Differential Step Functioning Research Articles

Comparison of Methods for Identifying Differential Step Functioning with Polytomous Item Response Data

ABSTRACT Much research has been devoted to identification of differential item functioning (DIF), which occurs when the item responses for individuals from two groups differ after they are conditioned on the latent trait being measured by the scale. There has been less work examining differential step functioning (DSF), which is present for polytomous items when the conditional likelihood of responses to specific categories differ between groups. DSF impacts estimation of the measured trait and reduces the effectiveness of standard DIF detection methods. The purpose of this simulation study was to extend upon earlier work by comparing several methods for detecting the presence of DSF in polytomous items, including an approach based on the lasso estimation of the generalized partial credit model. Results show that the lasso GPCM technique controlled the Type I error rate while yielding power rates somewhat lower than logistic regression and the MIMIC model, which were not able to control the Type I error rate in some conditions. An empirical example is also presented, and implications of this study for practice are discussed.

Detecting Differential Item Functioning and Differential Step Functioning Due to Differences that Should Matter

This study illustrates the use of differential item functioning (DIF) and differential step functioning (DSF) analyses to detect differences in item difficulty that are related to experiences of examinees, such as their teachers’ instructional practices, that are relevant to the knowledge, skill, or ability the test is intended to measure. This analysis is in contrast to the typical use of DIF or DSF to detect differences related to characteristics of examinees, such as gender, language, or cultural knowledge, that should be irrelevant. Using data from two forms of Ontario’s Grade 9 Assessment of Mathematics, analyses were performed comparing groups of students defined by their teachers’ instructional practices. All constructed-response items were tested for DIF using the Mantel Chi-Square, standardized Liu Agresti cumulative common log-odds ratio, and standardized Cox’s noncentrality parameter. Items exhibiting moderate to large DIF were subsequently tested for DSF. In contrast to typical DIF or DSF analyses, which inform item development, these analyses have the potential to inform instructional practice.

Tree-Based Global Model Tests for Polytomous Rasch Models.

Psychometric measurement models are only valid if measurement invariance holds between test takers of different groups. Global model tests, such as the well-established likelihood ratio (LR) test, are sensitive to violations of measurement invariance, such as differential item functioning and differential step functioning. However, these traditional approaches are only applicable when comparing previously specified reference and focal groups, such as males and females. Here, we propose a new framework for global model tests for polytomous Rasch models based on a model-based recursive partitioning algorithm. With this approach, a priori specification of reference and focal groups is no longer necessary, because they are automatically detected in a data-driven way. The statistical background of the new framework is introduced along with an instructive example. A series of simulation studies illustrates and compares its statistical properties to the well-established LR test. While both the LR test and the new framework are sensitive to differential item functioning and differential step functioning and respect a given significance level regardless of true differences in the ability distributions, the new data-driven approach is more powerful when the group structure is not known a priori-as will usually be the case in practical applications. The usage and interpretation of the new method are illustrated in an empirical application example. A software implementation is freely available in the R system for statistical computing.

Synchronization Motion Tracking Control of Multiple Underactuated Ships With Collision Avoidance

This paper presents a constructive method to design distributed controllers that force a group of $N$ underactuated ships under sea loads to perform a desired synchronization motion tracking task and guarantee no collisions between the ships. First, an observer is designed to globally exponentially estimate the sea loads. Then, the ship dynamics is transformed to an almost spherical form. Smooth or $p$ -times differentiable step functions are next introduced to ensure that the collision avoidance is only active when the ships are in their collision detection zones. For each ship, a dynamical variable is introduced to deal with relaxation of the reference path generation and difficulties caused by underactuation. The control design is based on the above developments, and Lyapunov and backstepping methods. The effectiveness of the proposed results is illustrated through simulations.

Using Mixed Methods to Interpret Differential Item Functioning

ABSTRACTAnalysis of differential item functioning (DIF) is often used to determine if cross-lingual assessments are equivalent across languages. However, evidence on the causes of cross-lingual DIF is still evasive. Expert appraisal is a qualitative method useful for obtaining detailed information about problematic elements in the different linguistic versions of items. In this article we propose and explore a mixed methods approach that integrates quantitative results from DIF analysis and qualitative findings from expert appraisal to discover reasons why items exhibit DIF across languages. First, polytomous DIF was analyzed in responses to the U.S. and Spanish version of scales from the PISA Student Questionnaire by Differential Step Functioning and Ordinal Logistic Regression. Items flagged by both methods were selected to be studied to interpret DIF causes. Secondly, experts were asked about non-comparable elements in items. Experts provided qualitative evidence on problematic issues (different interpretation patterns or response processes), that may have been the cause of the DIF. The integration of results from both methods was aimed at relating type of DIF to expert appraisal findings.

Path-tracking control of underactuated ships under tracking error constraints

This paper presents a constructive design of new controllers that force underactuated ships under constant or slow time-varying sea loads to asymptotically track a parameterized reference path, that guarantees the distance from the ship to the reference path always be within a specified value. The control design is based on a global exponential disturbance observer, a transformation of the ship dynamics to an almost spherical form, an interpretation of the tracking errors in an earth-fixed frame, an introduction of dynamic variables to compensate for relaxation of the reference path generation, p-times differentiable step functions, and backstepping and Lyapunov’s direct methods. The effectiveness of the proposed results is illustrated through simulations.

Net and Global Differential Item Functioning in PISA Polytomously Scored Science Items

The purpose of this study was to apply two types of Differential Item Functioning (DIF), net and global DIF, as well as the framework of Differential Step Functioning (DSF) to real testing data to investigate measurement invariance related to test language. Data from the Program for International Student Assessment (PISA)–2006 polytomously scored science items for four countries with different test languages were used, where French and English represented the reference languages. The findings showed that many items exhibited both types of DIF, although, in most cases, the results were inconsistent for the two source languages. In addition, net and global DIF tests did not always yield the same results depending on the DSF effect pattern. Furthermore, the DSF analysis provided valuable information over and above that provided by the net DIF analysis concerning the nature and the location of the DIF effect.

Modeling Item-Level and Step-Level Invariance Effects in Polytomous Items Using the Partial Credit Model

Measurement invariance is a common consideration in the evaluation of the validity and fairness of test scores when the tested population contains distinct groups of examinees, such as examinees receiving different forms of a translated test. Measurement invariance in polytomous items has traditionally been evaluated at the item-level, corresponding to what is broadly referred to as differential item functioning. However, recent advances in the study of measurement invariance in polytomous items has documented the value in examining invariance at the level of each step of the polytomous item, referred to as differential step functioning. To date, little documentation exists of methodology that can simultaneously evaluate both item-level and step-level invariance effects using a common parametric model. In this article, we describe how to use the partial credit model to simultaneously evaluate item-level and step-level invariance effects. A simulation study as well as a large empirical example are presented to demonstrate the use of this methodology applied to a large-scale administration of a translated test.

Formation control of underactuated ships with elliptical shape approximation and limited communication ranges

Based on the recent theoretical development for formation control of multiple fully actuated agents with an elliptical shape in Do (2012), this paper develops distributed controllers that force a group of N underactuated ships with limited communication ranges to perform a desired formation, and guarantee no collisions between any ships in the group. The ships are first fitted to elliptical disks for solving collision avoidance. A coordinate transformation is then proposed to introduce an additional control input, which overcomes difficulties caused by underactuation and off-diagonal terms in the system matrices. The control design relies on potential functions with the separation condition between elliptical disks and the smooth or p-times differentiable step functions embedded in.

Formation control of multiple elliptical agents with limited sensing ranges

This paper presents a design of cooperative controllers that force a group of N mobile agents with an elliptical shape and with limited sensing ranges to perform a desired formation. The controllers guarantee no collisions between any agents in the group. The desired formation can be stabilized at feasible reference trajectories with bounded time derivatives. The formation control design is based on an algebraic separation condition between ellipses, Lyapunov’s method, and smooth or p-times differentiable step functions. These functions are introduced and incorporated into novel potential functions to solve the collision avoidance problem without the need for switchings under the agents’ limited sensing ranges.

A Comparison of Adjacent Categories and Cumulative Differential Step Functioning Effect Estimators

The study of measurement invariance in polytomous items that targets individual score levels is known as differential step functioning (DSF). The analysis of DSF requires the creation of a set of dichotomizations of the item response variable. There are two primary approaches for creating the set of dichotomizations to conduct a DSF analysis: the adjacent categories approach, and the cumulative approach. To date, there is limited research on how these two approaches compare within the context of DSF, particularly as applied to a real data set. This study evaluated the results of a DSF analysis using both dichotomization schemes in order to determine if the two approaches yield similar results. The results revealed that the two approaches generally led to consistent results, particularly in the case where DSF effects were negligible. However, when significant DSF effects were present, the two approaches occasionally led to differing conclusions.

Flocking for Multiple Elliptical Agents With Limited Communication Ranges

In existing flocking and coordination control systems of multiple agents, an agent is considered as a single point or a circular disk. For agents with a long and narrow shape, fitting them to circular disks results in a problem with the large conservative area. This paper presents a design of distributed controllers for flocking of N mobile agents with an elliptical shape and with limited communication ranges. A separation condition for elliptical agents is first derived. Smooth or p-times differentiable step functions are then introduced. These functions and the separation condition between the elliptical agents are embedded in novel potential functions to design control algorithms for flocking. The controllers guarantee 1) no switchings in the controllers despite the agents' limited communication ranges; 2) no collisions between any agents; 3) asymptotic convergence of each agent's generalized velocity to a desired velocity; and 4) boundedness of the flock size, which is defined as the sum of all generalized distances between the agents, by a constant.

Formation control of multiple elliptic agents with limited sensing ranges

This paper presents a design of cooperative controllers that force a group of N mobile agents with an elliptic shape and limited sensing ranges to perform a desired formation and that guarantee no collisions between any agents in the group. The desired formation can be stabilized at feasible reference trajectories with bounded time derivatives. The formation control design is based on explicit algebraic separation conditions between ellipses, root conditions of cubic polynomials, the Lyapunov direct method, and smooth or p-times differentiable step functions. These functions are incorporated into novel potential functions to solve the collision avoidance problem without the need for switchings despite the agents' limited sensing ranges. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Formation control of underactuated ships with elliptical shape approximation and limited communication ranges

A method is presented to design distributed controllers that force a group of N underactuated ships with limited communication ranges to perform a desired formation, and guarantee no collisions between any ships in the group. The ships are first fitted to elliptical disks for solving collision avoidance. A separation condition for elliptical disks is then derived. Smooth or p -times differentiable step functions are next introduced to deal with the ships’ limited communication ranges. A coordinate transformation is then proposed to introduce an additional control input, which overcomes difficulties caused by underactuation and off-diagonal terms in the system matrices. The control design is based on novel potential functions with the separation condition between elliptical disks and the smooth or p -times differentiable step functions embedded in.

Formation control of multiple elliptical agents with limited sensing ranges

This paper presents a design of cooperative controllers that force a group of N mobile agents with an elliptical shape and with limited sensing ranges to perform a desired formation. The controllers guarantee no collisions between any agents in the group. The desired formation can be stabilized at feasible reference trajectories with bounded time derivatives. The formation control design is based on an algebraic separation condition between ellipses, Lyapunov’s method, and smooth or p -times differentiable step functions. These functions are introduced and incorporated into novel potential functions to solve the collision avoidance problem without the need for switchings under the agents’ limited sensing ranges.

Practical formation control of multiple underactuated ships with limited sensing ranges

This paper presents a constructive method to design cooperative controllers that force a group of N underactuated ships with limited sensing ranges to perform a desired formation, and guarantee no collisions between the ships. These ships do not have an independent actuator in the sway axis. The desired formation is stabilized at any sufficiently smooth reference trajectories, including fixed points and nonadmissible trajectories for the ships. The formation control design is based on several nonlinear coordinate changes, the transverse function approach, the backstepping technique, the Lyapunov direct method, and smooth and p -times differentiable step functions. These functions are introduced and incorporated into novel potential functions to solve the collision avoidance problem without the need of switchings despite the ships’ limited sensing ranges. Simulations illustrate the results.

Practical Formation Control of Multiple Unicycle-Type Mobile Robots with Limited Sensing Ranges

This paper presents a design of cooperative controllers that force a group of N unicycle-type mobile robots with limited sensing ranges to perform a desired tight formation and that guarantee no collisions between any robots in the group. The desired formation can be stabilized at any reference trajectories with bounded time derivatives. The formation control design is based on several nonlinear coordinate changes, the transverse function approach, the backstepping technique, the Lyapunov direct method, and smooth or p??times differentiable step functions. These functions are introduced and incorporated into novel potential functions to solve the collision avoidance problem without the need of switchings despite of the robots' limited sensing ranges. The proposed formation control system is applied to solve a gradient climbing problem.

Explaining Crossing DIF in Polytomous Items Using Differential Step Functioning Effects

Crossing, or intersecting, differential item functioning (DIF) is a form of nonuniform DIF that exists when the sign of the between-group difference in expected item performance changes across the latent trait continuum. The presence of crossing DIF presents a problem for many statistics developed for evaluating DIF because positive and negative conditional between-group differences in expected item score can cancel to yield a negligible between-group difference when aggregated across the latent trait continuum. As a result, many DIF detection statistics are insensitive to DIF that is crossing in nature. Currently, little is known about why crossing DIF occurs in polytomous items. This study demonstrates that crossing DIF in polytomous items can be explained by the presence of differential step functioning (DSF) effects that vary in sign across the steps of the polytomous item, suggesting that crossing DIF is caused by the presence of one or more secondary factors that yield a relative advantage for different groups at different steps of the polytomous item. Using this result, this study describes how to use the pattern of the DSF effects to (a) target the search for secondary traits responsible for a crossing DIF effect, (b) identify situations where certain DIF statistics will lead to the conclusion of no DIF despite a substantial crossing DIF effect, and (c) generate realistic crossing DIF effects in simulation research.

Distinguishing between Net and Global DIF in Polytomous Items

In this article, I address two competing conceptions of differential item functioning (DIF) in polytomously scored items. The first conception, referred to as net DIF, concerns between‐group differences in the conditional expected value of the polytomous response variable. The second conception, referred to as global DIF, concerns the conditional dependence of group membership and the polytomous response variable. The distinction between net and global DIF is important because different DIF evaluation methods are appropriate for net and global DIF; no currently available method is universally the best for detecting both net and global DIF. Net and global DIF definitions are presented under two different, yet compatible, modeling frameworks: a traditional item response theory (IRT) framework, and a differential step functioning (DSF) framework. The theoretical relationship between the IRT and DSF frameworks is presented. Available methods for evaluating net and global DIF are described, and an applied example of net and global DIF is presented.

Control of nonlinear systems with output tracking error constraints and its application to magnetic bearings

A constructive method is presented to design controllers that force the output of nonlinear systems in a strict feedback form to track a bounded and sufficiently smooth reference trajectory asymptotically. Under a suitable condition on the initial output tracking error, the proposed controllers guarantee the output tracking error within a symmetric or an asymmetric pre-specified limit range, and boundedness of all signals of the closed-loop system. A transformation is introduced to take care of the output tracking error constraint. Smooth and/or p-times differentiable step functions are proposed and incorporated in the output tracking error transformation to overcome difficulties due to the asymmetric limit range on the output tracking error. As a result, there are no switchings in the proposed controllers despite the asymmetric limit range. The proposed control design is then applied to design a tracking controller for active magnetic bearings as an illustrating application.