Covariance structure analysis or structural equation modeling is critical for political scientists measuring latent structural relationships, allowing for the simultaneous assessment of both latent and observed variables, alongside measurement error. Well-specified models are essential for theoretical support, balancing simplicity with optimal model fit. However, current approaches to improving model specification searches remain limited, making it challenging to capture all meaningful parameters and leaving models vulnerable to chance-based specification risks. To address this, we propose an improved Lagrange multiplier (LM) test incorporating stepwise bootstrapping in LM and Wald tests to detect omitted parameters. Monte Carlo simulations and empirical applications underscore its effectiveness, particularly in small samples and models with high degrees of freedom, thereby enhancing statistical fit.

This paper underscores the vital role of the chi-square test within political science research utilizing structural equation modeling (SEM). The ongoing debate regarding the inclusion of chi-square test statistics alongside fit indices in result presentations has sparked controversy. Despite the recognized limitations of relying solely on the chi-square test, its judicious application can enhance its effectiveness in evaluating model fit and specification. To exemplify this, we present three common scenarios pertinent to political science research where fit indices may inadequately address goodness-of-fit concerns, while the chi-square statistic can be effectively harnessed. Through Monte Carlo simulations, we examine strategies for enhancing chi-square tests within these scenarios, showcasing the potential of appropriately employed chi-square tests to provide a comprehensive model fit assessment. Our recommendation is to report both the chi-square test and fit indices, with a priority on precise model specification to ensure the trustworthiness of model fit indicators.

Most existing studies on the relationship between factor analysis (FA) and principal component analysis (PCA) focus on approximating the common factors by the first few components via the closeness between their loadings. Based on a setup in Bentler and de Leeuw (Psychometrika 76:461–470, 2011), this study examines the relationship between FA loadings and PCA loadings when specificities are treated as latent factors. In particular, we will examine the closeness between the two types of loadings when the number of observed variables (p) increases. Parallel to the development in Schneeweiss (Multivar Behav Res 32:375–401, 1997), an average squared canonical correlation (ASCC) is used as the criterion for measuring the closeness. We show that the ASCC can be partitioned into two parts, the first of which is a function of FA loadings and the inverse correlation matrix, and the second of which is a function of unique variances and the inverse correlation matrix of the observed variables. We examine the behavior of these two parts as p approaches infinity. The study gives a different perspective on the relationship between PCA and FA, and the results add additional insights on the selection of the two types of methods in the analysis of high dimensional data.

Climate change is a critical issue of our time, and its causes, pathways, and forecasts remain a topic of broader discussion. In this paper, we present a novel data driven pathway analysis framework to identify the key processes behind mean global temperature and sea level rise, and to forecast the magnitude of their increase from the present to 2100. Based on historical data and dynamic statistical modeling alone, we have established the causal pathways that connect increasing greenhouse gas emissions to increasing global mean temperature and sea level, with its intermediate links encompassing humidity, sea ice coverage, and glacier mass, but not for sunspot numbers. Our results indicate that if no action is taken to curb anthropogenic greenhouse gas emissions, the global average temperature would rise to an estimated 3.28 °C (2.46–4.10 °C) above its pre-industrial level while the global sea level would be an estimated 573 mm (474–671 mm) above its 2021 mean by 2100. However, if countries adhere to the greenhouse gas emission regulations outlined in the 2021 United Nations Conference on Climate Change (COP26), the rise in global temperature would lessen to an average increase of 1.88 °C (1.43–2.33 °C) above its pre-industrial level, albeit still higher than the targeted 1.5 °C, while the sea level increase would reduce to 449 mm (389–509 mm) above its 2021 mean by 2100.

This paper assesses the performance of regularized generalized least squares (RGLS) and reweighted least squares (RLS) methodologies in a confirmatory factor analysis model. Normal theory maximum likelihood (ML) and GLS statistics are based on large sample statistical theory. However, violation of asymptotic sample size is ubiquitous in real applications of structural equation modeling (SEM), and ML and GLS goodness-of-fit tests in SEM often make incorrect decisions on the true model. The novel methods RGLS and RLS aim to correct the over-rejection by ML and under-rejection by GLS. Proposed by Arruda and Bentler (2017), RGLS replaces a GLS weight matrix with a regularized one. Rediscovered by Hayakawa (2019), RLS replaces this weight matrix with one that derives from an ML function. Both of these methods outperform ML and GLS when samples are small, yet no studies have compared their relative performance. A confirmatory factor analysis Monte Carlo simulation study with normal and non-normal data was carried out to examine the statistical performance of these two methods at different sample sizes. Based on empirical rejection frequencies and empirical distributions of test statistics, we find that RLS and RGLS have equivalent performance when N≥70; whereas when N<70, RLS outperforms RGLS. Both methods clearly outperform ML and GLS with N≤400. Nonetheless, adopting mean and variance adjusted test proposed by Hayakawa (2019) for non-normal data, our results show that RGLS slightly outperforms RLS.

Covariance Structure Analysis (CSA) or Structural Equation Modeling (SEM) is critical for political scientists measuring latent structural relationships, allowing for the simultaneous assessment of both latent and observed variables, alongside measurement error. Well-specified models are essential for theoretical support, balancing simplicity with optimal model fit. However, current approaches to improving model specification searches remain limited, making it challenging to capture all meaningful parameters and leaving models vulnerable to chance-based specification risks. To address this, we propose an improved Lagrange Multipliers (LM) test incorporating stepwise bootstrapping in LM and Wald tests to detect omitted parameters. Monte Carlo simulations and empirical applications underscore its effectiveness, particularly in small samples and models with high degrees of freedom, thereby enhancing statistical fit.

Growth curve modeling is commonly used in psychological, educational, and social science research. The mainstream estimators for growth curve modeling are based on normal theory, but real data are unlikely to be exactly normally distributed. To improve estimation and inference with non-normal data, various estimators have been proposed. Among these estimators, the asymptotically distribution free (ADF) estimator does not need to rely on any distribution assumption but it is not efficient with small and modest sample sizes. We propose a distributionally weighted least squares DLS estimator in the growth curve modeling framework. DLS combines normal theory based and ADF based generalized least squares estimation to balance the information from the data and the normality assumption. Computer simulation results suggest that model-implied covariance-based DLS (DLSM) generally provides more accurate and efficient estimates than the examined alternative methods regardless of the distribution. In addition, the relative biases of standard error estimates and the Type I error rates of the Satorra-Bentler test statistic (T-SB) in DLSM were competitive with the classical methods including maximum likelihood and generalized least squares estimation. We illustrate how to implement DLSM and select the optimal tuning parameter by a bootstrap procedure in a real data example.

In structural equation modeling, researchers conduct goodness-of-fit tests to evaluate whether the specified model fits the data well. With nonnormal data, the standard goodness-of-fit test statistic T does not follow a chi-square distribution. Comparing T to chi(2)(df) can fail to control Type I error rates and lead to misleading model selection conclusions. To better evaluate model fit, researchers have proposed various robust test statistics, but none of them consistently control Type I error rates under all examined conditions. To improve model fit statistics for nonnormal data, we propose to use an unbiased distribution free weight matrix estimator ((Gamma) over cap (U)(DF)) in robust test statistics. Specifically, using normal theory based parameter estimates with (Gamma) over cap (U)(DF), we calculate various robust test statistics and robust standard errors. We conducted a simulation study to compare 63 existing robust statistic combinations with the 4 proposed robust statistics with (Gamma) over cap (U)(DF). The Satorra-Bentler statistic T-SB based on (Gamma) over cap (U)(DF) (T-SB(U)) provided acceptable Type I error rates at alpha = .01,.05, or .1 across all conditions (except a few cases with alpha = .01), regardless of the sample size and the distribution. T-SB(U) or T-MVA(2)U typically provided the smallest Anderson-Darling test values, showing the smallest distances between p-values and Uniform(0,1). We use a real data example to compare statistics with (Gamma) over cap (U)(DF) and that with (Gamma) over cap (ADF).

Chi-square tests based on maximum likelihood (ML) estimation of covariance structures often incorrectly over-reject the null hypothesis: Sigma = Sigma(theta) when the sample size is small. Reweighted least squares (RLS) avoids this problem. In some models, the vector of parameter must contain means, variances, and covariances, yet whether RLS also works in mean and covariance structures remains unexamined. This research extends RLS to mean and covariance structures, evaluating a generalized least squares function with ML parameter estimates. A Monte Carlo simulation study was carried out to examine the statistical performance of ML vs RLS with multivariate normal data. Based on empirical rejection frequencies and empirical averages of test statistics, this study shows that RLS performs much better than ML in mean and covariance structure models when sample sizes are small, whereas it does not perform better than ML to reject misspecified models.

This paper is aimed to determine the stakeholders' perception against a new forest management policy for stumpage sales in the Western Black Sea Region of Turkey. Ownership of country's forests (99.9%) belongs to the government and timber production and sales have been managed by The General Directorate of Forest (GDF) in Turkey. In the last decade, GDF applied the new stumpage policy to increase efficiency and decrease costs of management procedures because of the last policy's deficiency. In the case of stumpage policy, it is also important to consider benefits of this policy change to stakeholders. There are four main stakeholders in stumpage policy: forest management (GDF's technical personal, forest engineers), forest villagers (FV), forest cooperatives (FC), forest industry and logging contractors (FI). To analyze and evaluate these factors, this paper examines the perception and point of views of the stakeholders, using a large number of survey data. The structural equation modeling results show that the stumpage policy needs to be revised by the forest management due to stakeholders' negative perception related to technical, social and managerial aspects of the stumpage policy. This policy should be reconsidered to decrease the negative perception of the stakeholders and improved by the participatory approach by decision makers in Turkish forestry.

In real data analysis with structural equation modeling, data are unlikely to be exactly normally distributed. If we ignore the non-normality reality, the parameter estimates, standard error estimates, and model fit statistics from normal theory based methods such as maximum likelihood (ML) and normal theory based generalized least squares estimation (GLS) are unreliable. On the other hand, the asymptotically distribution free (ADF) estimator does not rely on any distribution assumption but cannot demonstrate its efficiency advantage with small and modest sample sizes. The methods which adopt misspecified loss functions including ridge GLS (RGLS) can provide better estimates and inferences than the normal theory based methods and the ADF estimator in some cases. We propose a distributionally weighted least squares (DLS) estimator, and expect that it can perform better than the existing generalized least squares, because it combines normal theory based and ADF based generalized least squares estimation. Computer simulation results suggest that model-implied covariance based DLS (DLSM) provided relatively accurate and efficient estimates in terms of RMSE. In addition, the empirical standard errors, the relative biases of standard error estimates, and the Type I error rates of the Jiang-Yuan rank adjusted model fit test statistic (T-JY) in DLSM were competitive with the classical methods including ML, GLS, and RGLS. The performance of DLSM depends on its tuning parameter a. We illustrate how to implement DLSM and select the optimal a by a bootstrap procedure in a real data example.

Sijtsma and Pfadt (Psychometrika, 2021) provide a wide-ranging defense for the use of coefficient alpha. Alpha is practical and useful when its limitations are acceptable. This paper discusses several methodologies for reliability, some new here, that go beyond alpha and were not emphasized by Sijtsma and Pfadt. Bentler’s (Psychometrika 33:335–345, 1968. https://doi.org/10.1007/BF02289328 ) combined factor analysis (FA) and classical test theory (CTT) model. FACTT provides a key conceptual foundation.

The Oxford Happiness Questionnaire (OHQ), Goal Orientation Questionnaire (GOQ), Revised Study Process Questionnaire (R-SPQ-2F), and Academic Volitional Strategy Inventory (AVSI) plus a brief demographics questionnaire were administered to 395 Thai, and 313 Australian undergraduate students to investigate cross-cultural differences in personality, motivation, learning styles and academic achievement (measured via GPAs). Equivalence of English- and Thai-language measures was ensured using a well-established standard translation-backtranslation procedure. Australian students exhibited higher AVSI scores, whereas Thai students scored more highly on Psychological Wellbeing, as well as on Study Approach, Self-Efficacy Enhancement, Stress Reducing Actions, and Negative-Based Incentives. Nevertheless, our findings provide some evidence that Asian and Western learning style stereotypes may be breaking down in the modern digitally connected world.

BACKGROUNDThe Centers for Medicare & Medicaid Services require that dialysis patients' health-related quality of life be assessed annually. The primary instrument used for this purpose is the Kidney Disease Quality of Life 36-Item Short-Form Survey (KDQOL-36), which includes the SF-12 as its generic core and 3 kidney disease-targeted scales: Burden of Kidney Disease, Symptoms and Problems of Kidney Disease, and Effects of Kidney Disease. Despite its broad use, there has been limited evaluation of KDQOL-36's psychometric properties.STUDY DESIGNSecondary analyses of data collected by the Medical Education Institute to evaluate the reliability and factor structure of the KDQOL-36 scales.SETTINGS & PARTICIPANTSKDQOL-36 responses from 70,786 dialysis patients in 1,381 US dialysis facilities that permitted data analysis were collected from June 1, 2015, through May 31, 2016, as part of routine clinical assessment.MEASUREMENTS & OUTCOMESWe assessed the KDQOL-36 scales' internal consistency reliability and dialysis facility-level reliability using coefficient alpha and 1-way analysis of variance. We evaluated the KDQOL-36's factor structure using item-to-total scale correlations and confirmatory factor analysis. Construct validity was examined using correlations between SF-12 and KDQOL-36 scales and "known groups" analyses.RESULTSEach of the KDQOL-36's kidney disease-targeted scales had acceptable internal consistency reliability (α=0.83-0.85) and facility-level reliability (r=0.75-0.83). Item-scale correlations and a confirmatory factor analysis model evidenced the KDQOL-36's original factor structure. Construct validity was supported by large correlations between the SF-12 Physical Component Summary and Mental Component Summary (r=0.40-0.52) and the KDQOL-36 scale scores, as well as significant differences on the scale scores between patients receiving different types of dialysis, diabetic and nondiabetic patients, and patients who were employed full-time versus not.LIMITATIONSUse of secondary data from a clinical registry.CONCLUSIONSThe study provides support for the reliability and construct validity of the KDQOL-36 scales for assessment of health-related quality of life among dialysis patients.

Basic growth curve models parameterize the mean and covariance structure of a set of repeated measures by latent factors that represent the polynomial influences of time. In practice it may be hard to choose the number of factors, i.e., the order of the polynomial. Simple calculations are proposed to estimate this order.

This chapter presents structural equation modeling as a tool for conducting research regarding how collections of variables may be related to each other as well as to a particular outcome or even multiple outcomes. Structural equation modeling refers to a collection of analytical techniques that can be used to model complex patterns of predictive relationships among a collection of both measured and latent variables. As a statistical tool, structural equation modeling combines the features of regression and factor analysis. The chapter offers conceptual illustrations and practical steps for carrying out structural equation modeling by describing mediation and moderation analyses in the context of music education research.

Statistical theories of goodness-of-fit tests in structural equation modeling are based on asymptotic distributions of test statistics. When the model includes a large number of variables or the population is not from a multivariate normal distribution, the asymptotic distributions do not approximate the distribution of the test statistics very well at small sample sizes. A variety of methods have been developed to improve the accuracy of hypothesis testing at small sample sizes. However, all these methods have their limitations, specially for nonnormal distributed data. We propose a Monte Carlo test that is able to control Type I error with more accuracy compared to existing approaches in both normal and nonnormally distributed data at small sample sizes. Extensive simulation studies show that the suggested Monte Carlo test has a more accurate observed significance level as compared to other tests with a reasonable power to reject misspecified models.

Black dialysis patients report better health-related quality of life (HRQOL) than White patients, which may be explained if Black and White patients respond systematically differently to HRQOL survey items.