Peter M. Bentler’s research while affiliated with University of California, Los Angeles and other places

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.

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.

This paper aims to advocate for a balanced approach to model fit evaluation in structural equation modeling (SEM). The ongoing debate surrounding chi-square test statistics and fit indices has been characterized by ambiguity and controversy. Despite the acknowledged limitations of relying solely on the chi-square test, its careful application can enhance its effectiveness in evaluating model fit and specification. To illustrate this point, we present three common scenarios relevant to social and behavioral science research using Monte Carlo simulations, where fit indices may inadequately address concerns regarding goodness-of-fit, while the chi-square statistic can offer valuable insights. Our recommendation is to report both the chi-square test and fit indices, prioritizing precise model specification to ensure the reliability of model fit indicators.

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.

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. [Word Count: 3,585]

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 generalized least squares (GLS) statistics are based on large sample statistical theory. However, ML and GLS goodness-of-fit tests often make incorrect decisions on the true model, when sample size is small. The novel methods RGLS and RLS aim to correct the over-rejection by ML and under-rejection by GLS. Both methods outperform ML and GLS when samples are small, yet no studies have compared their relative performance. A Monte Carlo simulation study was carried out to examine the statistical performance of these two methods. 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 for non-normal data, RGLS slightly outperforms RLS. Power analyses found that RLS generally showed small loss in power compared to ML and performed better than RGLS.

Climate change remains a top concern for the world, with its causes, pathways, and forecasts, still subject to debate. In this paper, we present a novel data driven pathway analysis framework to identify the key processes behind the mean global temperature and sea level rise, and to forecast the magnitude of the increases between now and 2100. Based on historical data and dynamic statistical modeling alone, we have confirmed the causal pathways from increased greenhouse gas emissions to increased global mean temperature and sea level, with its intermediate links including humidity, sea ice coverage and glacier volume, but not sunspot numbers. Our results indicate that, if no action is taken to rein in anthropogenic greenhouse gas emissions, the global average temperature is estimated to be 2.79°C higher than its pre-industrial level and the global sea level is expected to be 604 mm above its 2021 mean by 2100. However, if the global community would adhere to the greenhouse gas emission regulations outlined in the 2021 United Nations Conference on Climate Change (COP26), the global temperature would increase to a less threatening 1.58°C above its pre-industrial level, while the sea level increase would reduce to 455 mm above its 2021 mean.

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 χdf2 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 (Γ^DFU) in robust test statistics. Specifically, using normal theory based parameter estimates with Γ^DFU, 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 Γ^DFU. The Satorra–Bentler statistic TSB based on Γ^DFU (TSBU) provided acceptable Type I error rates at α=.01,.05, or .1 across all conditions (except a few cases with α=.01), regardless of the sample size and the distribution. TSBU or TMVA2U 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 Γ^DFU and that with Γ^ADF.

Chi-square tests based on maximum likelihood (ML) estimation of covariance structures often incorrectly over-reject the null hypothesis: Σ=Σθ 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.