1. 1crmda.KU.edu 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 Representing Time in Longitudinal Research: Assessment Lag as Moderator

2. 2crmda.KU.edu

3. 3 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

4. 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. 5 The Sequential Designs crmda.KU.edu

6. 6 Design Independent VariablesConfounded Effect Cohort- Sequential Age & Cohort Age x Cohort Interaction is confounded with Time Time- Sequential Age & Time Age x Time Interaction is confounded with Cohort Cross- Sequential Cohort & Time Cohort x Time Interaction is confounded with Age What’s Confounded? crmda.KU.edu

7. 7 Transforming to Accelerated Longitudinal crmda.KU.edu

8. 8 Accelerated Longitudinal Designs Fall 6 Spr 6 Fall 7 Spr 7 Fall 8 Spr 8 Fall 9 Grade 6 Grade 7 Grade 8 Grade crmda.KU.edu

9. 9 Accelerated Growth Curve Model (L13.1.GC.LevelCUBIC.Accelerated) Fall 6Spr.6Fall7Spr.7 Fall8Spr.8 Fall9 = = = = = = = = = = = = = = -4* 5* 0* -3* 0* 5* -1* 1* 0* -1* 1* -3* -2* -1* 0* 1* 2* 3* 1*  Inter-cept Linear  Quad-ratic  Cubic  Grade8 8=1 1* 0* Grade7 7=1 1* 0* = = ===== crmda.KU.edu

10. 10 Plot of Estimated Trends crmda.KU.edu

11. 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. 12 Transforming to Episodic Time crmda.KU.edu

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

14. T1T2 Age 1 student 2 3 4 5 6 7 8 01 2 Time 3 4 5 6 5;6 5;3 4;9 4;6 4;11 5;7 5;2 5;4 5;7 5;8 4;11 5;0 5;4 5;10 5;3 5;8 14crmda.KU.edu

15. T0T1 T2T3T4 T5T6 15crmda.KU.edu

16. T0T1 T2T3T4 T5T6 Intercept 1 1 1 1 1 1 1 16crmda.KU.edu

17. T0T1 T2T3T4 T5T6 Intercept 1 2 3 4 Growth 0 Linear growth 5 6 1 1 1 1 1 1 1 17crmda.KU.edu

18. T0T1 T2T3T4 T5T6 Intercept 1 1 1 Practice 1 Growth Constant Practice Effect 1 1 1 1 1 1 1 1 0 1 1 2 3 4 0 5 6 18crmda.KU.edu

19. T0T1 T2T3T4 T5T6 Intercept.45.35 PracticeGrowth Exponential Practice Decline 1 1 1 1 1 1 1 1 1 2 3 4 0 5 6.55.67.87 0 19crmda.KU.edu

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

21. 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 Developmental time-lag model 21crmda.KU.edu

22. 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. 23www.crmda.ku.edu Types of Change Effects

24. 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 24 Lag as Moderator (LAM) Models crmda.KU.edu

25. X1X1 X2X2 X3X3 X4X4 X5X5 XnXn Y1Y1 Y2Y2 Y3Y3 Y4Y4 Y5Y5 YnYn T1T1 T min T max 25 Variable Actual Assessments T2T2 X6X6 Y6Y6 X7X7 Y7Y7 X8X8 Y8Y8 X9X9 Y9Y9 XiXi YiYi XjXj YjYj crmda.KU.edu

26. X i is the focal predictor of outcome Y i Lag i can vary across persons b 1 describes the effect of X i on Y i when Lag i is zero b 2 describes the effect of Lag i on Y i when X i is zero b 3 describes change in the X i → Y i relationship as a function of Lag i 26 Multiple Regression LAM model crmda.KU.edu

27. 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. 27 An Empirical Example crmda.KU.edu

28. 28 Lag (Mean Centered) Effect of HOME T1 on MDI T2 HOME predicting MDI crmda.KU.edu

29. 29 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 Implications of LAM Models crmda.KU.edu

30. 30 Randomly Distributed Assessment X1X1 X2X2 X3X3 X4X4 X5X5 XnXn Y1Y1 YnYn T1T1 T begin T end T mid X6X6 X7X7 X8X8 X9X9 Y1Y1 Y1Y1 Y1Y1 Y1Y1 Y2Y2 Y2Y2 Y2Y2 Y2Y2 Y2Y2 Y3Y3 Y3Y3 Y3Y3 Y3Y3 Y3Y3 Y4Y4 Y4Y4 Y4Y4 Y4Y4 Y4Y4 Y5Y5 Y5Y5 Y5Y5 Y5Y5 Y5Y5 Y6Y6 Y6Y6 Y6Y6 Y6Y6 Y6Y6 Y7Y7 Y7Y7 Y7Y7 Y7Y7 Y7Y7 Y8Y8 Y8Y8 Y8Y8 Y8Y8 Y8Y8 Y9Y9 Y9Y9 Y9Y9 Y9Y9 Y9Y9 YnYn YnYn YnYn YnYn crmda.KU.edu

31. Early Communication Indicators

32. T-Scores Individual-likelihood Based Estimation –Allows individually varying values of time y it = α i + β i λ it + ε it –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. 38crmda.KU.edu 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 Thank You!

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.eduyhat@ttu.edu IMMAP (immap.educ.ttu.edu) Stats Camp (Statscamp.org) 39www.Quant.KU.edu