semPower: General power analysis for structural equation models

“In a power analysis, one of these quantities is computed as a function of the other quantities, giving rise to different types of power analyses (Faul et al., 2007).”

“…the model implementing the null hypothesis of interest is then fitted to the population variance–covariance matrix using the lavaan package (Rosseel, 2012), so that the population minimum F 0 along with the resulting model-implied variance covariance matrix ( ̂ 𝚺 ) and mean vector ( ̂ 𝛍…”

“For example, it has been suggested that a decline of ≥ .02 in the Mc associated with a metric in comparison to a configural invariance model may be considered a meaningful departure from measurement invariance (Cheung & Rensvold, 2002).”

“…of a measure that shows a direct relation to the non-centrality parameter, which is afforded by all non-centrality-based measures of fit such as the minimum of the employed fitting function itself or derived indices such as the RMSEA (because F 0 = df ∙ RMSEA 2 ; e.g., MacCallum et al., 1996).”

“When the comparison is performed against the saturated model, however, simulated power is often higher as a consequence of the well-documented positive bias of the model test when the assumptions are violated (e.g., Curran et al., 1996; Moshagen, 2012).”

“Existing tools providing analytical power analyses specifically for SEM (e.g., MacCallum et al., 1996, 2006; Preacher & Coffman, 2006) require provision of a non-centrality-based measure of effect such as the RMSEA and are thus limited to a model-free approach.”

“If the H0 model is correct, assuming multivariate normality, a sufficiently large sample size, and several mild regularity conditions (Browne, 1984; Yuan & Bentler, 2007), (1)”

“…= list(estimator = ' mlr ' ) leads to a simulated power estimate relying on a corrected test statistic that is asymptotically equivalent to the Yuan & Bentler (2000) statistic, whereas the analytic power estimate is still based on the asymptotically expected distributions based on ML.…”

“…approaches to statistical power necessarily allow for a model-based definition of the effect of interest, most existing tools implement post hoc power analyses only and are typically limited to particular model types (Kievit et al., 2018; Mulder, 2022; Schoemann et al., 2017; Zhang & Liu, 2018).”

“In SEM, maximizing the likelihood is equivalent to minimizing the maximum likelihood (ML) fitting function (Jöreskog, 1967), so the LRT can also be obtained by considering the sample-size-weighted difference in the minimized sample values of the fitting function for the H0 model ( ̂ F ) and the H1…”

“…of options providing either analytical (Jak et al., 2021; Miles, 2003; Z. Zhang & Yuan, 2018) or simulated power analysis (Muthén & Muthén, 2002; Wang & Rhemtulla, 2021) allow for a generic model-based definition of the effect, but these require the manual definition of the values of all…”

“…approaches to statistical power necessarily allow for a model-based definition of the effect of interest, most existing tools implement post hoc power analyses only and are typically limited to particular model types (Kievit et al., 2018; Mulder, 2022; Schoemann et al., 2017; Zhang & Liu, 2018).”

“…H0 model is invalid, however, the LRT—again assuming multivariate normality, a sufficiently large sample size, and several regularity conditions 1 (Satorra & Saris, 1985; Steiger et al., 1985)—asymptotically follows a noncentral χ 2 ( df , λ) distribution with non-centrality parameter λ and…”

“…however, the LRT—again assuming multivariate normality, a sufficiently large sample size, and several regularity conditions 1 (Satorra & Saris, 1985; Steiger et al., 1985)—asymptotically follows a noncentral χ 2 ( df , λ) distribution with non-centrality parameter λ and expected value df + λ, so…”

“For instance, in a simple two-wave cross-lagged panel model (CLPM; e.g., Newsom, 2015) involving four factors, each comprising three indicators loading by .50, N = 252 observations are required to yield 80% power on alpha = .05 to detect global mis-specifications corresponding to a root-mean-square…”

“Existing tools providing analytical power analyses specifically for SEM (e.g., MacCallum et al., 1996, 2006; Preacher & Coffman, 2006) require provision of a non-centrality-based measure of effect such as the RMSEA and are thus limited to a model-free approach.”

“The semPower 2 package is a massive improvement and extension over the first version of sem-Power (Moshagen & Erdfelder, 2016), which only provided analytical power analysis for common effect size metrics or based on a covariance matrix input (thereby offering model-based power analysis via rather…”

“Note that the minimized sample value of the fitting function, ̂ F , is a biased estimate of F 0 (McDonald, 1989) with an expected value of F 0 + df / ( N − 1).”

“…and extension over the first version of sem-Power (Moshagen & Erdfelder, 2016), which only provided analytical power analysis for common effect size metrics or based on a covariance matrix input (thereby offering model-based power analysis via rather cumbersome procedures; see Jobst et al., 2023).”

“When the comparison is performed against the saturated model, however, simulated power is often higher as a consequence of the well-documented positive bias of the model test when the assumptions are violated (e.g., Curran et al., 1996; Moshagen, 2012).”

“…approaches to statistical power necessarily allow for a model-based definition of the effect of interest, most existing tools implement post hoc power analyses only and are typically limited to particular model types (Kievit et al., 2018; Mulder, 2022; Schoemann et al., 2017; Zhang & Liu, 2018).”

“…M0 is the likelihood of the hypothesized (H0) model and L M1 is the likelihood of a more general (H1) model, where the H0 model must be statistically nested in the H1 model (i.e., the parameter space of the H0 model must be a subset of the parameter space of the H1 model; Bentler & Satorra, 2010).”

“Readers who are more comfortable with a graphical user interface are referred to Jak et al. (2021) for analytical and to Wang and Rhemtulla (2021) for simulated power analyses.”

“Therefore, semPower 2 interfaces four different data generation methods, namely Foldnes and Olsson (2016), Qu et al. (2020), Ruscio and Kaczetow (2008), and Vale and Maurelli (1983), each of which is associated with different multivariate distributions that may affect the empirically observed χ 2…”

“If the H0 model is correct, assuming multivariate normality, a sufficiently large sample size, and several mild regularity conditions (Browne, 1984; Yuan & Bentler, 2007), (1)”

“Of note, the actual effect in terms of F 0 not only is a function of the parameters that differ between the H0 and the H1 model, but also depends on all other parameters in the model (e.g., Moshagen & Auerswald, 2018; Wang & Rhemtulla, 2021).”

“Therefore, semPower 2 interfaces four different data generation methods, namely Foldnes and Olsson (2016), Qu et al. (2020), Ruscio and Kaczetow (2008), and Vale and Maurelli (1983), each of which is associated with different multivariate distributions that may affect the empirically observed χ 2…”

“…(KS) distance measures the discrepancy between the empirical and the reference cumulative distribution functions akin to the Kolmogorov–Smirnov test statistic, but relies on the average, rather than the maximum, absolute distance, KS = M ( F e ( x ) − F r ( x ) ) (see Yuan et al., 2007).”

“…(2008), and Vale and Maurelli (1983), each of which is associated with different multivariate distributions that may affect the empirically observed χ 2 -distributions differently (e.g., Auerswald & Moshagen, 2015; Foldnes & Grønneberg, 2015; Jobst et al., 2022) and requires different input.”

“…(2008), and Vale and Maurelli (1983), each of which is associated with different multivariate distributions that may affect the empirically observed χ 2 -distributions differently (e.g., Auerswald & Moshagen, 2015; Foldnes & Grønneberg, 2015; Jobst et al., 2022) and requires different input.”

“…and potentially error-prone, in particular when more complex models are considered and it is desired to define the model in terms of standardized parameters, which requires rather sophisticated knowledge concerning the proper definition of the residual variances (see e.g., Bader & Moshagen, 2022).”

“…approaches to statistical power necessarily allow for a model-based definition of the effect of interest, most existing tools implement post hoc power analyses only and are typically limited to particular model types (Kievit et al., 2018; Mulder, 2022; Schoemann et al., 2017; Zhang & Liu, 2018).”

“For MAR, semPower 2 implements the percentile method generating the maximum number of missing patterns (Zhang, 2021).”

“An a priori power analysis was conducted in RStudio (Posit Team, 2024) using the semPower package (Moshagen & Bader, 2023), and it indicated that 239 participants were necessary to observe medium (0.3) cross-lagged effects with good power (0.8) and that 2227 participants were necessary to observe…”

“A model-free power analysis was conducted using the semPower package (Moshagen & Bader, 2023) to determine the sample size for the global model fit criteria.”

“First, a power analysis for SEM was conducted using semPower[66], with the following parameters: desired power of 0.80, alpha level of 0.05, RMSEA as the effect measure, an effect size of 0.05, degrees of freedom (df) of 50, and 12 manifest variables.”

“Sample size was estimated using a web-frontend for “semPower 2” (an R-package providing a collection of functions to perform a-priori, post-hoc, and compromise power analyses for structural equation models (SEMs)) [35].”

“The “sem-POWER” tool was used to determine the minimum sample size required to detect the effect among the latent variables (Moshagen & Bader, 2024).”

“To determine an adequate sample size for our Structural Equation Modeling (SEM) analyses, we conducted an a priori power analysis using the semPower R package [75].”

“…of psychometric parameters, gtsummary [59] to obtain sum-mary tables, Hmisc [60] complementary functions for structural equation modeling, tidySEM [61], semPower [45], ad-hoc validity estimation of the sample size and flexible obtaining of tables that allow manipulation in other formats [62].”

“However, we achieved strong statistical power (1.00) to evaluate our CFA models in both studies through power analysis for structural equation models (Moshagen & Bader, 2024).”

“As this article reports a secondary analysis of data collected in studies yet to be published, we conducted a post-hoc power analysis using semPower (Moshagen & Bader, 2023 ), which showed that for a sample size of 213 with an alpha of .”

“Furthermore, by using an a priori sample size analysis, where a = 0.05, power = 0.8, and the target Goodness-of-fit Index ( GFI ) is at least 0.95, the estimated number of participants is equal to 89 [7].”

“Based on these parameters, the required minimum sample size was determined using R (version 4.4.1) and the semPower package, which is utilized for power analysis in structural equation modeling (Moshagen & Bader, 2023).”

“Sufficiency of the sample size for the SEM analysis was examined by conducting power analysis using the semPower package version 2.1.0 ( Moshagen & Bader, 2024).”

“However, a model-free a priori RMSEA-based power analysis [49] showed that a sample size of N = 104 was sufficient to detect misspecifications of a structure model involving 227 degrees of freedom (which corresponds to the original three-factor LAPS model with 23 manifest variables) at RMSEA effect…”

“Overall, the power for each CFA test (using the semPower package and following recommendations by Moshagen & Bader, 2023) was greater than 0.999.”

“A post hoc power analysis was conducted to evaluate the adequacy of the sample size for detecting the hypothesized model fit using root mean square error of approximation (RMSEA) = 0.05 as the effect size (Jobst et al., 2021; Moshagen & Bader, 2023).”

“We run a post-hoc power analysis using semPower (Moshagen & Bader, 2024).”

“An initial power analysis was conducted to select the minimum number of participants required to detect at least small effects, utilizing the semPower package in R (Moshagen and Bader 2023).”

“This technique integrates spatial dependencies directly into structural equation models, giving a comprehensive view of causal relationships while accounting for spatial autocorrelation [44], [45].”

“An a priori power analysis was conducted with semPOWER (Moshagen & Bader, 2024), which resulted in a minimum sample size of 89 participants assuming typical conditions for conducting structural equation modelling (Cronbach's α = .05, 1 − β = .80, RMSEA = .04, df = 707).”

“This model serves as a method for testing research indicators, which can be assessed through convergent and discriminant validity tests (Moshagen & Bader, 2024).”

“This study employs SPSS and AMOS 28 for the SEM analysis and apply the Maximum Likelihood method (see e.g., Hyvärinen et al. 2024; Moshagen and Bader 2024; Zyphur et al. 2023).”

“The pwr package (Champely, 2020) and WebPower (Zhang & Yuan, 2018) offer implementations for power analysis of many basic statistical tests in R (R Core Team, 2023), with extensions also available for linear structural equation models (SEM, see also Moshagen & Bader, 2024, for an implementation).”

“A power analysis using semPower in R (Moshagen & Bader, 2024) with a moderate effect size of .15, α of .05, assumed regression slope between latent variables of .20, and the smallest degrees of freedom for planned factor analysis models ( df = 12) revealed a sample size of at least 66 was adequate…”

“To ensure our sample size was adequate, we conducted a post hoc power analysis using the semPower package (Moshagen & Bader, 2023).”

“Using the semPower package in R (Moshagen & Bader, 2023), an a priori power analysis indicated that for a condition effect on a linear slope in the model of .05 (small) with 80% power and alpha of .05 with 145 degrees of freedom in the model, we would require 133 participants.”

“This index is one of the model fit indices that show the power of predicting the dependent variable (criterion) based on the independent variables (predictor) [95].”

“With a sample size as large as this, we estimated to be able to detect an effect size ꞵ ≥ .099 with 95% power and alpha = .05 (estimation with package semPower ; Moshagen & Bader, 2024; constant unidirectional causal effects, and constant autoregressive effects set to ꞵ = .070).”

“Furthermore, a G*Power analysis was conducted to justify the adequacy of the sample size (Moshagen and Bader, 2024).”

“All analyses were conducted using R packages dplyr, 39 fungible, 40 psych, 41 semPower, 42 and sjmisc.”

“…was carried out when applied under different method conditions (hair et al., 2019b). this evaluation includes power assessment parameters (Moshagen & Bader, 2024), robustness check tests using the Ramsey’s ReSet method (Ramsey, 1969; Sarstedt et al., 2020), and sensitivity analysis with…”

“Based on this criterion, our sample of 308 participants provided adequate statistical power (Moshagen & Bader, 2024).”

“In order to explore this concern, we conducted a post hoc power analysis using the semPower software and the analytical model-free power analysis (Moshagen & Bader, 2024).”

“The analysis was performed using R (version 4.4.2; R Core Team, 2023), and the packages semPower (version 2.1.1) 24 and lavaan (version 0.6–19).”

“A priori power analysis using the semPower package (Moshagen & Bader, 2023) indicated that 792 participants would provide adequate power (80”

“In addition, we calculated the prior sample size requirement using lavaan and semPower2 package within R (Moshagen & Bader, 2023), which recommended the number of participants of 190 with desired power of .80, alpha of .05, and factor correlation of .25 with 10 indicators for each factor.”

“Statistical power complements the beta error and gives the probability of rejecting a null hypothesis if this hypothesis is factually wrong (and thus to accept a true alternative hypothesis) (38).”