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.

`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")
```

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 Although some information are outdated, this provides a detailed description on generic model based power analysis: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

If you use `semPower`

in publications, please cite the package as follows:

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.