New trends in Inductive Developmental Systems Theory: Ergodicity, Idiographic Filtering and Alternative Specifications of Measurement Equivalence

Transcription

1 New trends in Inductive Developmental Systems Theory: Ergodicity, Idiographic Filtering and Alternative Specifications of Measurement Equivalence Peter Molenaar The Pennsylvania State University John Nesselroade University of Virginia University Park, October 31-November 2, 2011

2 The implicit assumption of ergodicity in standard analysis of inter-individual variation

3 Standard approach to statistical analysis in psychology: - Analysis of inter-individual variation (variation between subjects in a population of subjects; individual differences) - Strong assumption of homogeneity in (sub-)populations - Aimed at generalization to the state of affairs at the population level - Implicit assumption of applicability of results at the individual level of intra-individual variation

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6 Basic Question: Can results obtained in analyses of inter-individual variation be validly generalized to the subjectspecific level of intra-individual variation (and vice versa)? Molenaar, Measurement, 2004

7 Definition: A process is non-ergodic in case results of analysis of inter-individual variation do not generalize to the level of intra-individual change in time, and vice versa Equivalently: A process is ergodic in case results of analysis of inter-individual variation validly generalize to the level of intra-individual change in time, and vice versa

8 Ergodicity is not a generic property of homogeneous Hamiltonian systems (the primary class of candidate ergodic dynamic systems). Ergodicity is the weakest property in the ergodic hierarchy, including mixing and K-systems. Emch, G., & Liu, C. (2002). The logic of thermostatistical physics. Berlin: Springer.

9 Theorem (based on Birkhoff, 1931): A Gaussian process is non-ergodic if it is: - heterogeneous in time non-stationary (time-varying trends, variances, etc.) and/or - heterogeneous across subjects (subject-specific dynamics)

10 Immediate Consequence of Theorem: Developmental and Learning Processes have time-varying statistical characteristics, hence are heterogeneous in time (non-stationary). Consequently these processes are nonergodic and their analysis has to be based on time series of intra-individual change (time series analysis). Molenaar et al. (2009) Dev. Psych., 45,

11 Immediate Consequence of Theorem: Classical Test Theory is non-ergodic. In particular, the intra-individual means and variances (subjectspecific test reliabilities) are different from the inter-individual mean and variance (test reliability at group level). Molenaar, P.C.M. (2008). In: S.M. Hofer & D.F. Alwin (Eds.), Handbook of cognitive aging. Thousand Oaks: Sage,

12 Model heterogeneity across subjects (second criterion in Theorem) is invisible in standard factor analysis of inter-individual variation. Formal proof in: Kelderman & Molenaar (2007) Multivariate Behavioral Research, 42,

13 Standard 1-factor model (centered): y(i) = Λη(i) + ε(i), where Λ is fixed and ε(i) ~ N(0, Σ e ). The standard 1-factor model is proven to be indistinguishable from the random 1-factor model: y(i) = Λ(i)η(i) + ε(i) where Λ(i) ~ N(Λ, Σ l ) and ε(i) ~ N(0, Σ e(i) ).

14 The need for new perspectives on measurement equivalence in developmental science

15 Standard test for measurement equivalence Time 1 y 1 = L e 1 Time 2 y 2 = L e 2 Test: L 1 = L 2

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17 Tracking of airplanes by means of radar is based on a combination of Fourier analysis and (nonlinear) factor analysis (akin to beamforming). A given set of planes constitutes the latent factors, the loadings of which on the radar stations are timevarying due to changes in position. The standard approach to testing for measurement equivalence (i.e., asking whether the same planes are measured) would give nonsensical results.

18 Van der Waals Theory of Phase Transitions

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20 Some Aspects of Catastrophe Theory z(t)/ t = V[z(t), c]/ z V[z(t), c] is potential function with control vector c Canonical Forms Nonsingular Equilibria up to Diffeomorphisms: Morse Theorem Canonical Forms Singular Equilibria up to Diffeomorphisms : Thom s Theorem 20

21 Some Aspects of Catastrophe Theory z(t)/ t = V[z(t), x,y]/ z z unidimensional manifold x,y univariate control variables (e.g., P & T) Canonical Form (Thom s Theorem) is a cusp: z(t)/ t = z(t) 3 yz(t) -x 21

22 van der Maas & Molenaar, Psych. Review, 1992,

23 Both thermodynamic phase transitions and cognitive stage transitions are explained by singularities (catastrophes) in dynamic systems. The substance (e.g., water) or process (e.g., intelligence) undergoing a transition is identical and its transitions are described by a continuous dynamic equation (e.g., the cusp). Application of standard tests of measurement equivalence before, during and after the transition would give nonsensical results.

24 Both plane tracking and phase (stage) transitions concern (spatio-)temporal dynamic processes. Standard tests of invariance of factor loadings are ill-suited for testing measurement equivalence of dynamic (e.g., developmental) processes. The alternative test for measurement equivalence developed by Nesselroade et al. (2007) constitutes an important first step to arrive at more appropriate operationalizations of measurement equivalence in dynamic settings.

25 Epigenetic Origins of Heterogeneity across subjects Molenaar, P.C.M., Boomsma, D.I., & Dolan, C.V. (1993). A third source of developmental differences. Behavior Genetics, 23, Molenaar (2007). On the implications of the classical ergodic theorems: Analysis of developmental processes has to focus on intra-individual variation. Developmental Psychobiology, 50,

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29 Subject-Specific Heritabilities Molenaar, P.C.M. (2010) On the limits of standard quantitative genetic modeling of inter-individual variation: Extensions, ergodic conditions and a new genetic factor model of intra-individual variation. To appear in: K.E.Hood, C.T. Halpern, G. Greenberg, & R.M. Lerner (Eds.), Handbook of developmental science, behavior, and genetics. Malden, MA: Blackwell.

30 Longitudinal Genetic Factor Model (Inter-Individual Variation) Let y ijkmt denote the observed phenotypic score at the mth observed variable (m = 1,2,,M) for the jth member (j = 1,2) of the ith twin pair (i = 1,2,,N) of type k (k = 1 for MZ and k = 2 for DZ) at the tth measurement occasion t (t = 1,2,,T)

31 y ijkmt = mt ijkt + mt C ijkt + mt E ijkt + e ijkmt ijkt is the additive genetic factor score of the jth member of the ith twin pair of type k at measurement occasion t C ijkt is the common environmental factor score of the jth member of the ith twin pair of type k at measurement occasion t E ijkt is the specific environmental factor score of the jth member of the ith twin pair of type k at measurement occasion t

32 Longitudinal Evolution of Factor Scores ijkt = t,t-1 ijkt-1 + ijkt C ijkt = t,t-1 C ijkt-1 + ijkt E ijkt = t,t-1 E ijkt-1 + ijkt

33 Generic Longitudinal Factor Model Y 111 Y p11 Y 112 Y p12 A 1 (t) A 1 (t+1) 0.5 C 1 (t) C 1 (t+1) E 1 (t) Time 1 Time 2 E 1 (t+1) 1.0 A 2 (t) A 2 (t+1) C 2 (t) C 2 (t+1) E 2 (t) Y 121 Y p21 Y 122 Y p22 E 2 (t+1)

34 The general Longitudinal Genetic Factor Model is based on the strong assumption that all parameters (genetic, common and specific environmental factor loadings and lagged regression coefficients) are invariant across subjects.

35 Idiographic Filter is: - Based on analysis of intra-individual variation - Involves a new definition of measurement equivalence at the level of latent variables - Allows for subject-specific factor loadings Nesselroade, J.R., Gerstorf, D., Hardy, S.A., & Ram, N. (2007). Idiographic filters for psychological constructs. Measurement, 5,

36 Genetic Factor Model for Intra-Individual Variation (iface). Application to single MZ or DZ twin pair. y jmt = jm jt + jm C jt + jm E jkt + e jmt jt = j jkt-1 + jt C jt = j C jt-1 + jt E jt = j E jt-1 + jt All parameters in iface are subject-specific.

37 iface Y 11(t) Y p1(t) Y 11(t+1) Y p1(t+1) A 1 (t) A 1 (t+1) R C 1 (t) C 1 (t+1) R 1.0 E 1 (t) Time t Time t+1 E 1 (t+1) 1.0 A 2 (t) A 2 (t+1) C 2 (t) C 2 (t+1) E 2 (t) Y 12(t) Y p2(t) Y 12(t+1) Y p2(t+1) E 2 (t+1)

38 Preliminary Application iface to Multi-Lead EEG Data Obtained in Oddball Task

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41 DZ Twin Pair A (cor[a 1,A 2 ] =.40) Twin 1 a 2 c 2 e 2 res Cz Pz T T

42 DZ Twin Pair A (cor[a 1,A 2 ] =.40) Twin 2 a 2 c 2 e 2 res Cz Pz T T

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44 DZ Twin Pair A (cor[a 1,A 2 ] =.37) Twin 1 a 2 c 2 e 2 res C P C P

45 DZ Twin Pair A (cor[a 1,A 2 ] =.37) Twin 2 a 2 c 2 e 2 res C P C P

46 Heritability is high for a few leads which differ across subjects (Pz for twin 1; P3 and P4 for twin 2) The effects of common environment are high for leads neighboring the ones with high heritability (P3 and P4 for twin 1; Pz for twin 2), possibly due to A x C interaction (Molenaar et al., Genetic Epidemiology, 1990) Analogous results are obtained for other DZ twin pairs

47 Twin 1 Additive Genetic Common Environment Pz C4 P3 P4

48 Twin 2 Additive Genetic Common Environment Pz P3 P4

49 Application of iface to multi-lead EEG preliminary and can be generalized in several respects, including: - Alternative estimation techniques - Alternative model variants - Application to complete set of leads (19) - Application to separate ERP components - Frequency domain analysis

50 How to arrive at valid nomothethic laws in the presence of heterogeneous model structures

51 Simulated Data; Unified SEM Subject ROI1 ROI3 Legend: Contemp. ROI Effects Lagged ROI Effects Negative ROI2 ROI

52 Simulated Data; Unified SEM Subject ROI1 ROI3 Legend: Contemp. ROI Effects Lagged ROI Effects Negative ROI2 ROI

53 Simulated Data; Unified SEM Subject ROI1 ROI Legend: Contemp. ROI Effects Lagged ROI Effects Negative ROI2 ROI

54 Conclusion Multi-Subject Analysis All three subjects have exactly the same pattern and weights for their contemporaneous and lagged relationships, with only one exception: The timelagged relationship between ROI4 and ROI1 of Subject 3 is reversed with respect to Subjects 1 & 2. This overall subject-invariant (nomothetic) result with a single subject-specific detail is confirmed in a multisubject time series analysis

55 Simulated Data; Unified SEM Average ROI1 ROI Legend: Contemp. ROI Effects Lagged ROI Effects Negative.83 ROI ROI4.89

56 Miller et al., 2002, p Significant Activations During Episodic Retrieval

57 Successful Method to Estimate Common Model in the presence of Arbitrary Degrees of Model Heterogeneity Gates & Molenaar (submitted)

58 Gates, Molenaar, Hillary, & Slobonov. (2Gat011). NeuroImage. Extended Unified SEM (eusem) General form: Reduced to a lag of one:

59 Matlab Program for Identifying Group Model 1. Runs null model on each individual 2. Uses Lagrange Multiplier equivalents to identify which parameter, if opened, would optimally improve models for most individuals 3. Runs model across individuals with freed parameter 4. Repeats steps 2 and 3 5. Prunes beta estimates that are not significant Gates, Molenaar, & Rovine, In Progress

60 Matlab Program for Individual Maps 1. Begin with beta structure identified from the group analysis 2. Run semi-confirmatory iterative automatic search procedure which uses Lagrange Multiplier equivalents 3. Prunes beta estimates that are not significant

61 Recent Paper Examining Connectivity Methods: Smith et al., 2011 Used Balloon Model to simulate data Tested 38 analytic methods across 28 different simulations Contemporaneous correlation-based models worked best Models including only lagged, directed effects and models working on higher-order statistics did poorly Did not test any structural equation modeling (SEM) approaches

62 Connectivity Map for Simulations ROI5 ROI1 ROI2 ROI1 ROI4 ROI3 ROI5 ROI2 ROI4 ROI3

63 Findings Summarized: Effective Connectivity Approaches Ability to detect true connection: Smith Paper: under 20% Our Approach: 95% Ability to detect correct directionality: Smith Paper: top of 65% Our Approach: 94% Smith et al., 2011

64 I thank: In the USA: John Nesselroade Wayne Velicer Mike Rovine Nilam Ram Eric Loken Katie Gates In Europe: Dorret Boomsma Dirk Smit NSF grant