Envelope tests are a popular tool in spatial statistics, where they are used in goodness-of-fit testing. These tests graphically compare an empirical function $T(r)$ with its simulated counterparts from the null model. However, the type I error probability $\alpha$ is conventionally controlled for a fixed distance $r$ only, whereas the functions are inspected on an interval of distances $I$. In this study, we propose two approaches related to Barnard's Monte Carlo test for building global envelope tests on $I$:(1) ordering the empirical and simulated functions based on their $r$-wise ranks among each other, and (2) the construction of envelopes for a deviation test. These new tests allow the a priori selection of the global $\alpha$ and they yield $p$-values. We illustrate these tests using simulated and real point pattern data

No abstract

No abstract

Citation: Wiegand, T., P. Grabarnik, and D. Stoyan. 2016. Envelope tests for spatial point patterns with and without simulation. Ecosphere 7(6):e01365. 10. 1002/ecs2.1365 Abstract. Model testing is a central step of spatial point pattern analysis, which allows ecologists to judge if their data agree with ecological hypotheses. We present a simple and elegant solution of a challenging problem: the construction of a goodness-of-fit envelope test with prescribed significance level α. Our new Analytical Global Envelope (AGE) test is not restricted to the narrow frame of complete spatial randomness testing and its envelopes can be determined by mathematical calculations. This allows us to investigate the influence of key settings of the AGE test on the width of the envelope strip. To circumvent some assumptions of the simulation-free AGE test we present a corresponding Simulation-Based Global Envelope (SBGE) test. The envelope strip of the AGE and the SBGE test encircles the range of a summary function such as the pair correlation function under the null model, and it has the desired property that the null hypothesis can be rejected with significance level α if the empirical summary function wanders outside the envelopes. The AGE test can be applied under the mild conditions that the values of the summary functions under the null model are (approximately) normally distributed and are (approximately) independent for different distance bins r j . The SBGE test requires only the independence assumption. The width of the strip of the AGE envelopes scales for a broad range of point processes with 1/n, where n is the number of points. This casts doubt about attempts of goodness-of-fit testing with low n (say <100). The AGE and SBGE test operate with wider envelope strips than the classical "pointwise" test. Therefore, the pointwise test has to be considered as too liberal. Furthermore, we show that the width of the AGE/SBGE strip increases approximately with ln(b), where b is the number of distance bins. For example, the AGE/SBGE envelopes are for b = 20 more than 50% wider than the corresponding pointwise envelopes. Our study opens up new avenues to the test problem in point pattern statistics and the new AGE and SBGE tests can be widely applied in ecology to improve the practice in null model testing.

of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.