Todd D. Little University of Kansas Director, Quantitative Training Program

1. Representing Time in Longitudinal Research: Assessment Lag as Moderator Todd D. Little University of Kansas Director, Quantitative Training Program Director, Center for Research Methods and Data Analysis Director, Undergraduate Social and Behavioral Sciences Methodology Minor Member, Developmental Psychology Training Program crmda.KU.edu Colloquium presented 04-05-2013 @ Purdue University Special Thanks to Noel A. Card, James P. Selig, & Kristopher Preacher crmda.KU.edu

2. crmda.KU.edu

3. Overview • Conceptualizing and Representing Time in Longitudinal Research • B = ƒ(age) vs. Δ = ƒ(time) • The Accelerated Longitudinal Design • Developmental-Lag Model • The Lag as Moderator Model crmda.KU.edu 3

4. Validity Threats in Longitudinal Work • Threats to Validity • Maturation • In pre-post experiment effects may be due to maturation not the treatment • Most longitudinal studies, maturation is the focus. • Regression to the mean • Only applicable with measurement error • Instrumentation effects (factorial invariance) • Test-retest/practice effects (ugh) • Selection Effects • Sample Selectivity vs. Selective Attrition • Age, Cohort, and Time of Measurement are confounded • Sequential designs attempt to unconfound these. crmda.KU.edu

5. The Sequential Designs crmda.KU.edu

6. What’s Confounded? crmda.KU.edu

7. Transforming to Accelerated Longitudinal crmda.KU.edu

8. Accelerated Longitudinal Designs Grade crmda.KU.edu

9. Accelerated Growth Curve Model a2 a3 a1 a4 Linear Quad- ratic Inter- cept Cubic 3* 5* 0* 2* 5* 1* -3* -1* 1* -3* -2* 0* 1* 0* 1* -4* -1* -3* 1* 1* 0* 1* -1* 1* 1* -1* 1* 1* Fall 6 Spr. 6 Fall 7 Spr. 7 Fall 9 Fall 8 Spr. 8 = = = = = = = = Grade 8 = = = = = = = Grade 7 = = = = = = 1* 1* 7=1 8=1 0* 0* (L13.1.GC.LevelCUBIC.Accelerated) crmda.KU.edu

10. Plot of Estimated Trends crmda.KU.edu

11. Appropriate Time and Intervals • Age in years, months, days. • Experiential time: Amount of time something is experienced • Years of schooling, length of relationship, amount of practice • Calibrate on beginning of event, measure time experienced • Episodic time: Time of onset of a life event • Toilet trained, driver license, puberty, birth of child, retirement • Early onset, on-time, late onset: used to classify or calibrate. • Time since onset or time from normative or expected occurrence. • Measurement Intervals (rate and span) • How fast is the developmental process? • Intervals must be equal to or less than expected processes of change • Measurement occasions must span the expected period of change • Cyclical processes • E.g., schooling studies at yearly intervals vs. half-year intervals crmda.KU.edu

12. Transforming to Episodic Time crmda.KU.edu

13. Developmental time-lag model • Use 2-time point data with variable time-lags to measure a growth trajectory + practice effects (McArdle & Woodcock, 1997) crmda.KU.edu

14. Time Age student T1 T2 2 4 6 0 1 3 5 1 5;6 5;7 2 5;3 5;8 3 4;9 4;11 4 4;6 5;0 5 4;11 5;4 6 5;7 5;10 7 5;2 5;3 8 5;4 5;8 crmda.KU.edu

15. T0 T1 T2 T3 T4 T5 T6 crmda.KU.edu

16. Intercept 1 1 1 1 1 1 1 T0 T1 T2 T3 T4 T5 T6 crmda.KU.edu

17. Linear growth Intercept Growth 1 0 6 1 1 5 1 2 4 3 1 1 1 1 T0 T1 T2 T3 T4 T5 T6 crmda.KU.edu

18. Constant Practice Effect Intercept Growth Practice 0 1 0 6 1 1 1 5 1 1 2 4 3 1 1 1 1 1 1 1 1 T0 T1 T2 T3 T4 T5 T6 crmda.KU.edu

19. Exponential Practice Decline Intercept Growth Practice 0 1 0 6 1 1 1 5 .87 1 2 4 3 .67 1 1 .55 1 .45 .35 1 T0 T1 T2 T3 T4 T5 T6 crmda.KU.edu

20. The Equations for Each Time Point Constant Practice Effect Declining Practice Effect crmda.KU.edu

21. Developmental time-lag model • Summary • 2 measured time points are formatted according to time-lag • This formatting allows a growth-curve to be fit, measuring growth and practice effects crmda.KU.edu

22. Temporal Design • Changes (and causes) take time to Unfold • The ability to detect an effect depends on the measurement interval • The ability to model the shape of the effect requires adequate sampling of time intervals. • The ability to model the optimal effect requires knowing the shape in order to pick the optimal (peak) interval. • Lag within Occasion: the Lag as Moderator Model crmda.KU.edu

23. Types of Change Effects www.crmda.ku.edu

24. Lag as Moderator (LAM) Models • One possible way to address the issue of lag choice is to treat lag as a moderator • Following this approach lag is treated as a continuous variable that can vary across individuals crmda.KU.edu

25. Variable Actual Assessments X1 Y1 X2 Y2 X3 Y3 X4 Y4 X5 Y5 X6 Y6 X7 Y7 X8 Y8 X9 Y9 Xi Yi Xj Yj • • • Xn Yn T1 Tmin Tmax T2 crmda.KU.edu

26. Multiple Regression LAM model • Xiis the focal predictor of outcome Yi • Lagi can vary across persons • b1 describes the effect of Xi on Yiwhen Lagi is zero • b2 describes the effect of Lagi on Yi when Xi is zero • b3 describes change in the Xi →Yi relationship as a function of Lagi crmda.KU.edu

27. An Empirical Example • Data are from the Early Head Start (EHS) Research and Evaluation study (N = 1,823) • Data were collected at Time 1 when the focal children were approximately 14 months of age and again at Time 2 when the children were approximately 24 months of age • The average lag between Time 1 and Time 2 observations was 10.3 months with values ranging from 3.0 to 17.3 months • Measures: • The Home Observation for the Measurement of the Environment (HOME) assessed the quality of stimulation in the home at Time 1. • The Mental Development Index (MDI) from the Bayley Scales of Infant Development measured developmental status of children at Time 2. crmda.KU.edu

28. HOME predicting MDI Effect of HOMET1 on MDIT2 Lag (Mean Centered) crmda.KU.edu

29. Implications of LAM Models • Lag is embraced • LAM models allow us to model, not ignore, interactions of lag and hypothesized effects • Selecting/Sampling Lag is critical • Sampling only a single lag may limit generalizability • Theory Building • LAM models may yield a better understanding of relationships and richer theory regarding those relationships crmda.KU.edu

30. Randomly Distributed Assessment X1 Y1 Y1 Y1 Y1 Y1 Y2 Y2 Y2 Y2 Y2 X2 X3 Y3 Y3 Y3 Y3 Y3 Y4 Y4 Y4 Y4 Y4 X4 X5 Y5 Y5 Y5 Y5 Y5 Y6 Y6 Y6 Y6 Y6 X6 X7 Y7 Y7 Y7 Y7 Y7 X8 Y8 Y8 Y8 Y8 Y8 X9 Y9 Y9 Y9 Y9 Y9 • • • Xn Yn Yn Yn Yn Yn T1 Tbegin Tend Tmid crmda.KU.edu

31. Early Communication Indicators

32. T-Scores • Individual-likelihood Based Estimation • Allows individually varying values of time yit = αi+ βiλit + εit • Ages in months ((days/365)*12) were calculated and centered around locations of latent intercepts

33. T-Scores

34. Gestures

35. Vocalizations

36. Single Word Utterances

37. Multiple Word Utterances

38. Thank You! Todd D. Little University of Kansas Director, Quantitative Training Program Director, Center for Research Methods and Data Analysis Director, Undergraduate Social and Behavioral Sciences Methodology Minor Member, Developmental Psychology Training Program crmda.KU.edu Colloquium presented 04-06-2013 @ Purdue University crmda.KU.edu

39. Update Dr. Todd Little is currently at Texas Tech University Director, Institute for Measurement, Methodology, Analysis and Policy (IMMAP) Director, “Stats Camp” Professor, Educational Psychology and Leadership Email: yhat@ttu.edu IMMAP (immap.educ.ttu.edu) Stats Camp (Statscamp.org) www.Quant.KU.edu