Landscape Ecology

, Volume 31, Issue 1, pp 7–18 | Cite as

Cross-scale contradictions in ecological relationships

  • Kelly-Ann Dixon HamilEmail author
  • Basil V. Iannone III
  • Whitney K. Huang
  • Songlin Fei
  • Hao Zhang
Research Article



Not accounting for spatial heterogeneity in ecological analyses can cause modeled relationships to vary across spatial scales, specifically different levels of spatial resolution. These varying results hinder both the utility of data collected at one spatial scale for analyses at others and the determination of underlying processes.


To briefly review existing methods for analyzing data collected at multiple scales, highlight the effects of spatial heterogeneity on the utility of these methods, and to illustrate a practical statistical method to account for the sources of spatial heterogeneity when they are unknown.


Using simulated examples, we show how not accounting for the drivers of spatial heterogeneity in statistical models can cause contradictory findings regarding relationship direction across spatial scales. We then show how mixed effects models can remedy this multiscaling issue.


Ignoring sources of spatial heterogeneity in statistical models with coarse spatial scales produced contradictory results to the true underlying relationship. Treating drivers of spatial heterogeneity as random effects in a mixed effects model, however, allowed us to uncover this true relationship.


Mixed effects models is advantageous as it is not always necessary to know the influential explanatory variables that cause spatial heterogeneity and no additional data are required. Furthermore, this approach is well documented, can be applied to data having various distribution types, and is easily executable using multiple statistical packages.


Ecological fallacy MAUP Missing variables Multiscale Mixed effects models Spatial transmutation 



Funding was provided by a NSF Macrosystem Biology Grant # 1241932. We would like to thank Dr. Jianguo Wu for comments that greatly improved the overall quality of this paper.


  1. Araújo MB, Rozenfeld A (2014) The geographic scaling of biotic interactions. Ecography (Cop) 37:406–415Google Scholar
  2. Araújo MB, Thuiller W, Williams PH, Reginster I (2005) Downscaling European species atlas distributions to a finer resolution : implications for conservation planning. Glob Ecol Biogeogr 14:17–30CrossRefGoogle Scholar
  3. Argañaraz JP, Entraigas I (2014) Scaling functions evaluation for estimation of landscape metrics at higher resolutions. Ecol Inform 22:1–12CrossRefGoogle Scholar
  4. Azaele S, Cornell SJ, Kunin WE (2012) Downscaling species occupancy from coarse spatial scales. Ecol Appl 22:1004–1014PubMedCrossRefGoogle Scholar
  5. Bar-Massada A, Wood EM, Pidgeon AM, Radeloff VC (2012) Complex effects of scale on the relationships of landscape pattern versus avian species richness and community structure in a woodland savanna mosaic. Ecography (Cop) 35:393–411CrossRefGoogle Scholar
  6. Bates D, Maechler M, Bolker B, Walker S (2014) lme4: linear mixed-effects models using Eigen and S4Google Scholar
  7. Bombi P, D’Amen M (2012) Scaling down distribution maps from atlas data: a test of different approaches with virtual species. J Biogeogr 39:640–651CrossRefGoogle Scholar
  8. Bradter U, Kunin WE, Altringham JD, Thom TJ, Benton TG (2013) Identifying appropriate spatial scales of predictors in species distribution models with the random forest algorithm. Methods Ecol Evol 4:167–174CrossRefGoogle Scholar
  9. Buckley YM, Briese DT, Rees M (2003) Demography and management of the invasive plant species Hypericum perforatum. I. Using multi-level mixed-effects models for characterizing growth, survival and fecundity in a long-term data set. J Appl Ecol 40:481–493CrossRefGoogle Scholar
  10. Cleland DT, Avers PE, McNab WH, Jensen ME, Bailey RG, King T, Russell WE (1997) National Hierarchical Framework of Ecological Units. In: Boyce M, Haney A (eds) Ecosyst manag appl sustain for wildl resour. Yale University Press, New Haven, pp 181–200Google Scholar
  11. Cleland DT, Freeouf JA, Keys JE, Nowacki GJ, Carpenter CA, McNab WH (2007) Ecological subregions: sections and subsections for the conterminous United States. General Technical. Report WO-76. U.S. Department of Agriculture, Forest Service, Washington, D.C. Map, presentation scale 1:3,500,000; Albers equal area projection; coloredGoogle Scholar
  12. Cooper SD, Diehl S, Kratz K, Sarnelle O (1998) Implications of scale for patterns and processes in stream ecology. Aust J Ecol 23:27–40CrossRefGoogle Scholar
  13. Cressie NA (1993) Statistics for spatial data, revised. Wiley, New YorkGoogle Scholar
  14. Dark S, Bram D (2007) The modifiable areal unit problem (MAUP) in physical geography. Prog Phys Geogr 31:471–479CrossRefGoogle Scholar
  15. Dutilleul P, Legendre P (1993) Spatial heterogeneity against heteroscedasticity: an ecological paradigm versus a statistical concept. Oikos 66:152–171CrossRefGoogle Scholar
  16. Faraway J (2006) Extending the linear model with r: generalized linear, mixed effects and nonparametric regression models. Chapman & Hall/CRC Taylor & Francis Group, Boca RatonGoogle Scholar
  17. Fotheringham A (1989) Scale-independent spatial analysis. In: Goodchild M, Gopal S (eds) Accuracy spat databases. Taylor and Francis, London, pp 221–228Google Scholar
  18. Fotheringham A, Wong D (1991) The modifiable areal unit problem in statistical analysis. Environ Plan 23:1025–1044CrossRefGoogle Scholar
  19. Frazier AE (2014) A new data aggregation technique to improve landscape metric downscaling. Landscape Ecol 29:1261–1276CrossRefGoogle Scholar
  20. Fridley JD, Stachowicz JJ, Naeem S, Sax DF, Seabloom EW, Smith MD, Stohlgren TJ, Tilman D, Von Holle B (2007) The invasion paradox: reconciling pattern and process in species invasions. Ecology 88:3–17PubMedCrossRefGoogle Scholar
  21. Gotway CA, Young LJ (2002) Combining incompatible spatial data. J Am Stat Assoc 97:632–648CrossRefGoogle Scholar
  22. Gotway CA, Young LJ (2007) A geostatistical approach to linking geographically aggregated data from different sources. J Comput Graph Stat 16:115–135CrossRefGoogle Scholar
  23. Graf RF, Bollmann K, Suter W, Bugmann H (2005) The importance of spatial scale in habitat models: Capercaillie in the Swiss Alps. Landsc Ecol 20:703–717CrossRefGoogle Scholar
  24. Greven S, Kneib T (2010) On the behaviour of marginal and conditional AIC in linear mixed models. Biometrika 97(4):773–789CrossRefGoogle Scholar
  25. Hamer KC, Hill JK (2000) Scale-dependent effects of habitat disturbance on species richness in tropical forests. Conserv Biol 14:1435–1440CrossRefGoogle Scholar
  26. Heffernan JB, Soranno PA, Angilletta MJ, Buckley LB, Gruner DS, Keitt TH, Kellner JR, Kominoski JS, Rocha AV, Xiao J, Harms TK, Goring SJ, Koenig LE, McDowell WH, Powell H, Richardson AD, Stow C, Vargas R, Weathers KC (2014) Macrosystems ecology: Understanding ecological patterns and processes at continental scales. Front Ecol Environ 12:5–14CrossRefGoogle Scholar
  27. Holland JD, Bert DG, Fahrig L (2004) Determining the spatial scale of species’ response to habitat. Bioscience 54:227–233CrossRefGoogle Scholar
  28. Iannone BI, Potter KM, Dixon Hamil K, Huang W, Zhang H, Guo Q, Oswalt CM, Woodall CW, Fei S. Understanding biotic resistance to invasions across forests of the Eastern USA. Landscape Ecol current is. (this issue)Google Scholar
  29. Jackson HB, Fahrig L (2015) Are ecologists conducting research at the optimal scale? Glob Ecol Biogeogr 24:52–63CrossRefGoogle Scholar
  30. Jelinski D, Wu J (1996) The modifiable areal unit problem and implications for landscape ecology. Landscape Ecol 11:129–140CrossRefGoogle Scholar
  31. Keil P, Belmaker J, Wilson AM, Unitt P, Jetz W (2013) Downscaling of species distribution models: a hierarchical approach. Methods Ecol Evol 4:82–94CrossRefGoogle Scholar
  32. Kling A, Johnson A, O’Neill R (1991) Transmitation and functional representation of heterogeneous landscapes. Landscape Ecol 5:239–253CrossRefGoogle Scholar
  33. Laird NM, Ware JH (1982) Random-effects models for longitudinal data. Biometr Int Biometr Soc 38:963–974Google Scholar
  34. Levin SA (1992) The problem of pattern and scale in ecology: the Robert H. MacArthur award lecture. Ecology 73:1943–1967CrossRefGoogle Scholar
  35. Lloyd P, Palmer AR (1998) Abiotic factors as predictors of distribution in Southern African Bulbuls. Auk 115:404–411CrossRefGoogle Scholar
  36. McGill BJ (2010) Matters of scale. Science (80-) 328:575–576Google Scholar
  37. McPherson JM, Jetz W, Rogers DJ (2006) Using coarse-grained occurrence data to predict species distributions at finer spatial resolutions - possibilities and limitations. Ecol Modell 192:499–522CrossRefGoogle Scholar
  38. Müller S, Scealy JL, Welsh AH (2013) Model selection in linear mixed models. Stat Sci 28:135–167CrossRefGoogle Scholar
  39. O’Neill R, Rust B (1979) Aggregation error in ecological models. Ecol Modell 7:91–105CrossRefGoogle Scholar
  40. Openshaw S (1977) Optimal zoning systems for spatial interaction models. Environ Plan 9:169–184CrossRefGoogle Scholar
  41. Openshaw S (1983) The modifiable areal unit problem. Geo Books, Norwick, NorfolkGoogle Scholar
  42. Peters D, Bestelmeyer B, Turner M (2007) Cross-scale interactions and changing pattern-process relationships: consequences for system dynamics. Ecosystems 10:790–796CrossRefGoogle Scholar
  43. Pettorelli N, Laurance W, O’Brien T, Wegmann M, Nagendra H, Turner W (2014) Satellite remote sensing for applied ecologists: opportunities and challenges. J Appl Ecol 51:839–848CrossRefGoogle Scholar
  44. Pinheiro J, Bates D, DebRoy S, Sarkar D, R Core Team (2013) nlme: linear and nonlinear mixed effects models, R package version 3, p 57Google Scholar
  45. Powell KI, Chase JM, Knight TM (2013) Invasive plants have scale-dependent species-area relationships. Science 339(80):317–319Google Scholar
  46. R Core Team (2014) R: A language and environment for statistical computing. R Foundation for Statistical Computing, ViennaGoogle Scholar
  47. Renne IJ, Tracy BF, Rejmánek M (2003) The rich get richer: responses. Front Ecol Environ 1:122–123Google Scholar
  48. Robinson WS (1950) Ecological correlations and the behavior of individuals. Am Sociol Rev 15:351–357CrossRefGoogle Scholar
  49. Saura S, Castro S (2007) Scaling functions for landscape pattern metrics derived from remotely sensed data: Are their subpixel estimates really accurate? ISPRS J Photogramm Remote Sens 62:201–216CrossRefGoogle Scholar
  50. Selvin HC (1958) Durkheim’s suicide and problems of empirical research. Am J Sociol 63:607–619CrossRefGoogle Scholar
  51. Shea K, Chesson P (2002) Community ecology theory as a framework for biological invasions. Trends Ecol Evol 17:170–176CrossRefGoogle Scholar
  52. Soranno PA, Cheruvelil KS, Bissell EG, Bremigan MT, Downing JA, Fergus CE, Filstrup CT, Henry EN, Lottig NR, Stanley EH, Stow CA, Tan P, Wagner T, Webster KE (2014) Cross-scale interactions: quantifying multi-scaled cause-effect relationships in macrosystems. Front Ecol Environ 12:65–73CrossRefGoogle Scholar
  53. Stephens S, Koons D, Rotella J, Willey D (2003) Effects of habitat fragmentation on avian nesting success: a review of the evidence at multiple spatial scales. Biol Conserv 115:101–110CrossRefGoogle Scholar
  54. Stohlgren TJ, Barnett DT, Kartesz JT (2003) The rich get richer: patterns of plant invasions in the United States. Front Ecol Environ 1:11–14CrossRefGoogle Scholar
  55. Taylor BW, Irwin RE (2004) Linking economic activities to the distribution of exotic plants. Proc Natl Acad Sci U S A 51:17725–17730Google Scholar
  56. Turner MG (1989) Landscape ecology: the effect of pattern on process. Annu Rev Ecol Syst 20:171–197CrossRefGoogle Scholar
  57. Vaida F, Blanchard S (2005) Conditional Akaike information for mixed-effects models. Biometrika 92:351–370CrossRefGoogle Scholar
  58. Wakefield J, Lyons H (2010) Spatial aggregation and the ecological fallacy. In: Gelfand AE, Diggle P, Guttorp P, Fuentes M (eds) Handb spat stat. CRC Press, Stat, pp 541–558CrossRefGoogle Scholar
  59. Wheatley M, Johnson C (2009) Factors limiting our understanding of ecological scale. Ecol Complex 6:150–159CrossRefGoogle Scholar
  60. Wiens JA (1989) Spatial scaling in ecology. Funct Ecol 3:385–397CrossRefGoogle Scholar
  61. Wu J (2004) Effects of changing scale on landscape pattern analysis. Landscape Ecol 19:125–138CrossRefGoogle Scholar
  62. Wu J, Hobbs R (2002) Key issues and research priorities in landscape ecology: an idiosyncratic synthesis. Landscape Ecol 17:355–365CrossRefGoogle Scholar
  63. Wu J, Loucks O (1995) From balance of nature to hierarchical patch dynamics: a paradigm shift in ecology. Q Rev Biol 70:439–466CrossRefGoogle Scholar
  64. Wu J, Gao W, Tueller P (1997) Effects of changing spatial scale on the results of statistical analysis with landscape data: a case study. Geogr Inf Sci 3:30–41Google Scholar
  65. Zhang H, El-Shaarawi A (2010) On spatial skew-Gaussian processes and applications. Environmetrics 21:33–47Google Scholar
  66. Zuur AF, Ieno EN, Walker NJ, Savekiev AA, Smith GM (2009) Mixed Effects Models and Extensions in Ecology with R. Springer Science and Business Media, New YorkCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Kelly-Ann Dixon Hamil
    • 1
    Email author
  • Basil V. Iannone III
    • 2
  • Whitney K. Huang
    • 1
  • Songlin Fei
    • 2
  • Hao Zhang
    • 1
  1. 1.Department of StatisticsPurdue UniversityWest LafayetteUSA
  2. 2.Department of Forestry and Natural ResourcesPurdue UniversityWest LafayetteUSA

Personalised recommendations