Statistical boosting algorithms are renowned for their intrinsic variable selection and enhanced predictive performance compared to classical statistical methods, making them especially useful for complex models such as generalized additive models for location scale and shape (GAMLSS). Boosting this model class can suffer from imbalanced updates across the distribution parameters as well as long computation times. Shrunk optimal step lengths have been shown to address these issues. To examine the influence of socio-economic factors on the distribution of the number of antenatal care visits in Nigeria, we generalize boosting of GAMLSS with shrunk optimal step lengths to base-learners beyond simple linear models and to a more complex response variable distribution. In an extensive simulation study and in the application we demonstrate that shrunk optimal step lengths yield a more balanced regularization of the overall model and enhance computational efficiency across diverse settings, in particular in the presence of base-learners penalizing the size of the fit.
Generalized additive models for location, scale and shape (GAMLSS; Rigby and Stasinopoulos, 2005) have received increasing attention in recent years (Kneib, 2013;Klein, 2024), being able to model parameters beyond the mean and to incorporate a variety of effects. They extend generalized additive models (GAMs; Hastie and Tibshirani, 1990) in that all distribution parameters are modeled based on an additive predictor instead of one. Due to their ability to model the response variable distribution in a very flexible manner, GAMLSS are used in a variety of fields. In particular, GAMLSS are e.g. recommended for estimating child growth curves, where they are used to model the distribution of characteristics like height and weight as a function of age (World Health Organization, 2006). Beyond biostatistical settings, GAMLSS have also been applied in fields such as finance and for modeling weather and climate-related measurements (Ganegoda and Evans, 2012;Villarini et al., 2010).
While typically estimated using penalized maximum likelihood, alternative estimation techniques have emerged to take advantage of their particular strengths, one of which is statistical boosting (Mayr et al., 2012a;Thomas et al., 2018). In statistical boosting algorithms (Friedman, 2001;Bühlmann and Yu, 2003), regression models with a low degree of freedom are iteratively fitted to the current (pseudo-) residuals, resulting in an ensemble of weak learners as final model. Stopping the iterative updating scheme early allows the method to perform variable selection as well as coefficient shrinkage, often enhancing predictive performance compared to traditional methods. Because of this ability to yield sparse models, statistical boosting algorithms are particularly advantageous for complex model classes such as GAMLSS.
The increased model complexity, however, also poses challenges when boosting GAMLSS. While in general the non-cyclical algorithm introduced by Thomas et al. ( 2018) is favorable, it has been shown to be prone to imbalanced predictor updates and long computation times (Zhang et al., 2022;Daub et al., 2025). These issues arise when the potential updates of the different submodels are on different scales causing the selection procedure to favor updates of certain submodels over others. They originate from structural differences in the sizes of negative gradients (hence reflecting the structure of the respective likelihood) and, for example, occur in Gaussian location and scale model with a large variance (Zhang et al., 2022). One way to address such imbalances between submodels is to apply adaptive rather than fixed step lengths (Zhang et al., 2022;Daub et al., 2025), whereby the adaptive step lengths compensate for otherwise small update sizes and thus enable a fairer update selection. A more detailed explanation of this topic is provided in section 2.
To date, adaptive step lengths have only been applied for boosting GAMLSS with two-parameter response variable distributions and simple linear base-learners. Zhang et al. (2022) consider a Gaussian location and scale model with simple linear base-learners and propose using shrunk optimal step lengths to address the balancing issue. Daub et al. (2025) investigate a different type of adaptive step length, in addition to shrunk optimal step lengths, for negative binomial and Weibull distributed response variables. In this work, we generalize this approach with respect to two aspects. First and most importantly, a wider variety of base-learner types is considered, specifically for non-linear, categorical and spatial effects, extending beyond simple linear base-learners. Second, we apply the boosting algorithm to a zero-inflated negative binomial model for location, scale and shape (ZINB-GAMLSS), which has a more complex response variable distribution with substantial dependencies among its parameters.
The motivation of this extension is to enable a more sophisticated investigation of the relationship between antenatal care utilization in Nigeria and socio-economic, demographic and contextual characteristics of the mother. With 560 deaths per 100,000 live births reported in 2013, compared to 16 deaths per 100,000 live births in OECD member countries (World Health Organization et al., 2014), Nigeria is among the countries with the highest maternal mortality worldwide. Moreover, the country has an insufficient antenatal care utilization: 69.8% of the women receive fewer than the recommended minimum number of seven antenatal care visits for pregnancies without complications and 34.2% report having received no antenatal care at all (National Population Commission (NPC) [Nigeria] and ICF International, 2014;World Health Organization, 2016). Since antenatal care enables the identification of pregnancy-related and delivery-related risks and offers timely and appropriate interventions, strengthening the antenatal care utilization is regarded a key strategy for reducing maternal mortality (Wo
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