Modeling stunting risk with multicollinear categorical predictors using a CATPCA-based binary logistic spline framework

2.1 Data source

The study employs both simulation data and real-world data. The simulation dataset was generated for several sample sizes, where the binary response variable $Y$ follows a binomial distribution, and five categorical predictor variables were constructed to exhibit strong intercorrelations.

For the real data analysis, we used family stunting-risk data from South Sulawesi Province, Indonesia, collected in 2023. The dataset consists of 23,150 families obtained from the National population and family agency of south Sulawesi Province. The study variables are presented in Table 1.

2.2 Analysis method

2.2 CATPCA binary logistic spline regression

Let the binary response variable be $Y_i\in \left\{0,1\right\}$, for $i=1,\dots ,n$. Let $Z_{ki}^{\text{*}}$ denote the $k$-th principal component score for observation $i$, where $k=1,\dots ,m$. Each component is defined as in Eq. (2).

where ${\text{v}}_k$ is the eigenvector associated with the $k$-th component. To model the probability of success, define $\pi \left({\text{Z}}_{\text{i}}^{\text{*}}\right)=P\left({\text{Y}}_{\text{i}}=1\right)$. A nonparametric logistic regression model with truncated spline basis functions is employed:

$\pi \left(Z_i^{\text{*}}\right)=\frac{\text{exp}\left({\eta }_i\right)}{1+\text{exp}\left({\eta }_i\right)},$

with the linear predictor given by Eq. (3)

where $K_h$ denotes the $h$-th knot point $\left(h=1,\dots ,r\right)$, and

$\left(Z_{ki}^{\text{*}}-K_h{)}_{+}=\text{max}(0,Z_{ki}^{\text{*}}-K_h\right)$

is the truncated linear spline basis function. The coefficients ${\beta }_k$ represent parametric effects, while ${\gamma }_{kh}$ capture nonparametric spline effects (Islamiyati et al., 2023).

The number and location of knots are selected using the generalized cross validation (GCV) criterion to capture local variations between predictors and response. This reflects a structured transformation where categorical variables are converted into continuous representations for flexible modeling. Similar principles are observed in macromolecular and colloidal systems, where discrete components assemble into functional structures (Kim et al., 2003). CATPCA thus integrates heterogeneous categorical predictors into a unified framework prior to spline-based modeling.

3.1 Model parameter estimation

Heterogeneous or categorical predictor variables are first transformed into optimally quantified numerical variables through the CATPCA procedure, as described in Eq. (1), resulting in the quantified matrix ${\text{X}}^{\text{*}}$. From this matrix, a set of principal components ${\text{Z}}_{\text{k}}^{\text{*}},\text{for }k=1,\dots ,m,$ is extracted and used as predictors in the regression model. The binary response variable is denoted by ${\text{Y}}_{\text{i}}\in \left\{0,1\right\}$, for $i=1,\dots ,n$. The relationship between the predictors and the probability of success $\pi \left({\text{Z}}_{\text{i}}^{\text{*}}\right)=P\left({\text{Y}}_{\text{i}}=1\right)$ is modeled using a binary logistic regression with truncated linear spline basis functions. The model can be written as Eq. (4) below.

where ${\beta }_0$ is the intercept, ${\beta }_{k1}$ denotes the regression coefficient for the $k$-th principal component, ${\beta }_{k\left(1+h\right)}$ represents the coefficient associated with the truncated spline basis, and r denotes the number of knot points.

The following estimation procedure is derived from the proposed model using the maximum likelihood framework Eq. (5).

The corresponding log-likelihood function is in Eq. (6).

Let $Y={(y_1,y_2,\dots ,y_n)}^T$ be the response vector and let $\pi ={(\pi \left(Z_1^{\text{*}}\right),\dots ,\pi \left(Z_n^{\text{*}}\right))}^T$denote the vector of predicted probabilities. Furthermore, define $Z$ as the design matrix that includes a column of ones (intercept), the principal components ${\text{Z}}_{\text{k}}^{\text{*}}$, and the truncated spline basis terms ${\left({\text{Z}}_{\text{k}}^{\text{*}}-{\text{K}}_{\text{h}}\right)}_{+}$.

The first derivative (gradient or score function) of the log-likelihood can then be expressed in matrix form as Eq. (7)

To obtain the parameter estimates, the Newton–Raphson iterative method is applied. The update formula is given by Eq. (8)

where $\text{W}$ is a diagonal weight matrix with elements:

$w_i=\pi \left(Z_i^{\text{*}}\right)\left[1-\pi \left(Z_i^{\text{*}}\right)\right],i=1,\dots ,n.$

The iteration continues until convergence is achieved, yielding the optimal parameter estimates for the CATPCA-based binary logistic spline regression model.

3.2 Simulation study

In this study, a simulation was conducted using $n=250$observations, a binary response variable, and five categorical predictors, three of which were generated to exhibit strong intercorrelation $(r>0.6)$. The initial model exhibited variance inflation factor (VIF) values exceeding 15 and high coefficient variance. Although the deviance explained reached 68%, the model had a relatively large akaike information criterion (AIC) of 210.4.

For benchmark comparison, penalized logistic regression models, namely LASSO and ridge regression, were evaluated using cross-validated tuning parameters. Ridge regression yielded an AIC of 185.7 with 72.5% deviance explained, while LASSO achieved an AIC of 178.9 with 74.2% deviance explained. Although both methods improved model stability, they remain within a parametric framework and are less capable of capturing complex relationships compared to the proposed CATPCA-based spline model.

After applying CATPCA, the proposed spline-based logistic model showed substantial improvement in all evaluation metrics. The coefficient variances decreased substantially, and all VIF values fell below 4, indicating effective mitigation of multicollinearity. Model performance improved significantly, with the AIC decreasing to 142.6 and the deviance explained increased to 80.3%. To explicitly compare the performance of models with and without CATPCA, as well as that of penalized logistic alternatives, the results are summarized in Table 2. The proposed CATPCA-based spline model achieved the lowest AIC and the highest deviance explained among all competing models.

To assess robustness beyond in-sample criteria, 5-fold cross-validation was performed across all models. The CATPCA-based spline model achieved the highest accuracy (0.78) and area under the curve (0.82) with the lowest prediction error, indicating strong out-of-sample generalization performance. The improvement achieved through CATPCA can be interpreted as an optimization of the predictor space, where correlated variables are transformed into more stable representations. Similar optimization principles are observed in materials chemistry and solid-state systems, where structural refinement enhances functional properties and system stability.

3.3 Analysis of stunting data using the nonparametric logistic spline CATPCA regression model

The data analyzed consisted of 23,150 families at risk of stunting in South Sulawesi Province, Indonesia, of which 76.45% were not at risk, and 23.55% were at risk. Several predictor variables exhibited strong correlations, particularly among $X_4,X_5,X_6,$ and $X_7$ (e.g., $r_{46}=0.92$), indicating substantial multicollinearity. To address this, CATPCA was applied to transform categorical predictors into optimally scaled numerical variables. The dimensionality reduction resulted in five principal components $\left(Z_1^{\text{*}}-Z_5^{\text{*}}\right)$, explaining 81.45% of the total variance.

The relationship between quantified variables $X_j^{\text{*}}$ and principal components $Z_k^{\text{*}}$ is represented by loading values $L_{jk}$, indicating the strength of association. Loadings range from -1 to 1, with $\left|L_{jk}\right|\ge 0.50$ considered significant, as shown in Table 3.

Overall, the integration of CATPCA and binary logistic spline regression provides a flexible and informative framework. Environmental sanitation factors exhibit both parametric and nonparametric effects on stunting risk, whereas socioeconomic factors are primarily nonparametric, and demographic variables show predominantly parametric effects. The final model was constructed using the five CATPCA components, with the spline specification selected based on the minimum GCV criterion, as shown in Table 4.

Table 4 shows that the lowest GCV value (0.11) was obtained with a one-knot point. Accordingly, the optimal model selected was the CATPCA-based binary logistic spline regression model with one knot. The overall significance of the model was assessed using the Likelihood Ratio Test. The test statistic obtained was $D=87.84>{\chi }_{\left(10;0.05\right)}^2=18.31$ (p < 0.05). Thus, the model parameters collectively significant explanation for the variation in family stunting risk.

Furthermore, the significance of individual parameters was assessed using the Wald test. All coefficients were statistically significant at the 5% level$\text{ (}p<0.05),$ indicating that each predictor contributes meaningfully to the probability of a family being at risk of stunting. The resulting optimal one-knot model is given in Eq. (9).

Eq. (9) shows that the five principal components meaningfully explain the risk of family stunting (AIC = 165; accuracy = 86.41%). The first component is associated with representing household demographic structure, particularly factors related to women of reproductive age ($X_4^{\text{*}},X_5^{\text{*}},X_6^{\text{*}},$ and $X_7^{\text{*}}$). The third and fifth components primarily reflect environmental sanitation through access to drinking water and sanitation facilities ($X_2^{\text{*}}$ and $X_3^{\text{*}}$), while the fourth component captures variation related to children under two years of age and social support ($X_1^{\text{*}}$ and $X_9^{\text{*}}$). The second component represents socioeconomic conditions, including family welfare and assistance programs ($X_8^{\text{*}}$ and $X_9^{\text{*}}$).

Overall, the loading structure enhances model interpretability by identifying key dimensions of stunting risk. The results highlight complex relationships among demographic, environmental, and socioeconomic factors. Methodologically, the proposed framework integrates CATPCA and binary logistic spline regression to address multicollinearity, categorical heterogeneity, and nonlinear predictor effects simultaneously.