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**Analysis of growth curve data by using cubic smoothing splines.**
*(English)*
Zbl 1147.62051

Summary: Longitudinal data frequently arise in various fields of applied sciences where individuals are measured according to some ordered variable, e.g., time. A common approach used to model such data is based on the mixed models for repeated measures. This model provides an eminently flexible approach to modeling of a wide range of mean and covariance structures. However, such models are forced into a rigidly defined class of mathematical formulas which may not be well supported by the data within the whole sequence of observations.

A possible nonparametric alternative is a cubic smoothing spline, which is highly flexible and has useful smoothing properties. It can be shown that under normality assumptions, the solution of the penalized log-likelihood equation is the cubic smoothing spline, and this solution can be further expressed as a solution of a linear mixed model. It is shown here how cubic smoothing splines can be easily used in the analysis of complete and balanced data. The analysis can be greatly simplified by using the unweighted estimator studied in the paper. It is shown that if the covariance structure of the random errors belongs to a certain class of matrices, the unweighted estimator is the solution to the penalized log-likelihood function. This result is new in a smoothing spline context and it is not only confined to growth curve settings. The connection to mixed models is used in developing a rough testing of group profiles. Numerical examples are presented to illustrate the techniques proposed.

A possible nonparametric alternative is a cubic smoothing spline, which is highly flexible and has useful smoothing properties. It can be shown that under normality assumptions, the solution of the penalized log-likelihood equation is the cubic smoothing spline, and this solution can be further expressed as a solution of a linear mixed model. It is shown here how cubic smoothing splines can be easily used in the analysis of complete and balanced data. The analysis can be greatly simplified by using the unweighted estimator studied in the paper. It is shown that if the covariance structure of the random errors belongs to a certain class of matrices, the unweighted estimator is the solution to the penalized log-likelihood function. This result is new in a smoothing spline context and it is not only confined to growth curve settings. The connection to mixed models is used in developing a rough testing of group profiles. Numerical examples are presented to illustrate the techniques proposed.

### MSC:

62H12 | Estimation in multivariate analysis |

62G05 | Nonparametric estimation |

65C60 | Computational problems in statistics (MSC2010) |

62H15 | Hypothesis testing in multivariate analysis |

### Keywords:

covariance structures; longitudinal data; maximum likelihood; mixed models; penalized log-likelihood
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\textit{T. Nummi} and \textit{L. Koskela}, J. Appl. Stat. 35, No. 6, 681--691 (2008; Zbl 1147.62051)

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