Testing Latent Variable Distribution Fit in IRT Using Posterior Residuals

2020 ◽  
pp. 107699862095376
Author(s):  
Scott Monroe
Keyword(s):  
Latent Variable ◽  
Latent Trait ◽  
Item Parameter ◽  
Parameter Estimates ◽  
Fit Statistics ◽  
Item Fit ◽  
Sample Average ◽  
Variable Distribution ◽  
Standard Normal ◽  
Distribution Fit

This research proposes a new statistic for testing latent variable distribution fit for unidimensional item response theory (IRT) models. If the typical assumption of normality is violated, then item parameter estimates will be biased, and dependent quantities such as IRT score estimates will be adversely affected. The proposed statistic compares the specified latent variable distribution to the sample average of latent variable posterior distributions commonly used in IRT scoring. Formally, the statistic is an instantiation of a generalized residual and is thus asymptotically distributed as standard normal. Also, the statistic naturally complements residual-based item-fit statistics, as both are conditional on the latent trait, and can be presented with graphical plots. In addition, a corresponding unconditional statistic, which controls for multiple comparisons, is proposed. The statistics are evaluated using a simulation study, and empirical analyses are provided.

scholarly journals Summed Score Likelihood–Based Indices for Testing Latent Variable Distribution Fit in Item Response Theory

2017 ◽  
Vol 78 (5) ◽  
pp. 857-886 ◽  
Author(s):  
Zhen Li ◽  
Li Cai
Keyword(s):  
Item Response Theory ◽  
Item Response ◽  
Latent Variable ◽  
Item Parameter ◽  
Parameter Estimates ◽  
Test Statistics ◽  
Response Theory ◽  
Patient Reported ◽  
Variable Distribution ◽  
Distribution Fit

In standard item response theory (IRT) applications, the latent variable is typically assumed to be normally distributed. If the normality assumption is violated, the item parameter estimates can become biased. Summed score likelihood–based statistics may be useful for testing latent variable distribution fit. We develop Satorra–Bentler type moment adjustments to approximate the test statistics’ tail-area probability. A simulation study was conducted to examine the calibration and power of the unadjusted and adjusted statistics in various simulation conditions. Results show that the proposed indices have tail-area probabilities that can be closely approximated by central chi-squared random variables under the null hypothesis. Furthermore, the test statistics are focused. They are powerful for detecting latent variable distributional assumption violations, and not sensitive (correctly) to other forms of model misspecification such as multidimensionality. As a comparison, the goodness-of-fit statistic M2 has considerably lower power against latent variable nonnormality than the proposed indices. Empirical data from a patient-reported health outcomes study are used as illustration.

Examining Nonnormal Latent Variable Distributions for Non-Ignorable Missing Data

2021 ◽  
Vol 45 (3) ◽  
pp. 159-177
Author(s):  
Chen-Wei Liu
Keyword(s):  
Missing Data ◽  
Latent Variable ◽  
Bivariate Normal Distribution ◽  
Parameter Estimates ◽  
Missing Responses ◽  
Variable Distribution ◽  
Item Responses ◽  
Nonnormal Distribution ◽  
The Impact ◽  
Biased Estimates

Missing not at random (MNAR) modeling for non-ignorable missing responses usually assumes that the latent variable distribution is a bivariate normal distribution. Such an assumption is rarely verified and often employed as a standard in practice. Recent studies for “complete” item responses (i.e., no missing data) have shown that ignoring the nonnormal distribution of a unidimensional latent variable, especially skewed or bimodal, can yield biased estimates and misleading conclusion. However, dealing with the bivariate nonnormal latent variable distribution with present MNAR data has not been looked into. This article proposes to extend unidimensional empirical histogram and Davidian curve methods to simultaneously deal with nonnormal latent variable distribution and MNAR data. A simulation study is carried out to demonstrate the consequence of ignoring bivariate nonnormal distribution on parameter estimates, followed by an empirical analysis of “don’t know” item responses. The results presented in this article show that examining the assumption of bivariate nonnormal latent variable distribution should be considered as a routine for MNAR data to minimize the impact of nonnormality on parameter estimates.

Detection and Treatment of Careless Responses to Improve Item Parameter Estimation

2019 ◽  
Vol 44 (3) ◽  
pp. 309-341 ◽  
Author(s):  
Jeffrey M. Patton ◽  
Ying Cheng ◽  
Maxwell Hong ◽  
Qi Diao
Keyword(s):  
Parameter Estimation ◽  
Location Parameter ◽  
Item Parameter ◽  
Parameter Estimates ◽  
False Positive Error ◽  
Fit Statistics ◽  
Calibration Sample ◽  
Higher Power ◽  
Item Parameter Estimation ◽  
False Positive Error Rate

In psychological and survey research, the prevalence and serious consequences of careless responses from unmotivated participants are well known. In this study, we propose to iteratively detect careless responders and cleanse the data by removing their responses. The careless responders are detected using person-fit statistics. In two simulation studies, the iterative procedure leads to nearly perfect power in detecting extremely careless responders and much higher power than the noniterative procedure in detecting moderately careless responders. Meanwhile, the false-positive error rate is close to the nominal level. In addition, item parameter estimation is much improved by iteratively cleansing the calibration sample. The bias in item discrimination and location parameter estimates is substantially reduced. The standard error estimates, which are spuriously small in the presence of careless responses, are corrected by the iterative cleansing procedure. An empirical example is also presented to illustrate the proposed procedure. These results suggest that the proposed procedure is a promising way to improve item parameter estimation for tests of 20 items or longer when data are contaminated by careless responses.

scholarly journals Identifying Effortful Individuals With Mixture Modeling Response Accuracy and Response Time Simultaneously to Improve Item Parameter Estimation

2020 ◽  
Vol 80 (4) ◽  
pp. 775-807
Author(s):  
Yue Liu ◽  
Ying Cheng ◽  
Hongyun Liu
Keyword(s):  
Parameter Estimation ◽  
Response Time ◽  
Mixture Model ◽  
Response Times ◽  
Latent Trait ◽  
Item Parameter ◽  
Parameter Estimates ◽  
Response Accuracy ◽  
Parameter Recovery ◽  
Item Parameter Estimation

The responses of non-effortful test-takers may have serious consequences as non-effortful responses can impair model calibration and latent trait inferences. This article introduces a mixture model, using both response accuracy and response time information, to help differentiating non-effortful and effortful individuals, and to improve item parameter estimation based on the effortful group. Two mixture approaches are compared with the traditional response time mixture model (TMM) method and the normative threshold 10 (NT10) method with response behavior effort criteria in four simulation scenarios with regard to item parameter recovery and classification accuracy. The results demonstrate that the mixture methods and the TMM method can reduce the bias of item parameter estimates caused by non-effortful individuals, with the mixture methods showing more advantages when the non-effort severity is high or the response times are not lognormally distributed. An illustrative example is also provided.

IRT and MIRT Models for Item Parameter Estimation With Multidimensional Multistage Tests

2019 ◽  
Vol 45 (4) ◽  
pp. 383-402
Author(s):  
Paul A. Jewsbury ◽  
Peter W. van Rijn
Keyword(s):  
Item Response Theory ◽  
Item Response ◽  
Latent Variable ◽  
Large Scale ◽  
Real Data ◽  
Item Parameter ◽  
Practical Reasons ◽  
Parameter Estimates ◽  
Response Theory ◽  
Item Parameter Estimates

In large-scale educational assessment data consistent with a simple-structure multidimensional item response theory (MIRT) model, where every item measures only one latent variable, separate unidimensional item response theory (UIRT) models for each latent variable are often calibrated for practical reasons. While this approach can be valid for data from a linear test, unacceptable item parameter estimates are obtained when data arise from a multistage test (MST). We explore this situation from a missing data perspective and show mathematically that MST data will be problematic for calibrating multiple UIRT models but not MIRT models. This occurs because some items that were used in the routing decision are excluded from the separate UIRT models, due to measuring a different latent variable. Both simulated and real data from the National Assessment of Educational Progress are used to further confirm and explore the unacceptable item parameter estimates. The theoretical and empirical results confirm that only MIRT models are valid for item calibration of multidimensional MST data.

scholarly journals Plausible-Value Imputation Statistics for Detecting Item Misfit

2017 ◽  
Vol 41 (5) ◽  
pp. 372-387 ◽  
Author(s):  
R. Philip Chalmers ◽  
Victoria Ng
Keyword(s):  
Error Detection ◽  
Null Hypothesis ◽  
Latent Trait ◽  
Parametric Bootstrap ◽  
Detection Rates ◽  
Bootstrap Procedure ◽  
Fit Statistics ◽  
Item Fit ◽  
Secondary Analyses ◽  
Simulation Results

When tests consist of a small number of items, the use of latent trait estimates for secondary analyses is problematic. One area in particular where latent trait estimates have been problematic is when testing for item misfit. This article explores the use of plausible-value imputations to lessen the severity of the inherent measurement unreliability in shorter tests, and proposes a parametric bootstrap procedure to generate empirical sampling characteristics for null-hypothesis tests of item fit. Simulation results suggest that the proposed item-fit statistics provide conservative to nominal error detection rates. Power to detect item misfit tended to be less than Stone’s [Formula: see text] item-fit statistic but higher than the [Formula: see text] statistic proposed by Orlando and Thissen, especially in tests with 20 or more dichotomously scored items.

scholarly journals Asymptotically Corrected Person Fit Statistics for Multidimensional Constructs with Simple Structure and Mixed Item Types

2020 ◽  
Author(s):  
Maxwell Hong ◽  
Lizhen Lin ◽  
Alison Cheng
Keyword(s):  
Item Response ◽  
Latent Trait ◽  
Standard Normal Distribution ◽  
Item Response Model ◽  
Person Fit ◽  
Item Response Models ◽  
Fit Statistics ◽  
Standard Normal ◽  
Work Done ◽  
Item Types

Person fit statistics are frequently used to detect deviating behavior when assuming an item response model generated the data. A common statistic, $l_z$, has been shown in previous studies to perform well under a myriad of conditions. However, it is well known that $l_z$ does not follow a standard normal distribution when using an estimated latent trait. As a result, corrections of $l_z$, called $l_z^*$, have been proposed in the literature for specific item response models. We propose a more general correction that is applicable to many types of data, namely survey or tests with multiple item types and underlying latent constructs, which subsumes previous work done by others. In addition, we provide corrections for multiple estimators of $\theta$, the latent trait, including MLE, MAP and WLE. We provide analytical derivations that justifies our proposed correction, as well as simulation studies to examine the performance of the proposed correction with finite test lengths. An applied example is also provided to demonstrate proof of concept. We conclude with recommendations for practitioners when the asymptotic correction works well under different conditions and also future directions.

scholarly journals Interpretable Variational Graph Autoencoder with Noninformative Prior

2021 ◽  
Vol 13 (2) ◽  
pp. 51
Author(s):  
Lili Sun ◽  
Xueyan Liu ◽  
Min Zhao ◽  
Bo Yang
Keyword(s):  
Latent Variables ◽  
Latent Variable ◽  
Expert Knowledge ◽  
Structural Information ◽  
Standard Normal Distribution ◽  
Noninformative Prior ◽  
Latent Space ◽  
Distribution Parameters ◽  
Standard Normal ◽  
Low Dimensional

Variational graph autoencoder, which can encode structural information and attribute information in the graph into low-dimensional representations, has become a powerful method for studying graph-structured data. However, most existing methods based on variational (graph) autoencoder assume that the prior of latent variables obeys the standard normal distribution which encourages all nodes to gather around 0. That leads to the inability to fully utilize the latent space. Therefore, it becomes a challenge on how to choose a suitable prior without incorporating additional expert knowledge. Given this, we propose a novel noninformative prior-based interpretable variational graph autoencoder (NPIVGAE). Specifically, we exploit the noninformative prior as the prior distribution of latent variables. This prior enables the posterior distribution parameters to be almost learned from the sample data. Furthermore, we regard each dimension of a latent variable as the probability that the node belongs to each block, thereby improving the interpretability of the model. The correlation within and between blocks is described by a block–block correlation matrix. We compare our model with state-of-the-art methods on three real datasets, verifying its effectiveness and superiority.

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