Abstract
We introduce a new method for estimating the nonparametric regression curve for longitudinal data. This method combines two estimators: truncated spline and Fourier series. This estimation is completed by minimizing the penalized weighted least squares and weighted least squares. This paper also provides the properties of the new mixed estimator, which are biased and linear in the observations. The best model is selected using the smallest value of generalized cross-validation. The performance of the new method is demonstrated by a simulation study with a variety of time points. Then, the proposed approach is applied to a stroke patient dataset. The results show that simulated data and real data yield consistent findings.
Highlights
Nonparametric regression is a statistical method used if the data show an unknown regression curve. e strength of this method is its great flexibility since the data are used to find the form of its estimated regression curve without being influenced by subjective judgements [1]
E function fji, j 1, 2, . . . p, is an approximation using truncated spline functions and gki, k 1, 2, . . . , q, is that using Fourier series. e estimator μ is obtained through a two-step optimization, i.e., penalized weighted least squares (PWLS) and weighted least squares (WLS)
Lemma 1 presents the goodness of fit of the Fourier series component, Lemma 2 presents the penalty component, and Lemma 3 gives the solution for the truncated spline component. eorem 1 presents the PWLS optimization, and eorem 2 presents WLS optimization
Summary
Nonparametric regression is a statistical method used if the data show an unknown regression curve. e strength of this method is its great flexibility since the data are used to find the form of its estimated regression curve without being influenced by subjective judgements [1]. Montoya et al [2] conducted a simulation study to compare knot selection methods in a penalized regression spline model. Some previous studies that have been mentioned have limitations in that they can only be used for cross-sectional data To overcome this limitation, a longitudinal data model has been developed. E present study extends the use of the mixed truncated spline and Fourier series (MTSFS) model to larger sample sizes and various time point designs.
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