Issues in the Implementation of Longitudinal Growth Models for Student AchievementJoseph StevensUniversity of Oregon

Keith ZvochUniversity of Nevada-Las VegasNevada Center for Evaluation and Assessment

Presented at the Conference on Longitudinal Modeling of Student Achievement, University of Maryland, College Park, November, 2005.

© Stevens, 2005

Contact Information:

170 College of Education

5267 University of Oregon

Eugene, OR 97403

(541) 346-2445

stevensj@uoregon.edu

Presentation available at:

http://www.uoregon.edu/~stevensj/issues.ppt

No Child Left Behind NCLB requires the use of a cross-sectional, case

study design for the study of achievement and school effectiveness (Stevens, 2005)

How best to model student achievement and regularities in achievement by teacher or school?

Increasing interest in and development of longitudinal methods and modeling over several decades

Purposes Draw attention to certain research design issues in

the study of school effectiveness

Review things we know about the analysis of change

Describe issues in the design and use of longitudinal models and some empirical results

Discuss issues in the potential use of longitudinal modeling for accountability purposes

Design Issues

Status versus growth (Stevens, 2000)

State mandated test data on TerraNova Survey Plus for all middle schools in six NM School Districts from 1999-2001

36 Middle Schools; 5,544 students tested in grades 6, 7, and 8

Growth

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Analysis of Change “Investigators who ask questions regarding gain

scores would ordinarily be better advised to frame their questions in other ways.” Cronbach & Furby (1970)

“Problems in measuring change abound and the virtues in doing so are hard to find.” Linn & Slinde (1977)

Longitudinal research has been recognized as the “sine qua non of evaluation in nonexperimental settings” (Marco, 1974)

The Analysis of Change Height example Cross sectional comparisons do not measure

change effectively/accurately Cross sectional comparisons of cohorts

produce different results than analysis of change

Common to confuse longitudinal research hypotheses and language with cross sectional designs and results

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Analysis of Change Change versus variability (Nesselroade,

1991) Sources of individual differences (Cattell, 1966):

Stable individual differences Intraindividual variability - short-term, relatively

reversible change or fluctuation Intraindividual changes - long-term, relatively not

reversible

Interindividual differences in intraindividual growth

Student Growth Trajectories

Analysis of Change

What covariates are systematically related to interindividual differences in growth?

Aggregated Growth Teachers Schools Districts States

School Growth Trajectories

Design Issues

What are some design issues arising from the analysis of change literature?

Measurement Occasions Initial Starting Point Choice of initial starting point impacts model

results

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Number of Measurement Occasions Pre-post, two wave studies most common More is better

“Two waves of data are better than one, but maybe not much better” (Rogosa, 1995)

Shape of growth function Reliability of estimation

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Design Issues

Lag between treatment and outcome Treatment effects take time to realize The more distal the outcome measurement from

the treatment impact, the less the effect

Design Issues

Size or Length of Measurement Intervals Use shortest interval possible until temporal effect

established Intervals too short, cost, reactivity to

measurement, too little treatment effect Intervals too long, attrition, distal treatment

effects, inability to model growth function accurately

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Design Issues Attrition

Incomplete data How does attrition bias results?

Mobility At the student level, accountability for some,

opportunity to learn At the school level, accountability for all,

opportunity to teach Cohort Stability

Effects of AttritionZvoch & Stevens, 2005

An investigation of sample exclusion and student attrition effects on the longitudinal study of middle school mathematics performance

Mathematics achievement on state mandated TerraNova

24 middle schools Analytic Method: 2 and 3-level HLM growth

models

Effects of Attrition

One longitudinal cohort: Two samples Complete cohort sample: All students who

participated in accountability testing in 1998-99 (N = 6,098)

District accountability sample: A subset of the complete cohort sample – same school for three years, complete test data, no modified test administrations (N = 3,334)

Student Demographic Characteristics by Analytic Sample

_________________________________________________________________

Accountability Sample Complete Cohort Sample

(N = 3,334) (N = 6,098)

_________________________________________________________________

Student Characteristic Frequency Percent Frequency Percent

_________________________________________________________________

Female 1,710 51.3 3,016 49.5

Non-Anglo 1,797 53.9 3,536 58.0

English Language Learner 397 11.9 1,121 18.4

Free Lunch Recipient 1,170 35.1 2,628 43.1

Special Education 101 3.0 1,092 17.9

__________________________________________________________________

Note. Chi-square tests comparing the accountability sample (N = 3,334) with the group of students lost from the complete cohort sample (N = 2,764) on student background characteristics revealed that males and students in special populations were statistically over-represented in the group of excluded students (p < .01 in each comparison).

Two-Level Conditional Model Relating Individual Characteristics to Mathematics Achievement by Sample

__________________________________________________________________________ Accountability Sample Complete Cohort Sample (N = 3,334) (N = 6,098)___________________________________________________________________________

Fixed Effect Estimate SE t Estimate SE t___________________________________________________________________________Mean Achievement, β00 661.68 0.55 1200.18*** 648.60 0.44 1472.05***Biological Sex, β01 -0.11 1.10 -0.10 0.90 0.88 1.02Minority Status, β02 -12.42 1.25 -9.79*** -12.16 1.00 -12.09***Free Lunch Status, β03 -13.60 1.27 -10.75*** -14.21 0.98 -14.44***ELL Status, β04 -22.12 1.70 -12.99*** -21.78 1.27 -17.13***SPED Status, β05 -21.85 3.66 -5.97*** -34.97 1.31 -26.62***___________________________________________________________________________Mean Growth, β10 18.97 0.24 77.85*** 17.54 0.22 80.14***Biological Sex, β11 -2.96 0.51 -5.82*** -2.75 0.45 -6.16***Minority Status, β12 -0.78 0.59 -1.31 -2.12 0.52 -4.06***Free Lunch Status, β13 -0.85 0.61 -1.37 -1.69 0.53 -3.22**ELL Status, β14 -0.18 0.88 -0.20 -0.01 0.68 -0.01SPED Status, β15 1.10 1.51 0.73 -2.81 0.71 -

3.98***Initial Achievement Status, β16 0.06 0.01 4.42*** 0.02 0.01 2.25*___________________________________________________________________________*** p < .001; **p < .01; * p < .05

Three-Level Unconditional Model for Mathematics Achievement by Sample ______________________________________________________________________________

Accountability Sample Complete Cohort Sample

(N = 3,334) (N = 5,168) ______________________________________________________________________________

Fixed Effect Estimate SE t Estimate SE t______________________________________________________________________________ School Mean Achievement, γ000 659.19 2.97 222.29*** 648.96 3.09 209.84***School Mean Growth, γ100 18.44 0.92 20.12*** 17.64 0.87 20.16***______________________________________________________________________________

Variance VarianceRandom Effect Component df 2 Component df 2______________________________________________________________________________ Individual Achievement, r0ij 766.01 3310 8978.37*** 1132.02 4548 22708.86***Individual Growth, r1ij 27.98 3310 3858.62*** 44.04 4548 5736.01*** Level-1 Error, etij 313.12 361.59School Mean Achievement, u00j 202.73 23 704.61*** 222.42 23 855.44*** School Mean Growth, u10j 18.68 23 354.14*** 17.01 23 347.30***

Percentage of Variation Between Schools Achievement Status, π0ij 20.7 16.4Achievement Growth, π1ij 40.0 27.9______________________________________________________________________________Note. In this analysis, students who transferred schools within the district were dropped from the complete

cohort sample as these students could not be uniquely assigned to one school location (N = 930). *** p < .001

Effects of Attrition

Were cross-sample changes in school performance associated with the percentage of students from special populations excluded from the district accountability sample?

Figure 1. Cross-Sample School Achievement Mean Change in Mathematics as a Function of the Proportion of Students from Special Populations Excluded from the Accountability Sample

Figure 2. Cross-Sample School Growth Rate Change in Mathematics as a Function of the Proportion of Students

from Special Populations Excluded from the Accountability Sample

Conclusions Mathematics performance estimates differed

across two sample conditions District and school achievement higher and

student performance more similar in the restricted sample

Cross-sample school changes in student achievement closely related to the proportion of students from special student populations excluded

Cohort StabilityZvoch & Stevens, in press

An investigation of stability of cohorts from one year to the next

Mean achievement status and growth of students across cohorts

Changes in the achievement status and growth of students between student cohorts

Predictors of school achievement outcomes

Cohort Data StructureYear

Grade 99-00 00-01 01-02 02-03 03-04

3 C1 C2 C3

4 C1 C2 C3

5 C1 C2 C3

Note. Cohort 1 (N = 3,325), Cohort 2 (N = 3,347), Cohort 3 (N = 3,322); School N = 79

School Performance Indices Accountability Model

Outcome Focus

Across or Within Cohorts (Current

Performance)

Between Cohorts

(Improvement Over Time)

Status School Mean Achievement,

Percent Proficient

School Mean Achievement/

Proficiency Change

Growth School Mean Growth

School Mean Growth Change

Figure 1. Cross-cohort relationship between school mean achievement and school

mean growth in mathematics

High Mean Low Growth

High Mean High Growth

Low Mean Low Growth

Low Mean High Growth

Cohort Stability

Do the cross-cohort estimates of school performance vary with schools’ social context?

Figure 2. Cross-cohort school mean growth in mathematics as a function of the

percentage of free lunch recipients

0.00 0.25 0.50 0.75 1.00

Percentage of Free Lunch Recipients

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Figure 3. Cross-cohort school mean achievement in mathematics as a function of the

percentage of free lunch recipients

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To what degree do estimates of the mean achievement status and achievement growth of schools change with each successive student cohort?

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Cohort Stability

Do the between-cohort estimates of school improvement covary with cohort enrollment size?

Figure 7. Relationship between cohort-to-cohort changes in school mean growth and cohort enrollment size by school

Conclusions Cross-cohort performance differed by

outcome: mean achievement status or growth Cohort-to-cohort changes in student

performance also varied by outcome: change in school mean achievement or change in school mean growth

Across cohorts, schools’ social context was associated only with student achievement levels, not achievement growth

Changes in school performance were closely related to cohort enrollment size

Alternative Statistical Models for Longitudinal Analysis

Difference or Gain scores Residuals Growth Curve Models Latent Growth Curves Mixture Models Autoregressive Models

Alternative Statistical Models

Strengths and weaknesses of each model Results differ based on model used Models answer different questions Model complexity versus transparency

Measurement Issues

Carryover Effects (Learning, Sensitization, etc.) Need for Parallel Test forms

Measurement Issues: Scaling and Equating Do scales change over time?

Structure/dimensionality Units Standardization or scaling may mask change

Equating Vertical equating, usually one point in time across

grades/cohorts Need for longitudinal equating What spans can reasonably be equated? What content can reasonably be equated?

Measurement Issues: Reliability Regression to the mean in two wave studies Reliability of Difference Scores Measured Variables versus Latent Variables

Reliability Generalization Over time Interactions of time and other characteristics

Measurement Issues: Validity

Construct Equivalence over time Temporal Invariance

Measurement invariance Structural invariance

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Measurement Issues

Measurement instruments that are designed to measure cognitive growth (Collins, 1991)

True Developmental Scales

Measurement Issues: Validity

Messick (1989): Validity and consequences of alternative approaches

Pattern Matching (Shadish, Cook, & Campbell, 2002)

Riechardt (2000): study of plausible threats to validity of treatment effects

Measurement Issues: Validity Stevens (2005)

Three level HLM curvilinear growth models applied to state data.

23,469 sixth grade children took the state mandated TerraNova in 1999-00

Study includes the 23,296 sixth graders (99.3%) who took the mathematics subtest

These students were matched longitudinally to 7th, 8th, and 9th grade records for the years 2001, 2002, and 2003

Sample included only those students who were in their middle school for 2 or 3 years (17,596; 75.5% of students).

Schools with less than 5 students were also excluded (13 schools with a total of 24 students), resulting in an analytic sample of 242 schools (94% of schools) with 17,572 students.

This sample differs from the population in having about 1% more White and Hispanic, 1% fewer Native American, 1% fewer LEP and Special Education, and 2% fewer bilingual students.

Mathematics Achievement Predicted by Individual Characteristics _______________________________________________________________

Fixed Effect Coefficient SE t df _ p___ School Mean Achievement, γ000 663.54 1.28 513.86 241 < .001 White Student, γ010 14.62 0.77 18.88 241 < .001 LEP, γ020 -16.00 1.19 -13.50 241 < .001 Title 1 Student, γ030 -11.10 1.44 -7.71 241 < .001 Special Education, γ040 -33.09 1.88 -17.62 241 < .001 Modified Test, γ050 -16.83 2.63 -6.40 241 < .001 Free Lunch Student, γ060 -7.75 1.13 -6.85 241 < .001 Gender, γ070 -1.21 0.59 -2.03 241 .042

School Linear Growth, γ100 19.40 0.70 27.88 241 < .001 White Student, γ110 -1.20 0.64 -1.86 241 .062 LEP, γ120 0.70 1.13 0.60 241 .547 Title 1 Student, γ130 -2.58 0.95 -2.72 241 .007 Special Education, γ140 -2.16 1.67 -1.29 241 .196 Modified Test, γ150 -2.43 2.47 -0.99 241 .325 Free Lunch Student, γ160 -0.75 1.03 -0.73 241 .466 Gender, γ170 -4.68 0.59 -7.98 241 < .001_______________________________________________________________

Mathematics Achievement Predicted by Individual Characteristics (continued)______________________________________________________________Fixed Effect Coefficient SE t df p______________________________________________________________School Curvilinear Growth, γ200 -2.09 0.21 -9.78241 < .001 White Student, γ210 0.48 0.20 2.35 241 .019 LEP, γ220 -0.10 0.36 -0.27 241 .790 Title 1 Student, γ230 0.61 0.28 2.17 241 .030 Special Education, γ240 0.61 0.50 1.22 241 .224 Modified Test, γ250 -0.10 0.75 -0.14 241 .890 Free Lunch Student, γ260 0.26 0.33 0.79 241 .427 Gender, γ270 1.05 0.19 5.64 241 < .001______________________________________________________________School Level Level-1 Level-2 VarianceVariance Component Explained ______________________________________________________________

Mean Achievement, u00 242.78 184.89 23.8%Linear Growth, u10 41.46 30.68 26.0%Curvilinear Growth, u10 2.94 2.60 11.6%______________________________________________________________

Pattern matching: relationships between alternative measures of school effectiveness and confounding variables NCLB Proficiency (percent proficient or above using state determined cutpoint)

State rating of schools (weighted combination of proficiency score, attendance, dropout rates)

HLM Empirical Bayes (EB) intercept estimates

HLM EB slope estimates

r2 = .38, p < .001 r2 = .44, p < .001

School Rating

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Measurement Issues: Validity

Pattern Matching: Relation to schooling effects If schooling policies and practice impact student

learning they should emerge as correlates of growth

Growth measures are more sensitive to the effects of schooling than status measures (Bryk & Raudenbush, 1989)

Zvoch & Stevens (2005)

Study Purpose: To examine correlates of status and growth in mathematics achievement over a three year period.

Individual math achievement scores on the TerraNova were used from a longitudinal sample of middle school students in the sixth grade in 1998-99, seventh grade in 1999-00, and eighth grade in 2000-01

Study conducted in one urban school district in NM: 24 middle schools, over 20,000 students; 51% female, 49% male 47% Hispanic, 44% Anglo, 3% Native American, 3%

African American, 2% Asian, and 1% Other 17% of students were classified as ELL 17% special education 40% of students receive a free or reduced price lunch

Percent Free-Lunch (M = .49, SD = .28)

Mean Educational Level of Mathematics Staff (M = 17.61, SD = .58)

Mathematics Curricula (0 = Traditional Program, 1 = NSF Reform Curricula, 9 of the 24 middle schools (38%)

Pattern of results differed depending on whether status scores or growth scores were examined

Fixed Effect Coefficient SE t _________________________________________________________________ School Mean Achievement, γ000 650.18 1.51 429.77***

Percent Free Lunch, γ001 -0.22 0.05 -4.70***

Math Teacher Education, γ002 0.93 2.00 0.470.47

Math Curricula, γ003 -0.32 2.55 -0.13-0.13 School Mean Growth, γ100 18.96 0.83 22.75***

Percent Free Lunch, γ101 -0.02 0.02 -0.92-0.92 Math Teacher Education, γ102 3.29 1.08 3.06**

Math Curricula, γ103 -3.00 1.40 -2.14*

_________________________________________________________________Zvoch, K., & Stevens, J. J. (in press). Longitudinal effects of school

context and practice on mathematics achievement. Journal of Educational Research.

Using Longitudinal Models for Accountability SystemsDesign: Number of measurement occasions

Single occasion case studies Annual measurement interval, size of interval

Initial starting point, prior achievement

Some states cannot track students over time

Using Longitudinal Models for Accountability SystemsMeasurement: Need for new assessments designed to

measure cognitive growth Assessments sometimes vertically equated,

seldom longitudinally equated; Need for true developmental scales,

measurement invariance over time Construct equivalence over time

Using Longitudinal Models for Accountability Systems Attrition, Mobility, Cohort Effects

Need for further study How are school estimates biased? Can statistical adjustments be used?

School size, disaggregated group sizes and stability or bias

Summary and Conclusions Different designs, measures, and methods of

analysis are likely to provide different evaluations of student growth and school effectiveness

However, despite difficulties of longitudinal modeling, cross-sectional designs can not address fundamental issues of growth, change, and learning

Need for the inclusion of policy and practice variables

Importance of empirically validating assessment instruments and accountability systems

Bibliography

Bryk, A. S., & Raudenbush, S. W. (1987). Application of hierarchical linear models to assessing change. Psychological Bulletin, 101, 147-158.

Cattell, R. B. (1966). Patterns of change: Measurement in relation to state dimension, trait change, lability and process concepts. In R. B. Cattell (Ed.), Handbook of multivariate experimental psychology. Chicago: Rand McNally.

Collins, L. (1991). Measurement in longitudinal research. In L. M. Collins & J. L. Horn (Eds.), Best methods for the analysis of change: Recent advances, unanswered questions, future directions. Washington, DC: American Psychological Association.

Collins, L. M., & Horn, J. L. (1991). Best methods for the analysis of change: Recent advances, unanswered questions, future directions. Washington, DC: American Psychological Association.

Collins, L. M., & Sayer, A. (2001). New methods for the analysis of change. Washington, DC: American Psychological Association.

Cronbach, L. J., & Furby, L. (1970). How we should measure change—or should we? Psychological Bulletin, 74, 68-80.

CTB/McGraw-Hill (1997). TerraNova Technical Bulletin 1. Monterey, CA: Author.

Duncan, S. C., & Duncan, T. E. (1995). Modeling the processes of development via latent variable growth curve methodology. Structural Equation

Modeling: A Multidisciplinary Journal, 2(3), 187-213.

Ferrer, E., Hamagami, F., & McArdle, J. J. (2004). Teacher’s corner: Modeling latent growth curves with incomplete data using different types of structural equation modeling and multilevel software. Structural Equation Modeling: A Multidisciplinary Journal, 11(3), 452-483.

Linn, R.L., & Haug, C. (2002). Stability of school-building accountability scores and gains. Educational Evaluation and Policy Analysis, 24 (1), 29-36.

Linn, R. L., & Slinde, J. A. (1977). The determination of the significance of change between pre- and posttesting periods. Review of Educational Research, 47(1), 121-150.

Marco, G. L. (1974). A comparison of selected school effectiveness measures based on longitudinal data. Journal of Educational Measurement, 11, 225-234.

Meredith, W. & Horn, J. (2001). The role of factorial invariance in modeling growth and change. In L. M. Collins & A. Sayer (Eds.), New methods for the analysis of change. Washington, DC: American Psychological Association.

Messick, S. (1989). Validity. In R.L. Linn (Ed.), Educational Measurement (3rd Ed., pp. 13-103). New York: MacMillan.

Messick, S. (1994). The interplay of evidence and consequences in the validation of performance assessments. Educational Researcher, 23, 13-23.

Nesselroade, J. R. (1991). Interindividual differences in intraindividual change. In L. M. Collins & J. L. Horn (Eds.), Best methods for the analysis of change: Recent advances, unanswered questions, future directions. Washington, DC: American Psychological Association.

Raudenbush, S. W. (2001). Comparing personal trajectories and drawing causal inferences from longitudinal data. Annual Review of Psychology, 52, 501-525.

Raudenbush, S.W., Bryk, A.S., Cheong, Y.F., & Congdon, R.T. (2001). HLM 5: Hierarchical linear and nonlinear modeling. Chicago: Scientific Software International.

Raudenbush, S. W., & Willms, J. D. (1995). The estimation of school effects, Journal of Educational and Behavioral Statistics, 20 (4), pp. 307-35.

Reichardt, C. S. (2000). A typology of strategies for ruling out threats to validity. In L. Bickman (Ed.), Research design: Donald Campbell’s legacy (Vol. 2, pp. 89-115). Thousand Oaks, CA: Sage.

Rogosa, D. (1995). Myths about longitudinal research. In J.M. Gottman (Ed.). The analysis of change. Mahwah, NJ: Erlbaum.

Shadish, W. R., Cook, T. D., & Campbell, D. T. (2002). Experimental and quasi-experimental designs for generalized causal inference. Boston, MA: Houghton Mifflin Company.

Stevens, J. J. (2000). Educational Accountability Systems: Issues and Recommendations for New Mexico. Research report prepared for the New Mexico State Department of Education.

Stevens, J. J. (2005). The study of school effectiveness as a problem in research design. In R. Lissitz (Ed.), Value-added models in education: Theory and applications. Maple Grove, MN: JAM Press.

Stevens, J. J., Estrada, S., & Parkes, J. (2000). Measurement issues in the design of state accountability systems. Paper presented at the annual meeting of the AERA, New Orleans, LA.

Stone, C.A., & Lane, S. (2003). Consequences of a state accountability program: Examining relationships between school performance gains and teacher, student, and school variables. Applied Measurement in Education, 16, 1-26.

Teddlie, C., Reynolds, D., & Sammons, P. (2000). The methodology and scientific properties of school effectiveness research. In C. Teddlie, & D. Reynolds (Eds.), The International handbook of school effectiveness research. New York: FalmerPress.

Willett, J. B., & Sayer, A. G. (1994). Using covariance structure analysis to detect correlates and predictors of individual change over time, Psychological Bulletin, 116(2), 363-381.

Willett, J. B., Singer, J. D., & Martin, N. C. (1998). The design and analysis of longitudinal studies of development and psychopathology in context: Statistical models and methodological recommendations, Development and Psychopathology, 10, 395-426.

Zvoch, K., & Stevens, J. J. (in press). Successive Student Cohorts and Longitudinal Growth Models: An Investigation of Elementary School Mathematics Performance. Educational Policy Analysis Archives.

Zvoch, K., & Stevens, J. J. (in press). Longitudinal effects of school context and practice on mathematics achievement. Journal of Educational Research.

Zvoch, K., & Stevens, J. J. (2003). A multilevel, longitudinal analysis of middle school math and language achievement. Educational Policy Analysis Archives, 11 (20). (Available at: http://epaa.asu.edu/epaa/v11n20/).

Zvoch, K., & Stevens, J. J. (2005). Sample exclusion and student attrition effects in the longitudinal study of middle school mathematics performance. Educational Assessment, 10(2), 105-123.

Issues in the Implementation of Longitudinal Growth Models for Student AchievementJoseph Stevens & Keith Zvoch

170 College of Education, 5267 University of Oregon

Eugene, OR 97403, (541) 346-2445

stevensj@uoregon.edu

Presentation available at:

http://www.uoregon.edu/~stevensj/issues.ppt