semPower

semPower is an R-package that provides several functions to perform a-priori, compromise, and post-hoc power analyses for structural equation models (SEM).

(Very) basic functionality is also provided as a shiny app, which you can use online at https://sempower.shinyapps.io/sempower.

Installation

semPower can be installed via CRAN. The CRAN version often lags behind the development version, which can be installed as follows:

```

install.packages("devtools")

devtools::install_github("moshagen/semPower") ```

Manual

Find a detailed manual at https://moshagen.github.io/semPower/.

We also recommend this in-depth tutorial on power analyses in SEM using a previous version of semPower. Although some information are outdated, this provides a detailed description on generic model based power analysis:

Jobst, L., Bader, M., & Moshagen, M. (2023). A Tutorial on Assessing Statistical Power and Determining Sample Size for Structural Equation Models. Psychological Methods, 28, 207-221. https://doi.org/10.1037/met0000423 preprint

Citation

If you use semPower in publications, please cite the package as follows:

Moshagen, M., & Bader, M. (2024). semPower: General Power Analysis for Structural Equation Models. Behavior Research Methods, 56, 2901-2922. https://doi.org/10.3758/s13428-023-02254-7

Quick Examples for model-free power analyses

Determine the required sample size to detect misspecifications of a model (involving df = 100 degrees of freedom) corresponding to RMSEA = .05 with a power of 80% on an alpha error of .05:

ap <- semPower.aPriori(effect = .05, effect.measure = 'RMSEA', alpha = .05, power = .80, df = 100) summary(ap)

Determine the achieved power with a sample size of N = 1000 to detect misspecifications of a model (involving df = 100 degrees of freedom) corresponding to RMSEA = .05 on an alpha error of .05:

ph <- semPower.postHoc(effect = .05, effect.measure = 'RMSEA', alpha = .05, N = 1000, df = 100) summary(ph)

Determine the critical chi-square such that the associated alpha and beta errors are equal, assuming sample size of N = 1000, a model involving df = 100 degrees of freedom, and misspecifications corresponding to RMSEA = .05:

cp <- semPower.compromise(effect = .05, effect.measure = 'RMSEA', abratio = 1, N = 1000, df = 100) summary(cp)

Plot power as function of the sample size to detect misspecifications corresponding to RMSEA = .05 (assuming df = 100) on alpha = .05:

semPower.powerPlot.byN(effect = .05, effect.measure = 'RMSEA', alpha = .05, df = 100, power.min = .05, power.max = .99)

Plot power as function of the magnitude of effect (measured through the RMSEA assuming df = 100) at N = 500 on alpha = .05:

semPower.powerPlot.byEffect(effect.measure = 'RMSEA', alpha = .05, N = 500, df = 100, effect.min = .001, effect.max = .10)

Obtain the df of a model provided as lavaan model string (this requires the lavaan package):

lavModel <- ' f1 =~ x1 + x2 + x3 f2 =~ x4 + x5 + x6 ' semPower.getDf(lavModel)

Determine the required sample size to discriminate a model exhibiting an RMSEA of .04 on 44 df from a model with RMSEA = .05 on 41 df with a power of 80% on an alpha error of .05:

ap <- semPower.aPriori(effect = c(.04, .05), effect.measure = 'RMSEA', alpha = .05, power = .80, df = c(44, 41)) summary(ap)

See the manual for details.

Quick Examples for model-based power analyses

All the following examples determine the required sample size to detect the specified effect (a priori power analysis) with a power of 80% on alpha .05 and define the measurement model via the loadings argument. See the manual for details and for other ways to specify the measurement model.

CFA models

Determine sample size to detect that a correlation between the first and the second factor of at least .3 differs from zero: ``` Phi <- matrix(c( c(1, .3, .4, .5), c(.3, 1, .2, .6), c(.4, .2, 1, .1), c(.5, .6, .1, 1) ), ncol = 4, byrow = TRUE)

powerCFA <- semPower.powerCFA( # define type of power analysis type = 'a-priori', alpha = .05, power = .80, # define hypothesis Phi = Phi, nullEffect = 'cor = 0', nullWhich = c(1, 2), # define measurement model loadings = list( c(.7, .6, .5), c(.5, .8, .6), c(.7, .6, .5), c(.5, .8, .6)) ) summary(powerCFA) ```

Determine sample size to detect that the correlations between factor 1 and 2 (of .3) as well as between 3 and 4 (of .1) differ from each other: ``` Phi <- matrix(c( c(1, .3, .4, .5), c(.3, 1, .2, .6), c(.4, .2, 1, .1), c(.5, .6, .1, 1) ), ncol = 4, byrow = TRUE)

powerCFA <- semPower.powerCFA( # define type of power analysis type = 'a-priori', alpha = .05, power = .80, # define hypothesis Phi = Phi, nullEffect = 'corX = corZ', nullWhich = list(c(1, 2), c(3, 4)), # define measurement model loadings = list( c(.7, .6, .5), c(.5, .8, .6), c(.7, .6, .5), c(.5, .8, .6)) ) summary(powerCFA) ```

Determine sample size to detect that the correlations between factor 1 and 2 in group (of .3) differs from the one in group 2 (of .5): ``` Phi1 <- matrix(c( c(1, .3, .4, .5), c(.3, 1, .2, .6), c(.4, .2, 1, .1), c(.5, .6, .1, 1) ), ncol = 4, byrow = TRUE) Phi2 <- matrix(c( c(1, .5, .4, .5), c(.5, 1, .2, .6), c(.4, .2, 1, .1), c(.5, .6, .1, 1) ), ncol = 4, byrow = TRUE)

powerCFA <- semPower.powerCFA( # define type of power analysis type = 'a-priori', alpha = .05, power = .80, N = list(1, 1), # define hypothesis Phi = list(Phi1, Phi2), nullEffect = 'corA = corB', nullWhich = c(1, 2), # define measurement model loadings = list( c(.7, .6, .5), c(.5, .8, .6), c(.7, .6, .5), c(.5, .8, .6)) ) summary(powerCFA) ```

See the manual for more details.

Latent regression models

Determine sample size to detect that the first slope (of .2) differs from zero:

``` corXX <- matrix(c( c(1, .2, .6), c(.2, 1, .1), c(.6, .1, 1) ), ncol = 3, byrow = TRUE)

powerReg <- semPower.powerRegression( # define type of power analysis type = 'a-priori', alpha = .05, power = .80, # define hypothesis slopes = c(.2, .3, .4), corXX = corXX, nullEffect = 'slope = 0', nullWhich = 1, # define measurement model loadings = list( c(.7, .6, .5), c(.5, .8, .6), c(.7, .6, .5), c(.5, .8, .6)) ) summary(powerReg) ```

Determine sample size to detect that the first slope (of .2) differs from the third slope (of .4):

``` corXX <- matrix(c( c(1, .2, .6), c(.2, 1, .1), c(.6, .1, 1) ), ncol = 3, byrow = TRUE)

powerReg <- semPower.powerRegression( # define type of power analysis type = 'a-priori', alpha = .05, power = .80, # define hypothesis slopes = c(.2, .3, .4), corXX = corXX, nullEffect = 'slopeX = slopeZ', nullWhich = c(1, 3), # define measurement model loadings = list( c(.7, .6, .5), c(.5, .8, .6), c(.7, .6, .5), c(.5, .8, .6)) ) summary(powerReg) ```

Determine sample size to detect that the first slope in group 1 (of .2) differs from the first slope in group 2 (of .4):

``` corXX <- matrix(c( c(1, .2, .6), c(.2, 1, .1), c(.6, .1, 1) ), ncol = 3, byrow = TRUE)

powerReg <- semPower.powerRegression( # define type of power analysis type = 'a-priori', alpha = .05, power = .80, N = list(1, 1), # define hypothesis slopes = list(c(.2, .3, .4), c(.4, .3, .2)), corXX = corXX, nullEffect = 'slopeA = slopeB', nullWhich = 1, # define measurement model loadings = list( c(.7, .6, .5), c(.5, .8, .6), c(.7, .6, .5), c(.5, .8, .6)) ) summary(powerReg) ```

See the manual for more details.

Mediation models

Determine sample size to detect an indirect effect of at least .12 (= .3*.4) in a simple X -> M -> Y mediation based on an observed variable only model:

powerMed <- semPower.powerMediation( # define type of power analysis type = 'a-priori', alpha = .05, power = .80, # define hypothesis bYX = .25, bMX = .3, bYM = .4, nullEffect = 'ind = 0', # define observed only Lambda = diag(3) ) summary(powerMed)

Determine sample size to detect the indirect effect in group 1 (of .12) differs from the indirect effect in group 2 (of .25) in a simple X -> M -> Y mediation based on an observed variable only model:

powerMed <- semPower.powerMediation( # define type of power analysis type = 'a-priori', alpha = .05, power = .80, N = list(1, 1), # define hypothesis bYX = list(.25, .25), bMX = list(.3, .5), bYM = list(.4, .5), nullEffect = 'indA = indB', # define observed only Lambda = diag(3) ) summary(powerMed)

See the manual for more details.

Multigroup invariance

Determine sample size to detect metric-noninvariance across two groups of magnitude as defined through the different loadings:

powerMI <- semPower.powerMI( # define type of power analysis type = 'a-priori', alpha = .05, power = .80, N = list(1, 1), # define hypothesis comparison = 'configural', nullEffect = 'metric', # define measurement model loadings = list( # group 1 list( c(.7, .6, .5), c(.5, .8, .6), c(.7, .6, .5), c(.5, .8, .6)), # group 2 list( c(.6, .5, .4), c(.5, .8, .6), c(.6, .5, .4), c(.5, .8, .6)) ) ) summary(powerMI)

Determine sample size to detect that the latent means differ across groups:

powerMI <- semPower.powerMI( # define type of power analysis type = 'a-priori', alpha = .05, power = .80, N = list(1, 1), # define hypothesis comparison = c('loadings', 'intercepts'), nullEffect = c('loadings', 'intercepts', 'means'), # define measurement model (same for all groups) loadings = list( c(.7, .6, .5), c(.5, .8, .6), c(.7, .6, .5), c(.5, .8, .6)), # define indicator intercepts tau = list(rep(0, 12), rep(0, 12)), # define latent means Alpha = list( # group 1 rep(0, 4), # group 2 c(0.5, 0, 0.5, 0) ) ) summary(powerMI)

See the manual for more details and further hypotheses.

Longitudinal invariance

Determine sample size to detect metric-noninvariance across four measurement occasions of magnitude as defined through the different loadings:

powerLI <- semPower.powerLI( # define type of power analysis type = 'a-priori', alpha = .05, power = .80, # define hypothesis comparison = 'configural', nullEffect = 'metric', # define measurement model loadings = list( c(.7, .6, .5), # time 1 c(.6, .6, .5), # time 2 c(.5, .5, .4), # time 3 c(.4, .5, .4) # time 4 ), autocorResiduals = TRUE ) summary(powerLI)

See the manual for more details and further hypotheses.

Autoregressive models

Determine sample size to detect that the (wave-constant) lag-2 effects differ from zero in a 4-wave autoregressive model involving wave-constant lag-1 effects:

powerAutoreg <- semPower.powerAutoreg( # define type of power analysis 'a-priori', alpha = .05, power = .80, # define hypothesis nWaves = 4, autoregEffects = c(.6, .6, .6), lag2Effects = c(.2, .2), waveEqual = c('autoreg', 'lag2'), nullEffect = 'lag2=0', # define measurement model loadings = list( c(.5, .6, .7), c(.5, .6, .7), c(.5, .6, .7), c(.5, .6, .7) ), invariance = TRUE, autocorResiduals = TRUE ) summary(powerAutoreg)

Determine sample size to detect that the latent means differ across measurements in a 4 wave autoregressive model involving wave-constant lag-1 effects and wave-constant residual variances:

powerAutoreg <- semPower.powerAutoreg( # define type of power analysis 'a-priori', alpha = .05, power = .80, # define hypothesis nWaves = 4, autoregEffects = c(.6, .6, .6), variances = c(1, 1, 1, 1), means = c(0, .5, 1, .7), waveEqual = c('autoreg', 'var'), nullEffect = 'mean', # define measurement model loadings = list( c(.5, .6, .7), c(.5, .6, .7), c(.5, .6, .7), c(.5, .6, .7) ), standardized = FALSE, invariance = TRUE, autocorResiduals = TRUE ) summary(powerAutoreg)

See the manual for more details and further hypotheses.

ARMA models

Determine sample size to detect a that the lag-1 autoregressive effects differ across waves in a 10-wave ARMA model with wave-stable variances and moving average parameters :

powerARMA <- semPower.powerARMA( # define type of power analysis 'a-priori', alpha = .05, power = .80, # define hypothesis nWaves = 10, autoregLag1 = c(.5, .7, .6, .5, .7, .6, .6, .5, .6), mvAvgLag1 = rep(.3, 9), variances = rep(1, 10), waveEqual = c('var', 'mvAvg'), nullEffect = 'autoreg', # define measurement model loadings = rep(list(c(.6, .5, .6)), 10), invariance = TRUE, autocorResiduals = TRUE ) summary(powerARMA)

Same as above, but detect that the moving average parameters differ across waves:

powerARMA <- semPower.powerARMA( # define type of power analysis 'a-priori', alpha = .05, power = .80, # define hypothesis nWaves = 10, autoregLag1 = rep(.5, 9), mvAvgLag1 = c(.3, .4, .5, .3, .4, .5, .3, .4, .5), variances = rep(1, 10), waveEqual = c('var', 'autoreg'), nullEffect = 'mvAvg', # define measurement model loadings = rep(list(c(.6, .5, .6)), 10), invariance = TRUE, autocorResiduals = TRUE ) summary(powerARMA)

Same as above, but include (wave-constant) lag-2 effects and detect that the lag-2 autoregressive parameters differ from zero:

powerARMA <- semPower.powerARMA( # define type of power analysis 'a-priori', alpha = .05, power = .80, # define hypothesis nWaves = 10, autoregLag1 = rep(.5, 9), mvAvgLag1 = rep(.3, 9), autoregLag2 = rep(.2, 8), mvAvgLag2 = rep(.1, 8), variances = rep(1, 10), waveEqual = c('var', 'autoreg', 'mvAvg', 'autoregLag2', 'mvAvgLag2'), nullEffect = 'autoregLag2 = 0', # define measurement model loadings = rep(list(c(.6, .5, .6)), 10), invariance = TRUE, autocorResiduals = TRUE ) summary(powerARMA)

Same as above, but detect that residual variances differ across waves:

powerARMA <- semPower.powerARMA( # define type of power analysis 'a-priori', alpha = .05, power = .80, # define hypothesis nWaves = 10, autoregLag1 = rep(.5, 9), mvAvgLag1 = rep(.3, 9), variances = c(1, .8, .7, .6, .8, .7, .6, .8, .7, .6), waveEqual = c('mvAvg', 'autoreg'), nullEffect = 'var', # define measurement model loadings = rep(list(c(.6, .5, .6)), 10), invariance = TRUE, autocorResiduals = TRUE ) summary(powerARMA)

See the manual for more details and further hypotheses.

Cross-lagged panel models (with or without random intercept)

Determine sample size to detect a cross-lagged effect of X on Y of at least .10 in a two-wave CLPM:

powerCLPM <- semPower.powerCLPM( # define type of power analysis type = 'a-priori', alpha = .05, power = .80, # define hypothesis nullEffect = 'crossedX = 0', nWaves = 2, autoregEffects = c(.60, .70), crossedEffects = c(.10, .15), rXY = c(.3, .1), # define measurement model loadings = list( c(.7, .6, .5), c(.5, .8, .6), c(.7, .6, .5), c(.5, .8, .6)) ) summary(powerCLPM)

Same as above, but in a random-intercept CLPM involving 3 waves with observed variables only:

``` powerRICLPM <- semPower.powerRICLPM( # define type of power analysis type = 'a-priori', alpha = .05, power = .80, # define hypothesis nullEffect = 'crossedX = 0', nWaves = 3, autoregEffects = c(.60, .70), crossedEffects = c(.10, .15), rXY = c(.3, .1, .1), waveEqual = c('autoregX', 'autoregY', 'crossedX', 'crossedY'), # define measurement model Lambda = diag(6) ) summary(powerRICLPM)

```

Determine sample size to detect the cross-lagged effect of X on Y differs from the one of Y on X a two-wave CLPM:

powerCLPM <- semPower.powerCLPM( # define type of power analysis type = 'a-priori', alpha = .05, power = .80, # define hypothesis nullEffect = 'crossedX = crossedY', nWaves = 2, autoregEffects = c(.60, .70), crossedEffects = c(.10, .15), rXY = c(.3, .1), # define measurement model loadings = list( c(.7, .6, .5), c(.5, .8, .6), c(.7, .6, .5), c(.5, .8, .6)) ) summary(powerCLPM)

Determine sample size to detect the cross-lagged effect of X on Y in group 1 (of .10) differs from the one in group 2 (of .2):

powerCLPM <- semPower.powerCLPM( # define type of power analysis type = 'a-priori', alpha = .05, power = .80, N =list(1, 1), # define hypothesis nullEffect = 'crossedXA = crossedXB', nWaves = 2, autoregEffects = c(.60, .70), crossedEffects = list( # group 1 list(.10, .15), # group 2 list(.20, .15) ), rXY = c(.3, .1), # define measurement model loadings = list( c(.7, .6, .5), c(.5, .8, .6), c(.7, .6, .5), c(.5, .8, .6)) ) summary(powerCLPM)

See the manual for more details and for additional types of hypothesis.

LGCM models

Determine sample size to detect a that the mean of the slope factor differs from zero in a 3-wave LGCM:

powerLGCM <- semPower.powerLGCM( # define type of power analysis 'a-priori', alpha = .05, power = .80, # define hypothesis nWaves = 3, means = c(.5, .2), # i, s variances = c(1, .5), # i, s covariances = .25, nullEffect = 'sMean = 0', # define measurement model loadings = list( c(.6, .7, .5), c(.6, .7, .5), c(.6, .7, .5) ), autocorResiduals = TRUE ) summary(powerLGCM)

Same as above, but detect that the variance of the intercept factor differs from zero:

powerLGCM <- semPower.powerLGCM( # define type of power analysis 'a-priori', alpha = .05, power = .80, # define hypothesis nWaves = 3, means = c(.5, .2), # i, s variances = c(1, .5), # i, s covariances = .25, nullEffect = 'iVar = 0', # define measurement model loadings = list( c(.6, .7, .5), c(.6, .7, .5), c(.6, .7, .5) ), autocorResiduals = TRUE ) summary(powerLGCM)

Detect that the variance of a quadratic slope factor in a 4-wave LGCM differs from zero:

powerLGCM <- semPower.powerLGCM( # define type of power analysis 'a-priori', alpha = .05, power = .80, # define hypothesis nWaves = 4, quadratic = TRUE, means = c(.5, .2, .1), # i, s, s2 covariances = matrix(c( # i, s, s2 c(1, .2, .1), c(.2, .2, .05), c(.1, .05, .1) ), ncol = 3, byrow = TRUE), nullEffect = 's2Var = 0', # define measurement model loadings = list( c(.6, .7, .5), c(.6, .7, .5), c(.6, .7, .5), c(.6, .7, .5) ), autocorResiduals = TRUE ) summary(powerLGCM)

Detect that the intercept-slope covariance differs across groups in a two-group 3-wave LGCM:

powerLGCM <- semPower.powerLGCM( # define type of power analysis 'a-priori', alpha = .05, power = .80, N = list(1, 1), # define hypothesis nWaves = 3, means = c(.5, .2), variances = c(1, .5), covariances = list( c(.25), # group 1 c(.1)), # group 2 nullEffect = 'isCovA = isCovB', groupEqual = c('ivar', 'svar'), # define measurement model loadings = list( c(.6, .7, .5), c(.6, .7, .5), c(.6, .7, .5) ), autocorResiduals = TRUE ) summary(powerLGCM)

See the manual for more details and further hypotheses.

Simulated power analysis

Perform a simulated power-analysis with 500 replications and non-normal data with a population multivariate skewness of 10 and multivariate kurtosis of 200 to determine the sample size to detect that a correlation between the first and the second factor of at least .3 differs from zero: ``` Phi <- matrix(c( c(1, .3, .4, .5), c(.3, 1, .2, .6), c(.4, .2, 1, .1), c(.5, .6, .1, 1) ), ncol = 4, byrow = TRUE)

set.seed(1234) powerCFA <- semPower.powerCFA( # define type of power analysis type = 'a-priori', alpha = .05, power = .80, # define hypothesis Phi = Phi, nullEffect = 'cor = 0', nullWhich = c(1, 2), # define measurement model loadings = list( c(.7, .6, .5), c(.5, .8, .6), c(.7, .6, .5), c(.5, .8, .6)), # request simulated power analysis simulatedPower = TRUE, simOptions = list( nReplications = 500, type = 'mnonr', skewness = 10, kurtosis = 200 ))

summary(powerCFA) ```

See the manual for more details.