In this study, we develop a regression analysis method, namely, the Semiparametric Binary Logistic Regression (SBLR), by extending the classical logistic regression that integrates both parametric and nonparametric components, which allows it to simultaneously model linear and non-linear relationships. Here, to obtain the estimation of a nonparametric component in the form of a non-linear curve (sigmoid curve), we use the penalized spline, which is a smoothing technique used in the nonparametric approach due to its ability to produce smooth and adaptive curves for fluctuating data. In this smoothing technique, selecting the optimal smoothing parameters plays an important role in fitting the model. Commonly, this selection is based on the minimum value of ordinary Cross-Validation (CV) or Generalized Cross-Validation (GCV). However, these CV and GCV criteria cannot be used when the CV and GCV curves continuously decline and never rise; the minimum CV and GCV values would not be achieved because they are not directly applicable due to the non-quadratic nature of the log-likelihood function. Therefore, a Generalized Approximate Cross-Validation (GACV) criterion is used to address such cases. This distinguishes it from previous studies that used the CV or GCV criterion. In the application to real data, we define an SBLR model of Coronary Heart Disease (CHD) risk factors that can be used for prediction and interpretation purposes. The results of the study successfully demonstrate the efficacy of the proposed method in identifying critical non-linear thresholds for CHD risk factors, and it is statistically valid and highly effective for CHD risk prediction. In the future, we can use the results of this research as a basis of an early warning system, specifically alerting individuals with moderate stress levels and dietary habits exceeding the identified thresholds to be aware of the heightened probability of developing CHD. In addition, this research aligns with point three of the Sustainable Development Goals (SDGs), namely, premature mortality reduction from non-communicable diseases by 2030.
Chamidah et al. (Sat,) studied this question.