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Understanding Correlates of Change by Modeling Individual Differences in Growth

Published online by Cambridge University Press:  01 January 2025

David R. Rogosa  and
John B. Willett
David R. Rogosa*
Affiliation:
Stanford University
John B. Willett
Affiliation:
Stanford University
*
Requests for reprints shoulc be sent to David Rogosa, School of Education, Stanford University, Stanford CA 94305.
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Abstract

The study of correlates of change is the investigation of systematic individual differences in growth. Our representation of systematic individual differences in growth is built up in two parts: (a) a model for individual growth and, (b) a model for the dependence of parameters in the individual growth models on individual characteristics. First, explicit representations of correlates of change are constructed for various models of individual growth. Second, for the special case of initial status as a correlate of change, properties of collections of growth curves provide new results on the relation between change and initial status. Third, the shortcomings of previous approaches to the assessment of correlates of change are demonstrated. In particular, correlations of residual change measures with exogenous individual characteristics are shown to be poor indicators of systematic individual differences in growth.

Keywords

longitudinal analysismeasurement of change
Type
Original Paper
Information
Psychometrika , Volume 50 , Issue 2 , June 1985 , pp. 203 - 228
Copyright
Copyright © 1985 The Psychometric Society

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Footnotes

The research reported here was supported by a grant from The Spencer Foundation to the senior author and by the Study of Stanford and the Schools.

References

Anderson, J. E. (1939). The limitations of infant and preschool tests in the measurement of intelligence. Journal of Psychology, 8, 351–379.CrossRefGoogle Scholar
Anderson, T. W. (1960). Some stochastic process models for intelligence test scores. In Arrow, K. J., Karlin, S., & Suppes, P. (Eds.), Mathematical methods in the social sciences, Stanford, CA: Stanford University Press.Google Scholar
Bayley, N. (1949). Consistency and variability in the growth of intelligence from birth to eighteen years. Journal of Genetic Psychology, 75, 165–196.Google Scholar
Bayley, N. (1956). Individual patterns of development. Child Development, 27, 45–74.Google ScholarPubMed
Bayley, N. (1968). Behavioral correlates of mental growth: Birth to thirty-six years. American Psychologist, 23, 1–17.CrossRefGoogle Scholar
Berliner, D. C. (1976). Impediments to the study of teacher effectiveness. Journal of Teacher Education, 27, 5–13.CrossRefGoogle Scholar
Blomqvist, N. (1977). On the relation between change and initial value. Journal of the American Statistical Association, 72, 746–749.Google Scholar
Bloom, B. S. (1964). Stability and change in human characteristics, New York: Wiley.Google Scholar
Bohrnstedt, G. W. (1969). Observations on the measurement of change. In Borgatta, E. F. (Eds.), Sociological methodology 1969, San Francisco: Jossey-Bass.Google Scholar
Borich, G. D. (1979). Implications for developing teacher competencies from process-product research. Journal of Teacher Education, 30, 77–86.CrossRefGoogle Scholar
Bryk, A. S. (1977). An investigation of the effectiveness of alternative adjustment strategies in the analysis of quasi-experimental growth data, Cambridge: Harvard University, Graduate School of Education.Google Scholar
Bryk, A. S., & Weisberg, H. I. (1977). Use of the nonequivalent control group design when subjects are growing. Psychological Bulletin, 84, 950–962.CrossRefGoogle Scholar
Campbell, M. E. (1981). Statistical procedures with the law of initial values. Journal of Psychology, 108, 85–101.CrossRefGoogle Scholar
Coleman, J. S. (1968). The mathematical study of change. In Blalock, H. M., & Blalock, A. B. (Eds.), Methodology in social research, New York: McGraw-Hill.Google Scholar
Cook, T. D., & Campbell, D. T. (1979). Quasi-experimentation: Design and analysis issues for field settings, Skokie, IL: Rand McNally.Google Scholar
Cronbach, L. J., & Furby, L. (1970). How should we measure “change”—or should we?. Psychological Bulletin, 74, 68–80.CrossRefGoogle Scholar
Cronbach, L. J., & Snow, R. E. (1977). Aptitudes and instructional methods: A handbook for research on interactions, New York: Irvington.Google Scholar
Day, N. E. (1966). Fitting curves to longitudinal data. Biometrics, 22, 276–291.CrossRefGoogle ScholarPubMed
Fisk, P. R. (1967). Models of the second kind in regression analysis. Journal of the Royal Statistical Society (B), 29, 266–281.CrossRefGoogle Scholar
Furby, L. (1973). Interpreting regression toward the mean in developmental research. Developmental Psychology, 8, 172–179.CrossRefGoogle Scholar
Garside, R. F. (1956). The regression of gains upon initial scores. Psychometrika, 21, 67–77.CrossRefGoogle Scholar
Goldstein, H. (1979). The design and analysis of longitudinal studies, London: Academic Press.Google Scholar
Guire, K. E., & Kowalski, C. J. (1979). Mathematical description and representation of developmental change functions on the intra- and inter-individual levels. In Nesselroade, J. R., & Baltes, P. B. (Eds.), Longitudinal research in the study of behavior and development, New York: Academic Press.Google Scholar
Gulliksen, H. (1934). A rational equation of the learning curve based on Thorndike's law of effect. Journal of General Psychology, 11, 395–434.CrossRefGoogle Scholar
Guttman, R., & Guttman, L. (1965). A new approach to the analysis of growth patterns: The simplex structure of intercorrelations of measurements. Growth, 29, 219–232.Google Scholar
Harnquist, K. (1968). Relative changes in intelligence from 13 to 18. Scandinavian Journal of Psychology, 9, 50–82.CrossRefGoogle Scholar
Healy, M. J. R., & Goldstein, H. (1978). Regression to the mean. Annals of Human Biology, 5, 277–280.CrossRefGoogle ScholarPubMed
Hicklin, W. J. (1976). A model for mastery learning based on dynamic equilibrium theory. Journal of Mathematical Psychology, 13, 79–88.CrossRefGoogle Scholar
Hilgard, E. R. (1951). Methods and procedures in the study of learning. In Stevens, S. S. (Eds.), Handbook of Experimental Psychology, New York: Wiley.Google Scholar
Hui, S. L., & Berger, J. O. (1983). Empirical Bayes estimation of rates in longitudinal studies. Journal of the American Statistical Association, 78, 753–760.CrossRefGoogle Scholar
Hull, C. L. (1943). Principles of behavior, New York: Appleton-Century-Crofts.Google Scholar
Humphreys, L. G. (1960). Investigations of the simplex. Psychometrika, 25, 313–323.CrossRefGoogle Scholar
Jöreskog, K. G. (1970). Estimation and testing of simplex models. British Journal of Mathematical and Statistical Psychology, 23, 121–145.CrossRefGoogle Scholar
Keats, J. A. (1983). Ability measures and theories of cognitive development. In Wainer, H., Messick, S. (Eds.), Principles of modern psychological measurement: A festschrift for Frederick M. Lord, Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
Kessler, R. C. (1977). The use of change scores as criteria in longitudinal survey research. Quality and Quantity, 11, 43–66.CrossRefGoogle Scholar
Kessler, R. C., Greenberg, D. F. (1981). Linear panel analysis, New York: Academic Press.Google Scholar
Lacey, J. I. (1956). The evaluation of automatic responses: Toward a general solution. Annals of the New York Academy of Sciences, 67, 123–164.CrossRefGoogle Scholar
Lacey, J. I., & Lacey, B. C. (1962). The law of initial value in the longitudinal study of autonomic constitution: Reproducibility of autonomic responses and response patterns over a four year interval. Annals of the New York Academy of Sciences, 98, 1257–1290.CrossRefGoogle Scholar
Lewis, D. (1960). Quantitative methods in psychology, New York: McGraw-Hill.CrossRefGoogle Scholar
Linn, R. L., & Slinde, J. A. (1977). The determination of the significance of change between pre- and posttesting periods. Review of Educational Research, 47, 121–150.CrossRefGoogle Scholar
Lord, F. M. (1958). Further problems in the measurement of growth. Educational and Psychological Measurement, 18, 437–454.CrossRefGoogle Scholar
Lord, F. M. (1963). Elementary models for measuring change. In Harris, C. W. (Eds.), Problems in measuring change, Madison, WI: University of Wisconsin Press.Google Scholar
McCall, R. B., Appelbaum, M. I., & Hogarty, P. S. (1973). Developmental changes in mental performance. Monographs of the Society for Research in Child Development, 38 (No. 150).CrossRefGoogle ScholarPubMed
Medley, D. M. (1979). The effectiveness of teachers. In Peterson, P. L., & Walberg, H. J. (Eds.), Research on teaching: Concepts, findings and interpretations, Berkeley: McCutchan.Google Scholar
Messick, S. (1981). Denoting the base-free measure of change. Psychometrika, 46, 215–217.CrossRefGoogle Scholar
Nesselroade, J. R., Stigler, S. M., & Baltes, P. B. (1980). Regression toward the mean and the study of change. Psychological Bulletin, 88, 622–637.CrossRefGoogle Scholar
Nielsen, F., Rosenfeld, R. A. (1981). Substantive interpretations of differential equation models. American Sociological Review, 46, 159–174.CrossRefGoogle Scholar
O'Connor, E. F. (1972). Extending classical test theory to the measurement of change. Review of Educational Research, 42, 73–98.CrossRefGoogle Scholar
Preece, P. F. W. (1982). The fan-spread hypothesis and the adjustment for initial differences between groups in uncontrolled studies. Educational and Psychological Measurement, 42, 759–762.CrossRefGoogle Scholar
Rao, C. R. (1958). Some statistical methods for comparison of growth curves. Biometrics, 14, 1–17.CrossRefGoogle Scholar
Rao, C. R. (1965). The theory of least squares when the parameters are stochastic and its application to the analysis of growth curves. Biometrics, 52, 447–458.CrossRefGoogle Scholar
Rees, A. H., & Palmer, F. H. (1970). Factors related to change in mental test performance. Developmental Psychology Monographs, 3 (Part 2).CrossRefGoogle Scholar
Richards, F. J. (1959). A flexible growth function for empirical use. Journal of Experimental Botany, 10, 290–300.CrossRefGoogle Scholar
Roff, M. (1941). A statistical study of the development of intelligence test performance. Journal of Psychology, 11, 371–386.CrossRefGoogle Scholar
Rogosa, D. R., Brandt, D., & Zimowski, M. (1982). A growth curve approach to the measurement of change. Psychological Bulletin, 90, 726–748.CrossRefGoogle Scholar
Sagiv, A. (1979). General growth model for evaluation of an individual's progress in learning. Journal of Educational Psychology, 71, 866–881.CrossRefGoogle Scholar
Salemi, M. K., & Tauchen, G. E. (1982). Estimation of nonlinear learning models. Journal of the American Statistical Association, 77, 725–731.CrossRefGoogle Scholar
Sandland, R. L., & McGilchrist, C. A. (1979). Stochastic growth curve analysis. Biometrics, 35, 255–271.CrossRefGoogle Scholar
Seigel, D. G. (1975). Several approaches for measuring average rates of change for a second degree polynomial. The American Statistician, 29, 36–37.CrossRefGoogle Scholar
Soar, R. S. (1977). An integration of findings from four studies of teacher effectiveness. In Borich, G. D. (Eds.), The appraisal of teaching: Concepts and process, Reading, MA: Addison-Wesley.Google Scholar
Soar, R. S. (1979). Teacher behavior related to pupil growth. In Bennett, N., & McNamara, D. (Eds.), Focus on teaching, New York: Longman.Google Scholar
Sörbom, D. (1976). A statistical model for the measurement of change. In de Gruijter, D. N. M., & van der Kamp, L. J. T. (Eds.), Advances in psychological and educational measurement, New York: Wiley.Google Scholar
Sørensen, A. B., & Hallinan, M. T. (1977). A reconceptualization of school effects. Sociology of Education, 50, 273–289.CrossRefGoogle Scholar
Stake, R. E. (1961). Learning parameters, aptitudes and achievements. Psychometric Monograph, (No. 9).Google Scholar
Swamy, P. A. V. B. (1971). Statistical inference in a random coefficient regression model, New York: Springer-Verlag.CrossRefGoogle Scholar
Swamy, P. A. V. B. (1974). Linear models with random coefficients. In Zarembka, P. (Eds.), Frontiers in econometrics, New York: Academic Press.Google Scholar
Thomson, G. H. (1924). A formula to correct for the effect of errors of measurement on the correlation of initial values with gains. Journal of Experimental Psychology, 7, 321–324.CrossRefGoogle Scholar
Thorndike, R. L. (1966). Intellectual status and intellectual growth. Journal of Educational Psychology, 57, 121–127.CrossRefGoogle ScholarPubMed
Thurstone, L. L., & Ackerson, L. (1929). The mental growth curve for the Binet tests. Journal of Educational Psychology, 20, 569–583.CrossRefGoogle Scholar
Tucker, L. R. (1979). Comment on a note on a base-free measure of change. Psychometrika, 44, 357–357.CrossRefGoogle Scholar
Tucker, L. R., Damarin, F., & Messick, S. (1966). A base-free measure of change. Psychometrika, 31, 457–473.CrossRefGoogle ScholarPubMed
Tuma, N. B., & Hannan, M. T. (1984). Social dynamics: Models and Methods, New York: Academic Press.Google Scholar
Veldman, D., Brophy, J. (1974). Measuring teacher effects on pupil achievement. Journal of Educational Psychology, 66, 319–324.CrossRefGoogle Scholar
Walberg, H. J., & Tsai, S. L. (1983). Matthew effects in education. American Educational Research Journal, 20, 359–373.Google Scholar
Werts, C. E., & Hilton, T. L. (1977). Intellectual status and intellectual growth, again. American Educational Research Journal, 14, 137–146.CrossRefGoogle Scholar
Werts, C. E., & Linn, R. L. (1970). A general linear model for studying growth. Psychological Bulletin, 73, 17–22.CrossRefGoogle Scholar
Werts, C. E., Linn, R. L., & Jöreskog, K. G. (1977). A simplex model for analyzing academic growth. Educational and Psychological Measurement, 37, 745–756.CrossRefGoogle Scholar
Wiley, D. E., & Harnischfeger, A. (1973). Post hoc, ergo propter hoc: Problems in the attribution of change. Studies of Educative Processes (No. 7), Chicago: University of Chicago.Google Scholar
Wilder, J. (1957). The law of initial value in neurology and psychiatry. Journal of Nervous and Mental Disease, 125, 73–86.CrossRefGoogle ScholarPubMed
Wilder, J. (1965). Pitfalls in the methodology of the law of initial value. American Journal of Psychotherapy, 19, 577–584.CrossRefGoogle ScholarPubMed
Wishart, J. (1938). Growth rate determinations in nutrition studies with the bacon pig and their analysis. Biometrika, 30, 16–28.CrossRefGoogle Scholar
Wohlwill, J. F. (1980). Cognitive development in childhood. In Brim, O. G. Jr. & Kagan, J. (Eds.), Constancy and change in human development, Cambridge, MA: Harvard University Press.Google Scholar
Woodrow, H. (1940). Interrelations of measures of learning. Journal of Psychology, 10, 49–73.CrossRefGoogle Scholar
Woodrow, H. (1946). The ability to learn. Psychological Review, 53, 147–158.CrossRefGoogle ScholarPubMed