What is the shape of developmental change?

Abstract

Developmental trajectories provide the empirical foundation for theories about change processes during development. However, the ability to distinguish among alternative trajectories depends on how frequently observations are sampled. This study used real behavioral data, with real patterns of variability, to examine the effects of sampling at different intervals on characterization of the underlying trajectory. Data were derived from a set of 32 infant motor skills indexed daily during the first 18 months. Larger sampling intervals (2-31 days) were simulated by systematically removing observations from the daily data and interpolating over the gaps. Infrequent sampling caused decreasing sensitivity to fluctuations in the daily data: Variable trajectories erroneously appeared as step functions, and estimates of onset ages were increasingly off target. Sensitivity to variation decreased as an inverse power function of sampling interval, resulting in severe degradation of the trajectory with intervals longer than 7 days. These findings suggest that sampling rates typically used by developmental researchers may be inadequate to accurately depict patterns of variability and the shape of developmental change. Inadequate sampling regimes therefore may seriously compromise theories of development.

Figure 1

Idealized shapes of developmental change, with age shown on the X-axis and an index of behavioral expression or level of performance on the Y-axis. (a) Linear, (b) Accelerating, (c) Asymptotic, (d) Step-like, (e) S-shaped, (f), Variable, (g) Unsystematic, (h) Stair-climbing, (i) U-shaped, (j) Inverted-U-shaped.

Figure 2

Examples of developmental trajectories derived from daily data (black curves) for standing (balancing upright for ≥ 3s without holding a support) in two infants. (a) Trajectory that exhibits abrupt step-function from absent to present from one day to the next. Simulated monthly sampling (gray curve) results in an error in identifying the skill onset age, but does not distort the shape of the trajectory. (b) Variable trajectory, where skill vacillated 21 times between absent and present over the course of several weeks. Simulated monthly sampling (gray curve) misrepresents both the shape of the variable trajectory and the estimated onset age.

Figure 3

Effects of sampling interval on sensitivity to variability in developmental trajectories. (a) The number of observed transitions between absence and presence for one skill (standing). Each curve represents data for one of the 8 infants for whom we had a complete time series. Open symbols depict data when the skill was sampled daily; lines show data averaged across all possible phases at each of the 1- to 31-day sampling intervals. Note that the data point nearest the origin represents the stage-like data from infant #11, shown in Figure 2A. The other 7 data points show data for variable trajectories from other infants, including the top data point depicting infant #7, shown in Figure 2B. (b) Number of observed transitions, presented as in Figure 3A, for all 32 skills. The thick gray line represents the mean trajectory across all 261 time series. (c) The same data presented in Figure 3B expressed as a percentage of observed transitions recorded at daily intervals. The horizontal line at 100% represents the 41 time series with only 1 abrupt transition from absent to present (15.7% of all time series). Most time series consisted of variable trajectories when measured daily, but more than 75% of transitions were not detected when sampled at weekly intervals. (d) Distribution of R² values for inverse power functions fit to each of the 240 time series with multiple transitions. Most time series were best described by an inverse power function, indicating that modest increase in small sampling intervals (< 1 week) resulted in a sharp decline in the ability to detect transitions.

Figure 4

Effects of sampling interval on estimates of onset ages. (a) Neurally-inspired activation function and resulting estimate of the onset age applied to the daily data shown in Figure 1B for standing in infant #7. The onset age is determined by identifying the first instance of activity that exceeds a criterion threshold, then tracing the function back to the preceding period of inactivity. In this case, the function identifies an onset age at 501 days (shown as vertical dashed line). (b) Histograms showing errors in estimates of the onset age for one skill, standing, in all 8 of the infants for whom time series were available. Y-axis is expressed as a percentage of total estimates. Note that larger sampling intervals result in a greater range of errors, a general increase in the magnitude of errors, and a tendency for errors to be shifted toward later ages. (c) Number of days that estimates of onset ages deviated, either earlier or later, from estimates derived from daily sampling. Data are presented for all available skills for each child (261 time series) as a function of the sampling interval; the superimposed gray line shows the mean absolute error resulting from sampling at different intervals.

Figure 5

Simulated developmental trajectories (dark lines) generated by a simple Markov switching model. In each graph, the first 60 days represents a period where the behavior of interest is not yet expressed (p = 0), and the final 100 days represents a period of consistent expression in which the behavior occurs at a stable rate < 1. (a) A stage-like trajectory involving an abrupt transition from absence (extended through the first 120 days) to a high base rate of occurrence (p = .95) during the period of stable expression. (b) Trajectory involving an intervening acquisition period (from day 61 to day 120) before achieving a stable period with a high base rate (p = .95). During the acquisition period, behavior is generated by randomly switching between the early regime (absence) and the later period of stability (high base rate). (c) Trajectory involving an intervening acquisition period before a stable period with a lower base rate (p = .5). Regime switching occurs in the same way as in (b). In all three graphs, the thicker gray line shows a 15-day moving average that depicts the same data; in graphs (b) and (c), this smoothing function visually demarcates the variable acquisition period from the later period of stable expression.