# A Log-Linear Analytics Approach to Cost Model Regularization for Inpatient Stays through Diagnostic Code Merging

arXiv: https://arxiv.org/abs/2507.03843 · HTML: https://mlsanity.com/paper/2507.03843/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 55 references that are in the map.
- First posted on arXiv 5 Jul 2025.

## Builds on (most cited first, 6 of 6)

- [Surprises in High-Dimensional Ridgeless Least Squares Interpolation](https://mlsanity.com/paper/1903.08560/) (2019)
- [Implicit Regularization of Discrete Gradient Dynamics in Linear Neural Networks](https://mlsanity.com/paper/1904.13262/) (2019)
- [Traditional and Heavy-Tailed Self Regularization in Neural Network Models](https://mlsanity.com/paper/1901.08276/) (2019)
- [A Random Matrix Analysis of Random Fourier Features: Beyond the Gaussian Kernel, a Precise Phase Transition, and the Corresponding Double Descent](https://mlsanity.com/paper/2006.05013/) (2020)
- [The Implicit Regularization of Stochastic Gradient Flow for Least Squares](https://mlsanity.com/paper/2003.07802/) (2020)
- [Rethinking Parameter Counting in Deep Models: Effective Dimensionality Revisited](https://mlsanity.com/paper/2003.02139/) (2020)

## Abstract (arXiv)

Cost models in healthcare research must balance interpretability, accuracy, and parameter consistency. However, interpretable models often struggle to achieve both accuracy and consistency. Ordinary least squares (OLS) models for high-dimensional regression can be accurate but fail to produce stable regression coefficients over time when using highly granular ICD-10 diagnostic codes as predictors. This instability arises because many ICD-10 codes are infrequent in healthcare datasets. While regularization methods such as Ridge can address this issue, they risk discarding important predictors. Here, we demonstrate that reducing the granularity of ICD-10 codes is an effective regularization strategy within OLS while preserving the representation of all diagnostic code categories. By truncating ICD-10 codes from seven characters to six or fewer, we reduce the dimensionality of the regression problem while maintaining model interpretability and consistency. Mathematically, the merging of predictors in OLS leads to increased trace of the Hessian matrix, which reduces the variance of coefficient estimation. Our findings explain why broader diagnostic groupings like DRGs and HCC codes are favored over highly granular ICD-10 codes in real-world risk adjustment and cost models.