MIRES is a package for Measurement Invariance assessment using Random Effects and Shrinkage. This is the official implementation of the approach taken in Measurement Invariance Assessment with Bayesian Hierarchical Inclusion Modeling.
Unlike most other MI assessment methods, MIRES assumes all measurement parameters (loadings, intercepts, and residual SDs) can randomly vary across groups. Therefore, it is a fully mixed effects CFA model. If invariance is tenable, then the random effect SDs should all be effectively zero.
Unlike most mixed effects models, MIRES further regularizes the random effect SDs (RE-SDs) via an inclusion model. In sum, it includes an adaptive, dependent regularization model that shares regularization intensity across several RE-SDs. This allows the RE-SDs to rapidly shrink toward zero if they need to (i.e., if invariance is met), and to be largely untouched if they need to be non-zero (i.e., if invariance is not met). Consequently, MIRES can gain evidence in favor of invariance quicker than a mixed effects model otherwise could, especially with few groups.
Altogether, MIRES is a mixed effects CFA model, where the RE-SDs are also hierarchically modeled with a dependent regularizing prior (See Details for more information). When invariance is plausible, it will collapse into a fixed effects CFA model; when it is not, then a partially invariant or fully non-invariant model will be produced.
You can install the development version of MIRES from github with:
remotes::install_github("stephenSRMMartin/MIRES")or from CRAN with:
install.packages("MIRES")Simulated data are provided in the package.
This data has non-invariance in the loadings, and invariance elsewhere.
library(MIRES)
data(sim_loadings)
head(sim_loadings) x_1 x_2 x_3 x_4 x_5 x_6
1 1.7400023 1.3214588 0.6255326 0.901031733 0.9495002 1.2286485
2 -0.5640201 0.2203121 -1.4399299 -0.571124405 -0.9580103 -0.2434576
3 -1.1030768 -0.4088069 0.7182129 -0.009960373 -1.3954430 -0.1966013
4 -1.6139970 -1.3411869 -2.7745400 -0.818880762 -2.1656676 -1.1999327
5 -0.8192840 -0.9715812 -0.3293007 -0.538967762 -0.7168703 -0.4190734
6 0.3266671 0.8955085 2.4646536 0.596746521 1.5468213 0.4509530
x_7 x_8 group
1 0.3128417 1.5761853 1
2 -1.1585096 -1.1884185 1
3 -0.6639108 -0.6313458 1
4 -1.4482666 -2.3015851 1
5 -2.7528062 -1.0865613 1
6 3.1724797 2.1475378 1Fit and summarize the model.
fit <- mires(my_factor ~ x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8, group, sim_loadings, iter = 1000)
summary(fit)-----
MIRES model object
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Specification:
my_factor ~ x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8
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Configuration:
Factors: Univariate
Latent Mean and Variance Identification: Sum-to-zero, product-to-one.
Hierarchical inclusion model: Yes
Latent Scores Saved: No
Inclusion Model Prior Params: 0 , 0.25
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Fixed Effects
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Loadings
Item Mean Median SD L95 U95 Rhat
x_1 0.757 0.750 0.139 0.502 1.045 1.001
x_2 0.668 0.656 0.221 0.276 1.154 1.001
x_3 0.750 0.739 0.143 0.487 1.070 1.000
x_4 0.579 0.573 0.204 0.193 1.001 1.007
x_5 0.707 0.693 0.232 0.292 1.215 1.003
x_6 0.791 0.786 0.229 0.288 1.218 1.005
x_7 0.594 0.580 0.229 0.159 1.056 1.003
x_8 1.009 1.003 0.211 0.587 1.429 1.002
Residual Standard Deviations (Unlogged Scale)
Item Mean Median SD L95 U95 Rhat
x_1 0.754 0.750 0.061 0.652 0.891 1.001
x_2 0.743 0.740 0.052 0.633 0.839 1.001
x_3 0.750 0.747 0.051 0.653 0.854 1.004
x_4 0.695 0.692 0.051 0.596 0.791 1.001
x_5 0.687 0.685 0.051 0.590 0.788 1.001
x_6 0.710 0.709 0.057 0.594 0.819 1.002
x_7 0.720 0.715 0.051 0.624 0.819 1.005
x_8 0.788 0.785 0.064 0.679 0.932 1.001
Intercepts
Item Mean Median SD L95 U95 Rhat
x_1 -0.040 -0.040 0.083 -0.201 0.121 1.002
x_2 -0.029 -0.028 0.078 -0.190 0.114 1.003
x_3 -0.033 -0.035 0.086 -0.198 0.134 1.002
x_4 0.008 0.008 0.067 -0.112 0.152 1.001
x_5 -0.107 -0.107 0.079 -0.252 0.057 1.002
x_6 -0.014 -0.008 0.098 -0.222 0.160 1.005
x_7 -0.038 -0.037 0.091 -0.219 0.136 1.003
x_8 -0.025 -0.027 0.092 -0.202 0.156 1.002
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Measurement Invariance Assessment
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Random Effect Standard Deviations (Unlogged Scale)
Parameter Item Factor Mean Median SD L95 U95 Rhat BF01 BF10 Pr(SD <= 0.1| D) BF(SD <= 0.1)
Loading x_1 my_factor 0.219 0.192 0.156 0.000 0.504 1.008 2.107 0.475 0.232 3.046
Loading x_2 my_factor 0.437 0.401 0.181 0.182 0.768 1.002 0.001 729.233 0.000 0.000
Loading x_3 my_factor 0.225 0.192 0.162 0.000 0.523 1.003 1.933 0.517 0.216 2.777
Loading x_4 my_factor 0.388 0.352 0.180 0.104 0.767 1.002 0.017 58.707 0.007 0.071
Loading x_5 my_factor 0.460 0.425 0.191 0.161 0.836 1.002 0.004 244.913 0.001 0.010
Loading x_6 my_factor 0.473 0.429 0.207 0.146 0.903 1.003 0.023 42.880 0.002 0.025
Loading x_7 my_factor 0.467 0.430 0.185 0.173 0.827 1.001 0.002 480.630 0.000 0.000
Loading x_8 my_factor 0.407 0.375 0.193 0.076 0.798 1.003 0.131 7.628 0.021 0.222
Residual SD x_1 0.114 0.093 0.095 0.000 0.289 1.003 6.617 0.151 0.529 11.355
Residual SD x_2 0.092 0.073 0.076 0.000 0.236 1.001 7.097 0.141 0.641 18.052
Residual SD x_3 0.079 0.063 0.068 0.000 0.209 1.005 8.589 0.116 0.712 24.934
Residual SD x_4 0.083 0.067 0.074 0.000 0.218 0.999 9.658 0.104 0.697 23.202
Residual SD x_5 0.090 0.071 0.077 0.000 0.236 1.003 8.955 0.112 0.650 18.776
Residual SD x_6 0.100 0.077 0.089 0.000 0.264 1.000 6.815 0.147 0.617 16.287
Residual SD x_7 0.087 0.068 0.078 0.000 0.235 1.005 357.644 0.003 0.676 21.094
Residual SD x_8 0.084 0.066 0.076 0.000 0.225 1.002 9.266 0.108 0.688 22.294
Intercept x_1 0.094 0.072 0.089 0.000 0.258 1.001 8.213 0.122 0.647 18.571
Intercept x_2 0.091 0.074 0.077 0.000 0.238 1.003 8.168 0.122 0.636 17.703
Intercept x_3 0.096 0.077 0.079 0.000 0.245 1.003 6.632 0.151 0.622 16.601
Intercept x_4 0.065 0.050 0.061 0.000 0.173 0.999 13.225 0.076 0.803 41.211
Intercept x_5 0.092 0.074 0.078 0.000 0.233 1.004 7.707 0.130 0.644 18.329
Intercept x_6 0.126 0.104 0.100 0.000 0.316 1.001 5.705 0.175 0.482 9.426
Intercept x_7 0.139 0.121 0.100 0.000 0.328 1.001 2.978 0.336 0.396 6.629
Intercept x_8 0.093 0.072 0.085 0.000 0.251 1.001 173.931 0.006 0.642 18.131With three kinds of parameters (loadings, residual SDs, and intercepts), and J items, there are 3J RE-SDs: \(\sigma_p, p \in [0,\ldots,3J]\). The regularizing prior for each RE-SD, \(\sigma_p\) is defined as: \[ \sigma_p \sim \mathcal{N}^+(0, \exp(\tau_p)) \] If \(\exp(\tau_p)\) is very small, then the prior probability that \(\sigma_p = 0\) increases greatly, and the parameter corresponding to it is reduced to a fixed effect.
The probability that one parameter is invariant should be informed by other parameters with similar characteristics (e.g., they model the same item, or they are of the same type — Loading, intercept, or residual SD). Therefore, the regularization term for each \(\sigma_p\) can be dependent on others. We model the regularization terms \(\tau_p\) as: \[ \tau_p = \tau_0 + \tau_{\text{item}_p} + \tau_{\text{param}_p} + \lambda_p \] Therefore, there is a global scaling factor \(\tau_0\), an "item" effect (\(\tau_{\text{item}, p}), a "parameter" effect (\)_{, p}), and a unique effect \(\lambda_p\).
Therefore, if all parameters are invariant, \(\tau_0\) can decrease, and all RE-SDs can jointly approach zero. Conversely, if some loadings are invariant, then \(\tau_{\lambda}\) decreases, and the probability that other loadings are zero increases. If item \(j\)’s intercept and loading seems to vary, then \(\tau_j\) increases, and the probability that the item’s residual SD is invariant decreases. Finally, the \(\lambda_p\) terms provide an "escape hatch" of sorts, if a particular parameter uniquely needs to be invariant or non-invariant.