Scale effects in survey estimates of proportions and quantiles of per unit area attributes Steen Magnussen a,⇑ , Daniel Mandallaz b , Adrian Lanz c , Christian Ginzler c , Erik Næsset d , Terje Gobakken d a Natural Resources Canada, Canadian Forest Service, 506 West Burnside Road, Victoria, BC V8Z 1M5, Canada b Chair of Land Use Engineering, ETH Zurich, CH 8092 Zurich, Switzerland c Swiss Federal Research Institute, WSL, Zürcherstrasse 111, 8903 Birmensdorf ZH, Switzerland d Norwegian University of Life Sciences, P.O. Box 5003, NO1432 Ås, Norway article info Article history: Received 22 October 2015 Received in revised form 7 January 2016 Accepted 11 January 2016 Available online 20 January 2016 Keywords: Quantiles Area proportions Spatial support Spatial autocorrelation Scaled quantiles Scaled area proportions abstract Quantiles and proportions in a sampling distribution of a per unit area attribute (Y) depend on the spatial support (area) of employed survey plots. This is a nuisance for managers, and policy developers; in par- ticular when the underlying data have been collected with different spatial supports. Users of these statistics may wish to calibrate their estimates to a common scale of spatial support. The easiest way to do this is through scaling to a common plot size. We demonstrate a statistical method for upscaling. The method is illustrated in the context of a design-based forest inventory of a target attribute Y with a census of a co-located vector of auxiliary variables (X) correlated with Y. Two case studies from Norway and Switzerland confirmed significant and practically important scale effects in quantiles and propor- tions of above ground live tree biomass (Mg ha 1 ) and stem volume (m 3 ha 1 ). Upscaling requires an esti- mate of the spatial autocorrelation of Y given X at the scale of the original spatial support. We present an expedient method to this end. Our method affords estimation of scaled quantiles and proportions and assures consistency of sampling distribution across scales. Crown Copyright Ó 2016 Published by Elsevier B.V. All rights reserved. 1. Introduction Most per unit area forest survey results are summarized to means or totals (Schreuder et al., 1993, ch. 7; Köhl et al., 2006, ch. 2). Means and totals are generally easy to interpret, albeit with the caveat that the user should realize that the estimates have been scaled from the support area of a survey plot to a commonly accepted area unit (e.g. one hectare). For estimates of means and totals the scaling is unproblematic (Zeide, 1980; Kangas and Maltamo, 2006, ch. 4.1.1). With respect to estimates of quantiles and area proportions, the user should be aware of scale effects as these statistics depend on the sampling distribution of the attri- bute which in turn depends on the support area (size) of the survey plots (Smith, 1938; Correll and Cellier, 1987; Magnussen, 1989; Mandallaz and Ye, 1999; Mandallaz and Lanz, 2001; Gray, 2003; Duane et al., 2010; Hou et al., 2015). As plot size increases, the sampling distribution of per unit area attribute values becomes increasingly concentrated around the mean leading to changes quantiles and area proportions. Estimates of quantiles and area proportions are important for management and policy development purposes (Lagergren et al., 2006; Kurz et al., 2009; Yan et al., 2014). Questions like: ‘‘what is the proportion of forest land with per hectare volume (or biomass) above (below) a threshold value for economic exploitation?”; and ‘‘what is the upper (lower) limits of volume (biomass) in the 10% of the forest with the most (least) amount of per ha volume?”; can be answered from a sample-based estimate of the distribution function of per unit area volume (biomass). Yet a replication of the survey with a larger or smaller support area would generate differ- ent estimates. Hence users of statistics like quantiles and propor- tions of a per unit area survey attribute may rightly wish to remove these scale effects. Unfortunately, scale effects are an intrinsic property of quantiles and proportions of a per unit area attribute (Francisco and Fuller, 1991; Bellhouse and Stafford, 1999; Chen and Wu, 2002; del Mar Rueda et al., 2003). In forestry, the scale-size effect will depend on the specific forest, the species composition and structure, the attribute of interest, and the spatial autocorrelation in attribute values obtained from plots in close spatial proximity to each other (Smith, 1938; Correll and Cellier, 1987; Magnussen, 1989; Mandallaz and Ye, 1999; Mandallaz and Lanz, 2001). http://dx.doi.org/10.1016/j.foreco.2016.01.013 0378-1127/Crown Copyright Ó 2016 Published by Elsevier B.V. All rights reserved. ⇑ Corresponding author. E-mail address: Steen.magnussen@nrcan.gc.ca (S. Magnussen). Forest Ecology and Management 364 (2016) 122–129 Contents lists available at ScienceDirect Forest Ecology and Management journal homepage: www.elsevier.com/locate/foreco