Strong Tests of Developmental Ordering Hypotheses: Integrating Evidence From the Second Moment

Abstract

Developmental ordering is a fundamental prediction in developmental science. However, tests of ordering hypotheses are not generally available for continuously developing variables. One promising test of developmental ordering, the shape of the relationship between 2 variables, requires that changes in each underlying variable are captured equally well across the developmental span (measures are linearly related to the variables). If either measure is more sensitive to earlier or later developmental changes in the underlying variable, the shape of the relationship changes radically. The article demonstrates that the viable alternative hypotheses for an observed developmental relationship require specific types of nonlinearity in measurement and, therefore, have testable predictions for the residuals (the second moment). Ordering as evidence in developmental science is discussed.

Transfer of calibration between hand and foot: Functional equivalence and fractal fluctuations

“…The test set included rods of various materials and diameters, whereas the training set consisted only of wooden 1 The use of visual information is prone to individual differences (see Withagen & van Wermeskerken, 2009, for an example specific to visual feedback in dynamic touch) and nonlinearities (e.g., Steingrimsson, 2009;Stevens, 1957), both of which are at odds with conventional significance testing for mean differences (Dixon, 2005;Green & Luce, 1974;. It would be premature to compound, with two kinds of feedback, the variability due to multiple kinds of feedback before we have a better understanding of transfer in general (i.e., irrespective of accuracy).…”

Grammar and the Lexicon: Developmental Ordering in Language Acquisition

“…For example, Dixon (1998) showed how some developmental ordering hypotheses were eliminated by particular data patterns. In more recent work, Dixon (2005) showed that nonlinear mappings, such as the one discussed above, must create specific relationships between error and the predicted value of the measure, given standard ordinary least-squares (OLS) regression assumptions (Cohen, Cohen, West, & Aiken, 2003). Furthermore, the evaluation of these error patterns provides the tools to test directly whether a nonlinear mapping between the measure and the underlying variable is masking the true underlying relation among the constructs.…”

“…Thus, the only ways to obtain the observed nonlinear relationship when the lexicon and grammar are actually developing in synchrony are (1) for the measure of the lexicon to be a nonlinear, decelerating function of the underlying lexicon, or (2) for the measure of grammar to be a nonlinear, accelerating function of underlying grammar, or (3) for both those nonlinear mappings to occur. Dixon (2005) demonstrated that these types of nonlinear mappings between measures and their underlying constructs make predictions about the patterns that will be observed at the level of the residuals. Nonlinear mappings predict specific residual patterns.…”

“…However, given the types of measures researchers typically have available, testing these hypotheses has been notoriously difficult. The current article provides a detailed example of how a developmental ordering hypothesis can be tested using an approach presented by Dixon (2005), which we call the nonlinear-mapping approach. It also demonstrates how this approach can be extended for developmental systems that contain multiple influences, such that converging evidence is brought to bear on the issue.…”

Change in action: how infants learn to walk down slopes

“…The tide is shifting. Recent enthusiasm for microgenetic methods, individual functions, and within‐trial analyses in developmental psychology has led to new interest in describing learning trajectories and a stronger focus on understanding processes of change (Adolph & Robinson, 2008; Alibali & Goldin‐Meadow, 1993; Dixon, 2005; Siegler, 2006). With a microgenetic design, pre‐ and post‐test results are supplemented by data from the intervening sessions so as to reveal the shape of the underlying trajectory.…”