1. November 10, 2005SAMSI Longitudinal Working Group1 Computing Confidence Intervals for Predicting New Observations in the Linear Mixed Model Lloyd J. EdwardsKunthel By Department of Biostatistics, UNC-CH A. Jackson StennerGary L. Williamson Robert F. (Robin) Baker MetaMetrics, Inc.

2. November 10, 2005SAMSI Longitudinal Working Group2 Outline Introduction Basic Work with Growth Curves Prediction Error in the Mixed Linear Model New Software

3. November 10, 2005SAMSI Longitudinal Working Group3 Introduction MetaMetrics perspective –Unification of measurement –Characterization of measurement error –Life-span developmental approach –Fitting models to data vs. fitting data to models Longitudinal Working Group –Mutual interests (growth, mixed models, etc.) –Collaboration (theoretical, practical interests) –Summer GRA (production of new software)

4. November 10, 2005SAMSI Longitudinal Working Group4 Growth Curve Basics Growth Model –Multilevel formulation –Mixed Model Data Sets –NC –Palm Beach Example

5. November 10, 2005SAMSI Longitudinal Working Group5 Growth Model Multilevel formulation Level 1:L ti = 0i + 1i TIME ti + e ti Level 2: 0i = 00 + r 0i 1i = 10 + r 1i Mixed model formulation L ti = 00 + 10 TIME ti + r 0i + r 1i TIME ti + e ti

6. November 10, 2005SAMSI Longitudinal Working Group6

7. November 10, 2005SAMSI Longitudinal Working Group7 Prediction Scenarios for Two-Level Models Prediction and prediction intervals for: all observations in the data set one student in the data set, on future measurement occasions (given y i, X i, Z i ) a new student who is not in the data set

8. November 10, 2005SAMSI Longitudinal Working Group8 General Mixed Model Formulation Prediction Limits of the form:

9. November 10, 2005SAMSI Longitudinal Working Group9 Characterizing prediction error Distinctions –Simple linear case versus –Mixed Model analog versus

10. November 10, 2005SAMSI Longitudinal Working Group10 Characterizing prediction error Benefits –obtain best predicted status –state confidence limits for prediction –reduce apparent measurement error –consistent with a parametric form

11. November 10, 2005SAMSI Longitudinal Working Group11 New Software SAS IML Current features –Three prediction scenarios –Simple assumptions for error covariances –Restricted to two-level MLMs –Limited ability to incorporate covariates Available at: http://www.unc.edu/~kby/http://www.unc.edu/~kby/

12. November 10, 2005SAMSI Longitudinal Working Group12 Further Research Assumption of i.i.d. within-subject errors Literature suggests more complex error covariance structures. Chi and Reinsel (1989, JASA) extend to AR(1) errors We extend to general within-subject error covariance structure.

13. November 10, 2005SAMSI Longitudinal Working Group13 Closing Third Lexile National Reading Conference June 19-21, 2006 Developing Tomorrows Readers...Today http://www.Lexile.com