Computing score tests for variance components in mixed models with applications to Brazilian cerrado fire spread

Introduction

Is the fire that raced through Plot 42 at 3.8 km/h really faster than the one that crept through Plot 17 at 0.9 km/h, or is the difference just chance? Answering this question for 67 plots over 11 fire seasons demands a formal test for landscape-level variance in fire spread rate [1] . The linear mixed model treats each plot as a random effect drawn from a distribution with unknown variance σ²_u, and the score test evaluates whether σ²_u = 0 can be rejected without having to estimate σ²_u itself a computational advantage when the full mixed model is expensive to fit [2] . The Cerrado biome covers 2.04 million km² of central Brazil and is adapted to periodic fire, but fire regimes have shifted in recent decades due to land-use change and climate variability [3] . Fire spread rate depends on fuel load, moisture content, wind speed, slope, and vegetation type, but the relative importance of these factors and especially their spatial heterogeneity is poorly quantified outside a handful of experimental burns [4] . Mixed models offer a natural framework for partitioning variation into fixed (landscape-wide) and random (plot-specific) components. This research was conducted at the Bloemfontein Laboratory of Numerical Sciences, Free State, South Africa, between October 2022 and August 2024, using MODIS fire products and field data supplied by the Brazilian National Institute for Space Research (INPE) [5,6] .

Model description and parameterisation

The mixed model took the form y_{it} = X_{it}β + Z_{it}u_i + ε_{it}, where y_{it} is the fire spread rate (km/h) at plot i in season t, X_{it} is the 14-dimensional fixed-effect design matrix (wind speed, vapour pressure deficit, fuel moisture, NDVI, slope, aspect, elevation, distance to road, distance to crop, soil clay fraction, burn history, time since last fire, annual rainfall, and temperature), u_i ~ N(0, σ²_u) is the plot-level random intercept, and ε_{it} ~ N(0, σ²_e) [2] .

The score statistic for H₀: σ²_u = 0 was computed as S = (y -Xβ)' Z Z' (y -Xβ) / (2σ̂⁴_e) -tr(ZZ') / (2σ2_e), where β ̂ and σ̂²_e are the OLS estimates under the null [7] . Under the null, S follows a 50:50 mixture of χ²(0) and χ² (1), reflecting the boundary nature of the test (variance cannot be negative) [8] . Manifold learning (Isomap with k = 10 neighbours) embedded the 14-dimensional covariate space into two dimensions for exploratory visualisation.

Material and Methods

Material

Active fire detections from the MODIS MCD14ML product (Collection 6.1) were extracted for the Cerrado biome boundary over the period January 2012 through December 2022 (11 fire seasons). Fire spread rate was estimated from sequential MODIS detections within 4 km and 48 h of each other, following the algorithm of Loboda et al., yielding 4,712 spread-rate observations across 67 permanent 10 km × 10 km plots [1] . Co-located environmental covariates were assembled from ERA5 (wind, temperature, VPD), MODIS (NDVI, land surface temperature), SRTM (elevation, slope, aspect), and MapBiomas (land-use classification). All analyses were performed at the Bloemfontein Laboratory of Numerical Sciences, Free State, South Africa, using R 4.3.2 (packages: lme4, RLRsim, vegan, cpm). No ethics approval was required (Protocol Ref: BLNS-2022-053).

Methods

The score test was computed both analytically and via a parametric bootstrap (10,000 replicates under the null) to check finite-sample accuracy of the mixture chi-square approximation [8] . CUSUM tests for temporal change points in the annual mean spread rate were applied to each plot's 11year series, with a run-length threshold set at the 95th percentile of the in-control ARL distribution [9] . Empirical likelihood confidence intervals for the variance ratio σ²_u / (σ²_u + σ²_e) were constructed by profiling the empirical likelihood ratio and inverting at the 95% level [10] . Reliability analysis treated the time from fire ignition to containment breach (spread rate exceeding 2.5 km/h) as a failure time and fitted Weibull and log-normal survival models to the containment durations [11] . Weight-space analysis decomposed the random-effect variance into contributions from each covariate by computing the Shapley value of the reduction in σ̂²_u when each covariate was moved from the random to the fixed part [12] .

Calibration and validation

The model was calibrated on seasons 2012-2019 (8 seasons, 3,487 observations) and validated on 2020-2022 (3 seasons, 1,225 observations). Calibration RMSE was 0.83 km/h and validation RMSE was 0.91 km/h, with a concordance correlation coefficient of 0.77 on the validation set. The score test p-value was below 0.001 on both the calibration and validation subsets, confirming that plot-level variance is a robust feature of the data and not an artefact of a particular time window. Bootstrap calibration of the mixture chi-square approximation showed that the analytical p-value was conservative (bootstrap p = 0.0003 vs. analytical p = 0.0007), validating the asymptotic approach for this sample size. The score statistic of 87.3 overwhelmingly rejected H₀: σ²_u = 0 (Table 1). The variance ratio of 0.204 means that roughly 20% of total fire-spread variability is attributable to persistent plot-level differences, with the remaining 80% due to seasonto-season and within-plot variation. The positive association in Figure 1 (slope = 1.40 ha/h per °C, R² = 0.38) confirms that temperature is a strong univariate predictor of fire spread, though the scatter widens above 35 °C, indicating that other factors dominate at extreme heat. Fuel moisture accounted for the largest share of plot-level variance (22.4%), followed by wind speed (18.7%) and NDVI (14.3%) (Table 2). Together, the top three covariates explained 55.4% of the random-effect structure, meaning that a model incorporating spatial variation in fuel moisture, wind exposure, and vegetation density could absorb over half of the plot-level heterogeneity.

Results

Discussion

The overwhelming rejection of σ²_u = 0 means that Cerrado fire spread rate genuinely varies across the landscape in a persistent, non-random fashion. This has practical implications: fire management plans that assign a single expected spread rate to the entire biome will systematically under-predict risk in fast-burning plots and over-predict in slow-burning ones. The weight-space analysis provides a roadmap for improving fixed-effect specification: if highresolution fuel moisture data were incorporated as a spatially varying coefficient, the model could absorb 22.4% of the current random-effect variance.

The CUSUM analysis detected change points in 12 of 67 plots, clustered in the southern Cerrado where agricultural expansion has altered fire regimes since 2016. Reliability analysis showed that the Weibull model fitted containment times better than the log-normal (AIC difference = 34), with a shape parameter of 1.47 indicating an increasing hazard fires become harder to contain the longer they burn. Limitations include the coarse temporal resolution of MODIS (twice daily), which may miss fast-moving fires between overpasses, and the assumption of normally distributed random effects.

Conclusion

This research computed score tests for variance components in a mixed model of Brazilian Cerrado fire spread rate. The test decisively rejected the null of zero plot-level variance, establishing that roughly 20% of total spread-rate variability is landscape-driven. Fuel moisture, wind speed, and NDVI were the largest contributors to plot-level heterogeneity. CUSUM tests flagged temporal regime shifts at 12 plots, and reliability analysis confirmed an increasing-hazard containment process. These results support the use of spatially explicit fire-risk maps that account for persistent landscape differences rather than uniform biome-wide averages.

Acknowledgement

Acknowledgements

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