# Assessing Area under the Curve as an Alternative to Latent Growth Curve Modeling for Repeated Measures Zero-Inflated Poisson Data: A Simulation Study

## Abstract

**:**

## 1. Introduction

_{1}) multiplied by the sum of the time intervals is subtracted from the AUC equation to account for the baseline level. Note also that, for equal time intervals, the summation is the number of time points. This second AUC equation can permit scores below zero, as individuals may use less than they did at baseline. As such, it may be a better way to assess behavioral change even if it is not an area per se.

_{i}is the i

^{th}interval between consecutive time points.

## 2. Methods

#### 2.1. ZIP LGCM

#### 2.2. AUC

## 3. Results

_{2}on the intercept. Power decreased particularly for the binary model part and for sample sizes 100 and below, with power values no higher than 0.348 for the binary part growth factor (effect of X

_{2}on slope; n = 100) and 0.205 for the binary part growth factor (effect of X

_{1}on slope; n = 50).

#### 3.1. Area under the Curve

#### 3.2. Case Examples

## 4. Discussions

#### 4.1. Limitations and Conclusions

#### 4.2. Software

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Abbreviation | Definition |

GEE | General Estimating Equation |

LGCM | Latent Growth Curve Model |

ZIP | Zero-Inflated Poisson |

SEM | Structural Equation Modeling |

AUC | Area Under the Curve |

AUC—g | Area Under the Curve with respect to ground |

AUC—i | Area Under the Curve with respect to the increase |

ROC | Receiver Operating Characteristic |

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**Figure 2.**Histograms for the three area under the curve (AUC) measures for the N = 50 simulation. (Panel

**A**): AUC—i; (Panel

**B**): AUC—g; (Panel

**C**): LN AUC—g.

**Figure 3.**Observed values across three time points (observed trajectories; Panel

**A**) and calculated AUC values (Panel

**B**) for individual 10 from the N = 50 simulated data.

**Figure 4.**Observed values across three time points (observed trajectories; Panel

**A**) and calculated AUC values (Panel

**B**) for individual 18 from the N = 50 simulated data.

**Table 1.**Descriptive statistics and ZIP parameters for all four waves for the different sample sizes *.

N | Wave | Minimum | Maximum | Mean | SD | λ | π |
---|---|---|---|---|---|---|---|

500 | Time 1 | 0 | 83 | 1.93 | 7.419 | 29.45 | 0.93 |

Time 2 | 0 | 107 | 2.06 | 8.064 | 32.78 | 0.94 | |

Time 3 | 0 | 475 | 2.54 | 21.662 | 186.28 | 0.99 | |

Time 4 | 0 | 206 | 2.57 | 14.417 | 82.45 | 0.97 | |

250 | Time 1 | 0 | 83 | 2.17 | 8.341 | 33.23 | 0.93 |

Time 2 | 0 | 106 | 2.31 | 8.636 | 33.60 | 0.93 | |

Time 3 | 0 | 38 | 1.73 | 5.153 | 16.08 | 0.89 | |

Time 4 | 0 | 206 | 3.26 | 18.248 | 104.71 | 0.97 | |

100 | Time 1 | 0 | 63 | 2.15 | 8.964 | 38.524 | 0.94 |

Time 2 | 0 | 43 | 2.04 | 5.605 | 16.440 | 0.88 | |

Time 3 | 0 | 38 | 2.00 | 5.944 | 18.666 | 0.89 | |

Time 4 | 0 | 206 | 4.27 | 22.431 | 121.104 | 0.96 | |

50 | Time 1 | 0 | 63 | 2.14 | 9.165 | 40.391 | 0.95 |

Time 2 | 0 | 43 | 2.60 | 7.100 | 20.989 | 0.95 | |

Time 3 | 0 | 32 | 1.36 | 4.681 | 16.472 | 0.92 | |

Time 4 | 0 | 206 | 6.90 | 31.403 | 148.820 | 0.95 |

Count Intercept ^{1} | Binary Intercept ^{2} | Count Trend ^{3} | Binary Trend ^{4} | ||||||
---|---|---|---|---|---|---|---|---|---|

n | Measures ^{5} | X_{1} | X_{2} | X_{1} | X_{2} | X_{1} | X_{2} | X_{1} | X_{2} |

500 | Average | 1.0009 | −0.9993 | 0.5091 | −0.509 | 0.0998 | −0.1002 | 0.3545 | −0.3537 |

%Bias | 0.09 | −0.07 | 1.82 | 0.018 | −0.2 | 0.2 | 1.2857 | 1.0571 | |

MSE | 0.0032 | 0.0032 | 0.0331 | 0.331 | 0.0005 | 0.0005 | 0.106 | 0.011 | |

Coverage | 0.942 | 0.943 | 0.95 | 0.946 | 0.93 | 0.931 | 0.952 | 0.943 | |

Power | 1.00 | 1.00 | 0.829 | 0.846 | 0.992 | 0.992 | 0.939 | 0.933 | |

250 | Average | 1.0022 | −0.9996 | 0.5257 | −0.5203 | 0.1001 | −0.1007 | 0.3595 | −0.3609 |

%Bias | 0.22 | −0.04 | 5.14 | 4.06 | 0.1 | 0.7 | 2.7143 | 3.1143 | |

MSE | 0.0066 | 0.0067 | 0.0738 | 0.0732 | 0.001 | 0.0011 | 0.0229 | 0.0237 | |

Coverage | 0.934 | 0.939 | 0.943 | 0.949 | 0.924 | 0.925 | 0.948 | 0.945 | |

Power | 1.0 | 1.0 | 0.524 | 0.514 | 0.882 | 0.888 | 0.691 | 0.692 | |

100 | Average | 1.0039 | −1.004 | 0.5705 | −0.5599 | 0.1013 | −0.1015 | 0.3804 | −0.3846 |

%Bias | 0.39 | 0.4 | 14.1 | 11.98 | 1.3 | 1.5 | 8.6857 | 9.8857 | |

MSE | 0.0193 | 0.0187 | 0.2371 | 0.2274 | 0.0034 | 0.0032 | 0.0735 | 0.0737 | |

Coverage | 0.922 | 0.923 | 0.942 | 0.95 | 0.901 | 0.905 | 0.936 | 0.943 | |

Power | 1 | 1 | 0.203 | 0.189 | 0.543 | 0.543 | 0.347 | 0.348 | |

50 | Average | 1.0131 | −1.0123 | 5.2436 | −3.6967 | 0.1007 | −0.102 | −0.1756 | −0.18 |

%Bias | 1.31 | 1.23 | 948.72 | 639.34 | 0.7 | 2.00 | −150.171 | −48.571 | |

MSE | 0.0494 | 0.0473 | 25461.42 | 7943.048 | 0.0093 | 0.0092 | 1909.731 | 409.3025 | |

Coverage | 0.893 | 0.9 | 0.935 | 0.937 | 0.886 | 0.886 | 0.923 | 0.921 | |

Power | 0.986 | 0.985 | 0.112 | 0.111 | 0.334 | 0.343 | 0.205 | 0.204 |

^{1}Intercept factor for count part;

^{2}intercept factor for the binary part;

^{3}linear trend factor for the count part;

^{4}linear trend factor for the binary part;

^{5}X

_{1}and X

_{2}are the two continuous predictor variables.

n | AUC Measure | Mean | SD | Skewness | Kurtosis | Median | IQR ^{1} |
---|---|---|---|---|---|---|---|

500 | AUC—i | 1.061 | 25.589 | 12.146 | 240.570 | 0.00 | 2.00 |

AUC—g | 6.857 | 26.907 | 12.565 | 196.822 | 1.50 | 4.50 | |

LN AUC—g | 1.11 | 1.13 | 1.089 | 1.067 | 0.916 | ||

250 | AUC—i | 0.24 | 16.343 | −3.515 | 45.006 | 0.00 | 2.50 |

AUC—g | 6.756 | 20.107 | 7.818 | 77.213 | 1.50 | 5.00 | |

LN AUC—g | 1.158 | 1.135 | 0.991 | 0.709 | 0.916 | 1.79 | |

100 | AUC—i | 0.8 | 20.416 | −3.714 | 38.084 | 0.00 | 2.50 |

AUC—g | 7.25 | 18.484 | 4.754 | 26.398 | 1.50 | 4.50 | |

LN AUC—g | 1.164 | 1.181 | 1.087 | 0.708 | 0.916 | 1.70 | |

50 | AUC—i | 2.06 | 15.441 | 22.673 | 8.48 | 0.00 | 4.13 |

AUC—g | 8.48 | 23.780 | 18.467 | 1.105 | 1.250 | 4.13 | |

LN AUC—g | 1.105 | 1.251 | 1.340 | 1.535 | 0.805 | 1.63 |

^{1}IQR; interquartile range.

AUC—g ^{1} | AUC—i | ||||
---|---|---|---|---|---|

n | Measures | X_{1} | X_{2} | X_{1} | X_{2} |

500 | Average | 0.3282 | −0.4147 | 0.9069 | −1.4914 |

Bias | −0.243 | −0.072 | −2.274 | −0.441 | |

MSE | 0.0021 | 0.0018 | 1.3694 | 1.1977 | |

Coverage | 0.945 | 0.95 | 0.946 | 0.95 | |

Power | 1 | 1 | 0.134 | 0.283 | |

250 | Average | 0.2924 | −0.4504 | −0.5813 | 0.0029 |

Bias | −0.205 | 0.0889 | 1.6259 | −71 | |

MSE | 0.0045 | 0.004 | 1.1947 | 1.0654 | |

Coverage | 0.947 | 0.948 | 0.947 | 0.948 | |

Power | 0.991 | 1 | 0.091 | 0.051 | |

100 | Average | 0.3167 | −0.4294 | −2.9642 | −2.0807 |

Bias | −0.409 | −0.14 | 0.8918 | −0.54 | |

MSE | 0.0115 | 0.01 | 4.448 | 3.8672 | |

Coverage | 0.941 | 0.944 | 0.941 | 0.944 | |

Power | 0.845 | 0.986 | 0.313 | 0.195 | |

50 | Average | 0.4064 | −0.4795 | 0.1414 | −1.385 |

Bias | −0.392 | 0.1044 | −14.303 | 0.581 | |

MSE | 0.0226 | 0.021 | 4.7604 | 4.4243 | |

Coverage | 0.936 | 0.929 | 0.936 | 0.928 | |

Power | 0.802 | 0.924 | 0.069 | 0.125 |

^{1}Natural log-transformed.

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## Share and Cite

**MDPI and ACS Style**

Rodriguez, D.
Assessing Area under the Curve as an Alternative to Latent Growth Curve Modeling for Repeated Measures Zero-Inflated Poisson Data: A Simulation Study. *Stats* **2023**, *6*, 354-364.
https://doi.org/10.3390/stats6010022

**AMA Style**

Rodriguez D.
Assessing Area under the Curve as an Alternative to Latent Growth Curve Modeling for Repeated Measures Zero-Inflated Poisson Data: A Simulation Study. *Stats*. 2023; 6(1):354-364.
https://doi.org/10.3390/stats6010022

**Chicago/Turabian Style**

Rodriguez, Daniel.
2023. "Assessing Area under the Curve as an Alternative to Latent Growth Curve Modeling for Repeated Measures Zero-Inflated Poisson Data: A Simulation Study" *Stats* 6, no. 1: 354-364.
https://doi.org/10.3390/stats6010022