Asymptotic bias of normal-distribution-based maximum likelihood estimates of moderation effects with data missing at random

Moderation analysis is useful for addressing interesting research questions in social sciences and behavioural research. In practice, moderated multiple regression (MMR) models have been most widely used. However, missing data pose a challenge, mainly because the interaction term is a product of two or more variables and thus is a non-linear function of the involved variables. Normal-distribution-based maximum likelihood (NML) has been proposed and applied for estimating MMR models with incomplete data. When data are missing completely at random, moderation effect estimates are consistent. However, simulation results have found that when data in the predictor are missing at random (MAR), NML can yield inaccurate estimates of moderation effects when the moderation effects are non-null. Simulation studies are subject to the limitation of confounding systematic bias with sampling errors. Thus, the purpose of this paper is to analytically derive asymptotic bias of NML estimates of moderation effects with MAR data. Results show that when the moderation effect is zero, there is no asymptotic bias in moderation effect estimates with either normal or non-normal data. When the moderation effect is non-zero, however, asymptotic bias may exist and is determined by factors such as the moderation effect size, missing-data proportion, and type of missingness dependence. Our analytical results suggest that researchers should apply NML to MMR models with caution when missing data exist. Suggestions are given regarding moderation analysis with missing data.     

两种新的计算机化自适应测验在线标定方法

在线标定技术由于具有诸多优点而被广泛应用于计算机化自适应测验(CAT)的新题标定。Method A是想法最直接、算法最简单的CAT在线标定方法, 但它具有明显的理论缺陷--在标定过程中将能力估计值视为能力真值。将全功能极大似然估计方法(FFMLE)与“利用充分性结果”估计方法(ECSE)的误差校正思路融入Method A (新方法分别记为FFMLE-Method A和ECSE-Method A), 从理论上对能力估计误差进行校正, 进而克服Method A的标定缺陷。模拟研究的结果表明：(1)在大多数实验条件下, 两种新方法较Method A总体上可以改进标定精度, 且在测验长度为10的短测验上的改进幅度最大; (2)当CAT测验长度较短或中等(10或20题)时, 两种新方法的表现与性能最优的MEM已非常接近。当测验长度较长(30题)时, ECSE-Method A的总体表现最好、优于MEM; (3)样本量越大, 各种方法的标定精度越高。     

Balance Performance During Perturbed Standing Is Not Associated With Muscle Strength and Power in Young Adults

The authors investigate the ways in which varied postural responses to translating platform perturbations are associated with the variables of strength and power. Twenty-four physically active and 27 sedentary young adults were exposed to a set of postural perturbations at varied velocities (10 and 20 cm/s) and the respective accelerations (6.4 and 6.9 m/s²), constant distance (6 cm), and 4 directions of platform motion (forward, backward, left-lateral, and right-lateral). They also performed maximum voluntary isometric contraction (MVC) and chair rising/chair jumping tests. The analysis of variance revealed significant interaction effect for peak center of pressure displacement, direction by velocity: F_3,129 = 24.43, p = .002; and direction by acceleration: F_3,129 = 34.18, p = .001. There were no significant correlations between peak center of pressure displacements and peak force and peak rate of force development measured during MVC in either standing (r = .27–57) or sitting positions (r = .12–51) and peak power during chair jumping (r = .47–.59) in all participants. As such, only a small proportion of variance was explained (9–39%, 3–23%, and 23–41%, respectively). In conclusion, interaction effects indicate that the composition of stimuli strongly influences compensatory responses and this effect is more pronounced in sedentary than in physically active young adults. Nevertheless, the dynamic balance is not associated with muscle strength and power in either group.     

Profile-likelihood Confidence Intervals in Item Response Theory Models

Confidence intervals (CIs) are fundamental inferential devices which quantify the sampling variability of parameter estimates. In item response theory, CIs have been primarily obtained from large-sample Wald-type approaches based on standard error estimates, derived from the observed or expected information matrix, after parameters have been estimated via maximum likelihood. An alternative approach to constructing CIs is to quantify sampling variability directly from the likelihood function with a technique known as profile-likelihood confidence intervals (PL CIs). In this article, we introduce PL CIs for item response theory models, compare PL CIs to classical large-sample Wald-type CIs, and demonstrate important distinctions among these CIs. CIs are then constructed for parameters directly estimated in the specified model and for transformed parameters which are often obtained post-estimation. Monte Carlo simulation results suggest that PL CIs perform consistently better than Wald-type CIs for both non-transformed and transformed parameters.     

LGM模型中缺失数据处理方法的比较:ML方法与Diggle-Kenward选择模型

追踪研究中缺失数据十分常见。本文通过Monte Carlo模拟研究,考察基于不同前提假设的Diggle-Kenward选择模型和ML方法对增长参数估计精度的差异,并考虑样本量、缺失比例、目标变量分布形态以及不同缺失机制的影响。结果表明:(1)缺失机制对基于MAR的ML方法有较大的影响,在MNAR缺失机制下,基于MAR的ML方法对LGM模型中截距均值和斜率均值的估计不具有稳健性。(2)DiggleKenward选择模型更容易受到目标变量分布偏态程度的影响,样本量与偏态程度存在交互作用,样本量较大时,偏态程度的影响会减弱。而ML方法仅在MNAR机制下轻微受到偏态程度的影响。     

Item Response Theory with Estimation of the Latent Population Distribution Using Spline-Based Densities

The purpose of this paper is to introduce a new method for fitting item response theory models with the latent population distribution estimated from the data using splines. A spline-based density estimation system provides a flexible alternative to existing procedures that use a normal distribution, or a different functional form, for the population distribution. A simulation study shows that the new procedure is feasible in practice, and that when the latent distribution is not well approximated as normal, two-parameter logistic (2PL) item parameter estimates and expected a posteriori scores (EAPs) can be improved over what they would be with the normal model. An example with real data compares the new method and the extant empirical histogram approach.     

Loss of Information in Estimating Item Parameters in Incomplete Designs

In this paper, the efficiency of conditional maximum likelihood (CML) and marginal maximum likelihood (MML) estimation of the item parameters of the Rasch model in incomplete designs is investigated. The use of the concept of F-information (Eggen, 2000) is generalized to incomplete testing designs. The scaled determinant of the F-information matrix is used as a scalar measure of information contained in a set of item parameters. In this paper, the relation between the normalization of the Rasch model and this determinant is clarified. It is shown that comparing estimation methods with the defined information efficiency is independent of the chosen normalization. The generalization of the method to other models than the Rasch model is discussed. In examples, information comparisons are conducted. It is found that for both CML and MML some information is lost in all incomplete designs compared to complete designs. A general result is that with increasing test booklet length the efficiency of an incomplete design, compared to a complete design, is increasing, as is the efficiency of CML compared to MML. The main difference between CML and MML is seen in the effect of the length of the test booklet. It will be demonstrated that with very small booklets, there is a substantial loss in information (about 35%) with CML estimation, while this loss is only about 10% in MML estimation. However, with increasing test length, the differences between CML and MML quickly disappear.     

An Adaptive Algorithm for Pairwise Comparison‐based Preference Measurement

The Pairwise Comparison‐based Preference Measurement (PCPM) approach has been proposed for products featuring a large number of attributes. In the PCPM framework, a static two‐cyclic design is used to reduce the number of pairwise comparisons. However, adaptive questioning routines that maximize the information gained from pairwise comparisons promise to further increase the efficiency of this approach. This paper introduces a new adaptive algorithm for PCPM, which accounts for several response errors. The suggested approach is compared with an adaptive algorithm that was proposed for the Analytic Hierarchy Process as well as a random selection of pairwise comparisons. By means of Monte Carlo simulations, we quantify the extent to which the adaptive selection of pairwise comparisons increases the efficiency of the respective approach. Copyright © 2011 John Wiley & Sons, Ltd.     

A Statistical Hypothesis Testing Method for the Rank Ordering of the Priorities of the Alternatives in the Analytic Hierarchy Process

In analytic hierarchy process (AHP), a ratio scale (π₁, π₂, ⋯, π_t) for the priorities of the alternatives {T₁, T₂, ⋯, T_t} is used for a decision problem in which π_i/π_j is used to quantify the ratio of the priority of T_i to that of T_j. In a group decision‐making setup, the subjective estimates of π_i/π_j are obtained as entries of a pairwise comparison matrix for each member of the group. On the basis of these pairwise comparison matrices, one of the topics of interest in some situation is the total rank ordering of the priorities of the alternatives. In this article, a statistical method is proposed for testing a specific total rank ordering of the priorities of the alternatives. The method developed is then illustrated using numerical examples. Copyright © 2014 John Wiley & Sons, Ltd.     

Parent-child interaction during the first year of life in infants at elevated likelihood of autism spectrum disorder

Autism spectrum disorder (ASD) likely emerges from a complex interaction between pre-existing neurodevelopmental vulnerabilities and the environment. The interaction with parents forms a key aspect of an infant’s social environment, but few prospective studies of infants at elevated likelihood (EL) for ASD (who have an older sibling with ASD) have examined parent-child interactions in the first year of life. As part of a European multisite network, parent-child dyads of free play were observed at 5 months (62 EL infants, 47 infants at typical likelihood (TL)) and 10 months (101 EL siblings, 77 TL siblings). The newly-developed Parent-Infant/Toddler Coding of Interaction (PInTCI) scheme was used, focusing on global characteristics of infant and parent behaviors. Coders were blind to participant information. Linear mixed model analyses showed no significant group differences in infant or parent behaviors at 5 or 10 months of age (all ps≥0.09, d≤0.36), controlling for infant’s sex and age, and parental educational level. However, without adjustments, EL infants showed fewer and less clear initiations at 10 months than TL infants (p = 0.02, d = 0.44), but statistical significance was lost after controlling for parental education (p = 0.09, d = 0.36), which tended to be lower in the EL group. Consistent with previous literature focusing on parent-infant dyads, our findings suggest that differences between EL and TL dyads may only be subtle during the first year of life. We discuss possible explanations and implications for future developmental studies.