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).”

“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).”

“…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).”

“…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).”

“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.”

“…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).”

“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).”