Approximating Test Statistics Using Eigenvalue Block Averaging

We introduce and evaluate a new class of approximations to common test statistics in structural equation modeling. Such test statistics asymptotically follow the distribution of a weighted sum of i.i.d. chi-square variates, where the weights are eigenvalues of a certain matrix. The proposed eigenvalue block averaging (EBA) method involves creating blocks of these eigenvalues and replacing them within each block with the block average. The Satorra–Bentler scaling procedure is a special case of this framework, using one single block. The proposed procedure applies also to difference testing among nested models. We investigate the EBA procedure both theoretically in the asymptotic case, and with simulation studies for the finite-sample case, under both maximum likelihood and diagonally weighted least squares estimation. Comparison is made with 3 established approximations: Satorra–Bentler, the scaled and shifted, and the scaled F tests.     

Δ2 and χ2 I2 fit indices for structural equation models: A brief note of clarification

Marsh and Balla (1986) and Marsh, Balla, and McDonald (1988) proposed an index of fit called χ²I2, but McDonald and Marsh (1990) subsequently demonstrated that the index is biased and recommended that it not be used. Bollen (1989) independently proposed Δ2 which is the same as χ²I2 (hereafter referred to as χ²I2‐Δ2), indicating that it adjusts for sample size and degrees of freedom (df). Gerbing and Anderson (1992), apparently based on the assumption that the χ²I2‐Δ2 index is unbiased and appropriately corrects for df (penalizes a lack of parsimony), recommended its use, and the index is routinely presented by major computer programs (e.g., EQS and LISREL 8). However, a more critical evaluation of the χ²I2‐Δ2 index reveals that: (a) it is systematically biased (i.e., its value varies systematically with N) although the size of the bias may be small; (b) the adjustment for df is inappropriate in that it penalizes model parsimony instead of model complexity; and (c) the inappropriate penalty for model parsimony is larger for small N. Because of these undesirable properties, the χ²I2‐Δ2 index is not recommended for routine use.     

Pairwise deletion for missing data in structural equation models: Nonpositive definite matrices,parameter estimates,goodness of fit,and adjusted sample sizes

The use of sample covariance matrices constructed with pairwise deletion for data missing completely at random (S_PW) is addressed in a simulation study based on 3 sample sizes (n = 200, 500, 1,000) and 5 levels of missing data (%miss = 0, 1, 10, 25, and 50). Parameter estimates were unbiased, parameter variability was largely explicable in terms of the number of nonmissing cases, and no sample covariance matrices were nonpositive definite except when %miss was 50 and the sample size was 200. However, nominal χ² test statistics (and, thus, fit indices based on χ²s) were substantially biased by %miss and its interaction with N. Corrected χ²s based on the minimum, mean, and maximum number of nonmissing cases per measured variables and cases per covariance term (NPC) reduced but did not eliminate the bias. Empirically derived power functions did substantially better but may not generalize to other situations. Whereas the minimum NPC (the default in the SPSS version of LISREL) is probably better than most simple alternatives in many applications, the problem of how to assess fit for models fit to S_PWS has no simple solution; caution is recommended, and there is need for further research with more suitable methods for this problem.     

Item Selection and Ability Estimation Procedures for a Mixed-Format Adaptive Test

In this study we compared five item selection procedures using three ability estimation methods in the context of a mixed-format adaptive test based on the generalized partial credit model. The item selection procedures used were maximum posterior weighted information, maximum expected information, maximum posterior weighted Kullback-Leibler information, and maximum expected posterior weighted Kullback-Leibler information procedures. The ability estimation methods investigated were maximum likelihood estimation (MLE), weighted likelihood estimation (WLE), and expected a posteriori (EAP). Results suggested that all item selection procedures, regardless of the information functions on which they were based, performed equally well across ability estimation methods. The principal conclusions drawn about the ability estimation methods are that MLE is a practical choice and WLE should be considered when there is a mismatch between pool information and the population ability distribution. EAP can serve as a viable alternative when an appropriate prior ability distribution is specified. Several implications of the findings for applied measurement are discussed.     

Evaluation of a New Mean Scaled and Moment Adjusted Test Statistic for SEM

Recently a new mean scaled and skewness adjusted test statistic was developed for evaluating structural equation models in small samples and with potentially nonnormal data, but this statistic has received only limited evaluation. The performance of this statistic is compared to normal theory maximum likelihood and 2 well-known robust test statistics. A modification to the Satorra–Bentler scaled statistic is developed for the condition that sample size is smaller than degrees of freedom. The behavior of the 4 test statistics is evaluated with a Monte Carlo confirmatory factor analysis study that varies 7 sample sizes and 3 distributional conditions obtained using Headrick's fifth-order transformation to nonnormality. The new statistic performs badly in most conditions except under the normal distribution. The goodness-of-fit χ² test based on maximum-likelihood estimation performed well under normal distributions as well as under a condition of asymptotic robustness. The Satorra–Bentler scaled test statistic performed best overall, whereas the mean scaled and variance adjusted test statistic outperformed the others at small and moderate sample sizes under certain distributional conditions.     

Behavior of Asymptotically Distribution Free Test Statistics in Covariance Versus Correlation Structure Analysis

The asymptotically distribution free (ADF) method is often used to estimate parameters or test models without a normal distribution assumption on variables, both in covariance structure analysis and in correlation structure analysis. However, little has been done to study the differences in behaviors of the ADF method in covariance versus correlation structure analysis. The behaviors of 3 test statistics frequently used to evaluate structural equation models with nonnormally distributed variables, χ² test T_AGLS and its small-sample variants T_YB and T_F(AGLS) were compared. Results showed that the ADF method in correlation structure analysis with test statistic T_AGLS performs much better at small sample sizes than the corresponding test for covariance structures. In contrast, test statistics T_YB and T_F(AGLS) under the same conditions generally perform better with covariance structures than with correlation structures. It is proposed that excessively large and variable condition numbers of weight matrices are a cause of poor behavior of ADF test statistics in small samples, and results showed that these condition numbers are systematically increased with substantial increase in variance as sample size decreases. Implications for research and practice are discussed.     

Precise asymptotics in the law of the logarithm for random fields in Hilbert space

Consider the positive d-dimensional lattice d (d≥2) with partial ordering ≤, let {XK; K ∈ d } be i.i.d. random vari- ables taking values in a real separable Hilbert space (H, ||·||) with mean zero and covariance operator ∑, and set partial sums SN =∑K≤NXK, K , N ∈ d . Under some moment conditions, we obtain the precise asymptotics of a kind of weighted infinite series for partial sums SN as ε 0 by using the truncation and approximation methods. The results are related to the convergence rates of the law of the logarithm in Hilbert space, and they also extend the results of (Gut and Spǎtaru, 2003).     

Improving the Root Mean Square Error of Approximation for Nonnormal Conditions in Structural Equation Modeling

In this study, the authors investigated incorporating adjusted model fit information into the root mean square error of approximation (RMSEA) fit index. Through Monte Carlo simulation, the usefulness of this adjusted index was evaluated for assessing model adequacy in structural equation modeling when the multivariate normality assumption underlying maximum likelihood estimation is violated. Adjustment to the RMSEA was considered in 2 forms: a rescaling adjustment via the Satorra-Bentler rescaled goodness-of-fit statistic and a bootstrap adjustment via the Bollen and Stine adjusted model p value. Both properly specified and misspecifed models were examined. The adjusted RMSEA was evaluated in terms of the average index value across study conditions and with respect to model rejection rates under tests of exact fit, close fit, and not-close fit.     

Evaluating Model Fit With Ordered Categorical Data Within a Measurement Invariance Framework: A Comparison of Estimators

A paucity of research has compared estimation methods within a measurement invariance (MI) framework and determined if research conclusions using normal-theory maximum likelihood (ML) generalizes to the robust ML (MLR) and weighted least squares means and variance adjusted (WLSMV) estimators. Using ordered categorical data, this simulation study aimed to address these queries by investigating 342 conditions. When testing for metric and scalar invariance, Δχ² results revealed that Type I error rates varied across estimators (ML, MLR, and WLSMV) with symmetric and asymmetric data. The Δχ² power varied substantially based on the estimator selected, type of noninvariant indicator, number of noninvariant indicators, and sample size. Although some the changes in approximate fit indexes (ΔAFI) are relatively sample size independent, researchers who use the ΔAFI with WLSMV should use caution, as these statistics do not perform well with misspecified models. As a supplemental analysis, our results evaluate and suggest cutoff values based on previous research.     

The International Platform of Sexuality & Relationship Education

This study was an investigation of the additional risk conferred by the experience of psychogenic amnesia for memories of childhood sexual abuse (CSA) on the likelihood of becoming a victim of sexual assault in later life. A total of 210 community respondents completed a retrospective web-based trauma survey. The majority of respondents were female (74.3%) and their ages ranged from 16 to 65 years, with a mean age of 33 years. Chi-squared analysis revealed that survivors of CSA demonstrated significantly greater risk (58%, χ^2 = 44.461, p = 0.0005) of experiencing sexual assault in adolescence in comparison with their non-abused counterparts (13%). Furthermore, survivors who reported having been amnesic for their abuse-related memories demonstrated a higher rate of adolescent revictimisation (86%) than survivors who had retained continuous memories of their victimisation (48%, χ^2 = 8.626, p = 0.003). Overall, once-amnesic survivors of CSA demonstrated 6.6 times the risk of sexual assault and an eight-fold risk for rape during adolescence in comparison with their non-abused counterparts. It is proposed that the elevated risk conferred by amnesia for CSA might be mediated by two distinct pathways, both of which are associated with the use of dissociation as a defence mechanism.     

Maximum Likelihood Estimation: for the Multinomial Distribution

In a first course in mathematical statistics or categorical data analysis, the maximum likelihood estimators of the parameters of the multinomial distribution are often “derived” using calculus. However, the important step of verifying that the resulting estimators indeed maximize the likelihood is omitted or done incorrectly. In this paper, two simple methods of obtaining the maximum likelihood estimates are presented.     

修正的q-Stancu—Beta算子的加权统计逼近性质

基于q-整数的定义,引入了一类修正的q-Stancu—Beta算子,由算子的各阶矩量、中心矩量及加权光滑模结合统计逼近的概念,得到了算子的加权统计收敛定理,并研究了算子的局部逼近性质,得到了算子的局部逼近定理．     

Effect Size Measures for Differential Item Functioning in a Multidimensional IRT Model

This study adapted an effect size measure used for studying differential item functioning (DIF) in unidimensional tests and extended the measure to multidimensional tests. Two effect size measures were considered in a multidimensional item response theory model: signed weighted P‐difference and unsigned weighted P‐difference. The performance of the effect size measures was investigated under various simulation conditions including different sample sizes and DIF magnitudes. As another way of studying DIF, the χ² difference test was included to compare the result of statistical significance (statistical tests) with that of practical significance (effect size measures). The adequacy of existing effect size criteria used in unidimensional tests was also evaluated. Both effect size measures worked well in estimating true effect sizes, identifying DIF types, and classifying effect size categories. Finally, a real data analysis was conducted to support the simulation results.     

A Comparison of Item Fit Statistics for Mixed IRT Models

In this study we examined procedures for assessing model-data fit of item response theory (IRT) models for mixed format data. The model fit indices used in this study include PARSCALE's G² , Orlando and Thissen's S − X² and S − G² , and Stone's χ²* and G²* . To investigate the relative performance of the fit statistics at the item level, we conducted two simulation studies: Type I error and power studies. We evaluated the performance of the item fit indices for various conditions of test length, sample size, and IRT models. Among the competing measures, the summed score-based indices S − X² and S − G² were found to be the sensible and efficient choice for assessing model fit for mixed format data. These indices performed well, particularly with short tests. The pseudo-observed score indices, χ²* and G²* , showed inflated Type I error rates in some simulation conditions. Consistent with the findings of current literature, the PARSCALE's G² index was rarely useful, although it provided reasonable results for long tests.     

Confirmatory Factor Analysis of Ordinal Variables With Misspecified Models

Ordinal variables are common in many empirical investigations in the social and behavioral sciences. Researchers often apply the maximum likelihood method to fit structural equation models to ordinal data. This assumes that the observed measures have normal distributions, which is not the case when the variables are ordinal. A better approach is to use polychoric correlations and fit the models using methods such as unweighted least squares (ULS), maximum likelihood (ML), weighted least squares (WLS), or diagonally weighted least squares (DWLS). In this simulation evaluation we study the behavior of these methods in combination with polychoric correlations when the models are misspecified. We also study the effect of model size and number of categories on the parameter estimates, their standard errors, and the common chi-square measures of fit when the models are both correct and misspecified. When used routinely, these methods give consistent parameter estimates but ULS, ML, and DWLS give incorrect standard errors. Correct standard errors can be obtained for these methods by robustification using an estimate of the asymptotic covariance matrix W of the polychoric correlations. When used in this way the methods are here called RULS, RML, and RDWLS.     

关于考试成绩评估的一种思路

利用统计抽样方法绘制学生成绩直方图,采用极大似然估计法对学生考试成绩正态分布的参数进行估计,并运用χ2检验法检验该样本正态分布模型与实际情况的拟合程度;利用正态分布的性质和分布假设检验,对一次真实的考试成绩进行了检验分析,并利用Visual Basic语言编程实现了计算机软件的自动分析,得到了考试成绩评估的一般方法和结论。     

Kantorovich型q-Szász-Mirakjan-Schurer算子的加权统计逼近性质

基于q-积分的概念,研究了一类q-Szász-Mirakjan-Schurer算子的逼近性质,通过计算得到算子的各阶矩量及中心矩量,通过引入统计收敛定义得到算子的加权统计收敛定理,并通过加权光滑模研究了算子的局部逼近性质,得到了算子的局部逼近定理。     

A limit result for self-normalized random sums

Suppose {X, X _n ;n≥1} is a sequence i.i.d.r.v. withEX=0 andEX ²<∞. Shao (1995) proved a conjecture of Révész (1990); ifP(X=±1)=1/2, then

On Turán type inequality with doubling weights andA * weights

LetH _n be the set of real algebraic polynomials of degreen, whose zeros all lie in the interval [−1,1]. The well known Turán type inequalities tell us that forf(x)∈H _n , it holds ‖f′‖≥C√n‖f‖. This note deals with the weighted Turán type inequalities with the weights having inner singularities underL ^p norm for 0<p≤∞. Our results essentially extend the result of Wang and Zhou (2002), and the method used in this paper is simpler and more direct than that of Wang and Zhou (2002). The results and methods have their own values in approximation theory and computation.