Individual change and the timing and onset of important life events

Abstract

Researchers are often interested in studying how the timing of a specific event affects concurrent and future development. When faced with such research questions there are multiple statistical models to consider and those models are the focus of this paper as well as their theoretical underpinnings and assumptions regarding the nature of the effect of the event on the developmental process. We discuss three models, all variants of growth models specified within the multilevel modeling framework, which conceptualize the developmental process and the effect of the event in different ways. These models include the growth model with a time-invariant covariate, the growth model with a time-varying covariate, and the spline growth model. After discussing the models in detail, we applied these models to longitudinal data from the Berkeley Growth Study to examine cognitive changes during infancy and the effect of independent sitting on those changes. Results suggest that research conclusions depend on the model chosen and how certain results can be misconstrued unless the model accurately reflects the research questions. Recommendations and additional non-traditional models are discussed.

Keywords

Longitudinal growth model cognitive development change spline time-varying covariate

Developmentalists are inherently interested in how children, adolescents, and adults change in their physical attributes and psychological capabilities (e.g., cognitive ability, prosocial behavior) as well as the determinants and consequences of those changes. In examining developmental change there are instances where an important life event (or historical event) is thought to have a measurable effect on the developmental trajectory. For example, the onset of independent sitting and locomotion are two major early life transitions that are thought to bring about measureable psychological changes in the infant’s cognitive capabilities (Campos, Anderson, Barbu-Roth, Hubbard, Hertenstein, & Witherington, 2000; Soska, Adolph, & Johnson, 2010). A second example can be found in the adolescence literature, where the timing of pubertal development in females has been associated with important psychological changes, such as depression, conduct disorder, and the likelihood of engaging in risky sexual behavior (see Caspi, Lynam, Moffitt, & Silva, 1993; Deardorff, Gonzales, Christopher, Roosa, & Millsap, 2005; Dick, Rose, Viken, & Kaprio, 2000; Kim & Smith, 1999; Marceau, Ram, Houts, Grimm, & Susman, 2011). Examples of important life events in the adulthood literature include the birth of a child (Evans, Heron, Francomb, Oke, & Golding, 2001) and the death of a spouse or close family member (e.g., Murphy, 1982).

Many empirical studies have used different study designs and analytic approaches to test hypotheses regarding the effect a life event has on development. A major challenge to examining such effects is to untangle age effects from onset or timing effects. For example, the onset of independent sitting becomes more likely as infants get older and, at the same time, increases in age are associated with positive cognitive changes. Similar notions can be discussed when attempting to understand the effects of pubertal development in adolescence or the death of a spouse in adulthood. In an attempt to isolate the effect of the life event, researchers have traditionally collected cross-sectional data where age is held constant by the selection of participants and between-person differences in the timing of the event are associated with between-person differences in the outcome of interest. For example, Campos et al. (2000) described this approach to examining the effect of locomotion on children’s cognition as: “hold[ing] age constant and classify[ing] infants into those with and without locomotor experience” (Campos et al., 2000, p. 154). Here, differences in cognitive scores between the two groups are seen as a result of differences in locomotion (see also, Adolph & Tamis-LeMonda, 2014). Similarly, in studying the effect of pubertal development, researchers often restrict the age range of participants in an attempt to isolate the effect of pubertal development. For example, Quevedo, Benning, Gunnar, and Dahl (2009) carefully selected 12- to 13-year-olds, such that there were no age differences between children classified as pre-/early pubertal and children classified as mid-/late pubertal. Differences between the two groups would be attributed to differences in pubertal status and not age.

This cross-sectional approach to examining the effect of a life event on development, although commonly employed and logical, conflates between- and within-person effects. For example, researchers who find a positive association between pubertal status and externalizing behaviors using such a design are unable to determine if pubertal development had a measureable positive effect on the individual’s developmental progression in externalizing behavior (i.e., within-person effect) or if children who experienced puberty earlier had higher levels of externalizing behavior before and after puberty (i.e., between-person effect). In the latter, the event did not alter the individual’s change trajectory. The differences between the within-person and the between-person effect is illustrated in Figures 1A and 1B.

Figure 1.

(A) Visual depiction of longitudinal trajectories where the timing of the event is associated with between-person differences; (B) Visual depiction of longitudinal trajectories where the onset of the event is associated with within-person changes over time. Note: Illustrative Figure showing how cross-sectional differences at measurement can be attributable to (A) between-person differences or (B) within-person effects. Group 1 is a hypothetical early puberty group of individuals and Group 2 is a hypothetical late puberty group of individuals.

In each figure there are two hypothetical mean trajectories for externalizing behavior for an early puberty group (Group 1, solid black line) and a late puberty group (Group 2, solid gray line). Also in these plots are vertical lines indicating the onset of puberty for each group and the timing of the measurement occasion from a cross-sectional study where participants were selected based on age and either belonging to an early or late puberty group.

As seen in both figures, at the timing of the measurement occasion, the two groups differ in externalizing behaviors and pubertal status, with Group 1 having experienced puberty and Group 2 being pre-pubertal. Following the logic of cross-sectional studies using this approach, the cause of the differences in externalizing behaviors is the difference in the timing of puberty. However, evaluating the hypothetical mean trajectories, we see that the trajectories in Figure 1A are strictly linear and not affected by the onset of puberty. The differences in these trajectories arise from between-person differences in the intercept and slope that were present before the measurement occasion, as well as after. However, the hypothetical mean trajectories in Figure 1B show the expected within-person effects, such that there were no changes in the level of externalizing behavior prior to puberty (and no differences between the groups before the onset of puberty in either group), but after the onset of puberty there is a strong increase in externalizing scores for both groups. Thus, these trajectories represent what researchers often expect––that is, differences in externalizing scores between the two groups are attributable to the differences in the timing of puberty between the two groups.

The hypothetical mean trajectories presented in Figures 1A and 1B show how differences between groups in externalizing scores at the time of measurement can arise from between-person differences, within-person changes, or a combination of the two. Thus, the typical cross-sectional approach to studying the effect of an important life event on development can be problematic and potentially confuse between- and within-person effects. In order to distinguish between these types of effects, longitudinal data should be collected prior and subsequent to the event of interest. When such longitudinal data are collected, there are a variety of statistical models available to test specific theories regarding the nature of effects and we describe these models next, illustrate their use, and discuss their benefits and limitations. Throughout the introduction we refer back to our hypothetical example where changes in externalizing behavior are studied.

Statistical Models for Timing and Change

The models presented in this section focus on different ways to include the timing of an event into growth models (Bryk & Raudenbush, 1987; McArdle & Epstein, 1987; Meredith & Tisak, 1990; Preacher, Wichman, MacCallum, & Briggs, 2008; Singer & Willett, 2003), which are often used to study within-person change and between-person differences in change. Growth models can be fit within the multilevel and structural equation modeling (SEM) frameworks. Our presentation uses the multilevel notation, but identical models can be specified within the SEM framework. In the following descriptions, we discuss linear growth models for simplicity, but recognize that different within-person change models (e.g., quadratic, exponential, logistic) can be used depending on the nature of the within-person change process. The first model includes the timing of the event as a time-invariant covariate and relates the event timing to between-person differences in the intercept and slope (i.e., between-person effect). The second two models examine whether the timing of the event has an effect on the individual trajectories (i.e., within-person effect), but conceptualize the developmental process in different ways.

Growth Model with Timing as Time-Invariant Covariate

A first approach to examining how the timing of an event affects individual changes is to include the timing variable as a time-invariant predictor (covariate) of the intercept and slope of the linear growth model. This model can be written as

Level-1

y_{t n} = b_{0 n} + b_{1 n} \cdot a g e_{t n} + e_{t n}

Level-2

b_{0 n} = γ_{00} + γ_{01} \cdot X_{n} + d_{0 n}

b_{1 n} = γ_{10} + γ_{11} \cdot X_{n} + d_{1 n}

where $y_{t n}$ is the outcome of interest (e.g., externalizing behavior) measured at time t for individual n, $b_{0 n}$ is the intercept for individual n representing the true score for individual n when $a g e_{t n}$ is 0, $b_{1 n}$ is the linear slope for individual n representing the true amount of change for individual n for a one unit change in $a g e_{t n}$ , and $e_{t n}$ is a time-dependent residual. The time-dependent residual is assumed to follow a normal distribution ( $e_{t n} ~ N (0, σ_{e t}^{2})$ ), often with a time-invariant variance (i.e., $σ_{e t}^{2} = σ_{e}^{2}$ ; see Grimm & Widaman, 2010).

The individual intercept and slope ( $b_{0 n}$ and $b_{1 n}$ ) are the outcomes in the Level-2 Equation, where $X_{n}$ is the observed timing variable, which is a predictor of the between-person differences in the intercept and slope. In these equations, $γ_{00}$ is the predicted value of the intercept when the timing variable is 0, $γ_{01}$ is the effect of the timing variable on the intercept, $γ_{10}$ is the predicted value of the slope when the timing variable is 0, and $γ_{11}$ is the effect of the timing variable of the slope. Lastly, $d_{0 n}$ and $d_{1 n}$ are disturbance terms assumed to follow a multivariate normal distribution ( $d_{0 n}$ , $d_{1 n} ~ M V N ([\begin{matrix} 0 \\ 0 \end{matrix}], [\begin{matrix} σ_{0}^{2} \\ σ_{10} & σ_{1}^{2} \end{matrix}])$ ).

In the application of this model, the timing variable should be centered at a reasonable value to aid interpretation of $γ_{00}$ and $γ_{10}$ . For example, when studying the effect of differences in pubertal development, the timing variable can be centered at a value near the mean onset of puberty (e.g., ten years). Additionally, we note that the age variable ( $a g e_{t n}$ ) can also be centered, potentially, at the same value as the timing variable, to aid the interpretation of $γ_{00}$ .

From a theoretical standpoint, the growth model with timing as a covariate is a common cause model, such that the cause of the between-person differences in the intercept and slope are the same as or related to the cause of the differences in the timing of the event. The intercept in the growth model is typically centered prior to the event, so having the event predict the between-person differences in the intercept appears backwards unless there is a common underlying cause. For example, examining changes in externalizing behaviors before and after the onset of puberty with this model would suggest that the underlying cause of externalizing behavior changes are similar to the underlying cause of pubertal development––potentially related to the release of specific hormones. In describing the results from this model, it is important to emphasize that the onset of the event is not the mechanism, but the between-person differences in the timing of the event is a related mechanism.

The limitations of this approach are that the association between the timing of the life event and individual changes is a between-person association and that the timing variable should be observed for all participants. If the event occurred after the end of the study, the timing variable would be incomplete and those participants would potentially be dropped from the analysis. Such participants are dropped from the analysis when using the multilevel modeling framework and under certain model specifications when using the SEM framework.¹

Growth Model with Time-Varying Covariate

The second approach is a growth model where the onset of the event is treated as a time-varying covariate. In this approach, the onset of the event is organized as a dummy-coded variable that takes on the value of 0 prior to the event and 1 after the event. This model can be written as

Level-1

y_{t n} = b_{0 n} + b_{1 n} \cdot a g e_{t n} + γ_{20} \cdot Z_{t n} + e_{t n}

Level-2

b_{0 n} = γ_{00} + d_{0 n}

b_{1 n} = γ_{10} + d_{1 n}

where $y_{t n}$ is the outcome of interest (e.g., externalizing behavior) measured at time t for individual n, $Z_{t n}$ is the dummy-coded onset variable at time t for individual n, $b_{0 n}$ is the individual intercept for individual n representing the true score for individual n when $a g e_{t n}$ and $Z_{t n}$ are 0, $b_{1 n}$ is the individual slope for individual n representing the true amount of change for individual n for a one unit change in $a g e_{t n}$ holding $Z_{t n}$ constant, $γ_{20}$ is the effect of the onset of the event on the intercept holding $a g e_{t n}$ constant, and $e_{t n}$ is a time-dependent residual.

The model in Equation 2 is the typical way to include a time-varying covariate, where $γ_{20}$ is the effect of the event on the intercept; thus, $γ_{20}$ represents a jump in the trajectory when the event occurs. This can be visualized by the solid trajectory in Figure 2, where, in our hypothetical, changes in externalizing behavior are modeled as a function of age and the onset of puberty.

Figure 2.

Visual depiction of a spline model with (solid line) and without (dashed line) a change in level. Note: Illustrative Figure highlighting how the onset of puberty could have an immediate effect on the level of the individual trajectory (solid line) or an effect on the rate of change (dashed line).

The solid trajectory shows that externalizing behaviors are increasing with respect to age, but at the onset of puberty there is a jump in the trajectory such that scores are universally higher after the onset of puberty. It is important to note that in this specification, the rate of change is the same before and after the onset of puberty.

An alternative specification of the growth model with a time-varying covariate allows the onset of the event to alter the rate of change instead of the intercept. For this model, the product of the time-varying covariate and the timing metric centered at the event is included as a predictor, which can be written as

Level-1

y_{t n} = b_{0 n} + b_{1 n} \cdot a g e_{t n} + γ_{20} \cdot Z_{t n} \cdot (a g e_{t n} - X_{n}) + e_{t n}

Level-2

b_{0 n} = γ_{00} + d_{0 n}

b_{1 n} = γ_{10} + d_{1 n}

where $γ_{20}$ is the change in the individual slope of the trajectory after the life event and $X_{n}$ is the timing of the life event. The dashed line in Figure 2 is an example trajectory from this type of model where the rate of change in externalizing behaviors increases after the onset of puberty.

Lastly, Equations 2 and 3 can be combined to allow the onset of the event to have an effect on both the intercept and slope. Thus, there is an immediate jump (effect on the intercept) and a change in the individual rate (effect on the slope) that are associated with the onset of the event. This model can be written as

Level-1

y_{t n} = b_{0 n} + b_{1 n} \cdot a g e_{t n} + γ_{20} \cdot Z_{t n} + γ_{30} \cdot Z_{t n} \cdot (a g e_{t n} - X_{n}) + e_{t n}

Level-2

b_{0 n} = γ_{00} + d_{0 n}

b_{1 n} = γ_{10} + d_{1 n}

where $γ_{20}$ is the effect of the event on the intercept and $γ_{30}$ is the effect of the event on the slope. We reiterate that these effects are within-person effects and represent the effect of the onset of the event on the individual’s trajectory. Thus, there is an individual intercept and slope prior to the event ( $b_{0 n}$ and $b_{1 n}$ ) and a different individual intercept and slope after the event ( $b_{0 n} + γ_{20}$ and $b_{1 n} + γ_{30}$ ).

The major benefit of this approach is that the occurrence of the event is allowed to alter the within-person trajectory, which is in line with most researchers’ expectations. Thus, this model tends to align well with research questions. A secondary benefit is that participants can remain in the analysis if the event did not occur during the observation period, but it is known whether the event occurred prior to and subsequent to the observation period. The first limitation of this approach is that the effect of the onset of the event is typically specified to have a universal effect––the effect is the same for all participants; however, we note that this does not have to be the case (e.g., the effects $γ_{20}$ and $γ_{30}$ could vary over participants, but estimation may become difficult unless several measurement occasions are obtained before and after the event for participants). The second limitation is that non-continuous trajectories are possible because the onset of the event can alter the intercept (Equations 2 and 4), which may be theoretically unappealing.

Spline Growth Model

The spline, piecewise, or multiphase growth model (see Cudeck & Klebe, 2002; Ram & Grimm, 2007) is a common approach to study the effect of an event on an individual’s trajectory. In the spline growth model, individual changes prior and subsequent to the life event are jointly modeled with different, but connected, growth models. For example, a bi-linear spline growth model can be written as

Level-1

y_{t n} = b_{0 n} + b_{1 n} \cdot (a g e_{t n} - X_{n}) + b_{2 n} \cdot (m a x (a g e_{t n} - X_{n}, 0)) + e_{t n}

Level-2

b_{0 n} = γ_{00} + d_{0 n}

b_{1 n} = γ_{10} + d_{1 n}

b_{2 n} = γ_{20} + d_{2 n}

where $y_{t n}$ is the outcome of interest (e.g., externalizing behavior) measured at $a g e_{t n}$ for individual n, $X_{n}$ is the timing of the life event for individual n, $b_{0 n}$ is the intercept for individual n representing the true score for individual n at the event ( $a g e_{t n} - X_{n} = 0$ ), $b_{1 n}$ is the pre-event slope for individual n representing the true amount of change for individual n for a one unit change in $a g e_{t n}$ prior to the life event, $b_{2 n}$ is the individual difference in the slopes before and after the event, and $e_{t n}$ is a time-dependent residual. Thus, the post-event slope, or individual rate of change after the event, is $b_{1 n} + b_{2 n}$ . As above, the time-dependent residual is assumed to follow a normal distribution ( $e_{t n} ~ N (0, σ_{t e}^{2})$ ), often with a time-invariant variance (i.e., $σ_{t e}^{2} = σ_{e}^{2}$ ). The individual intercept and slopes ( $b_{0 n}$ , $b_{1 n}$ , and $b_{2 n}$ ) have means ( $γ_{00}$ , $γ_{10}$ , and $γ_{20}$ ) and disturbances ( $d_{0 n}$ , $d_{1 n}$ , and $d_{2 n}$ ) representing individual deviations around the means. These disturbance terms are assumed to follow a multivariate normal distribution ( $d_{0 n}$ , $d_{1 n}$ , $d_{2 n} ~ M V N ([\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} σ_{0}^{2} \\ σ_{10} & σ_{1}^{2} \\ σ_{20} & σ_{21} & σ_{2}^{2} \end{matrix}])$ ).

In this specification, the effect of the life event is embedded in $b_{2 n}$ . That is, the significance of $γ_{20}$ tests whether or not the pre-event and post-event slope means are equal and the significance of the variance of $b_{2 n}$ ( $σ_{2}^{2}$ ) tests whether or not the variances in pre-event and post-event slopes are equal. We note that spline models can be fit with different change trajectories before and after the knot point as well as with linear and nonlinear models in each phase (see Cudeck & Klebe, 2002).

Overall, the spline model and the time-varying covariate model in Equation 3, where the rate of change is allowed to differ before and after the life event, are very similar. The models become more similar if $γ_{20}$ in Equation 2 and $b_{2 n}$ in Equation 5 are either both random coefficients (allowed to vary over individuals) or estimated parameters (not allowed to vary over individuals). The similarities become more evident realizing that $Z_{t n} \cdot (a g e_{t n} - X_{n})$ in Equation 3 is equivalent to $(m a x (a g e_{t n} - X_{n}, 0))$ in Equation 5. The important difference between these two models is the timing metric. In Equation 3, the timing metric is age and in Equation 5, the timing metric is age relative to the life event. Thus, the choice between these two models should be driven by whether the primary underlying mechanism thought to produce individual change is age or age relative to the life event.

The main benefit of spline growth models is that the event is allowed to alter the within-person change trajectory. Additionally, the model allows for between-person differences in the effect of the event because $b_{2 n}$ , the individual change in the slope, is a random coefficient and therefore allowed to vary over individuals. One downside of these types of spline models is that the timing of the life event must be observed because of how it is incorporated into the timing metric of the growth model (i.e., cannot scale time with respect to the life event, if the life event has not occurred or life event is unknown). Spline models have been fit where an optimal knot point is estimated from the data (see Cudeck & Klebe, 2002); however, in these models there is not an a priori event that is thought to bring about measureable change in the within-person developmental process. A second limitation is that age effects should be equivalent before and after the life event. If age effects are not equivalent, then the significance of $γ_{20}$ and $σ_{2}^{2}$ may be attributable to nonlinear age effects instead of the effect of the life event.

From a theoretical standpoint, this type of spline model, where the timing metric is organized around a specific event, alters the underlying mechanism of the observed within-person changes. In Equations 1, 2, 3, and 4, age (or years relative to birth) was the timing metric and, as such, age was seen as the dominant driving force underlying the change process. In the spline model in Equation 5, age relative to the event is the dominant driving force underlying change. So, the theory should support the notion that the amount of time to the event and the amount of time from the event guides the change process instead of a typically used timing metric, such as age, time in study, or measurement occasion.

Illustrative Example

To illustrate the use of these models, we apply them to longitudinal cognitive data on infants and examine the effect of the timing and onset of the ability to sit alone for 30 seconds or more (referred to as independent sitting) on those cognitive changes. Early motor development in infants has been found to be associated with cognitive development (e.g., Acredolo, 1988; Bertenthal & Campos, 1990; Campos & Bertenthal, 1984) and changes in an infant’s motor skills increase their ability to explore and interact with their environment. This increased ability to interact with their environment may contribute to other areas of their development (Adolph, 1997; Gibson, 1988; Herbert, Gross, & Hayne, 2007). Independent sitting is an early motor ability that allows an infant to take a different perspective on their environment and allows them to more easily interact with stimuli with both hands.

Data

The data come from the Berkeley Growth Study (BGS), which was initiated by Nancy Bayley in 1928 to trace the normal intellectual, motor, and physical development through the first year (see Jones et al., 1971). The original participants of the BGS were selected as infants born in local hospitals in Berkeley, California.

The cognitive scores come from the California First-Year Mental Scale (CFYMS) (Bayley, 1933). The CFYMS was the precursor of the Bayley Scales of Infant Development and initially contained 115 items (129 were subsequently added bringing the total to 245). The scores analyzed here are the total scores representing the number of items correctly completed. The CFYMS was administered every month from 1 to 12 months of age. Motor data come from the California Infant Scale of Motor Development (CISMD) (Bayley, 1935), which was administered in conjunction with the CFYMS (i.e., monthly). A single item from the CISMD was selected for this illustration. The item asked whether the infant was able to sit alone for 30 seconds or more and the response was recorded as the infant’s age (in months) when they were first able to perform this skill. Valid responses were considered if the child performed the skill and was measured in the month prior (to be sure the infant could not perform the skill in the previous month). Scores on this item ranged from five to nine months with a mean of seven months.

Data from 19 infants were dropped because of incomplete data on the timing event––it was not known when these infants were able to sit for 30 seconds. Children dropped from analyses did not differ on CFYMS total scores at any age compared to children included in the analysis. Thus, bias in estimates attributable to incomplete data is minimal. Figure 3 is a longitudinal plot of the total scores from the CFYMS measured for the sample of 45 (22 females) participants. The individual cognitive scores show relatively strong increases as the infants get older and the rate of change appears to vary both within infants over time (nonlinear changes) and between individuals (variability in the rate of growth at any point in time).

Figure 3.

Longitudinal plot of observed trajectories for cognitive scores from the CFYMS from 1 through 12 months from the BGS. Note: N = 45, First-Year Mental Scale scores ranged from 1 to 106.

Analytic Techniques

The analyses began with the fitting of a variety of growth models to adequately capture individual changes in cognitive ability as measured by the CFYMS. A logistic growth model with two random coefficients was chosen. This version of the logistic model can be written as

y_{t n} = γ_{00} + b_{1 n} \cdot \frac{exp (γ_{20} \cdot (a g e_{t n} - b_{3 n}))}{1 + exp (γ_{20} \cdot (a g e_{t n} - b_{3 n}))} + e_{t n}

where $y_{t n}$ is the outcome measured at $a g e_{t n}$ for individual n, $γ_{00}$ is the lower asymptote, $b_{1 n}$ is the amount of change to the upper asymptote for individual n, $γ_{20}$ is the rate of approach to the asymptotic level, $b_{3 n}$ is the timing of accelerated changes for individual n, and $e_{t n}$ is the residual at $a g e_{t n}$ for individual n. The two random coefficients are assumed to follow a multivariate normal distribution with estimated means, variances, and a covariance $b_{1 n}, b_{3 n} ~ M V N ([\begin{matrix} γ_{10} \\ γ_{30} \end{matrix}], [\begin{matrix} σ_{1}^{2} \\ σ_{31} & σ_{3}^{2} \end{matrix}])$ , and the time-dependent residual is assumed to be normally distributed with mean 0 and estimated variance, $e_{t n} ~ N (0, σ_{e}^{2})$ .

After establishing this unconditional model, the logistic growth model was then fit with age of independent sitting, centered at seven months, as a time-invariant predictor of the intercept and slope. Next, three time-varying covariate models were fit following Equations 2, 3, and 4. Here, the logistic model was augmented by a dichotomous variable allowed to alter the intercept and/or by an exponential curve with the same rate parameter. The exponential curve allowed for the general shape of the curve to remain consistent and, at the same time, allow the rate of change to differ before and after independent sitting. Lastly, a spline model was fit with a logistic curve augmented by an exponential curve with the same rate parameter following the logic behind the time-varying covariate models. The Appendix contains the complete model specifications. All models were fit using PROC NLMIXED (Littell, Milliken, Stroup, Wolfinger, & Schabenberger, 2006) in SAS v. 9.3 (SAS Institute Inc., 2011). Programing code is available on the first author’s webpage.

Results

The results are organized around each model, beginning with the unconditional logistic growth model. Parameter estimates and fit information for the various models is contained in Table 1.

Table 1.

Parameter Estimates and Fit Statistics for Models Fit to the Longitudinal Data from the CFYMS.

	Logistic Growth Model	Between-Person Predictor	Time-Varying Covariate Models			Spline Model
	Logistic Growth Model	Between-Person Predictor	Level Effect	Rate of Change Effect	Level and Rate of Change	Spline Model
Fixed Effects
$γ_{00}$	−7.73 (−10.63 – −4.82)	−7.69 (−10.59 – −4.79)	−10.61 (−14.62 – −6.61)	−7.92 (−10.85 – −4.99)	−11.21 (−15.35 – −7.07)	−7.28 (−10.09 – −4.47)
$γ_{10}$	110.12 (105.42 – 114.81)	110.06 (105.40 – 114.73)	111.96 (106.21 – 117.72)	106.75 (100.56 – 112.95)	107.25 (100.24 – 114.27)	108.47 (102.04 – 114.91)
$γ_{11}$	—	−1.52 (−2.74 – −0.31)	—	—	—	—
$γ_{20}$	0.41 (0.38 – 0.43)	0.41 (0.38 – 0.43)	0.38 (0.35 – 0.41)	0.41 (0.39 – 0.44)	0.38 (0.35 – 0.42)	0.41 (0.39 – 0.44)
$γ_{30}$	5.81 (5.64 – 5.98)	5.81 (5.66 – 5.96)	5.70 (5.51 – 5.90)	5.65 (5.38 – 5.92)	5.43 (5.10 – 5.76)	−1.21 (−1.56 – −0.86)
$γ_{31}$	—	0.24 (0.13 – 0.35)	—	—	—	—
$γ_{40}$	—	—	2.46 (0.98 – 3.94)	2.88 (−0.69 – 6.46)	2.72 (1.24 – 4.20)	0.98 (−2.89 – 4.85)
$γ_{50}$	—	—	—	—	4.06 (0.43 – 7.69)	—
Random Effects
$σ_{1}^{2}$	11.27 (3.63 – 18.91)	9.04 (2.36 – 15.73)	13.04 (4.35 – 21.73)	10.74 (3.53 – 17.96)	12.14 (4.19 – 20.09)	10.74 (3.37 – 18.11)
$σ_{31}$	−0.38 (−0.93 – 0.17)	−0.03 (−0.47 – 0.42)	−0.28 (−0.86 – 0.30)	−0.32 (−0.86 – 0.22)	−0.20 (−0.76 – 0.36)	0.98 (−0.19 – 2.15)
$σ_{3}^{2}$	0.15 (0.07 – 0.23)	0.09 (0.04 – 0.15)	0.14 (0.06 – 0.21)	0.15 (0.07 – 0.22)	0.13 (0.06 – 0.21)	0.66 (0.36 – 0.95)
$σ_{e}^{2}$	9.71 (8.37 – 11.04)	9.70 (8.37 – 11.04)	9.53 (8.22 – 10.85)	9.68 (8.35 – 11.02)	9.49 (8.18 – 10.80)	9.70 (8.36 – 11.03)
Fit Statistics
-2LL	2818.8	2793.5	2807.9	2816.2	2803.0	2869.2
Akaike Information Criterion	2834.8	2813.5	2825.9	2834.2	2823.0	2887.2
Bayesian Information Criterion	2849.2	2831.5	2842.1	2850.5	2841.0	2903.4

Notes: Confidence intervals are contained within parentheses, N = 45, model parameters map onto specifications in the Appendix.

Logistic Growth Model

The parameters of the logistic growth model indicated that the sample had a predicted lower asymptote of −7.73 ( $γ_{00}$ ) and were expected to grow 110.12 ( $γ_{10}$ ) points from their lower to their upper asymptote, on average. The mean timing of accelerated changes was 5.81 ( $γ_{30}$ ) months (inflection point) and the rate of approach to the asymptote was 0.41 ( $γ_{20}$ ). From these model parameters we can derive estimates that are more interpretable. For example, the predicted mean score at 1 month was 5.78 points, the mean change from 1 to 12 months of age was 88.47 points, and the mean rate of change at the inflection point was 11.25 points per month.

There was significant variability in both the total amount of change from the lower asymptote ( $σ_{1}^{2} = 11.27$ , $t (43) = 2.97$ , p<.01) and the timing of accelerated changes ( $σ_{3}^{2} = 0.15$ , $t (43) = 3.75$ , p<.01), and the individual differences in these two aspects of change were not significantly related ( $σ_{31} = - 0.38$ , $t (43) = - 1.39$ , p=.17, $r_{31} = - 0.30$ ). The residual variance ( $σ_{e}^{2}$ ) was 9.71 and represents the variation in observed scores unexplained by the logistic growth model.

Logistic Growth Model with Timing of Event as Time-Invariant Covariate

The logistic growth model with the age of independent sitting as a between-person predictor was then fit. In this model, age of sitting was centered at seven months and included as a predictor of the amount of change to the upper asymptote and the timing of accelerated changes. The age of sitting was a significant predictor of both the total amount of change ( $γ_{11} = - 1.52, t (43) = - 2.53, p < .05$ ) and the timing of accelerated changes ( $γ_{11} = 0.24, t (43) = 4.35, p < .001$ ). Thus, infants who sat earlier were expected to show more overall growth, 1.52 points for every month earlier that they achieved independent sitting. Additionally, infants who sat earlier reached their peak rate of change earlier, 0.24 months earlier for every month earlier that they achieved independent sitting. Putting these results into context yields the following: An infant who was able to sit independently at 5 months was predicted to change 89.75 points from 1 to 12 months and change at a maximum rate of 11.57 points per month at 5.34 months, whereas an infant who was able to sit independently at 9 months was predicted to change 86.54 points from 1 to 12 months and change at a maximum rate of 10.94 points per month at 6.29 months.

Growth Models with Time-Varying Covariate

We fit three time-varying covariate models to examine the effect of the onset of independent sitting on the intercept, the rate of change, and the combination of the two. First, we fit a traditional logistic growth model with a time-varying covariate by specifying a logistic growth model and included the dummy-coded timing variable as a predictor of the CFYMS scores. The dummy-coded timing variable had a significant positive effect ( $γ_{40} = 2.46, t (43) = 3.35, p < .01$ ) suggesting that independent sitting had an immediate and positive effect on the infant’s CFYMS scores.

Second, we fit the time-varying covariate model with a product of the dummy-coded timing variable and an exponential function centered at the age of independent sitting. Thus, a logistic growth model was specified and augmented by an exponential curve after the age of independent sitting, which allows for a different rate of change after the event. The effect of the product of the dummy-coded timing variable and the exponential function was not significant ( $γ_{40} = 2.99, t (43) = 1.63, p = .11$ ). Thus, independent sitting did not significantly alter the individual rate of change in the logistic growth model.

Third, we fit the time-varying covariate model allowing for effects on both the intercept and rate of change. In this model, the effect of the dummy-coded timing variable ( $γ_{40} = 2.72, t (43) = 3.70, p < .001$ ) and the effect of the product between the dummy-coded timing variable and an exponential function ( $γ_{50} = 4.06, t (43) = 2.26, p < .05$ ) were both significant suggesting that infants’ CFYMS scores increased by 2.72 points after they achieved independent sitting and that their rate of change was also faster after they achieved independent sitting. In the month after independent sitting the rate of change was 0.32 points per month faster than expected from the logistic growth model.

Spline Growth Model

Prior to fitting the spline logistic growth model, we plotted the total scores with age in months relative to age at sitting ( $a g e_{t n} - X_{n}$ ) in Figure 4. As seen in this Figure, the new timing metric ranges from −8 to +7 months indicating that some infants were measured 8 months prior to sitting (infants who sat at 9 months) to 7 months after sitting (infants who sat at 5 months). As seen in this figure, the developmental pattern appeared approximately logistic, which led to our decision to fit a logistic-exponential spline model.

Figure 4.

Longitudinal plot of observed trajectories for cognitive scores from the CFYMS organized according to the onset of independent sitting. Note: The zero point on the x-axis is the age of independent sitting for all participants, N = 45, First-Year Mental Scale scores ranged from 1 to 106.

The logistic-exponential spline growth model was fit to the data. Allowing for between-person differences in the change in the rate of change after the life event led to convergence issues. We therefore fit a model where the variance of the change in the rate of change after the life event was set to 0. This model converged without issue. The change in the rate of change after the knot point was not significant ( $γ_{40} = 0.98, t (43) = 0.51, p = .61$ ) suggesting that there was not a change in the rate of change after the onset of independent sitting.

Discussion

Developmental researchers are often interested in studying change and testing hypotheses about within-person change, its determinants, and consequences. Here, we discussed several statistical models that can be used to examine whether and how a life event affects change trajectories. Understanding such processes is especially relevant in psychological research because of the multitude of hypotheses regarding how various life events are thought to affect our development.

The major challenge that arises when studying such effects is when the event (e.g., independent sitting) and the changes in the outcome of interest (e.g., cognitive changes) are both associated with the timing metric (e.g., age). For example, cognitive ability grows as infants get older and the likelihood of being able to sit independently for sustained periods of time also increases as infants get older. Attempting to isolate the association between independent sitting and cognitive changes is challenging unless longitudinal data are collected, and longitudinal data do not always provide a clear indication of the nature of these associations.

Summary of Results

Several models for longitudinal data were fit to evaluate the nature of the association between independent sitting and cognitive development. First, the between-person differences in the age of independent sitting was a predictor of the between-person differences in the growth function. The results indicated that children who were able to sit independently at an earlier age showed more overall cognitive growth and were changing more rapidly at an earlier age. These results suggest a common underlying cause of independent sitting and cognitive development. For our example, it may be that the parts of the brain that control motor development are partly responsible for cognitive development and that is why the between-person differences in the timing of independent sitting were associated with between-person differences in the growth of cognitive ability.

Second, a series of time-varying covariate growth models was fit, which allowed the onset of independent sitting to affect the within-person change trajectory in various ways. In the first of these models, the onset of independent sitting was found to have an immediate positive effect, such that cognitive scores were significantly higher after independent sitting and this advantage was maintained. In the second of these models, the onset of independent sitting was allowed to have an effect on the rate of growth. In this model, the onset of independent sitting was found to not alter the rate of cognitive growth. Finally, in the third model, the onset of independent sitting was allowed to have an immediate effect on cognitive scores as well as change the rate of growth. In this final model, the onset of sitting was found to have an immediate positive effect on cognitive scores as well as positively affect the rate of growth.

The third and final set of models were spline models where age was organized around independent sitting. Thus, in this set of models, the time metric for the growth model was age relative to the onset of independent sitting. In the spline models, allowing the rate of change to change after the onset of independent sitting did not significantly improve model fit. Thus, based on the spline growth model, the onset of sitting did not significantly alter the cognitive changes taking place.

Model Comparison, Evaluation, and Extension

The first consideration when choosing among the models discussed is theory. A model should be chosen that best allows the researcher to test their substantive theory. The first part of this process is to select a time metric for the growth model that is the dominant mechanism underlying the within-person changes. Here, we discussed age (or years since birth) and age relative to the onset of independent sitting (time to/from the event). These time-metrics are highly related, but conceptualize within-person change in different ways. If time relative to the event is thought to be the mechanism of change, then growth and spline growth models using this time metric are most reasonable (note that growth models with a time-invariant predictor and with a time-varying covariate can be fit using this time metric). If age or another time metric that does not revolve around the event is thought to guide the change process, then using growth and time-varying covariate models with this time metric are most appropriate.

The second consideration is the nature of the association between the timing of the event and the change process. The association could be between individuals, such that differences in the timing of the event are related to differences in parameter values of the growth curve, such as the intercept and rate of change. Alternatively, the association could be within individuals, such that the onset of the event alters the individual’s trajectory. Thus, the individual’s trajectory is different before and after the event. Furthermore, when considering within-person associations, researchers should evaluate whether the event altered the level and/or rate of change of the individual trajectory. Lastly, between-person and within-person associations can be simultaneously evaluated––the between-person differences in the timing of the event may be related to between-person differences in the trajectory and the onset of the event may affect the individual’s trajectory (see Curran & Bauer, 2011).

Returning to our illustrative example, early cognitive changes are thought to be primarily driven by age. Thus, the spline model with age relative to the onset of independent sitting is deemed inappropriate. Although, previous research on early locomotion suggests a within-person association between independent sitting and subsequent cognitive improvements, it is difficult to eliminate the possibility of between-person associations, such that early motor and early cognitive development have common determinants. Due to this, one final model was fit where the between-person differences in the timing of the event were included as predictors of the total amount of change and the timing of accelerated changes, and the onset of the event was allowed to affect the level and rate of change of the individual trajectory. In this final model, the timing of independent sitting was significantly associated with both the total amount of predicted growth and the timing of accelerated changes. Early independent sitting was associated with increases in cognitive growth ( $γ_{11} = - 1.41, t (43) = - 2.11, p < .05$ ) and earlier accelerated changes ( $γ_{31} = 0.20, t (43) = 3.44, p < .01$ ). Additionally, the onset of independent sitting was associated with an increased level in the individual trajectory ( $γ_{40} = 2.21, t (43) = 2.95, p < .01$ ), but not the rate of change ( $γ_{40} = 2.21, t (43) = 1.13, p = .27$ ). Thus, there were between- and within-person associations between the timing and onset of independent sitting and cognitive development. It is likely that there is some underlying common cause to motor and cognitive development in infants and that the onset of independent sitting brings about measureable increases in cognition.

Concluding Remarks

In conclusion, when encountering research questions about change and the timing of an event, we first recommend that researchers think carefully about the primary time metric guiding the within-person change process. Is time relative to the event (e.g., age relative to independent sitting) the most appropriate time metric? If so, then working with this time metric and determining whether a spline model fits the data better is the recommended analytic approach. If not, then using the original time metric and considering both the between-person and within-person effects as well as their combination is recommended to statistically test whether the onset of the event had an effect on the individual change process and/or the timing of the event was associated with between-person differences in the change trajectory.

Footnotes

Funding

This research has been supported by National Science Foundation Grant REAL-1252463 awarded to the University of Virginia, David Grissmer (PI) and Christopher Hulleman (Co-PI).

Note

Appendix

The following equations detail the specification of the models fit. In all cases, the residual variance was constrained to be equal across time $e_{t n} ~ N (0, σ_{e}^{2})$ , and random coefficients were specified to have a full unstructured covariance matrix $d_{1 n,} d_{3 n,} ~ N ([\begin{matrix} 0 \\ 0 \end{matrix}], [\begin{matrix} σ_{1}^{2} \\ σ_{31} & σ_{3}^{2} \end{matrix}])$ .

(1) Logistic growth model

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