Author Correction: Analytical power calculations for structural equation modeling: A tutorial and Shiny app

When Bayesian estimation is used to analyze Structural Equation Models (SEMs), prior distributions need to be specified for all parameters in the model. Many popular software programs offer default prior distributions, which is helpful for novel users and makes Bayesian SEM accessible for a broad audience. However, when the sample size is small, those prior distributions are not always suitable and can lead to untrustworthy results. In this tutorial, we provide a non-technical discussion of the risks associated with the use of default priors in small sample contexts. We discuss how default priors can unintentionally behave as highly informative priors when samples are small. Also, we demonstrate an online educational Shiny app, in which users can explore the impact of varying prior distributions and sample sizes on model results. We discuss how the Shiny app can be used in teaching; provide a reading list with literature on how to specify suitable prior distributions; and discuss guidelines on how to recognize (mis)behaving priors. It is our hope that this tutorial helps to spread awareness of the importance of specifying suitable priors when Bayesian SEM is used with small samples.

The default use of an alpha level of .05 is suboptimal for two reasons. First, decisions based on data can be made more efficiently by choosing an alpha level that minimizes the combined Type 1 and Type 2 error rate. Second, it is possible that in studies with very high statistical power, p values lower than the alpha level can be more likely when the null hypothesis is true than when the alternative hypothesis is true (i.e., Lindley’s paradox). In this article, we explain two approaches that can be used to justify a better choice of an alpha level than relying on the default threshold of .05. The first approach is based on the idea to either minimize or balance Type 1 and Type 2 error rates. The second approach lowers the alpha level as a function of the sample size to prevent Lindley’s paradox. An R package and Shiny app are provided to perform the required calculations. Both approaches have their limitations (e.g., the challenge of specifying relative costs and priors) but can offer an improvement to current practices, especially when sample sizes are large. The use of alpha levels that are better justified should improve statistical inferences and can increase the efficiency and informativeness of scientific research.

Micro-randomized trials (MRTs) are a novel experimental design for developing mobile health interventions. Participants are repeatedly randomized in an MRT, resulting in longitudinal data with time-varying treatments. Causal excursion effects are the main quantities of interest in MRT primary and secondary analyses. We consider MRTs where the proximal outcome is binary and the randomization probability is constant or time-varying but not data-dependent. We develop a sample size formula for detecting a nonzero marginal excursion effect. We prove that the formula guarantees power under a set of working assumptions. We demonstrate via simulation that violations of certain working assumptions do not affect the power, and for those that do, we point out the direction in which the power changes. We then propose practical guidelines for using the sample size formula. As an illustration, the formula is used to size an MRT on interventions for excessive drinking. The sample size calculator is implemented in R package MRTSampleSizeBinary and an interactive R Shiny app. This work can be used in trial planning for a wide range of MRTs with binary proximal outcomes.

Variance component linkage analysis is commonly used to map quantitative trait loci (QTLs) in general pedigrees. Large pedigrees are especially attractive for these studies because they provide greater power per genotyped individual than small pedigrees. We propose accurate and computationally efficient methods to calculate the analytical power of variance component linkage analysis that can accommodate large pedigrees. Our analytical power computation involves the approximation of the noncentrality parameter for the likelihood-ratio test by its Taylor expansions. We develop efficient algorithms to compute the second and third moments of the identical by descent (IBD) sharing distribution and enable rapid computation of the Taylor expansions. Our algorithms take advantage of natural symmetries in pedigrees and can accurately analyze many large pedigrees in a few seconds. We verify the accuracy of our power calculation via simulation in pedigrees with 2-5 generations and 2-8 siblings per sibship. We apply this proposed analytical power calculation to 98 quantitative traits in a cohort study of 6,148 Sardinians in which the largest pedigree includes 625 phenotyped individuals. Simulations based on eight representative traits show that the difference between our analytical estimation of the expected LOD score and the average of simulated LOD scores is less than 0.05 (1.5%). Although our analytical calculations are for a fully informative marker locus, in the settings we examined power was similar to what could be attained with a single nucleotide polymorphism (SNP) mapping panel (with >1 SNP/cM). Our algorithms for power analysis together with polygenic analysis are implemented in a freely available computer program, POLY.

Outcomes from studies assessing exposure often use multiple measurements. In previous work, using a model first proposed by Buonoccorsi (1991), we showed that combining direct (eg, biomarkers) and indirect (eg, self-report) measurements provides a more accurate picture of true exposure than estimates obtained when using a single type of measurement. In this article, we propose a tool for efficient design of studies that include both direct and indirect measurements of a relevant outcome. Based on data from a pilot or preliminary study, the tool, which is available online as a shiny app at https://michalbitan.shinyapps.io/shinyApp/, can be used to compute: (1) the sample size required for a statistical power analysis, while optimizing the percent of participants who should provide direct measures of exposure (biomarkers) in addition to the indirect (self-report) measures provided by all participants; (2) the ideal number of replicates; and (3) the allocation of resources to intervention and control arms. In addition we show how to examine the sensitivity of results to underlying assumptions. We illustrate our analysis using studies of tobacco smoke exposure and nutrition. In these examples, a near-optimal allocation of the resources can be found even if the assumptions are not precise.

BackgroundPancreatic ductal adenocarcinoma (PDAC) presents a significant challenge, with a 5-year survival rate of approximately 10%. Tumor heterogeneity contributes to the limited effectiveness of treatments. Several tumor and stroma molecular classifiers have attempted to clarify this heterogeneity with moderate agreement. Recognizing the complexity introduced by this extensive array of taxonomies, this study aims to develop a consensus molecular classifier by including both tumor and stroma features.MethodsWe analyzed mRNA expression data from 514 PDAC samples, applying batch correction, filtering out low-expression genes, and using variance-stabilizing transformation. Tumor and stroma profiles were classified with previously published systems, while stroma compartments were estimated through virtual microdissection. For each classifier, multiple machine learning models were trained and optimized, with the top performers used to assign subtypes. A consensus classifier was created by building subtype similarity networks and applying a Markov clustering algorithm, and robustness was evaluated through resampling. Associations between consensus classes and overall survival were examined using multivariate Cox models.ResultsThe results indicated that Elastic-Net emerged as the superior model. We identified two classes for tumor components (Consensus Classical and Consensus Non-classical) and stroma components (Consensus Normal-Immune and Consensus Activated-ECM). The consensus Random Forest achieved a balanced accuracy of 96.33 and 98.92%, respectively. Across cohorts, the PDAConsensus algorithm identified tumor subtypes with distinct prognostic value, with Consensus Non-classical tumors showing poorer survival than Consensus Classical tumors. Associations for stroma consensus classes were weaker and less consistent across datasets.ConclusionsWe developed a robust consensus classifier for PDAC that integrates tumor and stroma features. The classifier is accessible through the R package PDACMOC (PDACMolecularOmniClassifier, https://github.com/pavillos/PDACMOC, 10.5281/zenodo.17161373) and a Shiny app (https://pdacmoc.cnio.es/). This classifier offers a biologically grounded framework that integrates existing systems, allows for single-sample classification, and improves prognostic stratification. By enabling subtype-specific therapies and better patient stratification in clinical trials, it can help guide precision medicine and enhance outcomes in PDAC management.Supplementary InformationThe online version contains supplementary material available at 10.1186/s13073-025-01568-9.

Clustering methods are increasingly used in social science research. Generally, researchers use them to infer the existence of qualitatively different types of individuals within a larger population, thus unveiling previously "hidden" heterogeneity. Depending on the clustering technique, however, valid inference requires some conditions and assumptions. Common risks include not only failing to detect existing clusters due to a lack of power but also revealing clusters that do not exist in the population. Simple data simulations suggest that under conditions of sample size, number, correlation and skewness of indicators that are frequently encountered in applied psychological research, commonly used clustering methods are at a high risk of detecting clusters that are not there. Generally, this is due to some violations of assumptions that are not usually considered critical in psychology. The present article illustrates a simple R tutorial and a Shiny app (for those who are not familiar with R) that allow researchers to quantify a priori inferential risks when performing clustering methods on their own data. Doing so is suggested as a much-needed preliminary sanity check, because conditions that inflate the number of detected clusters are very common in applied psychological research scenarios.

The paper presents the results of analytical and numerical model calculations of elastic surface displacements caused by earthquakes and the processes of earthquake nucleation. The analytical calculations are focused on the model of a homogeneous elastic space; the numerical calculations also cover models of radially inhomogeneous distributions of elastic moduli in the crust and upper mantle described by the PREM model.

This paper proposes an accurate analytical calculation method for rotor eddy current loss of high-speed machines with the shield layer and retaining sleeve. The calculation model includes six subdomains, such as permanent magnet, shield layer, retaining sleeve, air gap, slot opening, winding. Considering the eddy current reaction of the permanent magnet, shield layer, and retaining sleeve, the vector magnetic potential equation derivation model for each subdomain is established, and the eddy current loss of the permanent magnet and shield layer caused by current harmonics and slot openings is accurately calculated. The model is used to study the factors affecting the rotor eddy current loss, including carrier ratio, modulation ratio, slot opening length, and thickness of permanent magnet and protective retaining sleeve. The results of model analysis and calculation are compared with the results of finite element and machine experiments to verify the correctness and regularity of model analysis and calculation.

To unravel how within-person psychological processes fluctuate in daily life, and how these processes differ between persons, intensive longitudinal (IL) designs in which participants are repeatedly measured, have become popular. Commonly used statistical models for those designs are multilevel models with autocorrelated errors. Substantive hypotheses of interest are then typically investigated via statistical hypotheses tests for model parameters of interest. An important question in the design of such IL studies concerns the determination of the number of participants and the number of measurements per person needed to achieve sufficient statistical power for those statistical tests. Recent advances in computational methods and software have enabled the computation of statistical power using Monte Carlo simulations. However, this approach is computationally intensive and therefore quite restrictive. To ease power computations, we derive simple-to-use analytical formulas for multilevel models with AR(1) within-person errors. Analytic expressions for a model family are obtained via asymptotic approximations of all sample statistics in the precision matrix of the fixed effects. To validate this analytical approach to power computation, we compare it to the simulation-based approach via a series of Monte Carlo simulations. We find comparable performances making the analytic approach a useful tool for researchers that can drastically save them time and resources.

An analytic electromagnetic calculation method for doubly fed induction generator (DFIG) in wind turbine system was presented. Based on the operation principles, steady state equivalent circuit and basic equations of DFIG, the modeling for electromagnetic calculation of DFIG was proposed. The electromagnetic calculation of DFIG was divided into three steps: the magnetic flux calculation, parameters derivation and performance checks. For each step, the detailed numeric calculation formulas were all derived. Combining the calculation formulas, the whole electromagnetic calculation procedure was established, which consisted of three iterative calculation loops, including magnetic saturation coefficient, electromotive force and total output power. All of the electromagnetic and performance data of DIFG can be calculated conveniently by the established calculation procedure, which can be used to evaluate the new designed machine. A 1.5 MW DFIG designed by the proposed procedure was built, for which the whole type tests including no-load test, load test and temperature rising test were carried out. The test results have shown that the DFIG satisfies technical requirements and the test data fit well with the calculation results which prove the correctness of the presented calculation method.

This paper investigated the effect of the tenon length on the strength and stiffness of the standard mortise and tenon joints, as well of the double mortise and tenon joints, that were bonded by poly(vinyl acetate) (PVAc) and polyurethane (PU) glues. The strength was analyzed by measuring applied load and by calculating ultimate bending moment and bending moment at the proportional limit. Stiffness was evaluated by measuring displacement and by calculating the ratio of applied force and displacement along the force line. The results were compared with the data obtained by the simplified static expressions and numerical calculation of the orthotropic linear-elastic model. The results indicated that increasing tenon length increased the maximal moment and proportional moment of the both investigated joints types. The analytically calculated moments were increased more than the experimental values for both joint types, and they had generally lower values than the proportional moments for the standard tenon joints, as opposed to the double tenon joints. The Von Mises stress distribution showed characteristic zones of the maximum and increased stress values. These likewise were monitored in analytical calculations. The procedures could be successfully used to achieve approximate data of properties of loaded joints.

Neutron scattering data for melt-crystallized polyethylene have been analyzed in order to clarify to what extent the chain folding is randomly reentrant. No attempt has been made to specify the molecular conformation in every detail, and the emphasis is on distinguishing between different classes of conformation. The most random folding corresponded to a model where the folding is imposed solely by the criterion of the chain segments moving the least possible distance during the crystallization process (a “freezing-in” model). This has been shown not to be compatible with published data. For this model analytic calculations are possible based on the projection of a three-dimensional Gaussian distribution onto a plane. A subunit model is then proposed which requires substantial local rearrangement of the chain as it folds during crystallization, but where the distribution of the subunits within the whole molecule is imposed by the preexisting Gaussian chain of the melt. Arguments based on space filling considerations are invoked, with the postulate of a surface structure which is neither crystalline nor truly amorphous. Anything approaching a random switchboard model (e.g., the freezing-in model which we consider) is contrary to both space filling considerations and to the comparison of observed and calculated neutron scattering. The analytical calculation which was performed for the freezing-in model was employed so as to simplify calculations for the subunit model. For scattering intensities over a wide range of scattering angle it is deduced that only the structure within the subunit need be considered. Numerical computer calculations involving only a small number of stems were then carried out for a number of different subunit structures, and some general features are noted which restrict the type of model which can explain the data. As in previously published analyses, a very high proportion of adjacent folds is not compatible with the results. A row model for the stems within a molecule can achieve good agreement, either with straight rows or with a certain amount of “stagger” incorporated. Up to about 40% of the folds could be adjacent. Models based on two-dimensional random walks did not give good agreement.

One of the principal objectives of modern production process is the improvement of quality level; this means also guaranteeing the required service life of different products and increase in their wear resistance. To perform this task, prediction of service life of fitted components is of crucial value, since with the development of production technologies and measuring devices it is possible to determine with ever increasing precision the data to be used also in analytical calculations. Having studied the prediction theories of wear process that have been developed in the course of time and can be classified into definite groups one can state that each of them has shortcomings that might strongly impair the results thus making unnecessary theoretical calculations. The proposed model for wear calculation is based on the application of theories from several branches of science to the description of 3D surface micro-topography, assessing the material’s physical and mechanical characteristics, substantiating the regularities in creation of the material particles separated during the wear process and taking into consideration definite service conditions of fittings.

The rotor eddy current losses of a high-speed permanent magnet synchronous motor reduce the efficiency of the motor and increase the temperature rise of the rotor. In severe cases, the permanent magnet of the rotor can be demagnetized, affecting the safe operation of the motor. In this paper, an analytical model of rotor eddy current losses calculation based on Maxwell equations is introduced. The eddy current losses of rotor structures, e.g., protection sleeve, shield layer and permanent magnets, are analyzed. The calculation results of rotor eddy current losses are compared with two-dimensional (2D) finite element analysis. The comparison verifies the accuracy and versatility of analytical calculation. Finally, the influences of the variables on eddy current losses in the derived model are analyzed. Based on the principle of minimizing eddy current losses, an optimized structure with copper layers covering both inner and outer surfaces of the protective sleeve is proposed. Furthermore, the optimized distribution parameter of copper film thickness is obtained.