Covariance pattern mixture models: Eliminating random effects to improve convergence and performance

Daniel McNeish; Jeffrey Harring

doi:10.3758/s13428-019-01292-4

Behavior Research Methods

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Covariance pattern mixture models: Eliminating random effects to improve convergence and performance

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Daniel McNeish
Jeffrey Harring

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First Online: 11 September 2019

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Abstract

Growth mixture models (GMMs) are prevalent for modeling unknown population heterogeneity via distinct latent classes. However, GMMs are riddled with convergence issues, often requiring researchers to atheoretically alter the model with cross-class constraints simply to obtain convergence. We discuss how within-class random effects in GMMs exacerbate convergence issues, even though these random effects rarely help answer typical research questions. That is, latent classes provide a discretization of continuous random effects, so including additional random effects within latent classes can unnecessarily complicate the model. These random effects are commonly included in order to properly specify the marginal covariance; however, random effects are inefficient for patterning a covariance matrix, resulting in estimation issues. Such a goal can be achieved more simply through covariance pattern models, which we extend to the mixture model context in this article (covariance pattern mixture models, or CPMMs). We provide evidence from theory, simulation, and an empirical example showing that employing CPMMs (even if they are misspecified) instead of GMMs can circumvent the computational difficulties that can plague GMMs, without sacrificing the ability to answer the types of questions commonly asked in empirical studies. Our results show the advantages of CPMMs with respect to improved class enumeration and less biased class-specific growth trajectories, in addition to their vastly improved convergence rates. The results also show that constraining the covariance parameters across classes in order to bypass convergence issues with GMMs leads to poor results. An extensive software appendix is included to assist researchers in running CPMMs in Mplus.

Keywords

Finite mixture modeling Convergence Growth mixture modeling Constraints Latent class analysis

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Notes

Appendix: Mplus code for fitting covariance pattern mixture models

To specify a covariance pattern matrix in general structural equation modeling software such as Mplus, the general strategy is to set the factor variances of the growth factors (e.g., intercept, slope) to 0 and then to apply the covariance pattern to the residual variances at each time point. Growth models in this framework are treated in a multivariate manner, such that each repeated measure is treated as a unique dependent variable. This can make the covariance pattern a little tricky to apply, because each element of the matrix must be programmed with a separate statement.

The challenge of specifying the covariance pattern depends on whether the residual variances at each time point are homogeneous (i.e., variances are equal for each repeated measure) or heterogeneous (i.e., the variances are uniquely estimated for each repeated measure). The covariance pattern is easier to specify with homogeneous variances; with heterogeneous variances, model constraints are necessary, to ensure that the correlations follow the desired pattern.

The following pages show the Mplus code for fitting covariance pattern mixture models with compound symmetry, first-order autoregressive, Toeplitz, unstructured covariance matrices, and a custom structure, as mentioned in the Discussion section. All code will assume that two classes are being extracted, but the number of classes can be changed without affecting the general idea of the code. To provide some context, we will use the NLSY data; these data can be downloaded from http://www.joophox.net/mlbook2/MLbook.htm or from the first author’s Open Science Framework page for this project (https://osf.io/yh6kf/). These data have four repeated measures, but the code can be extended to accommodate any number of repeated measures. We start with the homogeneous variance structures first, and then proceed to the heterogeneous variance structures.

Mplus code for covariance pattern mixture models is not appreciably different from standard growth mixture models. The primary difference stems from how the covariance matrices are modeled. Other options are not changed in Mplus when using a covariance pattern mixture model than for a standard growth mixture model.

Homogeneous variances, exchangeable structure

The exchangeable structure models all covariances as being equal, regardless of how far apart the measures are in time. It is also referred to as an exchangeable or equicorrelated structure. A covariance pattern model with an exchangeable structure is equivalent to a random-intercept model. Though parsimonious, in that all covariances are modeled with a single parameter, it can be too simplistic for some types of data in which the repeated measures are less related as they become further apart in time.

For the NLSY data, the marginal covariance matrix for this structure would be

$$ \left[\begin{array}{cccc}{\sigma}^2& & & \\ {}{\sigma}^2\rho & {\sigma}^2& & \\ {}{\sigma}^2\rho & {\sigma}^2\rho & {\sigma}^2& \\ {}{\sigma}^2\rho & {\sigma}^2\rho & {\sigma}^2\rho & {\sigma}^2\end{array}\right]. $$

Homogeneous variances, autoregressive structure

The autoregressive structure allows for observations to be less related as the observations grow further apart. To maintain parsimony, the autoregressive structure only estimates a single covariance parameter ρ. The covariance at lag 1 is equal to ρ, lag 2’s is equal to ρ², lag 3’s is equal to ρ³, and so forth. This structure requires model constraints to ensure that only the single ρ is estimated.

For the NLSY data, the marginal covariance matrix for this structure would be

$$ \left[\begin{array}{cccc}{\sigma}^2& & & \\ {}{\sigma}^2\rho & {\sigma}^2& & \\ {}{\sigma}^2{\rho}^2& {\sigma}^2\rho & {\sigma}^2& \\ {}{\sigma}^2{\rho}^3& {\sigma}^2{\rho}^2& {\sigma}^2\rho & {\sigma}^2\end{array}\right]. $$

Homogenous variances, Toeplitz structure

The main idea of a Toeplitz structure is that the covariances at equivalent lags are equal but that unequal lags are freely estimated. So, the covariances for time points that are one lag apart are constrained to be the same; the same is true for covariances of time points that are two lags apart. However, unlike in an autoregressive structure, the lag 1 covariance is completely unrelated to the lag 2 covariance (e.g., the lag 2 covariance is not the square of the lag 1 covariance, as with the autoregressive structure).

For the NLSY data, the marginal covariance matrix for this structure would be

$$ \left[\begin{array}{cccc}{\sigma}^2& & & \\ {}{\sigma}^2{\rho}_1& {\sigma}^2& & \\ {}{\sigma}^2{\rho}_2& {\sigma}^2{\rho}_1& {\sigma}^2& \\ {}{\sigma}^2{\rho}_3& {\sigma}^2{\rho}_2& {\sigma}^2{\rho}_1& {\sigma}^2\end{array}\right]. $$

Homogeneous variances, unstructured structure

An unstructured matrix is the most flexible but the least parsimonious option. Every element of the covariance matrix is uniquely estimated, so if there are many repeated measures or many latent classes, this can quickly lead to a large number of parameters. Reaching convergence with an unstructured matrix can be difficult and often requires a very large sample. If the model converges, it can sometimes provide a good initial idea for alternative structures that could be more parsimonious (e.g., if the estimates look approximately equal down the diagonals, perhaps an autoregressive or Toeplitz structure could provide a similar fit with fewer parameters).

For the NLSY data, the marginal covariance matrix for this structure would be

$$ \left[\begin{array}{cccc}{\sigma}^2& & & \\ {}{\sigma}_{21}& {\sigma}^2& & \\ {}{\sigma}_{31}& {\sigma}_{32}& {\sigma}^2& \\ {}{\sigma}_{41}& {\sigma}_{42}& {\sigma}_{43}& {\sigma}^2\end{array}\right]. $$

Homogeneous variances, custom structure

As we mentioned in the Discussion section, the flexibility of the structural equation modeling framework means that essentially any marginal structure can be specified. The observed correlation matrix of the four repeated measures is

$$ \left[\begin{array}{cccc}1& & & \\ {}.66& 1& & \\ {}.54& .78& 1& \\ {}.45& .76& .80& 1\end{array}\right]. $$

So let’s imagine that the (3, 2), (4, 2), and (4, 3) elements look close enough to constrain to be equal but the (2, 1), (3, 1), and (4, 1) elements should be uniquely estimated. The marginal covariance to be estimated would then be

$$ \left[\begin{array}{cccc}{\sigma}^2& & & \\ {}{\rho}_1& {\sigma}^2& & \\ {}{\rho}_3& {\rho}_2& {\sigma}^2& \\ {}{\rho}_4& {\rho}_2& {\rho}_2& {\sigma}^2\end{array}\right]. $$

This structure has only one additional parameter relative to the Toeplitz structure, but one could argue that the structure makes more efficient use of the parameters that are estimated (provided that the structure is roughly equal in each class).

General difference with heterogeneous variances

When the variances are uniquely estimated at each time point, the structure gets a little more complicated to ensure that the correlation for constrained off diagonals are in fact equal. If the same structures as the homogeneous case were used, the correlation could vary for constrained elements depending on the magnitude of the variance at the two time points of interest (this can be seen visual by allowing the variances to be uniquely estimated and requesting the STDXY standardized output in Mplus—the off-diagonal terms will be constrained but the correlations will be different to the extent that the variances differ). Therefore, the square root of the variances at each time point must be included in the off diagonal elements to appropriately scale these elements and ensure that all elements have equal correlations. Heterogeneous variances require a model constraint is nearly all cases so that elements are appropriately scaled.

Heterogeneous variances, compound symmetry

For the NLSY data, the marginal covariance matrix for this structure would be

Heterogeneous variances, autoregressive structure

For the NLSY data, the marginal covariance matrix for this structure would be

$$ \left[\begin{array}{cccc}{\sigma}_1^2& & & \\ {}{\rho \sigma}_2{\sigma}_1& {\sigma}_2^2& & \\ {}{\rho}^2{\sigma}_3{\sigma}_1& {\rho \sigma}_3{\sigma}_2& {\sigma}_3^2& \\ {}{\rho}^3\sigma 4{\sigma}_1& {\rho}^2{\sigma}_4{\sigma}_2& {\rho \sigma}_4{\sigma}_3& {\sigma}_4^2\end{array}\right]. $$

Heterogeneous variances, Toeplitz structure

For the NLSY data, the marginal covariance matrix for this structure would be

$$ \left[\begin{array}{cccc}{\sigma}_1^2& & & \\ {}{\rho}_1{\sigma}_2{\sigma}_1& {\sigma}_2^2& & \\ {}{\rho}_2{\sigma}_3{\sigma}_1& {\rho}_1{\sigma}_3{\sigma}_2& {\sigma}_3^2& \\ {}{\rho}_3\sigma 4{\sigma}_1& {\rho}_2{\sigma}_4{\sigma}_2& {\rho}_1{\sigma}_4{\sigma}_3& {\sigma}_4^2\end{array}\right]. $$

Heterogeneous variances, unstructured structure

For the NLSY data, the marginal covariance matrix for this structure would be

$$ \left[\begin{array}{cccc}{\sigma}_1^2& & & \\ {}{\sigma}_{21}& {\sigma}_2^2& & \\ {}{\sigma}_{31}& {\sigma}_{32}& {\sigma}_3^2& \\ {}{\sigma}_{41}& {\sigma}_{42}& {\sigma}_{43}& {\sigma}_4^2\end{array}\right]. $$

Because the unstructured structure does not set constraints in the off-diagonal, transitioning from homogeneous variances to heterogeneous variances does not have an effect on how the model is specified. That is, the heterogeneous variances do not affect the correlation of constrained elements because there are no constrained elements. The only difference to the above code for the homogeneous structure is that the parameter constraint labels (i.e., var1 and var2) need to be removed from the residual variances. Because there is no other difference, we will not write out all of the code again with this one minor alteration.

Heterogeneous variances, custom structure

Again using the observed correlation matrix of the four repeated measures, which is

$$ \left[\begin{array}{cccc}1& & & \\ {}.66& 1& & \\ {}.54& .78& 1& \\ {}.45& .76& .80& 1\end{array}\right]. $$

If we still wanted to constrain the (3, 2), (4, 2), and (4, 3) elements but freely estimate the (2, 1), (3, 1), and (4, 1) elements, while allowing the variances to be heterogeneous, the marginal covariance to be estimated would then be

$$ \left[\begin{array}{cccc}{\sigma}_1^2& & & \\ {}{\rho}_1{\sigma}_2{\sigma}_1& {\sigma}_2^2& & \\ {}{\rho}_3{\sigma}_3{\sigma}_1& {\rho}_2{\sigma}_3{\sigma}_2& {\sigma}_3^2& \\ {}{\rho}_4{\sigma}_4{\sigma}_1& {\rho}_2{\sigma}_4{\sigma}_2& {\rho}_2{\sigma}_4{\sigma}_3& {\sigma}_4^2\end{array}\right]. $$

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Daniel McNeish
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Jeffrey Harring
- 2

1.Arizona State UniversityTempeUSA
2.University of MarylandCollege ParkUSA

Covariance pattern mixture models: Eliminating random effects to improve convergence and performance

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