MIRES

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.

Installation

You can install the development version of MIRES from github with:

r remotes::install_github("stephenSRMMartin/MIRES")

or from CRAN with: r install.packages("MIRES")

Example

Simulated data are provided in the package.

This data has non-invariance in the loadings, and invariance elsewhere.

r 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 1

Fit and summarize the model.

r 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

Specification: myfactor ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x_8

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

Fixed Effects

Loadings Item Mean Median SD L95 U95 Rhat x1 0.757 0.750 0.139 0.502 1.045 1.001 x2 0.668 0.656 0.221 0.276 1.154 1.001 x3 0.750 0.739 0.143 0.487 1.070 1.000 x4 0.579 0.573 0.204 0.193 1.001 1.007 x5 0.707 0.693 0.232 0.292 1.215 1.003 x6 0.791 0.786 0.229 0.288 1.218 1.005 x7 0.594 0.580 0.229 0.159 1.056 1.003 x8 1.009 1.003 0.211 0.587 1.429 1.002

Residual Standard Deviations (Unlogged Scale) Item Mean Median SD L95 U95 Rhat x1 0.754 0.750 0.061 0.652 0.891 1.001 x2 0.743 0.740 0.052 0.633 0.839 1.001 x3 0.750 0.747 0.051 0.653 0.854 1.004 x4 0.695 0.692 0.051 0.596 0.791 1.001 x5 0.687 0.685 0.051 0.590 0.788 1.001 x6 0.710 0.709 0.057 0.594 0.819 1.002 x7 0.720 0.715 0.051 0.624 0.819 1.005 x8 0.788 0.785 0.064 0.679 0.932 1.001

Intercepts Item Mean Median SD L95 U95 Rhat x1 -0.040 -0.040 0.083 -0.201 0.121 1.002 x2 -0.029 -0.028 0.078 -0.190 0.114 1.003 x3 -0.033 -0.035 0.086 -0.198 0.134 1.002 x4 0.008 0.008 0.067 -0.112 0.152 1.001 x5 -0.107 -0.107 0.079 -0.252 0.057 1.002 x6 -0.014 -0.008 0.098 -0.222 0.160 1.005 x7 -0.038 -0.037 0.091 -0.219 0.136 1.003 x8 -0.025 -0.027 0.092 -0.202 0.156 1.002

Measurement Invariance Assessment

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 x1 myfactor 0.219 0.192 0.156 0.000 0.504 1.008 2.107 0.475 0.232 3.046 Loading x2 myfactor 0.437 0.401 0.181 0.182 0.768 1.002 0.001 729.233 0.000 0.000 Loading x3 myfactor 0.225 0.192 0.162 0.000 0.523 1.003 1.933 0.517 0.216 2.777 Loading x4 myfactor 0.388 0.352 0.180 0.104 0.767 1.002 0.017 58.707 0.007 0.071 Loading x5 myfactor 0.460 0.425 0.191 0.161 0.836 1.002 0.004 244.913 0.001 0.010 Loading x6 myfactor 0.473 0.429 0.207 0.146 0.903 1.003 0.023 42.880 0.002 0.025 Loading x7 myfactor 0.467 0.430 0.185 0.173 0.827 1.001 0.002 480.630 0.000 0.000 Loading x8 myfactor 0.407 0.375 0.193 0.076 0.798 1.003 0.131 7.628 0.021 0.222 Residual SD x1 0.114 0.093 0.095 0.000 0.289 1.003 6.617 0.151 0.529 11.355 Residual SD x2 0.092 0.073 0.076 0.000 0.236 1.001 7.097 0.141 0.641 18.052 Residual SD x3 0.079 0.063 0.068 0.000 0.209 1.005 8.589 0.116 0.712 24.934 Residual SD x4 0.083 0.067 0.074 0.000 0.218 0.999 9.658 0.104 0.697 23.202 Residual SD x5 0.090 0.071 0.077 0.000 0.236 1.003 8.955 0.112 0.650 18.776 Residual SD x6 0.100 0.077 0.089 0.000 0.264 1.000 6.815 0.147 0.617 16.287 Residual SD x7 0.087 0.068 0.078 0.000 0.235 1.005 357.644 0.003 0.676 21.094 Residual SD x8 0.084 0.066 0.076 0.000 0.225 1.002 9.266 0.108 0.688 22.294 Intercept x1 0.094 0.072 0.089 0.000 0.258 1.001 8.213 0.122 0.647 18.571 Intercept x2 0.091 0.074 0.077 0.000 0.238 1.003 8.168 0.122 0.636 17.703 Intercept x3 0.096 0.077 0.079 0.000 0.245 1.003 6.632 0.151 0.622 16.601 Intercept x4 0.065 0.050 0.061 0.000 0.173 0.999 13.225 0.076 0.803 41.211 Intercept x5 0.092 0.074 0.078 0.000 0.233 1.004 7.707 0.130 0.644 18.329 Intercept x6 0.126 0.104 0.100 0.000 0.316 1.001 5.705 0.175 0.482 9.426 Intercept x7 0.139 0.121 0.100 0.000 0.328 1.001 2.978 0.336 0.396 6.629 Intercept x8 0.093 0.072 0.085 0.000 0.251 1.001 173.931 0.006 0.642 18.131 ```

Limitations

• Currently only supports one-factor models. Multidimensional models will soon be supported.
• Assumes normality in the indicators. Alternative likelihoods are possible, but not yet implemented.
• Does not allow residual covariance.
• Does not yet support unregularized mixed models. Unregularized (i.e., constant, and independent) priors will soon be supported.

Details

With three kinds of parameters (loadings, residual SDs, and intercepts), and J items, there are 3J RE-SDs: $\sigmap, p \in [0,\ldots,3J]$. The regularizing prior for each RE-SD, $\sigmap$ is defined as: $$ \sigmap \sim \mathcal{N}^+(0, \exp(\taup)) $$ If $\exp(\taup)$ is very small, then the prior probability that $\sigmap = 0$ 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 $\sigmap$ can be dependent on others. We model the regularization terms $\taup$ as: $$ \taup = \tau0 + \tau{\text{item}p} + \tau{\text{param}p} + \lambdap $$ Therefore, there is a global scaling factor $\tau0$, an "item" effect ($\tau{\text{item}, p}), a "parameter" effect ($\tau{\text{param}, p}), and a unique effect $\lambda_p$.

Therefore, if all parameters are invariant, $\tau0$ 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 $\tauj$ increases, and the probability that the item's residual SD is invariant decreases. Finally, the $\lambdap$ terms provide an "escape hatch" of sorts, if a particular parameter uniquely needs to be invariant or non-invariant.