semPower.powerCFA {semPower}

semPower.powerCFA

Description

Convenience function for performing power analyses for CFA models to reject one of the following hypotheses: (a) a zero correlation between two factors, (b) the equality of two correlations between factors, or (c) the equality of a correlation between two factors across two or more groups. This requires the lavaan package.

Usage

semPower.powerCFA(
  type,
  comparison = "restricted",
  Phi = NULL,
  nullEffect = "cor = 0",
  nullWhich = NULL,
  nullWhichGroups = NULL,
  ...
)

Arguments

type

type of power analysis, one of 'a-priori', 'post-hoc', 'compromise'.

comparison

comparison model, one of 'saturated' or 'restricted' (the default). This determines the df for power analyses. 'saturated' provides power to reject the model when compared to the saturated model, so the df equal the one of the hypothesized model. 'restricted' provides power to reject the hypothesized model when compared to an otherwise identical model that just omits the restrictions defined in nullEffect, so the df equal the number of restrictions.

Phi

either a single number defining the correlation between exactly two factors or the factor correlation matrix. A list for multiple group models.

nullEffect

defines the hypothesis of interest, must be one of 'cor = 0' (the default) to test whether a correlation is zero, 'corX = corZ' to test for the equality of correlations, 'corA = corB' to test for the equality of a correlation across groups, and loading = 0 to test whether a loading is zero. Define the correlations to be set to equality in nullWhich and the groups in nullWhichGroups.

nullWhich

vector of size 2 indicating which element in Lambda should equal zero when nullEffect = 'loading = 0', or which factor correlation in Phi is hypothesized to equal zero when nullEffect = 'cor = 0', or to restrict to equality across groups when nullEffect = 'corA = corB', or list of vectors defining which correlations to restrict to equality when nullEffect = 'corX = corZ'. Can also contain more than two correlations, e.g., list(c(1, 2), c(1, 3), c(2, 3)) to set Phi[1, 2] = Phi[1, 3] = Phi[2, 3]. If omitted, the correlation between the first and the second factor is targeted, i. e., nullWhich = c(1, 2).

nullWhichGroups

for nullEffect = 'corA = corB', vector indicating the groups for which equality constrains should be applied, e.g. c(1, 3) to constrain the relevant parameters of the first and the third group. If NULL, all groups are constrained to equality.

...

mandatory further parameters related to the specific type of power analysis requested, see semPower.aPriori(), semPower.postHoc(), and semPower.compromise(), and parameters specifying the factor model. See details.

Details

This function performs a power analysis to reject various hypotheses arising in standard CFA models:

• nullEffect = 'loading = 0': Tests the hypothesis that a loading is zero.

• nullEffect = 'cor = 0': Tests the hypothesis that the correlation between two factors is zero.

• nullEffect = 'corX = corZ': Tests the hypothesis that two or more correlations between three or more factors are equal to each other.

• nullEffect = 'corA = corB': Tests the hypothesis that the correlation between two factors is equal in two or more groups (always assuming metric invariance).

For hypotheses regarding regression relationships between factors, see semPower.powerRegression(). For hypotheses regarding mediation effects, see semPower.powerMediation(). For hypotheses regarding measurement invariance, see semPower.powerMI().

Beyond the arguments explicitly contained in the function call, additional arguments are required specifying the factor model and the requested type of power analysis.

Additional arguments related to the definition of the factor model:

• Lambda: The factor loading matrix (with the number of columns equaling the number of factors).

• loadings: Can be used instead of Lambda: Defines the primary loadings for each factor in a list structure, e. g. loadings = list(c(.5, .4, .6), c(.8, .6, .6, .4)) defines a two factor model with three indicators loading on the first factor by .5, , 4., and .6, and four indicators loading on the second factor by .8, .6, .6, and .4.

• nIndicator: Can be used instead of Lambda: Used in conjunction with loadM. Defines the number of indicators by factor, e. g., nIndicator = c(3, 4) defines a two factor model with three and four indicators for the first and second factor, respectively. nIndicator can also be a single number to define the same number of indicators for each factor.

• loadM: Can be used instead of Lambda: Used in conjunction with nIndicator. Defines the loading either for all indicators (if a single number is provided) or separately for each factor (if a vector is provided), e. g. loadM = c(.5, .6) defines the loadings of the first factor to equal .5 and those of the second factor do equal .6.

So either Lambda, or loadings, or nIndicator and loadM need to be defined. If the model contains observed variables only, use Lambda = diag(x) where x is the number of variables.

Additional arguments related to the requested type of power analysis:

• alpha: The alpha error probability. Required for type = 'a-priori' and type = 'post-hoc'.

• Either beta or power: The beta error probability and the statistical power (1 - beta), respectively. Only for type = 'a-priori'.

• N: The sample size. Always required for type = 'post-hoc' and type = 'compromise'. For type = 'a-priori' and multiple group analysis, N is a list of group weights.

• abratio: The ratio of alpha to beta. Only for type = 'compromise'.

If a simulated power analysis (simulatedPower = TRUE) is requested, optional arguments can be provided as a list to simOptions:

• nReplications: The targeted number of simulation runs. Defaults to 250, but larger numbers greatly improve accuracy at the expense of increased computation time.

• minConvergenceRate: The minimum convergence rate required, defaults to .5. The maximum actual simulation runs are increased by a factor of 1/minConvergenceRate.

• type: specifies whether the data should be generated from a population assuming multivariate normality ('normal'; the default), or based on an approach generating non-normal data ('IG', 'mnonr', 'RC', or 'VM').

• missingVars: vector specifying the variables containing missing data (defaults to NULL).

• missingVarProp: can be used instead of missingVars: The proportion of variables containing missing data (defaults to zero).

• missingProp: The proportion of missingness for variables containing missing data (defaults to zero), either a single value or a vector giving the probabilities for each variable.

• missingMechanism: The missing data mechanism, one of MCAR (the default), MAR, or NMAR. The approaches generating non-normal data require additional arguments detailed below.

• nCores: The number of cores to use for parallel processing. Defaults to 1 (= no parallel processing). This requires the doSNOW package.

type = 'IG' implements the independent generator approach (IG, Foldnes & Olsson, 2016) approach specifying third and fourth moments of the marginals, and thus requires that skewness (skewness) and excess kurtosis (kurtosis) for each variable are provided as vectors. This requires the covsim package.

type = 'mnonr' implements the approach suggested by Qu, Liu, & Zhang (2020) and requires provision of Mardia's multivariate skewness (skewness) and kurtosis (kurtosis), where skewness must be non-negative and kurtosis must be at least 1.641 skewness + p (p + 0.774), where p is the number of variables. This requires the mnonr package.

type = 'RK' implements the approach suggested by Ruscio & Kaczetow (2008) and requires provision of the population distributions of each variable (distributions). distributions must be a list (if all variables shall be based on the same population distribution) or a list of lists. Each component must specify the population distribution (e.g. rchisq) and additional arguments (list(df = 2)).

type = 'VM' implements the third-order polynomial method (Vale & Maurelli, 1983) specifying third and fourth moments of the marginals, and thus requires that skewness (skewness) and excess kurtosis (kurtosis) for each variable are provided as vectors. This requires the semTools package.

Value

a list. Use the summary method to obtain formatted results. Beyond the results of the power analysis and a number of effect size measures, the list contains the following components:

Sigma

the population covariance matrix. A list for multiple group models.

mu

the population mean vector or NULL when no meanstructure is involved. A list for multiple group models.

SigmaHat

the H0 model implied covariance matrix. A list for multiple group models.

muHat

the H0 model implied mean vector or NULL when no meanstructure is involved. A list for multiple group models.

modelH0

lavaan H0 model string.

modelH1

lavaan H1 model string or NULL when the comparison refers to the saturated model.

simRes

detailed simulation results when a simulated power analysis (simulatedPower = TRUE) was performed.

See Also

semPower.genSigma() semPower.aPriori() semPower.postHoc() semPower.compromise()

Examples

## Not run: 
# get required N to detect a correlation of >= .2 between two factors
# with a power of 95% on alpha = 5%, where the factors are  
# measured by 5 and 6 indicators, respectively, and all loadings are equal to .5
powercfa <- semPower.powerCFA(type = 'a-priori',
                              Phi = .2, 
                              nIndicator = c(5, 6), loadM = .5,
                              alpha = .05, beta = .05)
# show summary
summary(powercfa)
# optionally use lavaan to verify the model was set-up as intended
lavaan::sem(powercfa$modelH1, sample.cov = powercfa$Sigma, 
sample.nobs = powercfa$requiredN, sample.cov.rescale = FALSE)
lavaan::sem(powercfa$modelH0, sample.cov = powercfa$Sigma, 
sample.nobs = powercfa$requiredN, sample.cov.rescale = FALSE)

# same as above, but determine power with N = 500 on alpha = .05
powercfa <- semPower.powerCFA(type = 'post-hoc',
                              Phi = .2, 
                              nIndicator = c(5, 6), loadM = .5,
                              alpha = .05, N = 500)

# same as above, but determine the critical chi-square with N = 500 so that alpha = beta
powercfa <- semPower.powerCFA(type = 'compromise',
                              Phi = .2, 
                              nIndicator = c(5, 6), loadM = .5,
                              abratio = 1, N = 500)
                              
# same as above, but compare to the saturated model 
# (rather than to the less restricted model)
powercfa <- semPower.powerCFA(type = 'a-priori',
                              comparison = 'saturated',
                              Phi = .2, 
                              nIndicator = c(5, 6), loadM = .5,
                              alpha = .05, beta = .05)
                              
# same as above, but provide a reduced loading matrix defining
# three indicators with loadings of .7, .6, and .5 on the first factor and
# four indicators with loadings of .5, .6, .4, .8 on the second factor 
powercfa <- semPower.powerCFA(type = 'a-priori',
                              Phi = .2, 
                              loadings = list(c(.7, .6, .5), 
                                              c(.5, .6, .4, .8)),
                              alpha = .05, beta = .05)

# detect that the loading of indicator 4 on the first factor differs from zero
Lambda <- matrix(c(
  c(.8, 0),
  c(.4, 0),
  c(.6, 0),
  c(.1, .5),
  c(0, .6),
  c(0, .7)
), ncol = 2, byrow = TRUE)
powercfa <- semPower.powerCFA(type = 'a-priori',
                              Phi = .2,
                              nullEffect = 'loading = 0', 
                              nullWhich = c(4, 1), 
                              Lambda = Lambda,
                              alpha = .05, beta = .05)


# get required N to detect a correlation of >= .3 between factors 1 and 3  
# in a three factor model. Factors are measured by 3 indicators each, and all loadings 
# on the first, second, and third factor are .5, .6, and .7, respectively.
Phi <- matrix(c(
  c(1.00, 0.20, 0.30),
  c(0.20, 1.00, 0.10),
  c(0.30, 0.10, 1.00)
), ncol = 3,byrow = TRUE)

powercfa <- semPower.powerCFA(type = 'a-priori',
                              Phi = Phi,
                              nullWhich = c(1, 3), 
                              nIndicator = c(3, 3, 3), loadM = c(.5, .6, .7),
                              alpha = .05, beta = .05)

# same as above, but ask for N to detect that 
# the correlation between factors 1 and 2 (of r = .2) differs from
# the correlation between factors 2 and 3 (of r = .3).
powercfa <- semPower.powerCFA(type = 'a-priori',
                              Phi = Phi,
                              nullEffect = 'corX = corZ',
                              nullWhich = list(c(1, 2), c(1, 3)), 
                              nIndicator = c(3, 3, 3), loadM = c(.5, .6, .7),
                              alpha = .05, beta = .05)
                              
# same as above, but ask for N to detect that all three correlations are unequal
powercfa <- semPower.powerCFA(type = 'a-priori',
                              Phi = Phi,
                              nullEffect = 'corX = corZ',
                              nullWhich = list(c(1, 2), c(1, 3), c(2, 3)), 
                              nIndicator = c(3, 3, 3), loadM = c(.5, .6, .7),
                              alpha = .05, beta = .05)
                              
# get required N to detect that the correlation between two factors
# in group 1 (of r = .2) differs from the one in group 2 (of r = .4). 
# The measurement model is identical for both groups:
# The first factor is measured by 3 indicators loading by .7 each, 
# the second factor is measured by 6 indicators loading by .5 each.
# Both groups are sized equally (N = list(1, 1)).
powercfa <- semPower.powerCFA(type = 'a-priori', 
                              nullEffect = 'corA = corB',
                              Phi = list(.2, .4), 
                              loadM = c(.7, .5), 
                              nIndicator = c(3, 6), 
                              alpha = .05, beta = .05, N = list(1, 1))

# request a simulated post-hoc power analysis with 500 replications.
set.seed(300121)
powercfa <- semPower.powerCFA(type = 'post-hoc',
                              Phi = .2, 
                              nIndicator = c(5, 6), loadM = .5,
                              alpha = .05, N = 500, 
                              simulatedPower = TRUE, 
                              simOptions = list(nReplications = 500))


## End(Not run)