The growth curve model: a review

A survey is given of papers which have influenced or have been influenced by the Growth Curve Model due to Potthoff & Roy (1964). The review covers, among others, methods of estimating parameters, the canonical version of the model, tests, extensions, incomplete data, Bayesian approaches and covariance structures.

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[5]  T. W. Anderson.  Asymptotically Efficient Estimation of Covariance Matrices with Linear Structure , 1973 .

[6]  Nan M. Laird,et al.  The Effect of Covariance Structure on Variance Estimation in Balanced Growth-Curve Models with Random Parameters , 1989 .

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[13]  Criteria for estimability in multivariate linear models , 1976 .

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[15]  A remark on a paper by misra , 1980 .

[16]  M. Kenward.  The use of fitted higher-order polynomial coefficients as covariates in the analysis of growth curves. , 1985, Biometrics.

[17]  T. Kariya.  Testing in the multivariate general linear model , 1985 .

[18]  K. Jöreskog.  Analysis of Covariance Structures , 1988 .

[19]  R. V. Foutz,et al.  Multiple comparisons in the randomization analysis of designed experiments with growth curve responses. , 1985, Biometrics.

[20]  Christine Waternaux,et al.  Methods for Analysis of Longitudinal Data: Blood-Lead Concentrations and Cognitive Development , 1989 .

[21]  T. Fearn.  A two-stage model for growth curves which leads to Rao's covariance adjusted estimators , 1977 .

[22]  D. Kleinbaum.  A generalization of the growth curve model which allows missing data , 1973 .

[23]  Helga Bunke,et al.  Statistical inference in linear models , 1987 .

[24]  Calyampudi Radhakrishna Rao,et al.  Linear Statistical Inference and its Applications , 1967 .

[25]  K. Jöreskog.  A general method for analysis of covariance structures , 1970 .

[26]  Tom Fearn,et al.  A Bayesian approach to growth curves , 1975 .

[27]  Takeaki Kariya,et al.  The General MANOVA Problem , 1978 .

[28]  C J Kowalski,et al.  Longitudinal data analysis. , 1974, Growth.

[29]  Hypothesis testing for two-group incomplete-data growth curve problems , 1987 .

[30]  D. C. Bowden,et al.  Tolerance Bands for Growth Curves , 1973 .

[31]  T. Kubokawa.  Two-stage procedures for parameters in a growth curve model , 1989 .

[32]  Erkki P. Liski.  Missing Data Problems in Growth Curves , 1984 .

[33]  J. Koziol,et al.  A distribution-free test for tumor-growth curve analyses with application to an animal tumor immunotherapy experiment. , 1981, Biometrics.

[34]  C. G. Khatri,et al.  A note on a manova model applied to problems in growth curve , 1966 .

[35]  A condition for null robustness , 1984 .

[36]  I. Hudson.  ASYMPTOTIC TESTS FOR GROWTH CURVE MODELS WITH AUTOREGRESSIVE ERRORS , 1983 .

[37]  W. Tan.  On testing correlation between growth curves , 1976 .

[38]  E. Vonesh,et al.  Efficient inference for random-coefficient growth curve models with unbalanced data. , 1987, Biometrics.

[39]  G. Zerbe,et al.  On Application of Growth Curve Techniques to Time Series Data , 1980 .

[40]  S. Lundbye-Christensen,et al.  A multivariate growth curve model for pregnancy. , 1991, Biometrics.

[41]  R. Helms,et al.  ML estimation for the multivariate normal distribution with general linear model mea1 and linear-structure covariance matrix; one-population, complete-data case , 1986 .

[42]  Discriminant functions from the dummy variable space for growth curve models , 1983 .

[43]  Calyampudi R. Rao,et al.  The theory of least squares when the parameters are stochastic and its application to the analysis of growth curves. , 1965, Biometrika.

[44]  P. Kulkarni.  Large sample estimation and prediction for explosive growth curve models , 1987 .

[45]  Calyampudi R. Rao,et al.  Some observations on multivariate statistical methods in anthropological research , 1961 .

[46]  A. Verbyla.  Analysis of repeated measures designs with changing covariates , 1988 .

[47]  Neil C. Schwertman,et al.  On the analysis of incomplete growth curve data, a Monte Carlo study of two nonparametric procedures , 1981 .

[48]  G. Reinsel,et al.  Models for Longitudinal Data with Random Effects and AR(1) Errors , 1989 .

[49]  T. Kariya,et al.  Nonnull and Optimality Robustness of Some Tests , 1985 .

[50]  H Swaminathan,et al.  Analysis Of Quasi-Experimental Time-Series Designs. , 1977, Multivariate behavioral research.

[51]  J. Berger,et al.  Empirical Bayes Estimation of Rates in Longitudinal Studies , 1983 .

[52]  R. C. Elston,et al.  203. Note: On Estimating Time-Response Curves , 1964 .

[53]  Ragnar Norberg,et al.  Hierarchical credibility: analysis of a random effect linear model with nested classification , 1986 .

[54]  S. Walker,et al.  A randomization test for comparison of groups of growth curves with different polynomial design matrices. , 1977, Biometrics.

[55]  Some Results for the Normal Manova Linear Functional Relationships Model. , 1983, Biometrical journal. Biometrische Zeitschrift.

[56]  Jack C. Lee.  Prediction and estimation of growth curves with special covariance structures , 1988 .

[57]  Erkki P. Liski,et al.  An APL-System for Analysing Multivariate Data , 1986 .

[58]  M. L. Eaton,et al.  MULTIVARIATE TESTS WITH INCOMPLETE DATA , 1983 .

[59]  E. Liski.  Estimation from incomplete data in growth curves models , 1985 .

[60]  W. H. Carter,et al.  A likelihood ratio test for a patterned covariance matrix in a multivariate growth-curve model. , 1984, Biometrics.

[61]  Y. Fujikoshi.  An Error Bound for an Asymptotic Expansion of the Distribution Function of an Estimate in a Multivariate Linear Model , 1985 .

[62]  Gregory C. Reinsel,et al.  Mean Squared Error Properties of Empirical Bayes Estimators in a Multivariate Random Effects General Linear Model , 1985 .

[63]  G. Koch,et al.  The Equivalence of Parameter Estimates from Growth Curve Models and Seemingly Unrelated Regression Models , 1985 .

[64]  G. Reinsel,et al.  Multivariate Repeated-Measurement or Growth Curve Models with Multivariate Random-Effects Covariance Structure , 1982 .

[65]  R. Woolson,et al.  Estimation of Growth Norms from Incomplete Longitudinal Data , 1987 .

[66]  Conditioning in the growth curve model , 1986 .

[67]  Karl G. Jöreskog,et al.  Statistical estimation of structural models in longitudinal-developmental investigations , 1979 .

[68]  I. Dunsmore.  Growth Curves in Two‐Period Change Over Models , 1981 .

[69]  G. Zerbe,et al.  On multiple comparisons in the randomization analysis of growth and response curves. , 1986, Biometrics.

[70]  N. Sugiura,et al.  Estimating common parameters of growth curve models , 1988 .

[71]  R. Muirhead.  Aspects of Multivariate Statistical Theory , 1982, Wiley Series in Probability and Statistics.

[72]  R. Gnanadesikan,et al.  Multivariate Analysis of Variance (MANOVA) , 1962 .

[73]  C. G. Khatri,et al.  On Certain Inequalities for Normal Distributions and their Applications to Simultaneous Confidence Bounds , 1967 .

[74]  S. Hui,et al.  Curve fitting for repeated measurements made at irregular time-points. , 1984, Biometrics.

[75]  K. R. Shah,et al.  On the unbiased estimation of fixed effects in a mixed model for growth curves , 1981 .

[76]  J. Ware,et al.  Random-effects models for longitudinal data. , 1982, Biometrics.

[77]  C. R. Rao,et al.  Some statistical methods for comparison of growth curves. , 1958 .

[78]  T. Fearn,et al.  The Chatham Blood Pressure Study. An application of Bayesian growth curve models to a longitudinal study of blood pressure in children. , 1979, International journal of epidemiology.

[79]  S. Hui,et al.  Multivariate slope ratio assay with repeated measurements. , 1985, Biometrics.

[80]  J. Koziol,et al.  Score and Wald tests for the multivariate growth curve model with missing data , 1988 .

[81]  John Wishart,et al.  GROWTH-RATE DETERMINATIONS IN NUTRITION STUDIES WITH THE BACON PIG, AND THEIR ANALYSIS , 1938 .

[82]  G. Box,et al.  Problems in the analysis of growth and wear curves. , 1950, Biometrics.

[83]  Dietrich Von Resen.  Moments of maximum likelihood estimators in the growth curve model , 1991 .

[84]  M-methods in growth curve analysis , 1986 .

[85]  C. Khatri.  Some Remarks on the Spherical Distributions and Linear Models , 1985 .

[86]  D. Rosen.  Maximum likelihood estimators in multivariate linear normal models , 1989 .

[87]  Yasunori Fujikoshi,et al.  On the asymptotic non-null distributions of the LR criterion in a general manova , 1974 .

[88]  T. W. Anderson,et al.  Maximum-likelihood estimation of the parameters of a multivariate normal distribution☆ , 1985 .

[89]  M. Hills.  A note on the analysis of growth curves. , 1968, Biometrics.

[90]  N. Laird,et al.  An algorithmic approach for the fitting of a general mixed ANOVA model appropriate in longitudinal settings , 1986 .

[91]  D. Rosen.  Moments for the inverted Wishart distribution , 1988 .

[92]  C. Khatri.  Some asymptotic inferential problems connected with elliptical distributions , 1988 .

[93]  D. F. Morrison,et al.  Multivariate Statistical Methods , 1968 .

[94]  B. Zehnwirth.  Hachemeister's Bayesian regression model revisited☆ , 1983 .

[95]  R. Nishii,et al.  ASYMPTOTIC COMPARISON OF THREE CONFIDENCE REGIONS IN ANALYSIS OF GROWTH CURVES , 1983 .

[96]  Jane F. Pendergast,et al.  Robust estimation in growth curve models , 1985 .

[97]  L. Erickson,et al.  Estimating and Testing Common Parameters for Some Multiresponse Models Associated With Microbial Growth and Bioenergetics , 1984 .

[98]  N. Laird,et al.  Maximum likelihood computations with repeated measures: application of the EM algorithm , 1987 .

[99]  On a gmanova model likelihood ratio test criterion , 1986 .

[100]  B. Sundt.  A hierarchical credibility regression model , 1979 .

[101]  R. C. Elston,et al.  Estimation of Time-Response Curves and Their Confidence Bands , 1962 .

[102]  Peter M. Hooper.  Sufficiency and Invariance in Confidence Set Estimation , 1982 .

[103]  L E Erickson,et al.  Maximum likelihood estimation of growth yields , 1982, Biotechnology and bioengineering.

[104]  M. Srivastava.  Multivariate data with missing observations , 1985 .

[105]  D. Rubin,et al.  Statistical Analysis with Missing Data. , 1989 .

[106]  Ted H. Szatrowski,et al.  Necessary and Sufficient Conditions for Explicit Solutions in the Multivariate Normal Estimation Problem for Patterned Means and Covariances , 1980 .

[107]  P. Sen,et al.  Nonparametric Methods in General Linear Models. , 1986 .

[108]  J. Koziol.  An introduction to multivariate growth curve analyses. , 1986, Growth.

[109]  Neil C. Schwertman,et al.  A Successive Differences Method for Growth Curves with Missing Data and Random Observation Times , 1986 .

[110]  T. Szatrowski.  Asymptotic Nonnull Distributions for Likelihood Ratio Statistics in the Multivariate Normal Patterned Mean and Covariance Matrix Testing Problem , 1979 .

[111]  R. Jennrich,et al.  Unbalanced repeated-measures models with structured covariance matrices. , 1986, Biometrics.

[112]  C. A. McGilchrist,et al.  Stochastic Growth Curve Analysis , 1979 .

[113]  D. M. Allen,et al.  Analysis of growth and dose response curves. , 1969, Biometrics.

[114]  G. Koch,et al.  Some Views on Parametric and Non-Parametric Analysis for Repeated Measurements and Selected Bibliography , 1980 .

[115]  Yasunori Fujikoshi,et al.  Asymptotic expansions of the non-null distributions of three statistics in GMANOVA , 1974 .

[116]  H. Weisberg,et al.  Empirical Bayes estimation of individual growth-curve parameters and their relationship to covariates. , 1983, Biometrics.

[117]  Muni S. Srivastava,et al.  An introduction to applied multivariate statistics , 1984 .

[118]  R. Potthoff,et al.  A generalized multivariate analysis of variance model useful especially for growth curve problems , 1964 .

[119]  Yasunori Fujikoshi,et al.  The likelihood ratio tests for the dimensionality of regression coefficients , 1974 .

[120]  W. N. Venables,et al.  An extension of the growth curve model , 1988 .

[121]  T. Caliński,et al.  A NOTE ON THE ANALYSIS OF COVARIANCE: EFFICIENCY OF CONCOMITANT VARIABLES , 1989 .

[122]  D. G. Kabe.  Some results for the GMANOVA model , 1975 .

[123]  Y. Fujikoshi.  Error bounds for asymptotic expansions of the distribution of the MLE in a GMANOVA model , 1987 .

[124]  C.G Khatri.  Testing some covariance structures under a growth curve model , 1973 .

[125]  Sums of squares and products matrices for a non-full ranks hypothesis in the model of potthoff and roy , 1977 .

[126]  M. Berger.  A Monte Carlo study of the power of alternative tests under the generalized Manova model , 1986 .

[127]  I. Olkin,et al.  Estimation for a regression model with an unknown covariance matrix , 1972 .

[128]  Gregory C. Reinsel,et al.  Estimation and Prediction in a Multivariate Random Effects Generalized Linear Model , 1984 .

[129]  Calyampudi R. Rao.  SOME PROBLEMS INVOLVING LINEAR HYPOTHESES IN MULTIVARIATE ANALYSIS , 1959 .

[130]  D. Bates,et al.  Newton-Raphson and EM Algorithms for Linear Mixed-Effects Models for Repeated-Measures Data , 1988 .

[131]  K. S. Madsen.  A Growth Curve Model for Studies in Morphometrics , 1977 .

[132]  Peter M. Hooper.  Simultaneous Estimation and Prediction Using the Expected Coverage Measure Criterion , 1988 .

[133]  Ronald W. Helms,et al.  Maximum likelihood estimates in the multivariate normal with patterned mean and covariance via the em algorithm , 1984 .

[134]  C. R. Rao,et al.  Least squares theory using an estimated dispersion matrix and its application to measurement of signals , 1967 .

[135]  Effects of the estimation of covariance matrix parameters in the generalized multivariate linear model , 1984 .

[136]  Y. Lee.  A note on Rao's reduction of Potthoff & Roy's generalized linear model , 1974 .

[137]  Y. Fujikoshi,et al.  Modern Multivariate Statistical Analysis: A Graduate Course and Handbook , 1987 .

[138]  A note on the analysis of the manova model and its application to growth curves , 1975 .

[139]  W. Flynn,et al.  A monte carlo study of alternative procedures for testing the hypothesis of parallelism for complete and incomplete growth curve data , 1985 .

[140]  Peter M. Hooper.  Simultaneous Interval Estimation in the General Multivariate Analysis of Variance Model , 1983 .

[141]  Ingram Olkin,et al.  Unbiasedness of Invariant Tests for Manova and Other Multivariate Problems , 1980 .

[142]  D. M. Allen,et al.  Computation of the estimated parameters and wald statistic for the generalized growth curve model , 1980 .

[143]  Gary O. Zerbe,et al.  Randomization Analysis of the Completely Randomized Design Extended to Growth and Response Curves , 1979 .

[144]  Line-segment confidence bands for repeated measures. , 1987, Biometrics.

[145]  Moments for a multivariate linear model with an application to the growth curve model , 1990 .

[146]  T. P. Burnaby.  Growth-Invariant Discriminant Functions and Generalized Distances , 1966 .

[147]  S. Chakravorti.  MAXIMUM LIKELIHOOD ESTIMATION FOR THE GROWTH CURVE MODEL WITH UNEQUAL DISPERSION MATRICES , 1975 .

[148]  Calyampudi R. Rao.  Recent trends of research work in multivariate analysis , 1972 .

[149]  Robert F. Woolson,et al.  Analysis of Incomplete Data from Longitudinal and Mixed Longitudinal Studies , 1978 .

[150]  A Generalization of the GMANOVA‐Model , 1987 .

[151]  On some tests in manocova model with different dispersion matrices , 1980 .

[152]  Optimal confidence regions in GMANOVA , 1986 .

[153]  M. Kenward.  The distribution of a generalized least squares estimator with covariance adjustment , 1986 .

[154]  Robert F. Woolson,et al.  Growth curve analysis of complete and incomplete longitudinal data , 1980 .

[155]  J. Ware.  Linear Models for the Analysis of Longitudinal Studies , 1985 .

[156]  S. Chakravorti.  On asymptotic properties of the maximum likelihood estimates of the general growth curve model , 1976 .

[157]  The greatest invariance-group of multivariate models , 1986 .

[158]  Peter M. Hooper.  Invariant Confidence Sets with Smallest Expected Measure , 1982 .

[159]  Generalized manova double linear hypothesis with double linear restrictions , 1975 .

[160]  E. Stanek,et al.  Growth curve models of repeated binary response. , 1988, Biometrics.

[161]  O B Allen,et al.  A guide to the analysis of growth curve data with special reference to SAS. , 1983, Computers and biomedical research, an international journal.

[162]  C. Lewis,et al.  ESTIMATION AND MODEL COMPARISONS FOR REPEATED MEASURES DATA , 1984 .

[163]  C R Rao,et al.  Simultaneous estimation of parameters in different linear models and applications to biometric problems. , 1975, Biometrics.

[164]  C. G. Khatri,et al.  Robustness study for a linear growth model , 1988 .

[165]  A. Kshirsagar,et al.  DIRECTION AND COLLINEARITY FACTORS OF WILK'S A ASSOCIATED WITH THE GROWTH CURVE MODEL , 1983 .

[166]  E. M. Carter,et al.  A growth-curve model approach to multivariate quantal bioassay , 1984 .

[167]  J. Miller,et al.  Explicit Maximum Likelihood Estimates from Balanced Data in the Mixed Model of the Analysis of Variance , 1980 .

[168]  T. Louis,et al.  General methods for analysing repeated measures. , 1988, Statistics in medicine.

[169]  Malay Ghosh,et al.  Nonparametric Methods in Longitudinal Studies , 1973 .

[170]  S. Chakravorti.  On some tests of growth curve model under Behrens-Fisher stituation , 1974 .

[171]  L.C.A. Corsten,et al.  Reconciliation of two different views on estimation of growth curve parameters , 1978 .

[172]  N. H. Timm.  2 Multivariate analysis of variance of repeated measurements , 1980 .

[173]  Takeaki Kariya,et al.  Robustness of Multivariate Tests , 1981 .

[174]  On the degree of polynomial in a general linear model , 1984 .

[175]  Arjun K. Gupta,et al.  Exact distribution of certain general test statistics in multivariate analysis , 1986 .

[176]  R. Woolson,et al.  Testing hypotheses for the growth curve model when the data are incomplete , 1982 .

[177]  Philip J. Byrne.  Multivariate analysis with a stationary time series covariance structure , 1988 .

[178]  Admissibility of Invariant Tests in the General Multivariate Analysis of Variance Problem , 1983 .

[179]  Neil C. Schwertman,et al.  Smoothing an indefinite variance-covariance matrix , 1979 .