Approximate Measurement Invariance by Factor Alignment

Estimates the group factor means and factor variances that make the measurement parameters as nearly invariant as possible across groups, following the alignment method of Asparouhov and Muthén (2014). The invariance ladder in measurement_invariance asks whether loadings and intercepts are exactly equal across groups. With many groups that hypothesis is essentially never true, the ladder stalls at configural invariance, and the comparison of factor means the researcher wanted never happens. Alignment takes the configural solution as given and searches for the group factor means and variances that concentrate the noninvariance in a few parameters instead of spreading it thinly over many, so that factor means stay comparable without claiming that exact invariance holds. Requires lavaan.

Usage

measurement_alignment(
  data,
  items,
  group,
  model = NULL,
  alignment = c("fixed", "free"),
  estimator = "ML",
  n_starts = 10,
  seed = NULL,
  epsilon = 0.01
)

Arguments

data: A data.frame holding the items and the grouping variable.
items: Character vector naming the indicator columns of data. Three or more items are required, and each must be numeric.
group: Single character string naming the grouping column of data. Two or more groups are required.
model: Optional lavaan model syntax for the measurement model, for the case where the default single factor over items is not what is wanted (a residual covariance, for example). The syntax must define exactly one latent variable and its indicators must be exactly items. The factor is standardized (mean 0, variance 1) in every group by this function, so do not set its scale in the syntax. When NULL (the default) the model f =~ item1 + item2 + ... is used.
alignment: Identification rule for the metric of the factor, either "fixed" (the default) or "free". Under "fixed" the first group's factor mean is held at 0 and its factor variance at 1. Under "free" the group factor means are constrained to average 0 and the group factor standard deviations to have a geometric mean of 1, so no group serves as the reference. The choice sets the metric the factor means and variances are reported on; see Details.
estimator: Estimator passed to lavaan for the configural model. Defaults to "ML"; "MLR" supplies the robust corrections.
n_starts: Number of starting values for the optimizer, a positive integer (default 10). The first start sets every factor mean to 0 and every factor variance to 1, the second uses a median based heuristic, and any remaining starts are random. The simplicity function has local minima, so several starts is the default rather than a refinement.
seed: Optional integer seed for the random starts. Defaults to NULL, which uses the caller's random number state and leaves it alone. When an integer is supplied the seed is set locally and the caller's state is restored on exit.
epsilon: Smoothing constant $\epsilon > 0$ inside the component loss function, default 0.01. Smaller values push the loss closer to $\sqrt{|x|}$ and make the surface harder to optimize; larger values smooth it at the cost of blurring the distinction between a few large differences and many small ones.

Value

A data.frame (class dmar_tbl) with one row per group and columns group (the group label), n (the number of cases the configural model used in that group), factor_mean (the estimated $\alpha_g$), and factor_variance (the estimated $\psi_g$). The rows follow the group ordering lavaan uses.

Attributes carry the rest of the solution:

aligned_loadings, aligned_intercepts: Matrices of the aligned $\lambda_{gi}$ and $\nu_{gi}$, groups in rows and items in columns.
configural_loadings, configural_intercepts: Matrices of the configural $\lambda^{0}_{gi}$ and $\nu^{0}_{gi}$, same layout.
simplicity_function: The achieved minimum of $F$.
simplicity_starts: The value of $F$ achieved from each starting value, in start order. A start the optimizer could not complete, or one that ended in the degenerate branch described in Details, is recorded as Inf and discarded.
converged: TRUE when the optimizer reported convergence at the retained solution.
n_starts: The number of starting values used.
n_optima: The number of distinct local minima the converged starts reached.
alignment: The identification rule used, "fixed" or "free".
epsilon: The smoothing constant used.
R2_loadings, R2_intercepts: Named numeric vectors, one entry per item, holding the per item $R^2$ invariance measures defined in Details.
R2_total: The same two measures pooled over items, as a named numeric vector with elements loadings and intercepts. This is the overall effect size of approximate invariance reported by Asparouhov and Muthén (2014).
item_loss: Matrix with one row per item and columns loadings, intercepts, and total, splitting the achieved simplicity function into per item contributions.
fit: The configural lavaan fit object.

Details

Alignment starts from the configural solution: the multiple group single factor model with every loading, intercept, and residual variance free across groups and the factor standardized (mean 0 and variance 1) in each group. Write $\lambda^{0}_{gi}$ and $\nu^{0}_{gi}$ for the resulting loading and intercept of item $i$ in group $g$. That solution is not unique: for any group factor means $\alpha_g$ and factor variances $\psi_g$ the reparameterized measurement parameters $$\lambda_{gi} = \lambda^{0}_{gi} / \sqrt{\psi_g}$$ $$\nu_{gi} = \nu^{0}_{gi} - \alpha_g \lambda^{0}_{gi} / \sqrt{\psi_g}$$ reproduce the observed means and covariances exactly and so fit the data identically. Alignment picks the member of that family whose measurement parameters are closest to invariant, by minimizing the total simplicity function $$F = \sum_i \sum_{g_1 < g_2} w_{g_1 g_2} f(\lambda_{g_1 i} - \lambda_{g_2 i}) + \sum_i \sum_{g_1 < g_2} w_{g_1 g_2} f(\nu_{g_1 i} - \nu_{g_2 i})$$ over $\alpha_g$ and $\psi_g$, with weights $w_{g_1 g_2} = \sqrt{N_{g_1} N_{g_2}}$ and the component loss function $$f(x) = \sqrt{\sqrt{x^2 + \epsilon}} = (x^2 + \epsilon)^{1/4}.$$

The fourth root is the substance of the method, not a technical detail. A squared loss would spread a fixed amount of noninvariance evenly over all the parameters, because halving two differences beats zeroing one. The fourth root is concave in $|x|$, so the marginal penalty falls as a difference grows: the criterion prefers a solution in which most parameters agree closely and a few disagree substantially, which is exactly the pattern of approximate invariance a researcher wants to find and report. The constant $\epsilon$ only rounds off the kink of $\sqrt{|x|}$ at zero so the criterion is differentiable. One consequence is worth knowing: each of the $2I$ terms in a group pair contributes at least $\epsilon^{1/4}$, so the smallest attainable value of $F$ is $2 I \epsilon^{1/4} \sum_{g_1 < g_2} w_{g_1 g_2}$, reached only when every aligned parameter is exactly invariant. The achieved value is comparable across runs on the same items, groups, and $\epsilon$, not across data sets.

Two constraints identify the solution. The scale of the factor is genuinely undetermined by $F$: multiplying every $\sqrt{\psi_g}$ and every $\alpha_g$ by the same constant leaves the aligned intercepts alone and shrinks the aligned loadings toward each other, so without a constraint the criterion is minimized by letting the factor variances run away. The location constraint plays the same role in the limiting case of exact invariance, where shifting all the factor means by a constant leaves $F$ unchanged. Under alignment = "fixed" the constraints are $\alpha_1 = 0$ and $\psi_1 = 1$. Under alignment = "free" they are $\sum_g \alpha_g = 0$ and $\prod_g \sqrt{\psi_g} = 1$, which treats the groups symmetrically and is the more natural choice when no group is a meaningful reference. The two rules give genuinely different solutions, not a relabeling of one another, because $F$ is not invariant to shifting the factor means.

The rule also sets the metric the answer is reported on, which matters when the estimates are compared with anything else. The constraint $\psi_1 = 1$ makes the first group's factor the common metric, so a reported factor mean of 0.4 says that group's factor mean is 0.4 of the first group's factor standard deviations above the first group's, and a reported factor variance of 1.3 says its factor variance is 1.3 times the first group's. Under "free" the common metric is the one in which the group factor standard deviations have a geometric mean of 1, and the factor means are deviations from their own average on that metric. Simulating data with known group factor means and then comparing them with the estimates requires dividing the generating means by the generating factor standard deviation of the reference group first.

The minimization runs optim with the BFGS method and the analytic gradient, over the factor means $\alpha_g$ and the log factor standard deviations $\log \sqrt{\psi_g}$, reduced to the coordinates the identification rule leaves free. The optimizer is run from n_starts starting values and the best solution is kept, because a single start is not safe. Two things can go wrong, and they are different problems.

The first is ordinary multimodality. When several loadings and intercepts are noninvariant the criterion has more than one local minimum, typically within a percent or so of each other in value but at different factor means. The number of distinct minima the starts reached is returned in the "n_optima" attribute and the value achieved from each start in "simplicity_starts". More than one is a signal to raise n_starts and to check whether the reported solution is stable.

The second is a degenerate branch, and it is the one that bites. Sending the factor variances of every group but the reference off to infinity drives those groups' aligned loadings to zero, which flattens the loading half of the criterion; the optimizer stops there and reports convergence with factor variances of $10^{36}$ or Inf. That is not an estimate, and on real data a meaningful share of random starts finds it. A start whose factor standard deviations or factor means leave a very wide sanity range (a factor of $10^4$ either way) is therefore discarded and its entry in "simplicity_starts" is Inf. If every start is discarded the function stops rather than return the degenerate solution.

Item level invariance is summarized with the $R^2$ measure of Asparouhov and Muthén (2014), which asks how much of the group to group variation in an item's configural parameter is accounted for by the estimated factor means and variances alone. Let $\bar{\lambda}_i$ and $\bar{\nu}_i$ be the aligned parameters of item $i$ averaged over groups. The factor means and variances by themselves imply the configural values $\sqrt{\psi_g} \bar{\lambda}_i$ and $\bar{\nu}_i + \alpha_g \bar{\lambda}_i$. Write $T_i$ for the sum over groups of the squared configural parameter of item $i$ and $E_i$ for the sum over groups of its squared departure from that implied value. Then $$R^2_i = 1 - E_i / T_i,$$ computed separately for the loadings and for the intercepts. A value of 1 for every item on the loadings is metric invariance, and on both loadings and intercepts is scalar invariance. The complementary view is the "item_loss" attribute, which splits the achieved simplicity function into the contribution of each item, so the items carrying the noninvariance can be named.

The criterion adds up raw parameter differences, so items on very different measurement scales do not contribute equally: an item whose raw variance is a hundred times another's dominates the sum. When the items are not already on a common metric, put them on one (standardizing them is the simple choice) before aligning.

The method is defined for two groups and is computed here for two, but it has little to offer there. With two groups the standard invariance ladder is tractable and interpretable, and the alignment criterion has only one group pair to work with. Alignment earns its keep when the number of groups makes exact invariance implausible and the ladder uninformative.

References

Asparouhov, T., & Muthén, B. (2014). Multiple-group factor analysis alignment. Structural Equation Modeling, 21(4), 495–508. doi:10.1080/10705511.2014.919210

Marsh, H. W., Guo, J., Parker, P. D., Nagengast, B., Asparouhov, T., Muthén, 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. doi:10.1037/met0000113

Muthén, B., & Asparouhov, T. (2014). IRT studies of many groups: The alignment method. Frontiers in Psychology, 5, Article 978. doi:10.3389/fpsyg.2014.00978

Muthén, 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. doi:10.1177/0049124117701488

Robitzsch, A. (2025). sirt: Supplementary item response theory models. R package version 4.2-133. https://CRAN.R-project.org/package=sirt

Author

Ken Kelley kkelley@nd.edu

Examples

# Five groups that differ in factor mean and factor variance, with two
# deliberately noninvariant measurement parameters (the loading of item
# 3 in group 2 and the intercept of item 5 in group 4). The first group's
# factor standard deviation is 1, which is the metric the default "fixed"
# rule reports on, so the estimates compare directly with the generating
# values below.
set.seed(113)
n_g <- c(400, 450, 380, 500, 420)
factor_mean <- c(0, 0.30, -0.50, 0.80, 0.20)
factor_sd   <- c(1, 1.20,  0.80, 1.10, 0.90)
Lambda <- matrix(0.8, nrow = 5, ncol = 6)
Nu     <- matrix(1.0, nrow = 5, ncol = 6)
Lambda[2, 3] <- 0.3
Nu[4, 5]     <- 1.8
d <- do.call(rbind, lapply(1:5, function(g) {
  eta <- rnorm(n_g[g], factor_mean[g], factor_sd[g])
  x <- sapply(1:6, function(i)
    Nu[g, i] + Lambda[g, i] * eta + rnorm(n_g[g], 0, 0.6))
  data.frame(x, cohort = paste0("cohort_", g))
}))
names(d)[1:6] <- paste0("x", 1:6)

out <- measurement_alignment(d, items = paste0("x", 1:6),
                             group = "cohort", seed = 113)
out                                   # recovered means and variances
#>  group    n   factor_mean factor_variance
#>  cohort_1 400 0           1              
#>  cohort_2 450 0.328       1.53           
#>  cohort_3 380 -0.54       0.581          
#>  cohort_4 500 0.861       1.28           
#>  cohort_5 420 0.213       0.783          
attr(out, "R2_loadings")              # item 3 stands out
#>        x1        x2        x3        x4        x5        x6 
#> 0.9999162 0.9976032 0.9079726 0.9994612 0.9996394 0.9983087 
attr(out, "R2_intercepts")            # item 5 stands out
#>        x1        x2        x3        x4        x5        x6 
#> 0.9999097 0.9989362 0.9968054 0.9995481 0.9525477 0.9992106 
attr(out, "item_loss")
#>    loadings intercepts    total
#> x1 1359.250   1371.846 2731.096
#> x2 1482.405   1470.454 2952.859
#> x3 2028.494   1381.886 3410.381
#> x4 1389.880   1418.496 2808.376
#> x5 1374.485   2399.982 3774.468
#> x6 1420.217   1427.319 2847.536

# Holzinger and Swineford's verbal tests across four groups formed by
# crossing school with sex. The raw tests are on very different scales,
# so they are standardized first (see Details).
data(holzinger_swineford)
hs <- holzinger_swineford
hs$school_sex <- interaction(hs$school, hs$sex, sep = ", ")
verbal <- c("t5_general_information", "t6_paragraph_comprehension",
            "t7_sentence", "t8_word_classification", "t9_word_meaning")
hs[verbal] <- scale(hs[verbal])
ma <- measurement_alignment(hs, items = verbal, group = "school_sex",
                            seed = 113)
ma
#>  group               n  factor_mean factor_variance
#>  Pasteur, Male       74 0           1              
#>  Pasteur, Female     82 0.0429      0.904          
#>  Grant-White, Male   72 0.608       0.802          
#>  Grant-White, Female 73 0.803       1.09           

# The broom verbs: one row per group, and the alignment summary.
generics::tidy(ma)
#>                  term  n factor_mean factor_variance
#> 1       Pasteur, Male 74  0.00000000       1.0000000
#> 2     Pasteur, Female 82  0.04288633       0.9043518
#> 3   Grant-White, Male 72  0.60761608       0.8023580
#> 4 Grant-White, Female 73  0.80291885       1.0856178
generics::glance(ma)
#>   n_groups alignment epsilon simplicity_function R2_loadings_mean
#> 1        4     fixed    0.01            1757.193        0.9942458
#>   R2_intercepts_mean  R2_total converged n_starts n_optima
#> 1          0.8839346 0.9345257      TRUE       10        1