Biometrics 67, 1518–1531 December 2011 DOI: 10.1111/j.1541-0420.2011.01604.x Variance Estimation for Systematic Designs in Spatial Surveys R. M. Fewster Department of Statistics, University of Auckland, Private Bag 92019, Auckland, New Zealand email: r.fewster@auckland.ac.nz Summary. In spatial surveys for estimating the density of objects in a survey region, systematic designs will generally yield lower variance than random designs. However, estimating the systematic variance is well known to be a diﬃcult problem. Existing methods tend to overestimate the variance, so although the variance is genuinely reduced, it is over-reported, and the gain from the more eﬃcient design is lost. The current approaches to estimating a systematic variance for spatial surveys are to approximate the systematic design by a random design, or approximate it by a stratiﬁed design. Previous work has shown that approximation by a random design can perform very poorly, while approximation by a stratiﬁed design is an improvement but can still be severely biased in some situations. We develop a new estimator based on modeling the encounter process over space. The new “striplet” estimator has negligible bias and excellent precision in a wide range of simulation scenarios, including strip-sampling, distance-sampling, and quadrat-sampling surveys, and including populations that are highly trended or have strong aggregation of objects. We apply the new estimator to survey data for the spotted hyena (Crocuta crocuta) in the Serengeti National Park, Tanzania, and ﬁnd that the reported coeﬃcient of variation for estimated density is 20% using approximation by a random design, 17% using approximation by a stratiﬁed design, and 11% using the new striplet estimator. This large reduction in reported variance is veriﬁed by simulation. Key words: Distance sampling; Encounter rate; Line transect sampling; Plot sampling; Poststratiﬁcation; Quadrat sam- pling; Strip sampling; Systematic sampling; Variance estimation. 1. Introduction Systematic survey designs are popular in spatial surveys such as strip sampling, quadrat sampling, and distance sampling from lines or points. The aim of these surveys is to esti- mate density of animals or plants (termed “objects”) in a deﬁned region. Systematic designs use a grid of equally spaced samplers—strips, lines, points, or quadrats—with a random start-point. They are easy to plan and implement in the ﬁeld, and they generally yield lower variance than random designs in which samplers are placed randomly and independently in the survey region. This is because random designs include realizations where several samplers fall by chance into high- density or low-density parts of the region, whereas systematic designs ensure even coverage of the region for all realizations. In many situations, systematic designs are also more precise than stratiﬁed designs (Cochran, 1946). The chief disadvantage of systematic designs is the dif- ﬁculty of estimating the improved variance. A systematic sample is based on only one random start-point, so the samplers are not independent replicates. Wolter (1984, 1985) highlighted three common approaches to systematic variance estimation for sampling a ﬁnite population in social statistics: 1. Random estimation, ignoring the problem of noninde- pendent samplers and using estimators derived for ran- dom designs; 2. Poststratiﬁcation, approximating the systematic design by a stratiﬁed design by grouping small sets of adja- cent samplers into strata, and using stratiﬁed variance estimators; 3. Modeling the process producing the ﬁnite population, for example by proposing a model for the correlation in response between adjacent members of the population. Similar ideas are used for spatial surveys. Most analyses ig- nore the problem (approach 1), but there is increasing recog- nition that this can be misleading. Millar and Olsen (1995), Simmonds and Fryer (1996), Kingsley (2000), and D’Orazio (2003) all used poststratiﬁcation (approach 2), and Fewster et al. (2009) extended this scheme to provide estimators for strip or line-transect sampling where line lengths are not equal. However, the poststratiﬁcation scheme is an ap- proximation and does not yield unbiased estimates for the variance. The aim of this article is to develop a new variance esti- mator for systematic spatial surveys. We create a model for the systematic variance, similar to approach 3 but exploit- ing the continuous nature of space. We show how the new variance estimator is applied to strip-sampling, line-transect distance-sampling, and quadrat or point-transect sampling surveys. We assess the estimator through a wide range of sim- ulations, reproducing those in two recent studies in which cor- rect variances were not always obtained (Fewster et al., 2009; Johnson, Laake, and Ver Hoef, 2010). We then apply the es- timator to distance-sampling data for spotted hyenas in the Serengeti National Park, Tanzania (Durant et al., in press), and show that the new estimator can make a dramatic impact on the standard error and conﬁdence interval width. This re- sult is veriﬁed by further simulations. All computations are 1518 C  2011, The International Biometric Society