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Keywords:
growth curve model; extended growth curve model; multivariate linear model
growth curve model; extended growth curve model; multivariate linear model
Summary:
The Extended Growth Curve Model (ECGM) is a multivariate linear model connecting different multivariate regression models in sample subgroups through common variance matrix. It has the form: \[ Y=\sum ^{k}_{i=1}X_iB_iZ_i^{\prime }+e, \quad \operatorname{vec}(e)\sim N_{n\times p}\left(0,\Sigma \otimes I_n\right). \] Here, matrices $X_i$ contain subgroup division indicators, and $Z_i$ corresponding regressors. If $k=1$, we speak about (ordinary) Growth Curve Model. The model has already its age (it dates back to 1964), but it has many important applications. That is why it is still intensively studied. Many articles investigating different aspects or special cases of the model appeared in recent years. We will try to summarize the progress done so far.
References:
[1] Åsenblad, N., Rosen, D. von: Analysis of cross-over designs via growth curve models. Journal of Statistical Planning and Inference 136 (2006), 475–497. DOI 10.1016/j.jspi.2004.06.061 | MR 2211351
[2] Bhattacharya, S., Basu, A., Bandyopadhyay, S.: Goodness-of-fit testing for exponential polynomial growth curves. Communications in Statistics – Theory and Methods 38 (2009), 340–363. MR 2510788 | Zbl 1159.62011
[3] Bochniak, A., Wesołowska-Janczarek, M.: On influence of variability in concomitant variables values on estimation of polynomial coefficients in growth curves models with concomitant variables changing in time and the same values for all experimental units. Colloquium Biometricum 40 (2010), 135–145.
[4] Fang, K. T., Wang, S. G., Rosen, D. von: Restricted expected multivariate least squares. Journal of Multivariate Analysis 97 (2006), 619–632. DOI 10.1016/j.jmva.2005.03.016 | MR 2236493
[5] Fujikoshi, Y., Rosen, D. von: LR tests for random-coefficient covariance structures in an extended growth curve model. Journal of Multivariate Analysis 75 (2000), 245–268. DOI 10.1006/jmva.2000.1907 | MR 1802550
[6] Hamid, J. S., Beyene, J., Rosen, D. von: A novel trace test for the mean parameters in a multivariate growth curve model. Journal of Multivariate Analysis 102 (2011), 238–251. DOI 10.1016/j.jmva.2010.09.001 | MR 2739112
[7] Heinen, M.: Analytical growth equations and their Genstat 5 equivalents. Netherlands Journal of Agricultural Science 47 (1999), 67–89.
[8] Hu, J.: Properties of the explicit estimators in the Extended Growth Curve Model. Statistics 44, 5 (2009), 477–492. DOI 10.1080/02331880903236884 | MR 2739406
[9] Hu, J., Yan, G.: Asymptotic normality and consistency of a two-stage generalized least squares estimator in the growth curve model. Bernoulli 14, 3 (2008), 623–636. DOI 10.3150/08-BEJ128 | MR 2537805 | Zbl 1155.62014
[10] Kanda, T., Ohtaki, M., Fujikoshi, Y.: Simultaneous confidence regions in an Extended Growth Curve Model with $k$ hierarchical within-individuals design matrices. Communications in Statistics – Theory and Methods 31, 9 (2002), 1605–1616. DOI 10.1081/STA-120013015 | MR 1925084 | Zbl 1009.62043
[11] Klein, D., Žežula, I.: On uniform correlation structure. In: Mathematical Methods In Economics And Industry, conference proceedings, Herl’any, Slovakia (2007), 94–100.
[12] Klein, D., Žežula, I.: The maximum likelihood estimators in the growth curve model with serial covariance structure. Journal of Statistical Planning and Inference 139 (2009), 3270–3276. DOI 10.1016/j.jspi.2009.03.011 | MR 2535199
[13] Kollo, T., Roos, A., Rosen, D. von: Aproximation of the distribution of the location parameters in the Growth Curve Model. Scandinavian Journal of Statistics 34 (2007), 499–510. DOI 10.1111/j.1467-9469.2006.00546.x | MR 2368795
[14] Kollo, T., Rosen, D. von: Advanced multivariate statistics with matrices. Springer, Dordrecht, 2005. MR 2162145
[15] Kollo, T., Rosen, D. von: Distribution and density approximation of the covariance matrix in the growth curve model. Statistics 35, 1 (2000), 1–22. DOI 10.1080/02331880108802722 | MR 1820821
[16] Kubáček, L.: Multivariate models with constraints confidence regions. Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 47 (2008), 83–100. MR 2482719 | Zbl 1165.62043
[17] Lin, S. H., Lee, J. C.: Exact tests in simple growth curve models and one-way ANOVA with equicorrelation error structure. Journal of Multivariate Analysis 84 (2003), 351–368. DOI 10.1016/S0047-259X(02)00060-X | MR 1965227 | Zbl 1014.62067
[18] Mentz, G. B., Kshirsagar, A. M.: Sum of profiles model with exchangeably distributed errors. Communications in Statistics – Theory and Methods 32, 8 (2003), 1591–1605. DOI 10.1081/STA-120022246 | MR 1996796 | Zbl 1184.62090
[19] Nummi, T.: Analysis of growth curves under measurement errors. Journal of Applied Statistics 27, 2 (2000), 235–243. DOI 10.1080/02664760021763 | MR 1743460 | Zbl 0941.62066
[20] Nummi, T., Koskela, L.: Analysis of growth curve data by using cubic smoothing splines. Journal of Applied Statistics 35, 6 (2008), 681–691. DOI 10.1080/02664760801923964 | MR 2516865 | Zbl 1147.62051
[21] Nummi, T., Möttönen, J.: On the analysis of multivariate growth curves. Metrika 52 (2000), 77–89. DOI 10.1007/s001840000063 | MR 1791926
[22] Ohlson, M., Andrushchenko, Z., Rosen, D. von: Explicit estimators under m-dependence for a multivariate normal distribution. Annals of the Institute of Statistical Mathematics 63 (2011), 29–42. DOI 10.1007/s10463-008-0213-1 | MR 2748932
[23] Ohlson, M., Rosen, D. von: Explicit estimators of parameters in the Growth Curve Model with linearly structured covariance matrices. Journal of Multivariate Analysis 101 (2010), 1284–1295. DOI 10.1016/j.jmva.2009.12.023 | MR 2595308
[24] Pihlak, M.: Approximation of multivariate distribution functions. Mathematica Slovaca 58, 5 (2008), 635–652. DOI 10.2478/s12175-008-0099-7 | MR 2434683 | Zbl 1195.62008
[25] Potthoff, R. F., Roy, S. N.: A generalized multivariate analysis of variance model useful especially for growth curve problems. Biometrika 51, 3-4 (1964), 313–326. MR 0181062 | Zbl 0138.14306
[26] Rao Chaganty, N.: Analysis of growth curves with patterned correlation matrices using quasi-least squares. Journal of Statistical Planning and Inference 117 (2003), 123–139. DOI 10.1016/S0378-3758(02)00362-2 | MR 2001145
[27] Rao Chaganty, N.: An alternative approach to the analysis of longitudinal data via generalized estimating equations. Journal of Statistical Planning and Inference 63 (1997), 39–54. DOI 10.1016/S0378-3758(96)00203-0 | MR 1474184
[28] Reinsel, G. C., Velu, R. P.: Reduced-rank growth curve models. Journal of Statistical Planning and Inference 114 (2003), 107–129. DOI 10.1016/S0378-3758(02)00466-4 | MR 1980874 | Zbl 1011.62056
[29] Roy, A., Khattree, R.: On implementation of a test for Kronecker product covariance structure for multivariate repeated measures data. Statistical Methodology 2 (2005), 297–306. DOI 10.1016/j.stamet.2005.07.003 | MR 2205602 | Zbl 1248.62092
[30] Rusnačko, R.: The comparison of two estimators of variance parameters in a special growth curve model. Forum Statisticum Slovacum 6, 5 (2010), 204–209.
[31] Satoh, K., Ohtaki, M.: Nonparametric growth curve model with local linear approximation. Communications in Statistics – Theory and Methods 35, 4 (2006), 641–648. DOI 10.1080/03610920500498790 | MR 2256245 | Zbl 1093.62046
[32] Srivastava, M.: Nested Growth Curve Models. Sankhyā A 64, 2 (2002), 379–408. MR 1981765 | Zbl 1192.62156
[33] Srivastava, M., Rosen, T. von, Rosen, D. von: Models with a Kronecker product covariance structure: estimation and testing. Mathematical Methods of Statistics 17, 4 (2008), 357–370. DOI 10.3103/S1066530708040066 | MR 2483463
[34] Srivastava, M., Rosen, D. von: Regression models with unknown singular covariance matrix. Linear Algebra and its Applications 354 (2002), 255–273. MR 1927661
[35] Vasdekis, V. G. S.: A comparison of REML and covariance adjustment method in the estimation of growth curve models. Communications in Statistics — Theory and Methods 37, 20 (2008), 3287–3297. MR 2467767
[36] Wawrzosek, J., Wesołowska-Janczarek, M.: Testability and estimability in multivariate linear normal model with various restrictions. Communications in Statistics – Theory and Methods 38 (2009), 828–841. MR 2522531
[37] Wesołowska-Janczarek, M.: Selected models and methods of parameter estimation in growth curves with concomitant variables. Colloquium Biometricum 39 (2009), 21–31.
[38] Wesołowska-Janczarek, M., Kolczyńska, E.: Comparison of two estimation methods in growth curve model with concomitant variables. Colloquium Biometricum 38 (2008), 135–149.
[39] Wong, C. S., Cheng, H.: Estimation in a growth curve model with singular covariance. Journal of Statistical Planning and Inference 97 (2001), 323–342. DOI 10.1016/S0378-3758(00)00220-2 | MR 1861157 | Zbl 1015.62056
[40] Wu, Q. G.: Existence conditions of the uniformly minimum risk unbiased estimators in extended growth curve models. Journal of Statistical Planning and Inference 69 (1998), 101–114. DOI 10.1016/S0378-3758(97)00119-5 | MR 1631157 | Zbl 0924.62057
[41] Wu, Q. G.: Some results on parameter estimation in extended growth curve models. Journal of Statistical Planning and Inference 88 (2000), 285–300. DOI 10.1016/S0378-3758(00)00084-7 | MR 1792046 | Zbl 0951.62045
[42] Wu, X. Y., Liang, H., Zou, G. H.: Unbiased invariant least squares estimation in a generalized growth curve model. Sankhyā A 71, 1 (2009), 73–93. MR 2579649 | Zbl 1193.62100
[43] Wu, H., Zhang J. T.: Local polynomial mixed-effects models for longitudinal data. Journal of the American Statistical Association 97, 459 (2002), 883–897. MR 1941417 | Zbl 1048.62048
[44] Wu, X. Y., Zou, G. H., Chen, J. W.: Unbiased invariant minimum norm estimation in generalized growth curve model. Journal of Multivariate Analysis 97 (2006), 1718–1741. DOI 10.1016/j.jmva.2006.05.007 | MR 2298885 | Zbl 1112.62054
[45] Wu, X. Y., Zou, G. H., Li, Y. F.: Uniformly minimum variance nonnegative quadratic unbiased estimation in a generalized growth curve model. Journal of Multivariate Analysis 100 (2009), 1061–1072. DOI 10.1016/j.jmva.2008.10.007 | MR 2498732 | Zbl 1157.62035
[46] Xu, L., Stoica, P., Li, J.: A diagonal growth curve model and some signal-processing applications. IEEE Transactions on Signal Processing 54, 9 (2006), 3363–3371. DOI 10.1109/TSP.2006.879296
[47] Xu, L., Stoica, P., Li, J.: A block-diagonal growth curve model. Digital Signal Processing 16 (2006), 902–912. DOI 10.1016/j.dsp.2006.05.005
[48] Yang, G. Q., Wu, Q. G.: Existence conditions for the uniformly minimum risk unbiased estimators in a class of linear models. Journal of Multivariate Analysis 88 (2004), 76–88. DOI 10.1016/S0047-259X(03)00058-7 | MR 2021861 | Zbl 1032.62063
[49] Ye, R. D., Wang, S. G.: Estimating parameters in Extended Growth Curve Models with special covariance structures. Journal of Statistical Planning and Inference 139 (2009), 2746–2756. DOI 10.1016/j.jspi.2008.12.012 | MR 2523663 | Zbl 1162.62055
[50] Yokoyama, T.: Estimation in a random effects model with parallel polynomial growth curves. Hiroshima Mathematical Journal 31 (2001), 425–433. MR 1870985 | Zbl 0989.62034
[51] Žežula, I.: Special variance structures in the growth curve model. Journal of Multivariate Analysis 97 (2006), 606–618. DOI 10.1016/j.jmva.2005.10.001 | MR 2236492 | Zbl 1101.62042
[52] Žežula, I., Klein, D.: Orthogonal decompositions in growth curve models. Acta et Commentationes Universitatis Tartuensis de Mathematica 14 (2010), 35–44. MR 2816617 | Zbl 1228.62065
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