Correcting Too Much or Too Little? The Performance of Three Chi-Square Corrections

This simulation study investigates the performance of three test statistics, T₁, T₂, and T₃, used to evaluate structural equation model fit under non normal data conditions. T₁ is the well-known mean-adjusted statistic of Satorra and Bentler. T₂ is the mean-and-variance adjusted statistic of Sattertwaithe type where the degrees of freedom is manipulated. T₃ is a recently proposed version of T₂ that does not manipulate degrees of freedom. Discrepancies between these statistics and their nominal chi-square distribution in terms of errors of Type I and Type II are investigated. All statistics are shown to be sensitive to increasing kurtosis in the data, with Type I error rates often far off the nominal level. Under excess kurtosis true models are generally over-rejected by T₁ and under-rejected by T₂ and T₃, which have similar performance in all conditions. Under misspecification there is a loss of power with increasing kurtosis, especially for T₂ and T₃. The coefficient of variation of the nonzero eigenvalues of a certain matrix is shown to be a reliable indicator for the adequacy of these statistics.     

Ensuring Positiveness of the Scaled Difference Chi-square Test Statistic

A scaled difference test statistic [(T)\tilde]_d\tilde{T}{}_{d} that can be computed from standard software of structural equation models (SEM) by hand calculations was proposed in Satorra and Bentler (Psychometrika 66:507–514, 2001). The statistic [(T)\tilde]_d\tilde{T}_{d} is asymptotically equivalent to the scaled difference test statistic [`(T)]<sub>d</sub>\bar{T}_{d} introduced in Satorra (Innovations in Multivariate Statistical Analysis: A Festschrift for Heinz Neudecker, pp. 233–247, 2000), which requires more involved computations beyond standard output of SEM software. The test statistic [(T)\tilde]<sub>d</sub>\tilde{T}_{d} has been widely used in practice, but in some applications it is negative due to negativity of its associated scaling correction. Using the implicit function theorem, this note develops an improved scaling correction leading to a new scaled difference statistic [`(T)]_d\bar{T}_{d} that avoids negative chi-square values.     

Empirically Corrected Rescaled Statistics for SEM with Small N and Large p

Survey data often contain many variables. Structural equation modeling (SEM) is commonly used in analyzing such data. With typical nonnormally distributed data in practice, a rescaled statistic T_rml proposed by Satorra and Bentler was recommended in the literature of SEM. However, T_rml has been shown to be problematic when the sample size N is small and/or the number of variables p is large. There does not exist a reliable test statistic for SEM with small N or large p, especially with nonnormally distributed data. Following the principle of Bartlett correction, this article develops empirical corrections to T_rml so that the mean of the empirically corrected statistics approximately equals the degrees of freedom of the nominal chi-square distribution. Results show that empirically corrected statistics control type I errors reasonably well even when N is smaller than 2p, where T_rml may reject the correct model 100% even for normally distributed data. The application of the empirically corrected statistics is illustrated via a real data example.     

A scaled difference chi-square test statistic for moment structure analysis

A family of scaling corrections aimed to improve the chi-square approximation of goodness-of-fit test statistics in small samples, large models, and nonnormal data was proposed in Satorra and Bentler (1994). For structural equations models, Satorra-Bentler's (SB) scaling corrections are available in standard computer software. Often, however, the interest is not on the overall fit of a model, but on a test of the restrictions that a null model sayM ₀ implies on a less restricted oneM ₁. IfT ₀ andT ₁ denote the goodness-of-fit test statistics associated toM ₀ andM ₁, respectively, then typically the differenceT _d =T ₀−T ₁ is used as a chi-square test statistic with degrees of freedom equal to the difference on the number of independent parameters estimated under the modelsM ₀ andM ₁. As in the case of the goodness-of-fit test, it is of interest to scale the statisticT _d in order to improve its chi-square approximation in realistic, that is, nonasymptotic and nonormal, applications. In a recent paper, Satorra (2000) shows that the difference between two SB scaled test statistics for overall model fit does not yield the correct SB scaled difference test statistic. Satorra developed an expression that permits scaling the difference test statistic, but his formula has some practical limitations, since it requires heavy computations that are not available in standard computer software. The purpose of the present paper is to provide an easy way to compute the scaled difference chi-square statistic from the scaled goodness-of-fit test statistics of modelsM ₀ andM ₁. A Monte Carlo study is provided to illustrate the performance of the competing statistics. This research was supported by the Spanish grants PB96-0300 and BEC2000-0983, and USPHS grants DA00017 and DA01070.     

Assessing Approximate Fit in Categorical Data Analysis

A family of Root Mean Square Error of Approximation (RMSEA) statistics is proposed for assessing the goodness of approximation in discrete multivariate analysis with applications to item response theory (IRT) models. The family includes RMSEAs to assess the approximation up to any level of association of the discrete variables. Two members of this family are RMSEA₂, which uses up to bivariate moments, and the full information RMSEA_n. The RMSEA₂ is estimated using the M₂ statistic of Maydeu-Olivares and Joe (2005, 2006), whereas for maximum likelihood estimation, RMSEA_n is estimated using Pearson's X² statistic. Using IRT models, we provide cutoff criteria of adequate, good, and excellent fit using the RMSEA₂. When the data are ordinal, we find a strong linear relationship between the RMSEA₂ and the Standardized Root Mean Squared Residual goodness-of-fit index. We are unable to offer cutoff criteria for the RMSEA_n as its population values decrease as the number of variables and categories increase.     

Mean Comparison: Manifest Variable Versus Latent Variable

Mean comparisons are of great importance in the application of statistics. Procedures for mean comparison with manifest variables have been well studied. However, few rigorous studies have been conducted on mean comparisons with latent variables, although the methodology has been widely used and documented. This paper studies the commonly used statistics in latent variable mean modeling and compares them with parallel manifest variable statistics. Our results indicate that, under certain conditions, the likelihood ratio and Wald statistics used for latent mean comparisons do not always have greater power than the Hotelling T² statistics used for manifest mean comparisons. The noncentrality parameter corresponding to the T² statistic can be much greater than those corresponding to the likelihood ratio and Wald statistics, which we find to be different from those provided in the literature. Under a fixed alternative hypothesis, our results also indicate that the likelihood ratio statistic can be stochastically much greater than the corresponding Wald statistic. The robustness property of each statistic is also explored when the model is misspecified or when data are nonnormally distributed. Recommendations and advice are provided for the use of each statistic. The research was supported by NSF grant DMS-0437167 and Grant DA01070 from the National Institute on Drug Abuse. We would like to thank three referees for suggestions that helped in improving the paper.     

The effect of covariate mean differences on the standard error and confidence interval for the comparison of treatment means

The use of covariates is commonly believed to reduce the unexplained error variance and the standard error for the comparison of treatment means, but the reduction in the standard error is neither guaranteed nor uniform over different sample sizes. The covariate mean differences between the treatment conditions can inflate the standard error of the covariate‐adjusted mean difference and can actually produce a larger standard error for the adjusted mean difference than that for the unadjusted mean difference. When the covariate observations are conceived of as randomly varying from one study to another, the covariate mean differences can be related to a Hotelling's T². Using this Hotelling's T² statistic, one can always find a minimum sample size to achieve a high probability of reducing the standard error and confidence interval width for the adjusted mean difference.     

Identifying the Source of Misfit in Item Response Theory Models

When an item response theory model fails to fit adequately, the items for which the model provides a good fit and those for which it does not must be determined. To this end, we compare the performance of several fit statistics for item pairs with known asymptotic distributions under maximum likelihood estimation of the item parameters: (a) a mean and variance adjustment to bivariate Pearson's X², (b) a bivariate subtable analog to Reiser's (1996) overall goodness-of-fit test, (c) a z statistic for the bivariate residual cross product, and (d) Maydeu-Olivares and Joe's (2006) M₂ statistic applied to bivariate subtables. The unadjusted Pearson's X² with heuristically determined degrees of freedom is also included in the comparison. For binary and ordinal data, our simulation results suggest that the z statistic has the best Type I error and power behavior among all the statistics under investigation when the observed information matrix is used in its computation. However, if one has to use the cross-product information, the mean and variance adjusted X² is recommended. We illustrate the use of pairwise fit statistics in 2 real-data examples and discuss possible extensions of the current research in various directions.     

What Causes the Mean Bias of the Likelihood Ratio Statistic with Many Variables?

Abstract

Survey data often contain many variables. Structural equation modeling (SEM) is commonly used in analyzing such data. However, conventional SEM methods are not crafted to handle data with a large number of variables (p). A large p can cause T_ml, the most widely used likelihood ratio statistic, to depart drastically from the assumed chi-square distribution even with normally distributed data and a relatively large sample size N. A key element affecting this behavior of T_ml is its mean bias. The focus of this article is to determine the cause of the bias. To this end, empirical means of T_ml via Monte Carlo simulation are used to obtain the empirical bias. The most effective predictors of the mean bias are subsequently identified and their predictive utility examined. The results are further used to predict type I errors of T_ml. The article also illustrates how to use the obtained results to determine the required sample size for T_ml to behave reasonably well. A real data example is presented to show the effect of the mean bias on model inference as well as how to correct the bias in practice.     

An effective fixed-point theorem in intuitionistic diagonalizable algebras

Summary Within the technical frame supplied by the algebraic variety of diagonalizable algebras, defined by R. Magari in [2], we prove the following:LetT be any first-order theory with a predicate Pr satisfying the canonical derivability conditions, including Löb's property. Then any formula inT built up from the propositional variables q, p₁, ..., p_n, using logical connectives and the predicate Pr, has the same fixed-points relative to q (that is, formulas (p₁ ..., p_n) for which for all p₁, ..., p_n T((p₁, ..., p_n), p₁, ..., p_n) (p₁, ..., p_n)) of a formula ^* of the same kind, obtained from in an effective way.Moreover, such ^* is provably equivalent to the formula obtained from substituting with ^* itself all the occurrences of q which are under Pr. In the particular case where q is always under Pr in , ^* is the unique (up to provable equivalence) fixedpoint of .Since this result is proved only assumingPr to be canonical, it can be deduced that Löb's property is, in a sense, equivalent to Gödel's diagonalization lemma.All the results are proved more generally in the intuitionistic case.The algebraization of the theories which express Theor, IXAllatum est die 19 Decembris 1975     

Thermophysical properties of substantially undercooled liquid Ti-Al-Nb ternary alloy measured by electromagnetic levitation

The thermophysical properties of undercooled liquid alloys at high temperature are usually difficult to measure by experiment. Here, we report the specific heat of liquid Ti₄₅Al₄₅Nb₁₀ ternary alloy in the undercooled state. By using electromagnetic levitation technique, a maximum undercooling of 287?K (0.15 T _L) is achieved for this alloy. Its specific heat is determined to be 32.72?±?2.51?J?mol^?1 K^?1 over a broad temperature range of 1578–2010?K.     

A note on the ω-incompleteness formalization

The paper studies two formal schemes related to -completeness.LetS be a suitable formal theory containing primitive recursive arithmetic and letT be a formal extension ofS. Denoted by (a), (b) and (c), respectively, are the following three propositions (where (x) is a formula with the only free variable x): (a) (for anyn) ( _T (n)), (b) _{T x} Pr _T (^–(x)^–) and (c) _T x(x) (the notational conventions are those of Smoryski [3]). The aim of this paper is to examine the meaning of the schemes which result from the formalizations, over the base theoryS, of the implications (b) (c) and (a) (b), where ranges over all formulae. The analysis yields two results overS : 1. the schema corresponding to (b) (c) is equivalent to ¬Cons _T and 2. the schema corresponding to (a) (b) is not consistent with 1-CON _T. The former result follows from a simple adaptation of the -incompleteness proof; the second is new and is based on a particular application of the diagonalization lemma.Presented byMelvin Fitting     

Effect of network connectivity on thermal properties of As30Te70– x Tl x glasses

Alternating differential scanning calorimetry (ADSC) studies were undertaken to investigate the effect of Tl addition on the thermal properties of As₃₀Te_{70– x} Tl _x (6?≤?x?≤?22 at%) glasses. These include parameters such as glass-transition temperature (T _g), changes in specific heat capacity (ΔC _p) and relaxation enthalpy (ΔH ^NR) at the glass transition. It was found that T _g of the glasses decreased with the addition of Tl, which is in contrast to the dependence of T _g in As–Te glasses on the addition of Al and In. The change in heat capacity ΔC _p through the glass transition was also found to decrease with increasing Tl content. The addition of Tl to the As–Te matrix may lead to a breaking of As–Te chains and the formation of Tl⁺Te^?AsTe_2/2 dipoles. There was no significant dependence of the change of relaxation enthalpy, through the glass transition, with composition.     

Robust step‐down tests for multivariate independent group designs

A composite step‐down procedure, in which a set of step‐down tests are summarized collectively with Fisher's combination statistic, was considered to test for multivariate mean equality in two‐group designs. An approximate degrees of freedom (ADF) composite procedure based on trimmed/Winsorized estimators and a non‐pooled estimate of error variance is proposed, and compared to a composite procedure based on trimmed/Winsorized estimators and a pooled estimate of error variance. The step‐down procedures were also compared to Hotelling's T² and Johansen's ADF global procedure based on trimmed estimators in a simulation study. Type I error rates of the pooled step‐down procedure were sensitive to covariance heterogeneity in unbalanced designs; error rates were similar to those of Hotelling's T² across all of the investigated conditions. Type I error rates of the ADF composite step‐down procedure were insensitive to covariance heterogeneity and less sensitive to the number of dependent variables when sample size was small than error rates of Johansen's test. The ADF composite step‐down procedure is recommended for testing hypotheses of mean equality in two‐group designs except when the data are sampled from populations with different degrees of multivariate skewness.     

Limited‐information goodness‐of‐fit testing of hierarchical item factor models

In applications of item response theory, assessment of model fit is a critical issue. Recently, limited‐information goodness‐of‐fit testing has received increased attention in the psychometrics literature. In contrast to full‐information test statistics such as Pearson’s X² or the likelihood ratio G², these limited‐information tests utilize lower‐order marginal tables rather than the full contingency table. A notable example is Maydeu‐Olivares and colleagues’M₂ family of statistics based on univariate and bivariate margins. When the contingency table is sparse, tests based on M₂ retain better Type I error rate control than the full‐information tests and can be more powerful. While in principle the M₂ statistic can be extended to test hierarchical multidimensional item factor models (e.g., bifactor and testlet models), the computation is non‐trivial. To obtain M₂, a researcher often has to obtain (many thousands of) marginal probabilities, derivatives, and weights. Each of these must be approximated with high‐dimensional numerical integration. We propose a dimension reduction method that can take advantage of the hierarchical factor structure so that the integrals can be approximated far more efficiently. We also propose a new test statistic that can be substantially better calibrated and more powerful than the original M₂ statistic when the test is long and the items are polytomous. We use simulations to demonstrate the performance of our new methods and illustrate their effectiveness with applications to real data.     

Hybridization-induced wide-band bipolaron model for high-Tc superconductors

Abstract

A microscopic theory based on the idea of hybridization-induced wide-band bipolarons is proposed to explain the occurrence of high-temperature superconductors such as La_2?x-Ba_xCuO₄, and La_2?xSr_xCuO₄. In our view, these are such that bipolarons are formed whose singlet ground state possesses a large binding energy (Δ) with bandwidth W (>Δ) as a result of two distinct electron-phonon mechanisms and two types of hybridization between Cu²⁺ d electrons and other electrons. This wide-band bipolaron gas has features similar to Cooper pairs and leads to a superconducting state with high T_c in which all occupied electron states participate. The large electron-phonon coupling and a large W both cause the transition temperature T_c, to rise spectacularly.     

Vortex dynamics in La1.86Sr0. 14CuO4 studied by magnetoresistivity

Abstract

Vortex dynamics in La_1.86Sr_0.14CuO₄ have been studied by the measurement of ρ_c ^//i (T, H), where ρ_c ^//i is the c-axis resistivity for H//i (i = c or a-b). We argue that, at temperatures higher than the irreversibility temperature T _irr, the usual vortex picture breaks down owing to the thermal motion of vortices, resulting in a T- and T _in-dependent anisotropic parameter γ. After taking into account the dependence of γ on T and T _irr, we show that at each given temperature we can rescale the ρ_c ^//a-b (T, H) data onto the corresponding ρ_c ^//c (T, H) curves. This scaling property clearly indicates that the Lorentz-force-free mechanism is responsible for ρ_c ^//a-b (T, H). Furthermore, we also show that the measured ρ_c ^//a-b (T, H) data can be explained in terms of the recently developed extended Josephson coupling model which is verified by rescaling ρ_c ^//a-b (T) data for various fields onto a single curve.     

Reliability of a Longitudinal Sequence of Scale Ratings

Reliability captures the influence of error on a measurement and, in the classical setting, is defined as one minus the ratio of the error variance to the total variance. Laenen, Alonso, and Molenberghs (Psychometrika 73:443–448, 2007) proposed an axiomatic definition of reliability and introduced the R _T coefficient, a measure of reliability extending the classical approach to a more general longitudinal scenario. The R _T coefficient can be interpreted as the average reliability over different time points and can also be calculated for each time point separately. In this paper, we introduce a new and complementary measure, the so-called R _Λ , which implies a new way of thinking about reliability. In a longitudinal context, each measurement brings additional knowledge and leads to more reliable information. The R _Λ captures this intuitive idea and expresses the reliability of the entire longitudinal sequence, in contrast to an average or occasion-specific measure. We study the measure’s properties using both theoretical arguments and simulations, establish its connections with previous proposals, and elucidate its performance in a real case study. The authors are grateful to J&J PRD for kind permission to use their data. We gratefully acknowledge support from Belgian IUAP/PAI network “Statistical Techniques and Modeling for Complex Substantive Questions with Complex Data.”     

Analytical derivation of the effective temperature for field-dependent hopping conductivity

The electric-field enhancement of hopping conductivity in amorphous solids can be described in terms of an effective temperature T _eff given by for hopping, at a temperature T and field F, within a uniform distribution of electronic states with localization length γ^?1. We have derived this expression analytically and find at low fields a value for C of 1/2^1/2, which is consistent with the F-independent value C = 0.67 previously obtained numerically for band-tail hopping. With increasing electric field, the decrease in C(F) matches the hopping transport characteristics in amorphous semiconductors (hydrogenated amorphous silicon and hydrogenated amorphous carbon nitride) better than previous numerical simulations.