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Modeling growth across repeated measures of individuals and evaluating predictors of growth can reveal developmental patterns and factors that affect those patterns. When growth follows a sigmoidal shape, the Logistic, Gompertz, and Richards nonlinear growth curves are plausible. These functions have parameters that specifically control the starting point, total growth, overall rate of change, and point of greatest growth. Variability in growth parameters across individuals can be explained by covariates in a mixed model framework. The purpose of this tutorial is to provide analysts a brief introduction to these growth curves and demonstrate their application. The 'saemix' package in R is used to fit models to simulated data to answer specific research questions. Enough code is provided in-text to describe how to execute the analyses with the complete code and data provided in Supplementary Materials.

Modeling growth trajectories is often of interest in the behavioral and social sciences. Nonlinear mixed effects growth models (NLMEGMs) allow for variation in the rate of growth for an individual across time, such that an individual’s growth may be characterized by periods of slower growth and periods of accelerated growth. While accurate modeling of the trajectory of within-individual growth is critical, answering questions about why individuals’ growth patterns vary from one another can provide insight into the characteristics associated with tempered or delayed development. NLMEGMs in a mixed model (also, multilevel or hierarchical modeling,

Nonlinear mixed effects growth models allow for the modeling of non-linear within-person change and between person differences in change (

Alternative methods of fitting curvilinear growth exist besides the Logistic, Gompertz, and Richards curves. Including polynomial terms (e.g., quadratic, cubic) for time can accomplish this. However, the parameters of the Logistic, Gompertz, and Richards curves allow for a more nuanced understanding of individual growth and the factors that may be associated with individual differences in growth. The researcher can ask substantively meaningful questions about specific aspects of growth that may not be possible with use of polynomial terms. Therefore, we focus on the Logistic, Gompertz, and Richards curves, describing each in terms of growth in student academic achievement.

Within the multilevel framework, growth parameters can be estimated to have fixed and random effects. If a growth parameter is constrained to be equal across all individuals, then the fixed effect is the estimated value of this single parameter for all individuals. If a growth parameter is allowed to vary across individuals, then a fixed and random effect are estimated for the growth parameter. The fixed effect is essentially the weighted average of the individual parameter estimates and the random effect contains a variance that estimates individual variability about the fixed effect. Variability in a parameter can then be explained by adding predictors to the model for each parameter.

Latent growth modeling is an alternative framework for estimating non-linear growth. Each framework has its benefits and challenges (

The focus of the present paper is within the multilevel modeling framework given 1) modeling longitudinal growth and evaluating predictors of this growth can be done in a straightforward manner in the multilevel modeling framework and 2) open-source programs exist for doing so.

The Logistic growth curve is defined as

Note that the parameters are allowed to vary across individuals (they each have an

Various modifications are possible. The researcher can evaluate predictors of each growth parameter. To do so, a predictor is added, for example, in the equation with

The Gompertz curve allows for asymmetric growth before and after the point of inflection. In an educational context, this would allow for a sharp increase as students enter school and then a more tempered approach to a maximum. The Gompertz function is

The Richards curve is more complex than either the Logistic or the Gompertz curve because the asymmetry is controlled by an additional parameter in the model. The functional form of the Richards curve is

There are several programs for fitting NLMEGMs.

The stochastic approximation expectation maximization algorithm uses Markov Chain Monte Carlo processes to determine the likelihood and the model parameters that maximize it in an iterative fashion (see

The starting values selected can have an impact on model estimates (

Like other mixed models fit with maximum likelihood, model comparisons can be made using Akaike information criterion (AIC;

The normalized prediction distribution error (NPDE) was developed for assessing non-linear mixed models.

We demonstrate the use of

For RQ1, we compare the fit of the three unconditional NLMEGMs, use the NPDE to evaluate Gompertz model fit, interpret model parameters, and display relevant output and plots. For RQ2, a conditional model is fit wherein the growth parameters allowed to vary are predicted by the dichotomous Sex variable. To answer RQ3, SES is added as a predictor. RQ4 requires evaluation of an interaction between Sex and SES; this is done by multiplying the SES and Sex of each child and adding this product as a third variable in the model. Model comparisons are needed throughout to evaluate model improvement; therefore, likelihood ratio tests are conducted and comparisons using AIC and BIC are made. There are limitations to our demonstration. First, the data are simulated; therefore, any data cleaning that would need to occur with actual data is not conducted. Second, diagnostics of model fit should be conducted at each step though we only demonstrate this with the unconditional model.

The Gompertz growth model is ultimately retained; therefore, the in-text example will demonstrate specification of the Gompertz growth model. Initial specifications of the Logistic and Richards functions needed for answering the first research question are available in

```
gompertz.model <- function(psi,id,x) {
t <- x[,1]
TtlGrwth <- psi[id,1]
Apprch <- psi[id,2]
Timing <- psi[id,3]
LwrAsy <- psi[id,4]
ypred <- LwrAsy+TtlGrwth*exp(-exp(-Apprch*(t-Timing)))
return(ypred)
}
```

The psi, id, and x components of the function are derived from the model and data objects that follow for each analysis. Note that the

`NLMEGM.options <- list(seed=1234, displayProgress=FALSE)`

Setting the seed allows for reproducibility. The estimation of model parameters is an iterative process; to see the iterations graphically and check convergence of the solution,

To answer RQ1 we must fit unconditional models using the data in the data file title

```
GompertzData.RQ1 <- saemixData(name.data = NLMEGMExData, header = TRUE,
name.group = c("ID"), name.predictors = c("time"), name.response = c("Achievement"),
name.X = "time")
```

We specify the data file and that it contains variable names (

After creating the data object, the model object is specified:

```
GompertzModel.RQ1 <- saemixModel(model=gompertz.model,
description = 'Gompertz', psi0=c(TtlGrwth=0,Apprch=0,Timing=0,LwrAsy=0),
covariance.model = matrix(c(1,1,1,0,1,1,1,0,1,1,1,0,0,0,0,0), ncol = 4, byrow = TRUE),
transform.par=c(0,0,0,0))
```

The model uses the previously created

To conduct the analysis:
`GompertzFit.RQ1 <- saemix(GompertzModel.RQ1,GompertzData.RQ1, NLMEGM.options)`

Abbreviated output is provided below. Note that the section

```
------------------------------------
---- Results ----
Fixed effects
Parameter Estimate SE CV(%)
TtlGrwth 5.220 0.055 1.1
Apprch 1.760 0.072 4.1
Timing 1.987 0.020 1.0
LwrAsy -0.066 0.015 22.7
a. 0.192 0.008 4.0
Variance of random effects
Parameter Estimate SE CV(%)
omega2.TtlGrwth 0.236 0.039 16
omega2.Apprch 0.405 0.072 18
omega2.Timing 0.034 0.006 16
cov.TtlGrwth.Apprch 0.234 0.043 18
cov.TtlGrwth.Timing -0.012 0.010 -87
cov.Apprch.Timing 0.002 0.014 648
Statistical criteria
Correlation matrix of random effects
omega2.TtlGrwth omega2.Apprch omega2.Timing
omega2.TtlGrwth 1.0000 0.756 -0.133
omega2.Apprch 0.756 1.0000 0.019
omega2.Timing -0.133 0.019 1.0000
Likelihood computed by linearisation
-2LL= 359.088
AIC= 381.088
BIC= 409.745
```

Recall, to answer RQ1 the results for the Logistic, Gompertz, and Richards growth models are compared. Information criterion values are useful for this purpose, and in

Model | -2LL | AIC | BIC |
---|---|---|---|

Logistic | 417.525 | 439.525 | 468.182 |

Gompertz | 359.088 | 381.088 | 409.745 |

Richards | 359.141 | 383.141 | 414.403 |

Each parameter estimate is derived iteratively; the plot of these iterations is used to evaluate convergence. See

Several diagnostic plots of model fit are available. First, simple plots of individual predictions overlayed on the observed values provides an indication of model fit. Such plots are presented in

Four plots that utilize the NPDE are evaluated (see

Though not necessary for answering the first research question, we review the parameter estimates in the

The

Three growth parameters were allowed to vary and correlate with one another. The

The remaining three research questions require variations of the code presented above. For each research question, the specific variation is provided in text with full code in

The

`name.covariates = c("Sex").`

To include Sex as a predictor of the Total Growth, Rate of Approach, and Timing parameters the following line to the

`covariate.model = matrix(c(1,1,1,0), ncol = 4, byrow = TRUE)`

The above line of code indicates that the Sex covariate is a predictor of the

Overall improvement in model fit after adding Sex was statistically significant (

The

`name.covariates = c("Sex", "SES")`

The command for creating the model object must be changed to include Sex as a predictor of the

`covariate.model = matrix(c(1,1,1,0,1,1,1,0), ncol = 4, byrow = TRUE)`

where the first set of four values

Overall improvement in model fit after adding SES to the model with Sex only as a predictor was statistically significant (

Predictor | Total growth | Rate of approach | Timing |
---|---|---|---|

Sex | 0.416 (0.079)*** | 0.676 (0.107)*** | 0.016 (0.040) |

SES | 0.253 (0.038)*** | 0.328 (0.053)*** | -0.012 (0.020) |

***

A test of moderation allows the investigator to determine if the relationship between a predictor and outcome is dependent on the value of a third variable. Moderation analyses can be used to determine for whom relationships hold or treatments are most effective. In growth modeling, moderation can be used to determine if, for example, the positive association between SES and the Total Growth parameter is equivalent for males and females or if for one sex the relationship is stronger. Next, the potential moderating role of Sex on the relationship between SES and Gompertz function parameters is investigated.

The

`name.covariates = c("Sex", "SES","SexSESmod")`

The command for creating the model object must be changed to include the interaction of Sex and SES as a predictor of Total Growth, Rate of Approach, and Timing. This is done within the

`covariate.model = matrix(c(1,1,1,0,1,1,1,0,1,1,1,0), ncol = 4, byrow = TRUE)`

Note that the first set of four values

Overall improvement in model fit after adding the interaction of Sex and SES to the previous model with Sex and SES as predictors was statistically significant (

Predictor | Total growth | Rate of approach | Timing |
---|---|---|---|

Sex | 0.414 (0.077)*** | 0.676 (0.100) *** | 0.015 (0.041) |

SES | 0.114 (0.064)* | 0.137 (0.065)* | -0.009 (0.033) |

Sex x SES | 0.216 (0.079)** | 0.362 (0.097)*** | -0.001 (0.041) |

*

NLMEGMs allow for the modeling of sigmoidal growth trajectories that resemble real-world phenomena. The parameters of NLMEGMs translate into easily understandable attributes of growth. Further extending the model to include predictors of these growth parameters can address specific hypotheses such as the existence of relationships between individual characteristics and growth or the effect of policy implementation on one or more attributes of growth. In this paper three non-linear growth trajectories were discussed and their accompanying parameters. This was followed by an example of how to use a specific package (

An initial concern for the analyst when planning to model non-linear growth trajectories is the framework in which the modeling should be conducted. Non-linear growth models can be fit as multilevel models or as latent growth models. The multilevel modeling framework was demonstrated here because other demonstrations require use of costly programs (e.g., SAS in

The demonstration utilized

The reader is encouraged to consider modeling non-linear growth trajectories to answer substantive questions about change over time. Extending beyond polynomial terms, sigmoidal trajectories may fit one’s data more effectively while providing the opportunity to test interesting hypotheses about components of growth. This paper contributes to the NLMEGMs literature by providing an accessible introduction to the models with a reproducible example.

The data for this article are freely available (see the

For this article the following Supplementary Materials are available via PsychArchives repository (for access see

Dataset for replicating example analyses.

Code for replicating all analyses.

The authors have no funding to report.

The authors have declared that no competing interests exist.

The authors have no additional (i.e., non-financial) support to report.