November 10, 2005 SAMSI Longitudinal Working Group 1

Computing Confidence Intervals for Predicting New Observations

in the Linear Mixed Model

Lloyd J. Edwards Kunthel ByDepartment of Biostatistics, UNC-CH

A. Jackson Stenner Gary L. Williamson

Robert F. (Robin) BakerMetaMetrics, Inc.

November 10, 2005 SAMSI Longitudinal Working Group 2

Outline

• Introduction

• Basic Work with Growth Curves

• Prediction Error in the Mixed Linear Model

• New Software

November 10, 2005 SAMSI Longitudinal Working Group 3

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)

November 10, 2005 SAMSI Longitudinal Working Group 4

Growth Curve Basics

• Growth Model– Multilevel formulation– Mixed Model

• Data Sets– NC– Palm Beach

• Example

November 10, 2005 SAMSI Longitudinal Working Group 5

Growth Model

Multilevel formulation

Level 1: Lti = 0i + 1iTIMEti + eti

Level 2: 0i = 00 + r0i

1i = 10 + r1i

Mixed model formulation

Lti = 00 + 10TIMEti + r0i + r1iTIMEti + eti

November 10, 2005 SAMSI Longitudinal Working Group 6

NC Longitudinal Data Analyses End-of-Grade Reading in Lexiles

Six-Wave Panel: Grades 3-8, 1998-2003 N=66,013

Two-Level Unconditional Linear Growth Model

Fixed Effect Estimate SE t prob Average initial status, 00 703.9 0.9 812.55 <.0001

Average rate of growth, 10 87.8 0.1 783.04 <.0001

Random Effect Variance SE z prob Level 1 (temporal variation) Within Student, ti 8941 24.6 363.35 <.0001

Level 2 (between students) Individual initial status, i0 44,859 273.0 164.32 <.0001

Individual rate of growth, i1 319 4.8 66.70 <.0001

November 10, 2005 SAMSI Longitudinal Working Group 7

Prediction Scenarios forTwo-Level Models

Prediction and prediction intervals for:• all observations in the data set

• one student in the data set, on future measurement occasions (given yi, Xi, Zi)

• a new student who is not in the data set

November 10, 2005 SAMSI Longitudinal Working Group 8

General Mixed ModelFormulation

Prediction Limits of the form:

iiiii ebZβXy

)ˆvar(ˆ 2/1, iipmi yyty

November 10, 2005 SAMSI Longitudinal Working Group 9

Characterizing prediction error

• Distinctions– Simple linear case

versus

– Mixed Model analog

versus

)ˆ( 0yVar )ˆ( 0Var

)ˆ( ii yyVar )ˆ(iiyVar bμ

November 10, 2005 SAMSI Longitudinal Working Group 10

Characterizing prediction error

• Benefits– obtain best predicted status– state confidence limits for prediction– reduce apparent measurement error– consistent with a parametric form

November 10, 2005 SAMSI Longitudinal Working Group 11

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/

November 10, 2005 SAMSI Longitudinal Working Group 12

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.

November 10, 2005 SAMSI Longitudinal Working Group 13

Closing

Third Lexile National Reading Conference

June 19-21, 2006

Developing Tomorrow’s Readers...Today

http://www.Lexile.com