Document Type: Original Paper


Department of Statistics Faculty of Mathematics and Natural Sciences Hasanuddin University, Makassar, Sulawesi Selatan, Indonesia.


Penalized spline criteria involve the function of goodness of fit and penalty, which in the penalty function contains smoothing parameters. It serves to control the smoothness of the curve that works simultaneously with point knots and spline degree. The regression function with two predictors in the non-parametric model will have two different non-parametric regression functions. Therefore, we propose the use of two smoothing parameters in the bi-variate predictor non-parametric regression model. We demonstrated its ability through longitudinal data simulation studies with a comparison of one smoothing parameter. It was done on several numbers of subjects with repeated measurements. The generalized cross validation value which is a measure of the model's ability is poured through the box plot. The results show that the use of two smoothing parameters is more optimal than one smoothing parameter. It was seen through a smaller generalized cross validation value on the use of two smoothing parameters. Application of blood sugar level data for patients with two smoothing parameters produced a penalized spline bi-variate predictor regression model with several segments of change patterns. There are five patterns at the time of treatment and blood pressure with the number of smoothing parameters is two, namely 0.39 and 0.73.


  1. Ruppert D., and Carrol R. J. Spatially-adaptive penalties for spline fitting. Aust. N. Z. J. Stat.., 42(2) : 205-223 (2000).
  2. Claeskens G., Kribovokova T., and Opsomer J. D. Asymptotic properties of penalized spline estimators. Biometrika, 96(3) : 529-544 (2009).
  3. Montoya L. E., Ulloa N., and Miller V. A simulation study comparing knot selection methods with equally spaced knot in a penalized regression spline. Int. J. Stat. Probab., 3(3): 96-110 (2014). 
  4. Lee T. C. M., and Oh H. S. Robust penalized regression spline fitting with application to additively mixed modeling. Comput. Stat., 22: 159-171 (2007). 
  5. Wang  B., Shi W., and Miao Z. Comparative analysis for robust penalized spline smoothing methods. Math. Probl. Eng., 2014(Article ID 642475) : 339-353 (2014). 
  6. Islamiyati A., Fatmawati, and Chamidah N. Estimation of covariance matrix on bi-response longitudinal data analysis with penalized spline regression. J. Phys.: Conf. Ser., 979: 012093 (2018). 
  7. Islamiyati A., Fatmawati, and Chamidah, N. Changes in blood glucose 2 hours after meals in type 2 diabetes patients based on length of treatment at Hasanuddin University Hospital, Indonesia.  Rawal Medical J., 45(1): 31-34 (2020).
  8. Zaherzadeh A., Rasekh A., and Babadi B. Diagnostic measures in ridge regression model with AR(1) errors under the stochastic linear restrictions. J. Sci. I. R. Iran, 29: 67-78 (2018).
  9. Mohammadi H., and Rasekh A. R. Liu Estimates and Influence Analysis in Regression Models with Stochastic Linear Restrictions and AR (1) Errors. J. Sci. I. R. Iran, 30(3): 271-285 (2019).
  10. Heckman N. E., and Ramsay J. O. Penalized regression with model-based penalties. Can. J. Stat., 28: 241-258 (2000).
  11. Yao, and Lee. Penalized spline models for functional principal component analysis. J. R. Stat. Soc. B., 68(1) : 3-25 (2006).
  12. Aydin D., and Yilmaz E. Modified spline regression based on randomly right-censored data: A comparative study. Commun. Stat-Simul. C., 47(9): 2587-2611 (2018).
  13. Chamidah N., and Lestari B. Spline estimator in homoscedastic multi-response nonparametric regression model in case of unbalanced number of observations. Far East J. Math. Sci.,100(9) : 1433-1453 (2016).
  14. Chamidah N., and Rifada M. Local linear estimation in bi-response semiparametric regression model for estimating median growth charts of children. Far East J. Math. Sci.,99(8): 1233-1244 (2016).
  15. Lestari B., Fatmawati, and Budiantara I. N. Estimation of regression function in multi-response nonparametric regression model using smoothing spline and kernel estimators. J. Phys.: Conf. Ser.,1097:012091 (2018).
  16. Chamidah N., Gusti K. H., Tjahjono E., and Lestari B. Improving of classification accuracy of cyst and tumor using local polynomial estimator. Telkomnika,17: 1492-1500 (2019).
  17. Lin X., and Zhang D. Inference in generalized additive mixed models by using smoothing splines. J. R. Stat. Soc. B., 61: 381-400 (1999).
  18. Ni X., Zhang D., and Zhang H. H. Variable selection for semiparametric mixed models in longitudinal studies. Biometrics, 66 : 79-88 (2010).
  19. Lai M. J., and Wang L. Bivariate penalized spline for regression. Stat. Sin., 23: 1399-1417 (2013).