Monte Carlo (MC) studies are experiments that repeatedly generate sample data from a population model [

24,

25]. The purpose of the repeated sample generation is to obtain empirical sampling distributions [

27]. MC methods are commonly used in latent variable modeling for multiple reasons, one of which is to examine the performance of specific modeling procedures under a particular set of known circumstances [

28]. Use of a population with a known structure allows for the investigation of fit index performance. To that end, we examined the utility of GMM fit indices under specific conditions that have been, or are likely to be, observed in investigations of the FE.

#### 2.1.1. Population Models

The population models from which samples were drawn for the current study all contained two classes of respondents defined by their growth patterns across three time points. We used two classes for the population models, because it is the simplest mixture model scenario, which is best placed to start when applying new models. Extensions to situations with more than two classes are easily generalizable from our presentation and have been documented elsewhere (e.g., [

23,

29,

30]). We used three time points because that is the minimum needed for a latent model [

17]. In one set of population models, both classes exhibited linear trends; in the other set of population models, both classes exhibited quadratic trends.

A typical growth model consists of two parts: a Level-1 component and Level-2 component [

31]. The Level-1 component is comprised of the observed variable (e.g., intelligence test scores) measured at the different time periods. The Level-1 portion of the growth model is given in

Equation 1.

where

${Y}_{ij}$ is the value of the outcome variable for person

i at time period

j,

${\lambda}_{0i}$ is the true intercept of the change trajectory for person

i,

${\lambda}_{1i}$ is the true linear slope of the change trajectory,

${\lambda}_{2i}$ is the true quadratic curvature of the change trajectory,

${\u03f5}_{ij}$ is the residual for person

i and

${T}_{j}$ is a variable indicating the time period.

The GMM version of the growth model is very similar to

Equation 1, but allows the model to differ by class. It is shown in

Equation 2.

where

k represents the class to which person

i belongs.

The variables used in

Equation 1 are not independent of each other, since they are observed on the same individuals at each time period. Thus, the Level-2 components are the unique individuals in the sample. The Level-2 model for the

k classes is given in

Equation 3.

where

${\gamma}_{00k}$ is the mean intercept for class

k,

${\zeta}_{0ik}$ is the intercept random error for person

i in class

k (with variance of

${\psi}_{00}$),

${\gamma}_{10k}$ is the mean of the linear slopes for class

k,

${\zeta}_{1ik}$ is the linear slope random error for person

i in class

k (with variance of

${\psi}_{11}$),

${\gamma}_{20k}$ is the mean of the quadratic terms for class

k and

${\zeta}_{2ik}$ is the quadratic slope random error for person

i in class

k (with a variance of

${\psi}_{22}$). For models that did not include the quadratic term,

${\gamma}_{20k}$ was set to zero, and the quadratic slope variance (

${\zeta}_{2ik}$) was removed.

Figure 1 shows a conceptual diagram of the GMM. For all models in the MC study, no linear or quadratic slope variances were estimated.

**Figure 1.**
Conceptual diagram of the linear and quadratic growth mixture model with Level-1 subscripts removed. Coefficients for dashed lines were set to zero for the linear model.

**Figure 1.**
Conceptual diagram of the linear and quadratic growth mixture model with Level-1 subscripts removed. Coefficients for dashed lines were set to zero for the linear model.

In all population models, the class trends were specified such that if classes were ignored, the change in scores across time resembled a typical FE (

i.e., a rise of approximately 0.30 IQ points a year). For example, the population version of the class-specific growth models are depicted in

Figure 2 along with the combined growth model if classes were ignored.

Figure 2a represents the model that included the linear term, while

Figure 2b represents the model that included the linear and quadratic terms. We refer to the conditions with the quadratic growth term as the crossing growth pattern conditions, while we refer to the conditions with only the linear growth term as the non-crossing growth pattern conditions.

We chose the two growth patterns, because they represented two extremes for having heterogeneity in the FE, which is a recommended strategy when initially investigating a phenomenon [

32]. Our choice does not indicate that we believe they are the only two growth patterns, only that they are two possible forms of heterogeneous growth that could underlie the FE.

**Figure 2.**
Class-specific and combined growth models. (**a**) Linear (non-crossing) growth model; (**b**) quadratic (crossing) growth model.

**Figure 2.**
Class-specific and combined growth models. (**a**) Linear (non-crossing) growth model; (**b**) quadratic (crossing) growth model.

#### 2.1.2. Design Factors

**Table 1.**
Design factors for the Monte Carlo study.

**Table 1.**
Design factors for the Monte Carlo study.
Factor | Level 1 | Level 2 |
---|

Class Prevalence (Class 1/Class 2) | 0.40/0.60 | 0.30/0.70 |

Sample Size | 200 | 800 |

Measure Reliability | 0.80 | 0.95 |

Growth Pattern | Linear (Not Crossed) | Quadratic (Crossed) |

The manipulated design factors included in the MC study were: (a) class prevalence, (b) sample size, (c) score reliability and (d) growth pattern. The levels for each factor are given in

Table 1. The design was fully crossed to yield 16 cells (

i.e., unique conditions). In all models, the intercept variance was set to 0.80. The reliability was specified as the ratio of the observed variable’s true variance (

i.e., variance explained by the model) to the total variance (

i.e., true variance plus residual variance) of variables at each time period. The reliability formula is given in

Equation 4.

where

${\psi}_{00}$ is the intercept variance in class

k,

${\psi}_{11}$ is the linear slope variance in class

k,

${\psi}_{22}$ is the quadratic term variance in class

k,

${\psi}_{01}$ is the covariance between the intercept and linear slope in class

k,

${\psi}_{02}$ is the covariance between the intercept and quadratic term in class

k,

${\psi}_{12}$ is the covariance between the linear slope and quadratic term in class

k,

${\lambda}_{t}$ is the time period indicator for time

t and

${\sigma}_{k}^{2}t$ is the residual variance at time

t in class

k.

For the current study, the linear slope and quadratic term variances were set to zero in both classes, as were the covariances. Furthermore, the intercept variances were set to 0.80 in both classes, and the residual variances were held constant across all time periods. Therefore,

Equation 4 reduces to

Equation 4a.

As an example, for a reliability of 0.95 for the linear (non-crossing) model, plug in the known values into

Equation 4a and solve for the unknown value,

${\sigma}^{2}$:

The population effect size for the GMMs is given in

Equation 5.

where

${\mu}_{D}$ is the mean of the difference scores,

${\sigma}_{D}$ is the standard deviation of the difference scores,

${\pi}_{k}$ is the prevalence of class

k,

${\mu}_{3k}$ is the mean of class

k at Time Period 3,

${\mu}_{1k}$ is the mean of class

k at Time Period 1 and

${\sigma}_{t}^{2}$ is the residual variance at time period

t.

As an example, the effect size for the linear (non-crossing) model with a reliability of 0.80 and a Class 2 prevalence and slope of 0.60 and 0.10, respectively, and a Class 1 prevalence and slope of 0.40 and 0.40, respectively is:

The population effect size for each cell of the design is given in

Table 2.

**Table 2.**
Population effect size for each cell of the design.

**Table 2.**
Population effect size for each cell of the design.
| | | Growth Pattern |
---|

Class Prevalence | Reliability | Sample Size | Crossing | Non-Crossing |
---|

${\pi}_{1}$ = 0.40, ${\pi}_{2}$ = 0.60 | 0.80 | 200 | 0.20 | 0.63 |

${\pi}_{1}$ = 0.40, ${\pi}_{2}$ = 0.60 | 0.80 | 800 | 0.20 | 0.63 |

${\pi}_{1}$ = 0.40, ${\pi}_{2}$ = 0.60 | 0.95 | 200 | 0.20 | 1.07 |

${\pi}_{1}$ = 0.40, ${\pi}_{2}$ = 0.60 | 0.95 | 800 | 0.20 | 1.07 |

${\pi}_{1}$ = 0.30, ${\pi}_{2}$ = 0.70 | 0.80 | 200 | 0.43 | 0.55 |

${\pi}_{1}$ = 0.30, ${\pi}_{2}$ = 0.70 | 0.80 | 800 | 0.43 | 0.55 |

${\pi}_{1}$ = 0.30, ${\pi}_{2}$ = 0.70 | 0.95 | 200 | 0.43 | 0.95 |

${\pi}_{1}$ = 0.30, ${\pi}_{2}$ = 0.70 | 0.95 | 800 | 0.43 | 0.95 |