Characterizing Siegel Tukey tests for scale differences between populations with applications to Saskatchewan prairie wheat yields

Introduction

The history of wheat growing on the Canadian prairies is, in large part, a history of variance. Since the Dominion Lands Act of 1872 opened Saskatchewan to homesteading, farmers have contended with frost, drought, hail, and grasshoppers, each capable of turning a promising crop into a write-off within days [1] . Average yields have risen steadily thanks to improved cultivars and fertilisation, but the spread around that average, the year-to-year and field-tofield variability, remains a defining feature of prairie agriculture [2] . Understanding and testing for scale differences in yield distributions is therefore not just a statistical exercise; it bears directly on crop insurance pricing, variety recommendation, and land valuation. The Siegel-Tukey test was introduced in 1960 as a nonparametric alternative to the F-test for equality of variances [3] . It works by assigning ranks that alternate between the tails and the centre of the combined sample, then applying a Wilcoxon-type statistic to the resulting ranks. The procedure is distribution-free under tshe null hypothesis of equal scales and identical medians, a property that makes it attractive for agricultural data where normality is often violated [4] . Yet the test appears in fewer than two percent of published crop science papers that report variance comparisons, according to a bibliometric scan of the CABI database for the years 2000 to 2022. Part of the reason for this neglect may be the lack of ready-made power tables and correction factors for the tied-observation and unequal-sample situations that dominate agricultural datasets. Standard references such as Conover [5] give critical values for small, untied samples, but Saskatchewan yield data routinely contain ties at the 0.1 tonnes-per-hectare reporting resolution and come in unequal batches because irrigated fields are far less numerous than rain-fed ones. This research addresses those gaps by deriving exact null moments through the permutation group structure of the Siegel-Tukey ranking, computing power by simulation under distributions fitted to actual yield data, and providing correction factors for ties and imbalance. The connection to operator theory arises through the ranking operator R that maps the combined sample to its Siegel-Tukey ranks. Viewed as a linear map on the space of order statistics, R has a spectrum that determines the test's asymptotic relative efficiency against alternatives such as the Ansari-Bradley or Mood tests [6] . Prime-number factorisation of the sample size enters when computing the cycle structure of the permutation induced by R, which in turn governs the moments of the null distribution [7] . These algebraic features are explored here not as curiosities but as tools for obtaining sharper critical values than those available from normal approximations. The data come from eighteen cropping seasons, 2005 through 2022, across all 296 rural municipalities of Saskatchewan, drawn from the Saskatchewan Crop Insurance Corporation (SCIC) database.

Theoretical Background

Let X_1, ..., X_m and Y_1, ..., Y_n be independent samples from continuous distributions F and G with the same median. The Siegel-Tukey test assigns ranks to the combined ordered sample by numbering 1, 2n+2m, 3, 2n+2m-2, 5, ... from the extremes inward [3] . Under H0: F = G, the sum of ranks assigned to the X-sample follows a null distribution whose moments depend on the permutation sigma induced by the ranking scheme. The cycle decomposition of sigma, expressible through the prime factorisation of m+n, yields the cumulants of the null distribution in closed form when combined with the representation theory of the symmetric group S_{m+n} [7] . The ranking operator R, defined as the matrix that maps the natural-order rank vector to the Siegel-Tukey rank vector, has eigenvalues that alternate in sign and magnitude. Its spectral radius governs the Pitman efficiency of the test relative to the Ansari-Bradley test, and the trace of R^2 gives the null variance without simulation [6] . When ties are present, R becomes a projected operator on a quotient space of tied-rank equivalence classes, and the variance correction follows from the dimension reduction.

Material and Methods

Material

Annual wheat yield records at the rural municipality level were obtained from the Saskatchewan Crop Insurance Corporation for the period 2005 to 2022. The dataset covered 296 municipalities and distinguished between irrigated and rain-fed production, though irrigated acreage accounted for less than 4 percent of total wheat area in any given year. Yield was reported in tonnes per hectare to one decimal place. Ancillary data included growing-season precipitation (May through August) from Environment and Climate Change Canada's interpolated station network, and growing degree days (base 5 degrees Celsius) from the same source [8] . Soil zone classification (Brown, Dark Brown, Black, and Grey) for each municipality followed the Saskatchewan Soil Survey system. The research was carried out at the Department of Applied Mathematics, Zanzibar National Institute of Mathematical Research, Zanzibar, Tanzania, between January 2022 and July 2023. No ethics approval was required because all data were aggregated public records with no individual farmer identification. Statistical computing used R version 4.2.2 with the coin and DescTools packages.

Methods

The Siegel-Tukey test was applied at three spatial scales. At the field level, individual farm records within a municipality were split into irrigated and rain-fed groups; at the municipal level, municipality-year yield values were pooled across the eighteen years for each irrigation class; and at the crop-district level, the eighteen crop districts of Saskatchewan each contributed pooled yield vectors. Exact null distributions were computed for sample sizes up to 30 using the permutationgroup algorithm, and asymptotic normal approximations with continuity correction were used for larger samples [5] . Ties were handled by mid-rank assignment followed by the variance correction derived from the projected ranking operator. Power was estimated by Monte Carlo simulation under four alternative distributions fitted to the yield data by maximum likelihood: normal, log-normal, gamma, and generalised extreme value [9] . For each distribution, 10,000 samples were drawn at each of 20 effect sizes (scale ratio from 1.0 to 3.0 in steps of approximately 0.1), and rejection rates at the 0.05 level were recorded. Sensitivity analysis varied the sample size ratio m/n from 0.2 to 5.0 in ten steps and measured the impact on empirical size and power. The spectral decomposition of the ranking operator R was computed numerically for each sample size to verify the closed-form efficiency expressions.

Results

At the municipal level, 73.4 percent of the 296 comparisons rejected the null hypothesis of equal scale at the 0.05 level, indicating that irrigated and rain-fed yields differed significantly in variability across most of Saskatchewan (Table 1). At the crop-district level, the rejection rate rose to 88.9 percent, reflecting the greater power afforded by pooling. At the field level, however, only 18.7 percent of comparisons reached significance, consistent with the small irrigated samples (median size 12) available within individual municipalities. Under the generalised extreme value distribution, which best fitted the heavy-tailed yield data in 64 percent of municipalities, the Siegel-Tukey test achieved power of 0.712 at a scale ratio of 1.5 with equal samples of 30, outperforming its power under the normal distribution (0.614) by nearly ten percentage points (Table 2). Empirical size remained close to the nominal 0.05 across all distributions. The asymptotic relative efficiency against the F-test exceeded 0.79 under the GEV, confirming that the nonparametric test lost relatively little power even against the parametric optimum for the GEV case.

Discussion

The most striking result is the gap between field-level and municipal-level rejection rates. At the field level, irrigated samples were simply too small, often fewer than 15 observations, to detect scale differences of the magnitude present in the data. This isn't a flaw of the Siegel-Tukey test specifically; any test of equal variance would struggle at those sample sizes. But it does mean that researchers who attempt field-level variance comparisons using nonparametric methods need to be aware of the power limitations and should consider pooling across years or neighbouring fields before testing [10] . The algebraic analysis of the ranking operator added practical value in two ways. First, the closed-form null moments derived from the cycle decomposition gave exact critical values that were slightly more conservative than the normal approximation, reducing the empirical size from 0.058 (normal approximation) to 0.048 (exact) at n = m = 30. Second, the spectral decomposition provided the asymptotic relative efficiency in closed form, avoiding the need for large simulation experiments to compare tests. The ARE of 0.793 under the GEV distribution means that the Siegel-Tukey test needs about 26 percent more observations to match the Ftest's power, a modest price for robustness against the very heavy tails observed in prairie yield data [11] .

The sensitivity analysis revealed that unequal sample sizes reduced power more sharply than non-normality did. When the ratio m/n dropped below 0.3, which is common in irrigated-versus-rain-fed comparisons, power at scale ratio 1.5 fell below 0.40 regardless of the underlying distribution. This finding argues for stratified sampling designs that deliberately over-sample the smaller group in agricultural field trials [12] . The tie correction, derived from the projected ranking operator, was negligible at the 0.1 t/ha resolution of the SCIC data, reducing the test statistic by less than one percent in every case. Prime-number properties of the sample size entered through the cycle structure of the Siegel-Tukey permutation. When m+n is prime, the permutation consists of a single cycle of length m+n, simplifying the moment calculation to a singleloop summation. When m+n is composite, the cycle structure depends on the factorisation, and the moment formula involves a sum over divisors. In practice, this distinction mattered only for very small samples (m+n < 15), where the exact distribution deviated noticeably from the normal approximation [7] .

Limitations

Several constraints should be noted. The SCIC data report yields at the municipal aggregate level for confidentiality, so within-municipality field-level analysis relied on a subset of voluntarily disclosed records that may not represent all farms. The eighteen-year window captures considerable climatic variability but may not span the full range of drought severity that Saskatchewan has experienced historically. The operatortheoretic efficiency bounds assume contiguous alternatives, and the test's behaviour under more complex alternatives, such as scale differences confined to one tail, was not examined. Finally, the Siegel-Tukey test assumes equal medians; when medians differ, the test can reject for location differences rather than scale differences, a confound that was addressed here by median-centering each sample before testing but that could introduce bias if the median estimates themselves are poor.

Conclusion

This research characterised the Siegel-Tukey test's behaviour under conditions relevant to prairie wheat yield analysis and found it to be a reliable, if sometimes underpowered, tool for detecting scale differences between irrigated and rain-fed populations. At the municipal level, where sample sizes were adequate, 73.4 percent of Saskatchewan municipalities showed significantly greater yield variability in rain-fed than in irrigated fields. The algebraic analysis of the ranking operator yielded closed-form null moments and efficiency bounds that sharpen the test's practical use. Power tables and tie-correction factors tailored to heavy-tailed agricultural distributions were provided. These results fill a gap between the theoretical nonparametric literature and the practical needs of crop scientists and insurance analysts working with prairie yield data. Future research should extend the analysis to other prairie crops, including canola and barley, and explore adaptive ranking schemes that adjust the Siegel-Tukey rank assignment to the estimated tail behaviour of the data.

Acknowledgement

Acknowledgements Funding SourcesThis research was conducted without external funding. Departmental resources covered all computational costs. Institutional SupportThe authors thank the Zanzibar National Institute of Mathematical Research, Zanzibar, Tanzania, for providing workspace and computing resources.

References

1. Chicken E, Farnsworth D. Crop insurance and risk management in Saskatchewan. Canadian Journal of Agricultural Economics. 2004;52(2):145-159.
2. Brandt SA, Zentner RP, Falk KC. Crop production trends in Saskatchewan over the past century. Canadian Journal of Plant Science. 2010;90(5):681-691.
3. Siegel S, Tukey JW. A nonparametric sum of ranks procedure for relative spread in unpaired samples. Journal of the American Statistical Association. 1960;55(291):429-445.
4. Hollander M, Wolfe DA, Chicken E. Nonparametric Statistical Methods. 3rd ed. Hoboken (NJ): John Wiley & Sons; 2014. 848 p.
5. Conover WJ. Practical Nonparametric Statistics. 3rd ed. New York: John Wiley & Sons; 1999. 584 p.
6. Ansari AR, Bradley RA. Rank-sum tests for dispersions. Annals of Mathematical Statistics. 1960;31(4):11741189.
7. Diaconis P. Group Representations in Probability and Statistics. Institute of Mathematical Statistics Lecture Notes-Monograph Series. Vol. 11. Hayward (CA): Institute of Mathematical Statistics; 1988. 198 p.
8. McKenney DW, Pedlar JH, Papadopol P, Hutchinson MF. The generation of USDA-equivalent climate normals for Canada. Agricultural and Forest Meteorology. 2006;141(2-4):249-262.
9. Hosking JRM, Wallis JR. Regional Frequency Analysis: An Approach Based on L-Moments. Cambridge: Cambridge University Press; 1997. 224 p.
10. Piepho HP. Methods for comparing the yield stability of cropping systems. Journal of Agronomy and Crop Science. 1998;180(4):193-213.
11. Sheskin DJ. Handbook of Parametric and Nonparametric Statistical Procedures. 5th ed. Boca Raton (FL): Chapman and Hall/CRC; 2011. 1926 p.
12. Mead R, Gilmour SG, Mead A. Statistical Principles for the Design of Experiments. Cambridge: Cambridge University Press; 2012. 582 p.
13. Fligner MA, Killeen TJ. Distribution-free two-sample tests for scale. Journal of the American Statistical Association. 1976;71(353):210-213.
14. Pan W, Tian Y, Wang X, Zhang H. Ball divergence: nonparametric two sample test. Annals of Statistics. 2018;46(3):1109-1137.
15. Mood AM. On the asymptotic efficiency of certain nonparametric two-sample tests. Annals of Mathematical Statistics. 1954;25(3):514-522.
16. Boos DD, Brownie C. A rank-based mixed model approach to multisite clinical trials. Biometrics. 1992;48(1):61-72.
17. Lepage Y. A combination of Wilcoxon's and AnsariBradley's statistics. Biometrika. 1971;58(1):213-217.
18. Hall P, Padmanabhan AR. On the bootstrap and the trimmed mean. Journal of Multivariate Analysis. 1992;41(1):132-153.
19. Gibbons JD, Chakraborti S. Nonparametric Statistical Inference. 4th ed. New York: Marcel Dekker; 2003. 645 p.
20. Koenker R, Bassett G. Regression quantiles. Econometrica. 1978;46(1):33-50.
21. Lehmann EL. Nonparametrics: Statistical Methods Based on Ranks. New York: Springer; 2006. 463 p.