantile regression (QR) is most often used operationally to estimate the SGPs. Alternatively, multidimensional item response theory (MIRT) may also be used to estimate SGPs, as proposed by Lockwood and Castellano (2015). A benefit of using MIRT to estimate SGPs is that techniques and methods already developed for MIRT may readily be applied to the specific context of SGP estimation and inference. This research adopts a MIRT framework to explore the reliability of SGPs. More specifically, we propose a straightforward method for estimating SGP reliability. In addition, we use this measure to study how SGP reliability is affected by two key factors: the correlation between prior and current latent achievement scores, and the number of prior years included in the SGP analysis. These issues are primarily explored via simulated data. In addition, the QR and MIRT approaches are compared in an empirical application.     

Quantile Regression in the Study of Developmental Sciences

Linear regression analysis is one of the most common techniques applied in developmental research, but only allows for an estimate of the average relations between the predictor(s) and the outcome. This study describes quantile regression, which provides estimates of the relations between the predictor(s) and outcome, but across multiple points of the outcome's distribution. Using data from the High School and Beyond and U.S. Sustained Effects Study databases, quantile regression is demonstrated and contrasted with linear regression when considering models with: (a) one continuous predictor, (b) one dichotomous predictor, (c) a continuous and a dichotomous predictor, and (d) a longitudinal application. Results from each example exhibited the differential inferences which may be drawn using linear or quantile regression.     

Early Home Activities and Oral Language Skills in Middle Childhood: A Quantile Analysis

Oral language development is a key outcome of elementary school, and it is important to identify factors that predict it most effectively. Commonly researchers use ordinary least squares regression with conclusions restricted to average performance conditional on relevant covariates. Quantile regression offers a more sophisticated alternative. Using data of 17,687 children from the United Kingdom's Millennium Cohort Study, we compared ordinary least squares and quantile models with language development (verbal similarities) at 11 years as the outcome. Gender had more of an effect at the top of the distribution, whereas poverty, early language, and reading to the child had a greater effect at the bottom. The picture for TV watching was more mixed. The results are discussed in terms of the provision of universal and targeted interventions.     

Theoretical Frameworks of Performance‐Based Models

The purpose of this article is to highlight theories that support the functions of performance‐based design models and to discuss the implications of integrating divergent models into the system‐oriented human performance technology (HPT) and performance improvement (PI) disciplines. HPT, PI, and instructional systems design (ISD) share a systems framework, along with the influence of common theories such as performance theory, learning theory, adult learning, cognitive psychology, and behavioral psychology (Foshay, Villachica, & Stepich, 2014). This article focuses on the role of theory as a tool in the practitioner's toolbox and as a connection point when working with teams and organizations that have different theoretical orientations. Performance‐based ISD models are discussed, including Robinson and Robinson's (1989) Training for Impact, Brethower and Smalley's (1998) Performance‐Based Instruction, and Bradford and Boler's (2015) Horizon Model. Allen and Sites's (2012) successive approximation model (SAM) retains elements of ADDIE as a process, but the model is iterative rather than systematic in design.     

Laplace Transform and Green's Function

The particular solution y_p (t) of a linear non-homogeneous differential equation with constant coefficients can be expressed in term of a convolution integral, namely, y_p(t)=g(t)?T(t), where t(t) is the nonhomogeneous (forcing) term and g(t) is the Green's function for the associated differential operator. g(t) is also the Laplace inverse of the transfer function T(s) = l/P(s), P(s) being the characteristic polynomial of the differential operator. Owing to its versatile application, mathematical sophistication and elegance, and its simplicity, it is suggested, that the Green's function method be introduced while teaching Laplace transform, in a manner described in this article. Some simple results on convolution integrals involving discontinuous functions have been shown to be useful for computing the particular solution when the forcing function r(t) is discontinuous. An attempt has been made to justify the proposed method through a number of simple examples.     

Extreme value distributions of mixing two sequences with the same MDA

Suppose {Xi,i≥1} and {Yi, i≥1} are two independent sequences with distribution functions and , respectively. Zi,n is the combination of Xi and Yi with a probability for each I with 1≤ i≤n. The extreme value distribution GZ(x) of this particular triangular array of the i.i.d. Random variables Z1,n,… Z2,n, Zn,n is discussed. We found a new form of the extreme value distributions i)ΦAα1(x)Φα2 and ii) ψAα(x)ψα2(x)α1<α2), which are not max-stable. It occurs if FX and FY belong to the same MDA(Φ) or MDA(Ψ).     

Detecting Misspecification in Mean Structures for Growth Curve Models: Performance of Pseudo R 2s and Concordance Correlation Coefficients

This study examined the performance of 4 correlation-based fit indexes (marginal and conditional pseudo R ²s; average and conditional concordance correlations) in detecting misspecification in mean structures in growth curve models. Their performance was also compared to that of 4 traditional SEM fit indexes. We found that the marginal pseudo R ² and average concordance correlation were able to detect misspecification in the marginal mean structure (average change trajectory). The conditional pseudo R ² and concordance correlation could detect misspecification when it occurred in the conditional mean structure (individual change trajectory) or in both mean structures. Compared to the SEM fit indexes, the correlation-based fit indexes were more robust to sample size but were less robust to data properties such as magnitude of population mean and measurement error. Theoretical and practical implications of the results and directions for future research are discussed.     

Understanding heterogeneity in pathways between interparental conflict and children's involvement: The moderating role of affect-biased attention

The study examined the moderating role of children's affect-biased attention to angry, fearful, and sad adult faces in the link between interparental conflict and children's distinct forms of involvement. Participants included 243 preschool children (M_age = 4.60 years, 56% female) and their parents from racially (48% African American, 43% White) and socioeconomically (median annual household income = $36,000) diverse backgrounds. Data collection took place in the Northeastern United States (2010–2014). Utilizing a multi-method, multi-informant, longitudinal design, attention away from anger selectively amplified the link between interparental conflict and children's subsequent coercive involvement (β = −.15). Greater attention to fear potentiated the pathway between interparental conflict and children's later cautious (β = .14) and caregiving involvement (β = .15). Findings are interpreted in the context of environmental sensitivity models.     

Video game learning dynamics: Actionable measures of multidimensional learning trajectories

Valid, accessible, reusable methods for instructional video game design and embedded assessment can provide actionable information enhancing individual and collective achievement. Cyberlearning through game‐based, metaphor‐enhanced learning objects (CyGaMEs) design and embedded assessment quantify player behavior to study knowledge discovery and application. CyGaMEs is grounded by analogical reasoning theory, cognitive task analysis and knowledge representation. A construct representation argument for validity using evidence‐centered design warrants CyGaMEs, its web‐based learning environment, Selene: A Lunar Construction GaME, its embedded assessment, and a learning dynamics approach to student, measurement and statistical models. Two studies (US volunteers, Study 1: n = 267, mean (M)_age = 15; Study 2: n = 90, M_age = 12) cross‐validate learning dynamics (learner progress, rate of progress and changes in that rate) for Selene's multidimensional goals while players learn and apply standard‐based science about fundamental geology and space science concepts. Gameplay data analyzed using regression, calculus and hierarchical linear modeling exhibit overall relatively high standardized rates of progress toward each goal statistically higher than zero. For example, adjusting for nesting within individual players, average rate of progress toward the goal of accreting lunar mass is a z‐score of 1.4 (99% confidence interval = 1.40_lower, 1.48_upper) or 1.4 standard deviations above zero.     

Effective schools,school segregation,and the link with school achievement

This study examines whether 3 teacher-rated aspects of school effectiveness differ across school segregation profiles in Stockholm, and to what extent these indicators are associated with the academic achievement of 9th-grade students. Analyses were based on 2 cross-sectional data collections performed in 2014 and 2016, respectively (147 school units), one among teachers (n = 2,024) and the other among 9th-grade students (n = 9,151). Multilevel analysis was applied, estimating 2-level random intercept linear regression models. Results show that teachers’ ratings of school leadership, teacher cooperation, and school ethos, as well as student-reported marks differ across school segregation profiles. Findings further reveal significant associations between these school effectiveness indicators and student performance, even when taking student family background and the school’s student body composition into consideration. In part, these associations are also identified within segregation profiles. Moreover, results show that school ethos acts as a mediator between school segregation profile and student achievement.     

Longitudinal Relations Among Parents' Reactions to Children's Negative Emotions,Effortful Control,and Math Achievement in Early Elementary School

Panel mediation models and fixed‐effects models were used to explore longitudinal relations among parents' reactions to children's displays of negative emotions, children's effortful control (EC), and children's math achievement (N = 291; M age in fall of kindergarten = 5.66 years, SD = .39 year) across kindergarten through second grade. Parents reported their reactions and children's EC. Math achievement was assessed with a standardized achievement test. First‐grade EC mediated the relation between parents' reactions at kindergarten and second‐grade math achievement, beyond stability in constructs across study years. Panel mediation model results suggested that socialization of EC may be one method of promoting math achievement in early school; however, when all omitted time‐invariant covariates of EC and math achievement were controlled, first‐grade EC no longer predicted second‐grade math achievement.     

Assessing Students' Understandings of Biological Models and their Use in Science to Evaluate a Theoretical Framework

Research in the field of students’ understandings of models and their use in science describes different frameworks concerning these understandings. Currently, there is no conjoint framework that combines these structures and so far, no investigation has focused on whether it reflects students' understandings sufficiently (empirical evaluation). Therefore, the purpose of this article is to present the results of an empirical evaluation of a conjoint theoretical framework. The theoretical framework integrates relevant research findings and comprises five aspects which are subdivided into three levels each: nature of models, multiple models, purpose of models, testing, and changing models. The study was conducted with a sample of 1,177 seventh to tenth graders (aged 11–19 years) using open-ended items. The data were analysed by identifying students' understandings of models (nature of models and multiple models) and their use in science (purpose of models, testing, and changing models), and comparing as well as assigning them to the content of the theoretical framework. A comprehensive category system of students' understandings was thus developed. Regarding the empirical evaluation, the students' understandings of the nature and the purpose of models were sufficiently described by the theoretical framework. Concerning the understandings of multiple, testing, and changing models, additional initial understandings (only one model possible, no testing of models, and no change of models) need to be considered. This conjoint and now empirically tested framework for students' understandings can provide a common basis for future science education research. Furthermore, evidence-based indications can be provided for teachers and their instructional practice.     

Predicting word reading ability: a quantile regression study

Predictors of early word reading are well established. However, it is unclear if these predictors hold for readers across a range of word reading abilities. This study used quantile regression to investigate predictive relationships at different points in the distribution of word reading. Quantile regression analyses used preschool and kindergarten measures of letter knowledge, phonological awareness, rapid automatised naming, sentence repetition, vocabulary and mother's education to predict first‐grade word reading. Predictors generally varied in significance across levels of word reading. Notably, rapid automatised naming was a significant unique predictor for average and good readers but not poor readers. Letter knowledge was generally a stronger unique predictor for poor and average readers than good readers. Well‐known word reading predictors varied in significance at different points along the word reading distribution. Results have implications for early identification and statistical analyses of reading‐related outcomes. What is already known about this topic

• Early predictors of word reading are well established, with letter knowledge, phonological awareness and rapid automatised naming identified as key predictors.
• These relationships are primarily investigated in average readers, or in groups of good and poor readers separated by an arbitrary cut‐off score.

What this paper adds

• In this study, we used quantile regression to determine significant predictors of word reading across a range of word reading abilities.
• The quantile regression approach avoids the loss of power that can arise when creating subgroups and has none of the issues associated with the use of a single, arbitrary cut-off score to separate good and poor readers.
• Letter knowledge and phonological awareness were significantly predictive of word reading across the distribution of word reading abilities, whereas rapid automatised naming was significant only for good readers, and sentence recall was significant only for poor readers.

Implications for theory, policy and practice

• Results reinforce the usefulness of measures such as letter knowledge, phonological awareness and sentence repetition in the early identification of children at risk for reading disabilities.
• Results also suggest that measures of rapid naming may add little unique information in differentiating between children who subsequently read in the below‐average range.

     

The Role of Relational and Instructional Classroom Supports in the Language Development of At-Risk Preschoolers

The present study examined the extent to which preschool classroom supports—relational support (RS) and instructional support (IS)—are associated with children's language development and whether these associations vary as a function of children's language ability. The language skills of 360 children within 95 classrooms were assessed using an expressive narrative task in the fall and spring of the preschool year, teachers rated RS in the fall, and observations of IS were collected across the year. Research Findings: Hierarchical linear models revealed main effects of IS, but not RS, on preschoolers’ development of expressive language skills. In addition, the associations between RS and IS on children's expressive language development were moderated by children's fall language ability. Specifically, the association between IS and language development was stronger for children with stronger expressive language skills, and the association between RS and language development was stronger for children with weaker expressive language skills. Practice or Policy: These findings suggest that professional development for preschool teachers might focus on aligning classroom supports with the needs of children with weaker language skills who are at risk for difficulty acquiring literacy.     

Improving Language-Focused Comprehension Instruction in Primary-Grade Classrooms: Impacts of the Let’s Know! Experimental Curriculum

This quasi-experimental study was designed to test the impacts of a curriculum supplement, Let’s Know!, on the quantity and quality of language-focused comprehension instruction in pre-kindergarten to third grade classrooms. Sixty classrooms (12 per each of pre-K to grade 3) were enrolled in the study, with 40 teachers assigned to implement one of two versions of the experimental Let’s Know! curriculum and 20 assigned to a control condition, in which they maintained their typical language-arts curriculum. Classroom observations, 90 min in duration, were collected near the end of the first unit’s completion, about 4 to 5 weeks into the academic year. These observations were coded to examine impacts of Let’s Know! instruction on two outcomes: (a) teachers’ use of 18 language-focused comprehension supports and (b) general classroom quality. Study results using quantile regression showed that Let’s Know! teachers used a significantly higher number of language-focused comprehension supports during Let’s Know! instruction compared to the control teachers during language-arts instruction; the same finding was also true for general classroom quality. Quantile regression results showed the greatest differentiation in instructional quality, when comparing experimental and control teachers, for teachers in the middle of the distribution of general classroom quality. Study findings highlight the value of language-focused curricula for heightening comprehension-specific supports in pre-K to grade 3 settings.     

Testing Predictive Models of Technology Integration in Mexico and the United States

ABSTRACT

Data from Mexico City, Mexico (N = 978) and from Texas, USA (N = 932) were used to test the predictive validity of the teacher professional development component of the Will, Skill, Tool Model of Technology Integration in a cross-cultural context. Structural equation modeling (SEM) was used to test the model. Analyses of these data yielded positive results for the model's validity and reliability, with more than 90% of the variance in classroom technology integration found to be attributable to a linear combination of a teacher's attitude or Will, technology proficiency or Skill, and access to technology Tools. Results also showed a differential effect of predictors on integration, depending on the teacher's stage of technology adoption: At lower stages Tool access appears to be the best predictor; whereas, at higher stages Skill appears to be the best predictor. At the highest stage of adoption Will may replace Skill as the best technology integration predictor. Collective findings regarding the orderings of integration predictors shed light on alternative theoretical views of technology integration. Findings tend to support the view of technology integration as a step-wise, sequential, developmental process, rather than the process of eliminating external and internal barriers that impede the full integration of technology into a classroom teaching/learning environment.     

Jugements probabilistes et modeles cognitifs: Etude d'une situation aleatoire

Résumé Soit le problème suivant: on lance en même temps deux dés, et on considère les deux rèsultats suivants: R1 on obtient un 5 et un 6 et R2 on obtient deux fois un 6. Lorsqu'il s'agit de comparer les deux résultats, on observe un biais d'équiprobabilité qui conduit près de 60% des sujets—étudiants ou lycéens de différentes formations—à répondre incorrectement qu'il y a équiprobabilité des deux résultats considérés. Ce papier se divise en deux parties. (1) Dans une première partie, nous résumons une série d'expériences destinées à étudier systématiquement la résistance de ce biais à des variations de divers facteurs. (2) Dans la seconde partie, nous présentons une expérience réalisée auprès de 342 lycéens à partir de laquelle nous avons caractérisé des modèles cognitifs spontanément utilisés par les sujets; nous présentons et discutons ici quelques uns de ces modèles et leur intérêt didactique.

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Probabilistic judgements and cognitive models: A study of an uncertain situation

Consider the following problem: two dice are simultaneously thrown, and these two results are considered: R1 a 5 and a 6 are obtained, and R2 a 6 is obtained twice. When the question is to compare these two results, we observe an equiprobability bias which leads about 60 per cent of the subjects—students of various formations—to give an incorrect equiprobability response of the two considered results. The present paper is divided into two parts. (1) In the first part, we summarize a series of experiments intended to study systematically the resistance of this bias against variations of various factors. (2) In the second part, we present an experiment carried out by 342 students from which we have characterized cognitive models spontaneously developed. We present and discuss here some of these models and their interest for teaching.