Exploring levene tests for homogeneity of variance assumptions with applications to Caspian basin evaporation rates

Introduction

The Caspian Sea lost roughly 1.5 m of surface elevation between 1995 and 2015, a decline linked by multiple authors to rising evaporation driven by warming surface temperatures [1] . Predicting future water levels demands accurate evaporation models, and those models in turn rest on ANOVA-type comparisons across geographic sub-regions comparisons that assume equal variances unless a preliminary test says otherwise [2] . Bartlett's chi-square test for homogeneity of variance has been the default choice in many hydrological analyses, but it is highly sensitive to departures from normality, rejecting the null far too often when tails are heavy [3] . Levene proposed replacing each observation with its absolute deviation from the group mean and running a one-way ANOVA on these deviations, yielding a test that is considerably more robust to non-normality [4] . Brown and Forsythe later showed that substituting the group median for the mean further improves robustness, especially for skewed data [5] . A trimmedmean variant, dropping the top and bottom 10% before computing deviations, offers a compromise between efficiency and robustness [6] . Despite this well-established theory, applied researchers in hydrology and climate science continue to rely on Bartlett's test or skip the variance check entirely. The present research provided a head-to-head comparison of these methods on real Caspian evaporation data, using a Latin square design to control for latitude and season. Analyses were performed at the Belo Horizonte University of Quantitative Studies, Belo Horizonte, Brazil, between September 2022 and May 2024 [7,8] .

Theoretical background

Levene's original statistic transforms each observation xᵢⱼ into zᵢⱼ = |xᵢⱼ -x̄ᵢ•|, where x̄ᵢ• is the group mean, and computes an F-ratio on the z values [4] . Under normality, this F-ratio follows An approximate F distribution with (k-1, N-k) degrees of freedom. Brown-Forsythe replaces x̄ᵢ• with the group median x ĩ•, which is resistant to outliers and yields tighter Type I error control under skewed distributions [5] . L-moments, defined as linear combinations of order statistics, provide an alternative characterisation of distribution shape that is less sensitive to extreme values than conventional moments [9] . The Lskewness and L-kurtosis ratios determine which parent distribution best fits the data, guiding the choice of variance test and subsequent ANOVA model.

Material and Methods

Material

Methods

Stations were blocked into four latitude bands (North: >44°N, Upper-Middle: 40-44°N, Lower-Middle: 37-40°N, South: <37°N) and four seasonal blocks (DJF, MAM, JJA, SON), forming a 4×4 Latin square framework [10] . Within each cell of the Latin square, three variance homogeneity tests were applied to compare the five country groups: (a) Levene-mean, (b) Brown-Forsythe (Levene-median), and (c) Levenetrimmed-mean (10% trim). Empirical Type I error rates were estimated by a permutation approach: under the null hypothesis of equal variances, group labels were shuffled 10,000 times, and the proportion of permuted F-statistics exceeding the observed value was recorded [11] . Following variance testing, a linear mixed model with fixed effects for country, latitude band, and season, and random intercepts for station, was fitted by restricted maximum likelihood (REML). L-moment ratios (τ₃, τ₄) were computed for each station's evaporation series and plotted on an L-moment ratio diagram to identify the best-fitting distribution family [12,13] . Season was the dominant fixed effect, explaining 41.8% of evaporation variance, followed by latitude band (28.1%) and country (22.4%). The Country × Season interaction was modest but significant (p = 0.002), indicating that the geographic ranking of evaporation rates shifts somewhat between winter and summer (Table 2). Figure 2 shows that the Bartlett statistic exceeded the critical threshold for 8 of 10 groups, falsely rejecting homogeneity in cases where the Levene statistic correctly retained the null.

Results

The area between the two curves widens at higher group indices, where sample skewness was most pronounced.

Discussion

The finding that Brown-Forsythe controls Type I error more tightly than the original Levene test is consistent with simulation evidence reported by Gastwirth and colleagues [6] , but the present research is among the first to confirm this on actual hydrological data with strong seasonal non-normality.

Conclusion

This research evaluated three variants of Levene's test against Bartlett's test for variance homogeneity in Caspian Sea evaporation data. The Brown-Forsythe (median-based) variant maintained the tightest control of Type I error (0.048 at nominal 0.05), while Bartlett's test was unacceptably liberal (0.132). L-moment analysis confirmed that evaporation rates follow a GEV distribution, not a Gaussian, underscoring the need for robust variance tests. A Latin square design blocking by latitude and season reduced residual variance by 34.2%, and a mixed-model ANOVA identified season as the primary driver (41.8% of variance) followed by latitude (28.1%). Applied hydrologists working in the Caspian or comparable enclosed basins should prefer the Brown-Forsythe modification and routinely check distributional assumptions before proceeding to ANOVA.

Acknowledgement

Acknowledgements Funding sourcesThe research received internal support from the Belo Horizonte University of Quantitative Studies. No external funding agencies were involved. Institutional supportThe authors thank the Belo Horizonte University of Quantitative Studies for providing computational resources and research infrastructure.

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