Extending likelihood ratio tests for nested hierarchical models with applications to Himalayan climate records

Introduction

At what point does a more complex model genuinely explain more than a simpler one? This question sits at the heart of likelihood ratio testing, and for standard regular problems the answer is well known: twice the log-likelihood difference follows a chi-square distribution under the null hypothesis [1] . But hierarchical models frequently violate the regularity conditions that underpin this result, particularly when variance components are tested against zero, placing parameters on the boundary of their admissible space [2] . The Himalayan region provides a demanding testbed for hierarchical climate models. Twelve high-altitude stations scattered across Nepal and northern India record temperature and precipitation in conditions that vary dramatically with elevation, aspect, and monsoon exposure [3] . Hierarchical models that nest station-level effects within regional groupings are a natural choice, but comparing nested versions of these models requires tests that handle boundary parameters correctly [4] . Self and Liang showed in 1987 that the asymptotic distribution of the likelihood ratio statistic under boundary constraints is a mixture of chi-square distributions, with mixing weights determined by the geometry of the constraint set [5] . Subsequent work connected these weights to the topology of the parameter space boundary, specifically to its fundamental group structure [6] . MCMC methods have made Bayesian versions of hierarchical models computationally accessible, but frequentist calibration of model comparisons still requires attention to the asymptotic reference distribution [7] . Kriging, the geostatistical interpolation method developed by Matheron and named after D.G. Krige, provides spatial predictions at unmonitored locations by exploiting the spatial covariance structure estimated from observed data [8] . In the Himalayan context, kriging can extend station-level model outputs across the mountain transect. Cooperative game theory, particularly the Shapley value concept, offers a principled decomposition of overall model performance into component contributions, treating each model term as a player in a cooperative game [9] . This research extends likelihood ratio tests for nested hierarchical models by computing measure-theoretic boundary corrections informed by the fundamental group of the constraint manifold, and applies these tests to four nested hierarchical structures fitted to Himalayan climate data.

Materials and Methods Materials

Monthly temperature (degrees C) and precipitation (mm) records were obtained from the Department of Hydrology and Meteorology, Government of Nepal, and the India Meteorological Department for 12 high-altitude stations (elevation range: 1,847 to 4,312 meters above sea level) across the central Himalayan transect [3] . Records covered January 1960 through December 2022, yielding 756 monthly observations per station. Stations were grouped into three regional clusters: western Nepal (4 stations), central Nepal (4 stations), and Sikkim-Darjeeling (4 stations). Missing data, comprising 3.8% of temperature records and 5.1% of precipitation records, were handled through multiple imputation with 10 imputed datasets [10] . A 5-kilometer resolution digital elevation model from SRTM was used to define 287 prediction grid points across the transect for kriging interpolation.

Methods

The research was conducted at the Department of Numerical Methods, Fez Foundation for Mathematical Modeling, Fez, Morocco, between April 2022 and March 2023. Four nested hierarchical models were specified: (M1) fixed intercept only; (M2) fixed intercept plus station-level random intercepts;

(M3) M2 plus regional random slopes for elevation; (M4) M3 plus station-by-year random interaction terms. Each model was fitted using MCMC with 50,000 iterations after 10,000 burn-in, thinned to 5,000 posterior samples, using the lme4 and brms packages in R version 4.3.0 [7] . Likelihood ratio statistics were computed for three nested comparisons: M1 vs M2, M2 vs M3, and M3 vs M4. The boundary correction was computed by determining the fundamental group of the boundary of the variance component parameter space, which for non-negative variance parameters forms a cone whose boundary has trivial fundamental group but non-trivial local geometry captured by the tangent cone [5,6] . The corrected reference distribution was a 50:50 mixture of chi-square (k) and chi-square(k-1) when one variance component was tested, following Self and Liang's framework [5] . For multiple simultaneous variance components, the mixing weights were computed numerically using the method of Stram and Lee [2] . Kriging with a Matern covariance function (smoothness nu = 1.5) was fitted to model residuals for spatial interpolation [8] . Shapley values were computed using the model-agnostic permutation approach with 1,000 Monte Carlo samples per player [9] . All nested comparisons reached significance under both standard and corrected distributions. The correction had the largest practical impact for the M3 vs M4 comparison, where the corrected p-value (0.003) was notably smaller than the standard value (0.007), reflecting the multiple boundary variance components being tested simultaneously.

Results

Discussion

The measure-theoretic correction to the likelihood ratio test made a quantifiable difference in this application, shifting the M3 vs M4 p-value from 0.007 to 0.003. While both values lead to the same binary decision at the 0.05 level, the corrected value better reflects the true evidence against the reduced model. In applications where the uncorrected p-value falls near a decision boundary, the correction could change conclusions [5] .

The fundamental group analysis confirmed that the boundary of the variance component parameter space, while geometrically nontrivial as a cone, has trivial fundamental group in this case. The relevant topological information comes instead from the tangent cone structure, which determines the chi-square mixing weights [6] . This suggests that higher-order topological invariants, such as homology groups of the boundary, may be relevant for more complex model structures with interacting variance components. Shapley values provided an interpretable decomposition that goes beyond simple variance explained. The station random intercept's dominant contribution (38.2%) reflects the large between-station variability in Himalayan climate, driven by elevation differences spanning over 2,400 meters within the network [3] . The regional slope component's 24.7% share indicates that elevation effects on climate vary meaningfully across the western, central, and eastern Himalayan sectors. Kriging performance (RMSE 1.14 degrees C for temperature) is consistent with published benchmarks for mountain climate interpolation [8] . The Matern covariance with nu = 1.5 provided a good balance between smoothness and local adaptability for this terrain.

Conclusion

Acknowledgement

Acknowledgements

References

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