Estimating rate of change for nonlinear trajectories in the framework of individual measurement occasions: A new perspective on growth curves

Abstract

Researchers are often interested in examining between-individual differences in within-individual processes. If the process under investigation is tracked for a long time, its trajectory may show a certain degree of nonlinearity, so that the rate of change is not constant. A fundamental goal of modeling such nonlinear processes is to estimate model parameters that reflect meaningful aspects of change, including the parameters related to change and other parameters that shed light on substantive hypotheses. However, if the measurement occasion is unstructured, existing models cannot simultaneously estimate these two types of parameters. This article has three goals. First, we view the change over time as the area under the curve (AUC) of the rate of change versus time ($r-t$) graph. Second, using the instantaneous rate of change midway through a time interval to approximate the average rate of change during that interval, we propose a new specification to describe longitudinal processes. In addition to obtaining the individual change-related parameters and other parameters related to specific research questions, the new specification allows for unequally spaced study waves and individual measurement occasions around each wave. Third, we derive the model-based interval-specific change and change from baseline, two common measures to evaluate change over time. We evaluate the proposed specification through a simulation study and a real-world data analysis. We also provide OpenMx and Mplus 8 code for each model with the novel specification.

Appendices

Appendix A: Derivation of rate of change, interval-specific change, and change from baseline

This section provides the detailed derivation for the mean and variance of the rate of change ($dy_{ij}$ in the LBGM or $dy_{ij\_\text {mid}}$ in each parametric nonlinear LCSM), interval-specific change ($\delta _{ij}$), and change from baseline ($\Delta _{ij}$).

A.1 Derivation of the mean and variance of rate of change

For the LBGM, we have $dy_{ij}=\eta _{1i}\times \gamma _{j-1}$ ($j=2, \dots , J$) from Eq. 4. Suppose $f: \mathcal {R}\rightarrow \mathcal {R}$ is a function,¹⁴ which takes a point $\eta _{i}\in \mathcal {R}$ as input and produces $f(\eta _{i})\in \mathcal {R}$ as output. By the delta method, the mean and variance of the rate of change of the LBGM can be expressed as $\mu _{dy_{ij}}=\mu _{\eta _{1}}\times \gamma _{j-1}\quad (j=2,\dots ,J)$ and $\phi _{dy_{ij}}=\psi _{11}\times \gamma _{j-1}^{2}\quad (j=2,\dots ,J)$, respectively. Similarly, the mean and variance of the rate of change of each parametric LCSM can be expressed as

• Quadratic function:

  $$\begin{aligned} \mu _{dy_{ij\_\text {mid}}}= & {} \mu _{\eta _{1}}+2\times \mu _{\eta _{2}}\times t_{ij\_\text {mid}}\quad (j=2,\dots ,J), \\ \phi _{dy_{ij\_\text {mid}}}= & {} \psi _{11}+4\times \psi _{22}\times t^{2}_{ij\_\text {mid}}\\{} & {} +4\times \psi _{12}\times t_{ij\_\text {mid}}\quad (j=2,\dots ,J), \end{aligned}$$

• Negative exponential function:

  $$\begin{aligned} \begin{aligned}&\mu _{dy_{ij\_\text {mid}}}=b\times \mu _{\eta _{1}}\times \exp (-b\times t_{ij\_\text {mid}})\quad (j=2,\dots ,J), \\&\phi _{dy_{ij\_\text {mid}}}\!=\!\psi _{11}\times [b\times \exp (-b\times t_{ij\_\text {mid}})]^{2}\!\!\!\!\quad (j=2,\dots ,J), \end{aligned} \end{aligned}$$

• Jenss–Bayley function:

  $$\begin{aligned} \mu _{dy_{ij\_\text {mid}}}= & {} \mu _{\eta _{1}}+c\times \mu _{\eta _{2}}\\{} & {} \times \exp (c\times t_{ij\_\text {mid}})\quad (j=2,\dots ,J), \\ \phi _{dy_{ij\_\text {mid}}}= & {} \psi _{11}+\psi _{22}\times [c\times \exp (c\times t_{ij\_\text {mid}})]^{2}+2\\{} & {} \!\times \psi _{12}\!\times c\times \exp (c\!\times t_{ij\_\text {mid}})\!\!\!\!\quad (j=2,\dots ,J). \end{aligned}$$

As we can see from the above equations, the mean value and variance of the rate of change of the LBGM are fixed at each study wave, while these parameters from a parametric LCSM are individual-specific in the framework of individual measurement occasions since they are functions of the middle point of each time interval. Therefore, in practice with individual measurement occasions, one may set $t_{ij\_\text {mid}}$ as the average value of the middle of two consecutive measurement times across all individuals to simplify the calculation of the mean and variance of the rate of change for the parametric LCSMs.

A.2 Derivation of the mean and variance of interval-specific change

It is straightforward to derive the mean value and variance for each interval-specific change from the corresponding value of rate of change. Specifically, the mean and variance of each interval-specific change for the nonparametric LCSM can be expressed as $\mu _{\delta y_{ij}}=\mu _{dy_{ij}}\times (t_{ij}-t_{i(j-1)})\quad (j=2,\dots ,J)$ and $\phi _{\delta y_{ij}}=\psi _{11}\times \gamma ^{2}_{j-1}\times (t_{ij}-t_{i(j-1)})^{2}\quad (j=2,\dots ,J)$, respectively. Similarly, for a parametric LCSM, the mean and variance of each interval-specific change can be expressed as $\mu _{\delta y_{ij}}=\mu _{dy_{ij\_\text {mid}}}\times (t_{ij}-t_{i(j-1)})\quad (j=2,\dots ,J)$ and $\phi _{\delta y_{ij}}=\phi _{dy_{ij\_\text {mid}}}\times (t_{ij}-t_{i(j-1)})^{2}\quad (j=2,\dots ,J)$, respectively. Similar to the rate of change, these values of interval-specific change are also individual-specific in the framework of individual measurement occasions.

A.3 Derivation of the mean and variance of change from baseline

Based on the parameters related to interval-specific change above, we are able to derive the mean value of change from baseline at each post-baseline point. In particular, the mean value of change from baseline at each post-baseline $t_{ij}$ for a parametric or nonparametric LCSM can be expressed as $\mu _{\Delta y_{ij}}=\sum _{j=2}^{j}\mu _{dy_{ij\_\text {mid}}}\times (t_{ij}-t_{i(j-1)})$ and $\mu _{\Delta y_{ij}}=\sum _{j=2}^{j}\mu _{dy_{ij}}\times (t_{ij}-t_{i(j-1)})$, respectively. The variance of change from baseline at each post-baseline point can be expressed as

• Nonparametric function:

  $$\begin{aligned} \phi _{\Delta y_{ij}}=\psi _{11}\times \big (\sum _{j=2}^{j}\gamma _{j-1}\times (t_{ij}-t_{i(j-1)})\big )^{2}\quad (j=2,\dots ,J), \end{aligned}$$

• Quadratic function:

  $$\begin{aligned} \phi _{\Delta y_{ij}}= & {} \psi _{11}\times \big (\sum ^{j}_{j=2}(t_{ij}-t_{i(j-1)})\big )^{2}+4\times \psi _{22}\\{} & {} \times \big (\sum ^{j}_{j=2}t_{ij\_\text {mid}}(t_{ij}-t_{i(j-1)})\big )^{2} \\{} & {} +4\times \psi _{12}\\{} & {} \times \sum ^{j}_{j=2}t_{ij\_\text {mid}}(t_{ij}-t_{i(j-1)})^{2}\quad (j=2,\dots ,J), \end{aligned}$$

• Negative exponential function:

  $$\begin{aligned} \phi _{\Delta y_{ij}}= & {} \psi _{11}\times \big (b\times \sum ^{j}_{j=2}\exp (-b\times t_{ij\_\text {mid}})\\{} & {} \times (t_{j}-t_{j-1})\big )^{2}\quad (j=2,\dots ,J), \end{aligned}$$

• Jenss–Bayley function:

  $$\begin{aligned} \phi _{\Delta y_{ij}}= & {} \psi _{11}\times \big (\sum ^{j}_{j=2}(t_{ij}-t_{i(j-1)})\big )^{2}+\psi _{22}\\{} & {} \times \big (\sum ^{j}_{j=2}c\times \exp (c\times t_{ij\_\text {mid}})\times (t_{ij}-t_{i(j-1)})\big )^{2}\\{} & {} +2\times \psi _{12}\\{} & {} \times \big (\sum ^{j}_{j=2}c\times \exp (c\times t_{ij\_\text {mid}})\\{} & {} \times (t_{ij}-t_{i(j-1)})^{2}\big )\quad (j=2,\dots ,J). \end{aligned}$$

Table 10 Summary of performance metrics of nonparametric latent change score model (latent basis growth model)

The mean values and variances of the change from baseline are also individual-specific values as the rate of change and interval-specific change.

To summarize, the mean values and variances of rate of change, interval-specific change, and change-from-baseline values are individual-specific values due to individual measurement occasions. In practice, there are two ways to summarize these values. First, one may plot the mean value and variance for a latent variable of interest (as we did for the mean values of change from baseline in Fig. 4). Second, it is also possible to obtain approximated values of the mean and variance of a latent variable at each point by fixing individual time points to a specific value (i.e., the mean time point of a study wave across all individuals), as we did for the means and variances of rate of change in Tables 7-9.

Appendix B: Data generation and simulation step

For each condition of each model listed in Table 3, we carried out the simulation study according to the following steps:

• 1. Generate growth factors for the LBGM and each parametric LGCM using the R package MASS (Venables and Ripley 2002),

• 2. Generate the time structure with J waves $t_{j}$ as specified in Table 3 and allow for disturbances around each wave $t_{ij}\sim U(t_{j}-\Delta , t_{j}+\Delta )$ ($\Delta =0.25$) to have individual measurement occasions,

• 3. Calculate factor loadings of each individual for the LBGM and each parametric LGCM, which are functions of the individual measurement occasions and the additional growth coefficient(s) (when applicable),

• 4. Calculate the values of the repeated measurements based on the growth factors (and additional growth coefficients), factor loadings, and residual variance,

• 5. Implement each LCSM and the corresponding LGCM (if applicable), estimate the parameters, and construct the corresponding $95\%$ Wald confidence intervals,

• 6. Repeat the above steps until achieving 1, 000 convergent solutions.

Appendix C: Detailed simulation results

C.1 Performance metrics

In this section, we examine the performance metrics of each parameter under all conditions for each LCSM, including relative bias, empirical SE, relative RMSE, and empirical coverage of the nominal $95\%$ confidence interval. For each parameter of each model, we calculated each performance measure across 1000 replications under each condition and summarized the values of each performance metric across all conditions into the corresponding median and range. In general, the models with the novel specification are capable of providing unbiased and accurate point estimates with target coverage probabilities. We provide these summaries in Tables 10-13.

Table 11 Summary of performance metrics of quadratic latent change score model and corresponding latent growth curve model

Table 12 Summary of performance metrics of negative exponential latent change score model and corresponding latent growth curve model

Table 13 Summary of performance metrics of Jenss–Bayley latent change score model and corresponding latent growth curve model

Table 10 provides the summary of the nonparametric LCSM with the proposed specification. The LBGM was able to generate unbiased point estimates and small empirical SEs. The magnitude of the relative biases of all parameters in the model was less than 0.02. In addition, except for the parameters related to the intercept, the magnitude of the parameters’ empirical SEs was less than 0.19 (the empirical SE of the intercept mean and variance were below 0.38 and 2.67, respectively, while the empirical SE of the covariance between the intercept and shape factor was below 0.43). In addition, the estimates from the LBGM were accurate: the magnitude of the relative RMSE of all parameters was lower than 0.29. In addition, the CPs of all parameters under all conditions that we considered in the simulation were around 0.95, indicating that the $95\%$ confidence interval generated by the LBGM covered the population value at the target level under each condition.

Table 11 provides the summary of the four performance metrics for the quadratic LCSM with the proposed model specification. The performance of the quadratic LCSM was also satisfactory. Specifically, the magnitude of the relative biases and the relative RMSE of all parameters was less than 0.03 and 0.53, respectively. Moreover, except for the parameters related to the intercept, the magnitude of the empirical SE was below 0.25. Additionally, the CPs of all parameters under all conditions were sufficiently close to 0.95.

Table 12 lists the summary of the performance metrics of the negative exponential LCSM with the proposed specification. In general, the model performed satisfactorily. In particular, the magnitude of the relative biases of most parameters was below 0.01, although the relative bias of the mean and variance of the vertical distance was slightly larger, reaching 0.03 and 0.06, respectively. In addition, we noticed that the CP of the mean value of the vertical distance was not satisfied. Through further investigation, we found that the negative exponential LCSM worked better under the conditions with the small ratio of the growth rate (i.e., $b=0.4$) and the long study duration with more records in the early stage (i.e., ten repeated measurements with unequally spaced waves).

We provide the summary of the performance metrics for the Jenss–Bayley LCSM with the proposed specification in Table 13. The performance of this model was generally satisfactory. It can be seen from the table that the estimates of the parameters related to the asymptotic slope and vertical distance were not ideal, but they were still acceptable (i.e., the relative bias of these estimates was still less than $10\%$). With further examination, we noticed that the Jenss–Bayley LCSM performed better under the conditions with more repeated measurements, the time structure of unequally spaced study waves, and the larger sample size.

C.2 Model comparison

This section compares the model performance between each parametric LCSM and the corresponding LGCM based on the four measures and information criteria, including Akaike’s Information Criteria (AIC) and Bayesian Information Criteria (BIC). We provide the summary of the performance metrics of quadratic, exponential, and Jenss–Bayley LGCM in Tables 11, 12, and 13, respectively. Under each condition, we noticed that the point estimates of each replication’s quadratic LCSM and LGCM stayed consistent up to the fourth decimal place. Therefore, except for one cell, the summary tables of the performance metrics of the two models were the same. Additionally, the estimated likelihood values of the two models were the same in all replications under all conditions up to four decimals, and so were the AIC and BIC. This is what we expect: as shown in Eq. 9 and Fig. 2b, the rate of change of the quadratic function has a linear relationship with the time t, so the instantaneous slope halfway through a time interval is identical to the ARC in that period. To this end, the estimates of the quadratic LCSM are the same as those from the corresponding LGCM.

For the negative exponential function and Jenss–Bayley function, the LGCM outperformed the corresponding LCSM, especially in terms of parameter estimation related to the vertical distance (and the slope of linear asymptote of the Jenss–Bayley function). Specifically, although these point estimates from the LGCMs and LCSMs can be considered unbiased (i.e., relative biases are less than $10\%$), the bias from the LGCMs was still relatively smaller. As shown in Tables 12 and 13, the negative exponential LCSM and Jenss–Bayley LCSM tended to overestimate¹⁵ the vertical distance and slightly underestimate the additional coefficient (i.e., b or c). In addition, the Jenss–Bayley LCSM could underestimate the slope of the linear asymptote. This is not surprising. The negative exponential and Jenss–Bayley functions are growth curves with decreasing deceleration (i.e., negative acceleration). Therefore, the instantaneous slope halfway through a period is numerically smaller than the ARC in the time interval. For this reason, both models are likely to overestimate $d_{ij\_{\text {mid}}}$ to satisfy the specified functions. For the negative exponential LCSM, this results in an overestimated vertical distance,.¹⁶

As shown in Eq. 11, $d_{ij\_{\text {mid}}}$ of the Jenss–Bayley function consists of the linear asymptote slope and negative exponential term, and the estimation of the two terms is supposed to be complementary. So for the Jenss–Bayley function, an overestimated vertical distance led to an underestimated linear slope. In addition, since these point estimates from the LGCM were less biased, the CPs generated by the LGCM covered the population values better than the corresponding CPs from the LCSM. In addition, among all 24 conditions of the negative exponential (Jenss–Bayley) function, there were 13 (8) conditions where the difference in the BIC across all replications is less than 6.¹⁷ For the remaining conditions, at least $63.0\%$ ($87.9\%$) replications reported a BIC difference of less than 6. It suggests that there is no strong evidence for model preference between the LCSM and the corresponding LGCM under most replications for the simulation study.