This paper is concerned with two theoretically and empirically important issues in longitudinal research: (1) identifying correlates and predictors of change and (2) discerning patterns of change. Two traditional methods of change measurement—the residualized observed difference and the residualized gain score—are discussed. A general structural equation model for measuring residualized true change and studying patterns of true growth or decline is described. This approach allows consistent and efficient estimation of the degree of interrelation ship between residualized change in a repeatedly assessed psychological construct and other variables, such as studied/presumed correlates and predictors of growth or decline on the latent dimension. Sub stantively interesting patterns of change on the trait level, such as regression to the mean, over- crossing, and fan-spreading, can be discerned. The model is useful in research situations in which it is of theoretical and empirical concern to identify those variables that correlate with, or can be used to predict, such patterns of true growth or decline that deviate from a group-specific trend in longitudinally-measured psychological constructs. The approach is illustrated using data from a cog nitive intervention study of plasticity in fluid intel ligence of aged adults (Baltes, Dittmann-Kohli, & Kliegl, 1986).
Baltes, P.B., Dittmann-Kohli, F., & Kliegl, R. (1986). Reserve capacity of the elderly in aging-sensitive tests of fluid intelligence: Replication and extension. Psychology and Aging, 1, 172-177.
2.
Baltes, P.B., & Willis, S.L. (1982). Plasticity and enhancement of intellectual functioning in old age: Penn State's adult development and enrichment project (ADEPT. In F. I. M. Craik & S. Trehub (Eds.), Aging and cognitive processes (pp. 353-389). New York: Plenum.
3.
Belsky, J. (1990). The psychology of aging: Theory, research, and interventions. Pacific Grove CA: Brooks/Cole .
4.
Bentler, P.M. (1978). The interdependence of theory, methodology, and empirical data: Causal analysis as an approach to construct validation. In D. B. Kandel (Ed.), Longitudinal drug research (pp. 267-302). New York: Wiley .
5.
Bentler, P.M. (1989). EQS: Structural equation program manual. Los Angeles: BMDP Statistical Software .
6.
Bereiter, C. (1963). Some persisting dilemmas in the measurement of change. In C. W. Harris (Ed.), Problems in measuring change (pp. 3-20). Madison: University of Wisconsin.
7.
Blieszener, R., Willis, S.L., & Baltes, P.B. (1981). Training research in aging on the fluid ability of inductive reasoning. Journal of Applted Developmental Psychology2. 247-265.
8.
Bollen, K.A. (1989). Structural equations with latent variables. New York: Wiley.
9.
Bond, L. (1979). On the base-free measure of change. Psychometnka, 44, 351-355.
10.
Browne, M.W., & Du Toit, S.H.C. (1991). Models for learning data . In L. M. Collins & J. L. Horn (Eds.), Best methods for the analysts of change (pp. 47-68). Washington DC: American Psychological Association.
11.
Browne, M.W., & Mels, G. (1990). RAMONA PC: User's guide [Computer program and manual]. Columbus: Department of Psychology, Ohio State University.
12.
Burr, J., & Nesselroade, J.R. (1990). Change measurement. In A. von Eye (Ed.), Statistical methods in longitudinal research: Principles and structuring change (Vol. 1, pp. 3-34). San Diego CA: Academic Press.
13.
Cavanaugh, J.C. (1990). Adult development and aging. Belmont CA: Wadsworth.
14.
Chernoff, H. (1954). On the distribution of the likelihood ratio. Annals of Mathematical Statistics, 25, 573-578.
15.
Cochran, W.G. (1968). Errors of measurement in statistics. Technometrics, 10, 637-666.
16.
Cronbach, L.J., & Furby, L. (1970). How should we "measure" change—or should we? Psychological Bulletin, 74, 68-80.
17.
Cudeck, R. (1989). Analyzing correlation matrices using covariance structure models. Psychological Bulletin, 105, 317-327.
18.
Fraser, C. (1992). COSAN: User's guide [Computer program and manual] . Faculty of Education, University of New England. Armidale, NSW, Australia.
19.
Gulliksen, H. (1950). Theory of mental tests. New York: Wiley.
20.
Harlow, L.L. (1991). Comments on "Models for learning data." In L. M. Collins & J. L. Horn (Eds.), Best methods for the analysis of change (pp. 69-74). Washington DC: American Psychological Association.
21.
Healey, M.J.R., & Goldstein, H. (1978). Regression to the mean. Annals of Human Biology, 5, 277-280.
22.
Huba, G.J., & Bentler, P.M. (1982). On the usefulness of latent variable causal modeling in testing theories of naturally occurring events: A rejoinder to Martin. Journal of Personality and Social Psychology, 43, 604-611.
23.
Isaac, P.D. (1970). Linear regression, structural relations, and measurement error. Psychological Bulletin, 74, 213-218.
24.
Jamieson, J., & Howk, S. (1992). The law of initial values: A four factor theory . International Journal of Psychophysiology, 12, 53-61.
25.
Jin, P. ( 1992). Toward reconceptualization of the law of initial values . Psychological Bulletin, 111, 176-184.
26.
Jöreskog, K.G., & Sörbom, D. (1988). LISREL 7: A guide to the program and its applications . Chicago IL: SPSS Scientific Software .
27.
Kendall, M.G., & Stuart, A.S. (1962). The advanced theory of statistics (Vol. 2). London: Griffin .
28.
Lacey, J.I., & Lacey, B.C. (1962). The law of initial values in the longitudinal study of autonomic constitution. Annals of the New York Academy of Sciences, 28, 1257-1290.
29.
Linn, R.L., & Slinde, J.A. (1977). The determination of the significance of change between pre- and posttesting periods. Review of Educational Research, 49, 121-150.
30.
Lord, F.M. (1960). Large-sample covariance analysis when the control variable is fallible. Journal of the American Statistical Association, 55, 307-321.
31.
Lord, F.M. (1963). Elementary models for measuring change. In C. W. Harris (Ed.), Problems in measuring change (pp. 21-38). Madison: University of Wisconsin Press.
32.
Lord, F.M., & Novick, R. (1968). Statistical theories of mental test scores. Reading MA: Addison-Wesley.
33.
Manning, W.H., & DuBois, P.H. (1962). Correlational methods in research on human learning . Perceptual and Motor Skills, 15, 287-321.
34.
McArdle, J.J. (1991). Comments on "Latent variable models for studying differences and change." In L. M. Collins & J. L. Horn (Eds.), Best methods for the analysis of change (pp. 164-169). Washington DC: American Psychological Association.
35.
McArdle, J.J., & Aber, M.S. (1990). Patterns of change within latent variable structural equation models. In A. von Eye (Ed.), Statistical methods in longitudinal research: Principles and structuring change (Vol. 1, pp. 151-224). San Diego CA: Academic Press.
36.
McArdle, J.J., & Anderson, E. (1990). Latent variable growth models for research on aging. In J. E. Birren & K. W. Schaie (Eds.), Handbook of the psychology of aging (3rd ed., pp. 22-44). New York: Academic Press.
37.
McArdle, J.J., & Epstein, D. (1987). Latent growth curves within developmental structural equation models. Child Development, 58, 110-133.
McCallum, R.C., Roznowski, M., & Necowitz, L.B. (1992). Model modifications in covariance structure analysis: The problem of capitalization on chance. Psychological Bulletin, 111, 490-504.
40.
Meredith, W. (1991). Latent variable models for studying differences and change. In L. M. Collins & J. L. Horn (Eds.), Best methods for the analysis of change (pp. 149-163). Washington DC: American Psychological Association.
41.
Messick, S. (1981). Denoting the base-free measure of change. Psychometrika, 46, 215-217.
42.
Muthén, B.O. (1987). LISCOMP: Analysis of linear structural equations with a comprehensive measurement model [Computer program and manual]. Mooresville IN: Scientific Software.
43.
Muthén, B.O. (1991). Analysis of longitudinal data using latent variable models with varying parameters. In L. M. Collins & J. L. Horn (Eds.), Best methods for the analysis of change (pp. 1-17). Washington DC : American Psychological Association.
44.
Nesselroade, J.R. (1991). Interindividual differences in intraindividual change. In L. M. Collins & J. L. Horn (Eds.), Best methods for the analysis of change (pp. 92-105). Washington DC: American Psychological Association.
45.
Nesselroade, J.R., Stigler, S.M., & Baltes, P.B. (1980). Regression toward the mean and the study of change . Psychological Bulletin, 88, 622-637.
46.
Rao, C.R. (1965). Linear statistical inference and its applications . New York: Wiley.
47.
Raykov, T. (1992a). Base-free measurement of change: A structural equation modeling approach. Zeitschrift fuer Psychologie, 200, 79-86.
48.
Raykov, T. (1992b). Structural models for studying correlates and predictors of change. Australian Journal of Psychology, 44, 101-112.
49.
Raykov, T. (1992c). On structural models for analyzing change. Scandinavian Journal of Psychology, 33, 247-265.
50.
Raykov, T., Tomer, A., & Nesselroade, J.R. (1991). Reporting structural equation modeling results in psychology and aging: Some proposed guidelines. Psychology and Aging, 6, 499-503.
51.
Rindskopf, D. (1983). Parameterizing inequality constraints on unique variances in linear structural models. Psychometrika, 48, 73-83.
52.
Rindskopf, D. (1984). Using phantom and imaginary latent variables to parameterize constraints in linear structural models. Psychometrika , 49, 37-47.
53.
Rogosa, D.R., Brandt, D., & Zimowski, M. (1982). A growth curve approach to the measurement of change. Psychological Bulletin, 90, 726-748.
54.
Rogosa, D.R., & Willett, J.B. (1983). Demonstrating the reliability of the difference score in the measurement of change. Journal of Educational Psychology, 20, 335-343.
55.
Rogosa, D.R., & Willett, J.B. (1985). Understanding correlates of change by modeling individual differences in growth. Psychometrika, 50, 203-228.
56.
SAS Institute. (1990). SAS Technical Report P-200 SASlSTAT Software: CALIS and LOGISTIC Procedures. Release 604 . Cary NC: SAS Institute.
57.
Schaie, K.W., & Hertzog, C. (1985). Measurement in the psychology of adulthood and aging. In J. E. Birren & K. W. Schaie (Eds.), Handbook of the psychology ofaging (2nd ed., pp. 61-92). New York: Van Nostrand Reinhold.
58.
Shapiro, A. (1985). Asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints. Biometrika, 72, 133-144.
59.
Steiger, J.H. (1989). EzPATH: A supplementary module for SYSTAT and SYGRAPH [Computer program]. Evanston IL: SYSTAT.
60.
Tisak, J., & Meredith, W. (1989). Exploratory longitudinal factor analysis in multiple populations. Psychometrika, 54, 261-281.
61.
Tisak, J., & Meredith, W. (1990a). Longitudinal factor analysis. In A. von Eye (Ed.), Statistical methods in longitudinal research: Principles and structuring change (Vol. 1, pp. 125-150). San Diego CA: Academic Press.
62.
Tisak, J., & Meredith, W. (1990b). Descriptive and associative developmental models . In A. von Eye (Ed.), Statistical methods in longitudinal research: Time series and categorical longitudinal data (Vol. 2, pp. 387-406). San Diego CA: Academic Press.
63.
Tucker, L.R. (1979). Comment on a note on a base-free measure of change . Psychometrika, 44, 357.
64.
Tucker, L.R., Damarin, F., & Messick, S. (1966). A base-free measure of change. Psychometrika, 31, 457-473.
65.
Webster, H., & Bereiter, C. (1963). The relationship of changes measured by mental test scores. In C. W. Harris (Ed.), Problems in measuring change (pp. 39-59). Madison: University of Wisconsin Press.
66.
Wilder, J. (1965). Pitfalls in the methodology of the law of initial value. American Journal of Psychology, 19, 577-584.
