# An Investigation of Growth Mixture Models for Studying the Flynn Effect

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Mixture Models

#### 1.2. Current Study

## 2. Method

#### 2.1. Monte Carlo Study

#### 2.1.1. Population Models

**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 2.**Class-specific and combined growth models. (

**a**) Linear (non-crossing) growth model; (

**b**) quadratic (crossing) growth model.

#### 2.1.2. Design Factors

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) |

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 |

#### 2.2. Data Generation

#### 2.3. Fit Indices to Aid in the Growth Mixture Model Selection

#### 2.3.1. Absolute Model Fit

#### 2.3.2. Lo–Mendell–Rubin Likelihood Ratio Test

#### 2.3.3. Relative Fit Indices

Index | Abbreviation | Formula |
---|---|---|

Akaike information criteria | AIC | $-2logL+2p$ |

Consistent AIC | CAIC | $-2logL+p[log(n)+1]$ |

Corrected AIC | AICc | AIC $+\frac{\left(2\right(p+1\left)\right(p+2\left)\right)}{(n-p-2)}$ |

Bayesian information criteria | BIC | $-2logL+plog\left(n\right)$ |

Sample size adjusted BIC | SSBIC | $-2logL+plog\left[\frac{(n+2)}{24}\right]$ |

Draper’s information criterion | DIC | $-2logL+p\left(log\left(\frac{n}{2\pi}\right)\right)$ |

#### 2.3.4. Classification Certainty

#### 2.4. Analysis

#### 2.5. Software

## 3. Results

#### 3.1. Monte Carlo Study

Accuracy | |
---|---|

Fit Index | (% Correct) |

Akaike information criterion (AIC) | 57.1 |

Corrected Akaike information criterion (AICc) | 66.7 |

Consistent Akaike information criterion (CAIC) | 99.9 |

Bayesian information criterion (BIC) | 99.8 |

Sample size adjusted BIC (SSBIC) | 79.0 |

Draper’s information criterion (DIC) | 96.8 |

Integrated classification likelihood with BIC approximation (ICL-BIC) | 89.5 |

Lo–Mendell–Rubin likelihood ratio test (LMR) | 54.4 |

Fit Index | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Class Prevalence | Reliability | Sample Size | AIC | AICc | CAIC | BIC | SSBIC | DIC | ICL-BIC | LMR |

Crossing Growth Pattern | ||||||||||

π_{1} = 0.40, π_{2} = 0.60 | 0.80 | 200 | 60 | 75 | 100 | 100 | 66 | 96 | 100 | 76 |

π_{1} = 0.40, π_{2} = 0.60 | 0.80 | 800 | 58 | 63 | 100 | 100 | 96 | 100 | 100 | 73 |

π_{1} = 0.40, π_{2} = 0.60 | 0.95 | 200 | 60 | 76 | 100 | 100 | 67 | 96 | 100 | 78 |

π_{1} = 0.40, π_{2} = 0.60 | 0.95 | 800 | 60 | 64 | 100 | 100 | 96 | 100 | 100 | 73 |

π_{1} = 0.30, π_{2} = 0.70 | 0.80 | 200 | 58 | 74 | 100 | 100 | 64 | 96 | 100 | 78 |

π_{1} = 0.30, π_{2} = 0.70 | 0.80 | 800 | 58 | 62 | 100 | 100 | 97 | 100 | 100 | 74 |

π_{1} = 0.30, π_{2} = 0.70 | 0.95 | 200 | 58 | 74 | 100 | 100 | 66 | 96 | 100 | 79 |

π_{1}= 0.30, π_{2} = 0.70 | 0.95 | 800 | 60 | 63 | 100 | 100 | 97 | 100 | 100 | 73 |

Non-Crossing Growth Pattern | ||||||||||

π_{1} = 0.40, π_{2} = 0.60 | 0.80 | 200 | 50 | 66 | 100 | 99 | 58 | 92 | 66 | 9 |

π_{1} = 0.40, π_{2} = 0.60 | 0.80 | 800 | 59 | 63 | 100 | 100 | 96 | 99 | 73 | 13 |

π_{1} = 0.40, π_{2} = 0.60 | 0.95 | 200 | 55 | 68 | 100 | 100 | 60 | 93 | 77 | 24 |

π_{1} = 0.40, π_{2} = 0.60 | 0.95 | 800 | 62 | 65 | 100 | 100 | 96 | 99 | 94 | 80 |

π_{1} = 0.30, π_{2} = 0.70 | 0.80 | 200 | 50 | 66 | 100 | 100 | 57 | 93 | 68 | 9 |

π_{1} = 0.30, π_{2} = 0.70 | 0.80 | 800 | 57 | 61 | 100 | 100 | 96 | 99 | 74 | 17 |

π_{1} = 0.30, π_{2} = 0.70 | 0.95 | 200 | 50 | 65 | 100 | 99 | 58 | 93 | 85 | 29 |

π_{1} = 0.30, π_{2} = 0.70 | 0.95 | 800 | 60 | 64 | 100 | 100 | 94 | 99 | 97 | 84 |

#### 3.1.1. Relative Fit Indices

#### 3.1.2. Classification Certainty

#### 3.1.3. Lo–Mendell–Rubin Likelihood Ratio Test

#### 3.2. National Intelligence Tests

Variable | Class 1 | Class 2 | Combined |
---|---|---|---|

Membership | |||

n | 89 (24.7%) | 272 (75.4%) | 361 |

Female | 43 (48.3%) | 169 (62.1%) | |

Age | |||

12 Years | 12 (13.5%) | 35 (12.9%) | |

13 Years | 66 (74.2%) | 155 (57.0%) | |

14 Years | 11 (12.4%) | 82 (30.2%) | |

NIT Scores | |||

Time 1 | 236.0 (49.4) | 238.2 (46.7) | 237.6 (47.3) |

Time 2 | 266.4 (36.0) | 264.4 (36.1) | 264.9 (36.0) |

Time 3 | 223.0 (19.4) | 287.9 (24.1) | 271.9 (36.2) |

**Table 7.**Summary of model fit for the models with two–five-classes from the National Intelligence Test data.

Fit Index | ||||||||
---|---|---|---|---|---|---|---|---|

Classes | AIC | AICc | CAIC | BIC | SSBIC | DIC | ICL-BIC | LMR p |

2 | 11,043 | 11,019 | 11,097 * | 11,086 * | 11,051 | 11,091 * | 11,301 * | 0.09 |

3 | 11,029 | 10,997 | 11,102 | 11,087 | 11,040 | 11,094 | 11,361 | 0.60 |

4 | 11,023 * | 10,983 | 11,116 | 11,097 | 11,036 * | 11,105 | 11,439 | 0.34 |

5 | 11,024 | 10,976 * | 11,136 | 11,113 | 11,040 | 11,124 | 11,544 | 0.49 |

**Figure 3.**Class-specific and combined growth patterns from matched respondents collected from three waves of Estonian National Intelligence Test scores.

**Figure 4.**Distribution of National Intelligence Test scores for the three matched Estonian cohorts (1934, 1998 and 2006). The plotted score is the sum across the ten sub-tests.

## 4. Discussion

#### 4.1. Design Factors

#### 4.2. Consequences of Incorrect Model Selection

#### 4.3. Additional Considerations

#### 4.4. Comparison with Previous Flynn Effect Research Using the Estonian NIT

#### 4.5. Recommendations

## Author Contributions

## Conflicts of Interest

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Morgan, G.B.; Beaujean, A.A. An Investigation of Growth Mixture Models for Studying the Flynn Effect. *J. Intell.* **2014**, *2*, 156-179.
https://doi.org/10.3390/jintelligence2040156

**AMA Style**

Morgan GB, Beaujean AA. An Investigation of Growth Mixture Models for Studying the Flynn Effect. *Journal of Intelligence*. 2014; 2(4):156-179.
https://doi.org/10.3390/jintelligence2040156

**Chicago/Turabian Style**

Morgan, Grant B., and A. Alexander Beaujean. 2014. "An Investigation of Growth Mixture Models for Studying the Flynn Effect" *Journal of Intelligence* 2, no. 4: 156-179.
https://doi.org/10.3390/jintelligence2040156