sirt (version 3.9-4)

invariance.alignment: Alignment Procedure for Linking under Approximate Invariance

Description

The function invariance.alignment performs alignment under approximate invariance for $G$ groups and $I$ items (Asparouhov & Muthen, 2014; Byrne & van de Vijver, 2017; DeMars, xxxx; Finch, 2016; Fischer & Karl, 2019; Flake & McCoach, 2018; Kim et al., 2017; Marsh et al., 2018; Muthen & Asparouhov, 2014, 2018; Pokropek, Davidov & Schmidt, 2019). It is assumed that item loadings and intercepts are previously estimated as a unidimensional factor model under the assumption of a factor with zero mean and a variance of one.

The function invariance_alignment_constraints postprocesses the output of the invariance.alignment function and estimates item parameters under equality constraints for prespecified absolute values of parameter tolerance.

The function invariance_alignment_simulate simulates a one-factor model for multiple groups for given matrices of $\nu$ and $\lambda$ parameters of item intercepts and item slopes (see Example 6).

The function invariance_alignment_cfa_config estimates one-factor models separately for each group as a preliminary step for invariance alignment (see Example 6). Sampling weights are accommodated by the argument weights.

Usage

invariance.alignment(lambda, nu, wgt=NULL, align.scale=c(1, 1),
    align.pow=c(.5, .5), eps=1e-3, psi0.init=NULL, alpha0.init=NULL, center=FALSE,
    optimizer="optim", fixed=NULL, ...)
# S3 method for invariance.alignment
summary(object, digits=3, file=NULL, ...)
invariance_alignment_constraints(model, lambda_parm_tol, nu_parm_tol )
# S3 method for invariance_alignment_constraints
summary(object, digits=3, file=NULL, ...)
invariance_alignment_simulate(nu, lambda, err_var, mu, sigma, N)
invariance_alignment_cfa_config(dat, group, weights=NULL, verbose=FALSE, ...)

Arguments

lambda

A $G\times I$ matrix with item loadings

nu

A $G\times I$ matrix with item intercepts

wgt

A $G\times I$ matrix for weighing groups for each item

align.scale

A vector of length two containing scale parameter $a_{\lambda }$ and $a_{\nu }$ (see Details)

align.pow

A vector of length two containing power $p_{\lambda }$ and $p_{\nu }$ (see Details)

eps

A parameter in the optimization function

psi0.init

An optional vector of initial ${\psi }_0$ parameters

alpha0.init

An optional vector of initial ${\alpha }_0$ parameters

center

Logical indicating whether estimated means and standard deviations should be centered.

optimizer

Name of the optimizer chosen for alignment. Options are "optim" (using stats::optim) or "nlminb" (using stats::nlminb).

fixed

Logical indicating whether SD of first group should be fixed to one. If fixed=FALSE, the product of all SDs is set to one. If NULL, then fixed is automatically chosen by default. For many groups, fixed=FALSE is chosen.

object

Object of class invariance.alignment

digits

Number of digits used for rounding

file

Optional file name in which summary should be sunk

…

Further optional arguments to be passed

model

Model of class invariance.alignment

lambda_parm_tol

Parameter tolerance for $\lambda$ parameters

nu_parm_tol

Parameter tolerance for $\nu$ parameters

err_var

Error variance

mu

Vector of means

sigma

Vector of standard deviations

N

Vector of sample sizes per group

dat

Dataset with items

group

Vector containing group indicators

weights

Optional vector of sampling weights

verbose

Logical indicating whether progress should be printed

Value

A list with following entries

pars

Aligned distribution parameters

itempars.aligned

Aligned item parameters for all groups

es.invariance

Effect sizes of approximate invariance

lambda.aligned

Aligned ${\lambda }_{ig,1}$ parameters

lambda.resid

Residuals of ${\lambda }_{ig,1}$ parameters

nu.aligned

Aligned ${\nu }_{ig,1}$ parameters

nu.resid

Residuals of ${\nu }_{ig,1}$ parameters

Niter

Number of iterations for $f_{\lambda }$ and $f_{\nu }$ optimization functions

fopt

Minimum optimization value

align.scale

Used alignment scale parameters

align.pow

Used alignment power parameters

Details

For $G$ groups and $I$ items, item loadings ${\lambda }_{ig0}$ and intercepts ${\nu }_{ig0}$ are available and have been estimated in a 1-dimensional factor analysis assuming a standardized factor.

The alignment procedure searches means ${\alpha }_{g0}$ and standard deviations ${\psi }_{g0}$ using an alignment optimization function $F$. This function is defined as $$F=\sum _i\sum _{g_1<g_2}{w}_{i,g1}{w}_{i,g2}{f}_{\lambda }({\lambda }_{i{g}_1,1}-{\lambda }_{i{g}_2,1})+\sum _i\sum _{g_1<g_2}{w}_{i,g1}{w}_{i,g2}{f}_{\nu }({\nu }_{i{g}_1,1}-{\nu }_{i{g}_2,1})$$ where the aligned item parameters ${\lambda }_{ig,1}$ and ${\nu }_{ig,1}$ are defined such that

$${\lambda }_{ig,1}={\lambda }_{ig0}/{\psi }_{g0}\text{and}{\nu }_{ig,1}={\nu }_{ig0}-{\alpha }_{g0}{\lambda }_{ig0}/{\psi }_{g0}$$ and the optimization functions are defined as $$f_{\lambda }(x)=|x/a_{\lambda }{|}^{p_{\lambda }}\approx [(x/a_{\lambda }{)}^2+\epsilon {]}^{p_{\lambda }/2}\text{and}{f}_{\nu }(x)=|x/a_{\nu }{]}^{p_{\nu }}\approx [(x/a_{\nu }{)}^2+\epsilon {]}^{p_{\nu }/2}$$ using a small $\epsilon >0$ (e.g. .001) to obtain a differentiable optimization function. For $p_{\nu }=0$ or $p_{\lambda }=0$, the optimization function essentially counts the number of different parameter and mimicks a $L_0$ penalty which is zero iff the argument is zero and one otherwise. It is approximated by $$f(x)=2/(1+\exp (-\gamma \sqrt{x^2+\epsilon })-1$$ (Oelker & Tutz, 2017).

For identification reasons, the product ${\mathrm{\Pi }}_g{\psi }_{g0}$ of all group standard deviations is set to one. The mean ${\alpha }_{g0}$ of the first group is set to zero.

Note that Asparouhov and Muthen (2014) use $a_{\lambda }=a_{\nu }=1$ (which can be modified in align.scale) and $p_{\lambda }=p_{\nu }=0.5$ (which can be modified in align.pow). In case of $p_{\lambda }=2$, the penalty is approximately $f_{\lambda }(x)=x^2$, in case of $p_{\lambda }=0.5$ it is approximately $f_{\lambda }(x)=\sqrt{|x|}$. Note that sirt used a different parametrization in versions up to 3.5. The $p$ parameters have to be halved for consistency with previous versions (e.g., the Asparouhov & Muthen parametrization corresponds to $p=.25$; see also Fischer & Karl, 2019, for an application of the previous parametrization).

Effect sizes of approximate invariance based on $R^2$ have been proposed by Asparouhov and Muthen (2014). These are calculated separately for item loading and intercepts, resulting in $R_{\lambda }^2$ and $R_{\nu }^2$ measures which are included in the output es.invariance. In addition, the average correlation of aligned item parameters among groups (rbar) is reported.

Metric invariance means that all aligned item loadings ${\lambda }_{ig,1}$ are equal across groups and therefore $R_{\lambda }^2=1$. Scalar invariance means that all aligned item loadings ${\lambda }_{ig,1}$ and aligned item intercepts ${\nu }_{ig,1}$ are equal across groups and therefore $R_{\lambda }^2=1$ and $R_{\nu }^2=1$ (see Vandenberg & Lance, 2000).

References

Asparouhov, T., & Muthen, B. (2014). Multiple-group factor analysis alignment. Structural Equation Modeling, 21(4), 1-14. 10.1080/10705511.2014.919210

Byrne, B. M., & van de Vijver, F. J. R. (2017). The maximum likelihood alignment approach to testing for approximate measurement invariance: A paradigmatic cross-cultural application. Psicothema, 29(4), 539-551. 10.7334/psicothema2017.178

DeMars, C. E. (2019). Alignment as an alternative to anchor purification in DIF analyses. Structural Equation Modeling, xxx(x), xxx-xxx. 10.1080/10705511.2019.1617151

Finch, W. H. (2016). Detection of differential item functioning for more than two groups: A Monte Carlo comparison of methods. Applied Measurement in Education, 29,(1), 30-45, 10.1080/08957347.2015.1102916

Fischer, R., & Karl, J. A. (2019). A primer to (cross-cultural) multi-group invariance testing possibilities in R. Frontiers in Psychology | Cultural Psychology, 10:1507. 10.3389/fpsyg.2019.01507

Flake, J. K., & McCoach, D. B. (2018). An investigation of the alignment method with polytomous indicators under conditions of partial measurement invariance. Structural Equation Modeling, 25(1), 56-70. 10.1080/10705511.2017.1374187

Kim, E. S., Cao, C., Wang, Y., & Nguyen, D. T. (2017). Measurement invariance testing with many groups: A comparison of five approaches. Structural Equation Modeling, 24(4), 524-544. 10.1080/10705511.2017.1304822

Marsh, H. W., Guo, J., Parker, P. D., Nagengast, B., Asparouhov, T., Muthen, B., & Dicke, T. (2018). What to do when scalar invariance fails: The extended alignment method for multi-group factor analysis comparison of latent means across many groups. Psychological Methods, 23(3), 524-545. 10.1037/met0000113

Muthen, B., & Asparouhov, T. (2014). IRT studies of many groups: The alignment method. Frontiers in Psychology | Quantitative Psychology and Measurement, 5:978. 10.3389/fpsyg.2014.00978

Muthen, B., & Asparouhov, T. (2018). Recent methods for the study of measurement invariance with many groups: Alignment and random effects. Sociological Methods & Research, 47(4), 637-664. 10.1177/0049124117701488

Oelker, M. R., & Tutz, G. (2017). A uniform framework for the combination of penalties in generalized structured models. Advances in Data Analysis and Classification, 11(1), 97-120. 10.1007/s11634-015-0205-y

Pokropek, A., Davidov, E., & Schmidt, P. (2019). A Monte Carlo simulation study to assess the appropriateness of traditional and newer approaches to test for measurement invariance. Structural Equation Modeling, 26(5), 724-744. 10.1080/10705511.2018.1561293

Vandenberg, R. J., & Lance, C. E. (2000). A review and synthesis of the measurement invariance literature: Suggestions, practices, and recommendations for organizational research. Organizational Research Methods, 3, 4-70. 10.1177/109442810031002s

See Also

For IRT linking see also linking.haberman or TAM::tam.linking.

For modeling random item effects for loadings and intercepts see mcmc.2pno.ml.

Examples

# NOT RUN {
#############################################################################
# EXAMPLE 1: Item parameters cultural activities
#############################################################################

data(data.activity.itempars, package="sirt")
lambda <- data.activity.itempars$lambda
nu <- data.activity.itempars$nu
Ng <-  data.activity.itempars$N
wgt <- matrix( sqrt(Ng), length(Ng), ncol(nu) )

#***
# Model 1: Alignment using a quadratic loss function
mod1 <- sirt::invariance.alignment( lambda, nu, wgt )
summary(mod1)

#****
# Model 2: Different powers for alignment
mod2 <- sirt::invariance.alignment( lambda, nu, wgt,  align.pow=c(.5,1),
              align.scale=c(.95,.95))
summary(mod2)

# compare means from Models 1 and 2
plot( mod1$pars$alpha0, mod2$pars$alpha0, pch=16,
    xlab="M (Model 1)", ylab="M (Model 2)", xlim=c(-.3,.3), ylim=c(-.3,.3) )
lines( c(-1,1), c(-1,1), col="gray")
round( cbind( mod1$pars$alpha0, mod2$pars$alpha0 ), 3 )
round( mod1$nu.resid, 3)
round( mod2$nu.resid,3 )

# L0 penalty
mod2b <- sirt::invariance.alignment( lambda, nu, wgt,  align.pow=c(0,0),
              align.scale=c(.3,.3))
summary(mod2b)

#****
# Model 3: Low powers for alignment of scale and power
# Note that setting increment.factor larger than 1 seems necessary
mod3 <- sirt::invariance.alignment( lambda, nu, wgt, align.pow=c(.5,.75),
            align.scale=c(.55,.55), psi0.init=mod1$psi0, alpha0.init=mod1$alpha0 )
summary(mod3)

# compare mean and SD estimates of Models 1 and 3
plot( mod1$pars$alpha0, mod3$pars$alpha0, pch=16)
plot( mod1$pars$psi0, mod3$pars$psi0, pch=16)

# compare residuals for Models 1 and 3
# plot lambda
plot( abs(as.vector(mod1$lambda.resid)), abs(as.vector(mod3$lambda.resid)),
      pch=16, xlab="Residuals lambda (Model 1)",
      ylab="Residuals lambda (Model 3)", xlim=c(0,.1), ylim=c(0,.1))
lines( c(-3,3),c(-3,3), col="gray")
# plot nu
plot( abs(as.vector(mod1$nu.resid)), abs(as.vector(mod3$nu.resid)),
      pch=16, xlab="Residuals nu (Model 1)", ylab="Residuals nu (Model 3)",
      xlim=c(0,.4),ylim=c(0,.4))
lines( c(-3,3),c(-3,3), col="gray")

# }
# NOT RUN {
#############################################################################
# EXAMPLE 2: Comparison 4 groups | data.inv4gr
#############################################################################

data(data.inv4gr)
dat <- data.inv4gr
miceadds::library_install("semTools")

model1 <- "
    F=~ I01 + I02 + I03 + I04 + I05 + I06 + I07 + I08 + I09 + I10 + I11
    F ~~ 1*F
    "

res <- semTools::measurementInvariance(model1, std.lv=TRUE, data=dat, group="group")
  ##   Measurement invariance tests:
  ##
  ##   Model 1: configural invariance:
  ##       chisq        df    pvalue       cfi     rmsea       bic
  ##     162.084   176.000     0.766     1.000     0.000 95428.025
  ##
  ##   Model 2: weak invariance (equal loadings):
  ##       chisq        df    pvalue       cfi     rmsea       bic
  ##     519.598   209.000     0.000     0.973     0.039 95511.835
  ##
  ##   [Model 1 versus model 2]
  ##     delta.chisq      delta.df delta.p.value     delta.cfi
  ##         357.514        33.000         0.000         0.027
  ##
  ##   Model 3: strong invariance (equal loadings + intercepts):
  ##       chisq        df    pvalue       cfi     rmsea       bic
  ##    2197.260   239.000     0.000     0.828     0.091 96940.676
  ##
  ##   [Model 1 versus model 3]
  ##     delta.chisq      delta.df delta.p.value     delta.cfi
  ##        2035.176        63.000         0.000         0.172
  ##
  ##   [Model 2 versus model 3]
  ##     delta.chisq      delta.df delta.p.value     delta.cfi
  ##        1677.662        30.000         0.000         0.144
  ##

# extract item parameters separate group analyses
ipars <- lavaan::parameterEstimates(res$fit.configural)
# extract lambda's: groups are in rows, items in columns
lambda <- matrix( ipars[ ipars$op=="=~", "est"], nrow=4,  byrow=TRUE)
colnames(lambda) <- colnames(dat)[-1]
# extract nu's
nu <- matrix( ipars[ ipars$op=="~1"  & ipars$se !=0, "est" ], nrow=4,  byrow=TRUE)
colnames(nu) <- colnames(dat)[-1]

# Model 1: least squares optimization
mod1 <- sirt::invariance.alignment( lambda=lambda, nu=nu )
summary(mod1)
  ##   Effect Sizes of Approximate Invariance
  ##          loadings intercepts
  ##   R2       0.9826     0.9972
  ##   sqrtU2   0.1319     0.0526
  ##   rbar     0.6237     0.7821
  ##   -----------------------------------------------------------------
  ##   Group Means and Standard Deviations
  ##     alpha0  psi0
  ##   1  0.000 0.965
  ##   2 -0.105 1.098
  ##   3 -0.081 1.011
  ##   4  0.171 0.935

# Model 2: sparse target function
mod2 <- sirt::invariance.alignment( lambda=lambda, nu=nu, align.pow=c(.5,.5) )
summary(mod2)
  ##   Effect Sizes of Approximate Invariance
  ##          loadings intercepts
  ##   R2       0.9824     0.9972
  ##   sqrtU2   0.1327     0.0529
  ##   rbar     0.6237     0.7856
  ##   -----------------------------------------------------------------
  ##   Group Means and Standard Deviations
  ##     alpha0  psi0
  ##   1 -0.002 0.965
  ##   2 -0.107 1.098
  ##   3 -0.083 1.011
  ##   4  0.170 0.935

#############################################################################
# EXAMPLE 3: European Social Survey data.ess2005
#############################################################################

data(data.ess2005)
lambda <- data.ess2005$lambda
nu <- data.ess2005$nu

# Model 1: least squares optimization
mod1 <- sirt::invariance.alignment( lambda=lambda, nu=nu, align.pow=c(2,2) )
summary(mod1)

# Model 2: sparse target function and definition of scales
mod2 <- sirt::invariance.alignment( lambda=lambda, nu=nu, control=list(trace=2) )
summary(mod2)

#############################################################################
# EXAMPLE 4: Linking with item parameters containing outliers
#############################################################################

# see Help file in linking.robust

# simulate some item difficulties in the Rasch model
I <- 38
set.seed(18785)
itempars <- data.frame("item"=paste0("I",1:I) )
itempars$study1 <- stats::rnorm( I, mean=.3, sd=1.4 )
# simulate DIF effects plus some outliers
bdif <- stats::rnorm(I, mean=.4, sd=.09)+( stats::runif(I)>.9 )* rep( 1*c(-1,1)+.4, each=I/2 )
itempars$study2 <- itempars$study1 + bdif
# create input for function invariance.alignment
nu <- t( itempars[,2:3] )
colnames(nu) <- itempars$item
lambda <- 1+0*nu

# linking using least squares optimization
mod1 <- sirt::invariance.alignment( lambda=lambda, nu=nu )
summary(mod1)
  ##   Group Means and Standard Deviations
  ##          alpha0 psi0
  ##   study1 -0.286    1
  ##   study2  0.286    1

# linking using powers of .5
mod2 <- sirt::invariance.alignment( lambda=lambda, nu=nu, align.pow=c(1,1) )
summary(mod2)
  ##   Group Means and Standard Deviations
  ##          alpha0 psi0
  ##   study1 -0.213    1
  ##   study2  0.213    1

# linking using powers of .25
mod3 <- sirt::invariance.alignment( lambda=lambda, nu=nu, align.pow=c(.5,.5) )
summary(mod3)
  ##   Group Means and Standard Deviations
  ##          alpha0 psi0
  ##   study1 -0.207    1
  ##   study2  0.207    1

#############################################################################
# EXAMPLE 5: Linking gender groups with data.math
#############################################################################

data(data.math)
dat <- data.math$data
dat.male <- dat[ dat$female==0, substring( colnames(dat),1,1)=="M"  ]
dat.female <- dat[ dat$female==1, substring( colnames(dat),1,1)=="M"  ]

#*************************
# Model 1: Linking using the Rasch model
mod1m <- sirt::rasch.mml2( dat.male )
mod1f <- sirt::rasch.mml2( dat.female )

# create objects for invariance.alignment
nu <- rbind( mod1m$item$thresh, mod1f$item$thresh )
colnames(nu) <- mod1m$item$item
rownames(nu) <- c("male", "female")
lambda <- 1+0*nu

# mean of item difficulties
round( rowMeans(nu), 3 )

# Linking using least squares optimization
res1a <- sirt::invariance.alignment( lambda, nu, align.scale=c( .3, .5 ) )
summary(res1a)

# Linking using optimization with absolute value function (pow=.5)
res1b <- sirt::invariance.alignment( lambda, nu, align.scale=c( .3, .5 ),
                align.pow=c(1,1) )
summary(res1b)

#-- compare results with Haberman linking
I <- ncol(dat.male)
itempartable <- data.frame( "study"=rep( c("male", "female"), each=I ) )
itempartable$item <- c( paste0(mod1m$item$item),  paste0(mod1f$item$item) )
itempartable$a <- 1
itempartable$b <- c( mod1m$item$b, mod1f$item$b )
# estimate linking parameters
res1c <- sirt::linking.haberman( itempars=itempartable )

#-- results of sirt::equating.rasch
x <- itempartable[ 1:I, c("item", "b") ]
y <- itempartable[ I + 1:I, c("item", "b") ]
res1d <- sirt::equating.rasch( x, y )
round( res1d$B.est, 3 )
  ##     Mean.Mean Haebara Stocking.Lord
  ##   1     0.032   0.032         0.029

#*************************
# Model 2: Linking using the 2PL model
I <- ncol(dat.male)
mod2m <- sirt::rasch.mml2( dat.male, est.a=1:I)
mod2f <- sirt::rasch.mml2( dat.female, est.a=1:I)

# create objects for invariance.alignment
nu <- rbind( mod2m$item$thresh, mod2f$item$thresh )
colnames(nu) <- mod2m$item$item
rownames(nu) <- c("male", "female")
lambda <- rbind( mod2m$item$a, mod2f$item$a )
colnames(lambda) <- mod2m$item$item
rownames(lambda) <- c("male", "female")

res2a <- sirt::invariance.alignment( lambda, nu, align.scale=c( .3, .5 ) )
summary(res2a)

res2b <- sirt::invariance.alignment( lambda, nu, align.scale=c( .3, .5 ),
                align.pow=c(1,1) )
summary(res2b)

# compare results with Haberman linking
I <- ncol(dat.male)
itempartable <- data.frame( "study"=rep( c("male", "female"), each=I ) )
itempartable$item <- c( paste0(mod2m$item$item),  paste0(mod2f$item$item ) )
itempartable$a <- c( mod2m$item$a, mod2f$item$a )
itempartable$b <- c( mod2m$item$b, mod2f$item$b )
# estimate linking parameters
res2c <- sirt::linking.haberman( itempars=itempartable )

#############################################################################
# EXAMPLE 6: Data from Asparouhov & Muthen (2014) simulation study
#############################################################################

G <- 3  # number of groups
I <- 5  # number of items
# define lambda and nu parameters
lambda <- matrix(1, nrow=G, ncol=I)
nu <- matrix(0, nrow=G, ncol=I)

# define size of noninvariance
dif <- 1

#- 1st group: N(0,1)
lambda[1,3] <- 1+dif*.4; nu[1,5] <- dif*.5

#- 2nd group: N(0.3,1.5)
gg <- 2 ; mu <- .3; sigma <- sqrt(1.5)
lambda[gg,5] <- 1-.5*dif; nu[gg,1] <- -.5*dif
nu[gg,] <- nu[gg,] + mu*lambda[gg,]
lambda[gg,] <- lambda[gg,] * sigma

#- 3nd group: N(.8,1.2)
gg <- 3 ; mu <- .8; sigma <- sqrt(1.2)
lambda[gg,4] <- 1-.7*dif; nu[gg,2] <- -.5*dif
nu[gg,] <- nu[gg,] + mu*lambda[gg,]
lambda[gg,] <- lambda[gg,] * sigma

# define alignment scale
align.scale <- c(.2,.4)   # Asparouhov and Muthen use c(1,1)
# define alignment powers
align.pow <- c(.5,.5)   # as in Asparouhov and Muthen

#*** estimate alignment parameters
mod1 <- sirt::invariance.alignment( lambda, nu, eps=.01, optimizer="optim",
            align.scale=align.scale, align.pow=align.pow, center=FALSE )
summary(mod1)

#--- find parameter constraints for prespecified tolerance
cmod1 <- sirt::invariance_alignment_constraints(model=mod1, nu_parm_tol=.4,
            lambda_parm_tol=.2 )
summary(cmod1)

#############################################################################
# EXAMPLE 7: Similar to Example 6, but with data simulation and CFA estimation
#############################################################################

#--- data simulation

set.seed(65)
G <- 3  # number of groups
I <- 5  # number of items
# define lambda and nu parameters
lambda <- matrix(1, nrow=G, ncol=I)
nu <- matrix(0, nrow=G, ncol=I)
err_var <- matrix(1, nrow=G, ncol=I)

# define size of noninvariance
dif <- 1
#- 1st group: N(0,1)
lambda[1,3] <- 1+dif*.4; nu[1,5] <- dif*.5
#- 2nd group: N(0.3,1.5)
gg <- 2 ;
lambda[gg,5] <- 1-.5*dif; nu[gg,1] <- -.5*dif
#- 3nd group: N(.8,1.2)
gg <- 3
lambda[gg,4] <- 1-.7*dif; nu[gg,2] <- -.5*dif
#- define distributions of groups
mu <- c(0,.3,.8)
sigma <- sqrt(c(1,1.5,1.2))
N <- rep(1000,3) # sample sizes per group

#* simulate data
dat <- sirt::invariance_alignment_simulate(nu, lambda, err_var, mu, sigma, N)
head(dat)

#--- estimate CFA models
pars <- sirt::invariance_alignment_cfa_config(dat[,-1], group=dat$group)
print(pars)

#--- invariance alignment
# define alignment scale
align.scale <- c(.2,.4)
# define alignment powers
align.pow <- c(.5,.5)
mod1 <- sirt::invariance.alignment( lambda=pars$lambda, nu=pars$nu, eps=.01,
            optimizer="optim", align.scale=align.scale, align.pow=align.pow, center=FALSE)
#* find parameter constraints for prespecified tolerance
cmod1 <- sirt::invariance_alignment_constraints(model=mod1, nu_parm_tol=.4,
            lambda_parm_tol=.2 )
summary(cmod1)

#--- estimate CFA models with sampling weights

#* simulate weights
weights <- stats::runif(sum(N), 0, 2)
#* estimate models
pars2 <- sirt::invariance_alignment_cfa_config(dat[,-1], group=dat$group, weights=weights)
print(pars2$nu)
print(pars$nu)
# }