Testing Spatial Symmetry Using Contingency Tables Based on Nearest Neighbor Relations

Table 6

Presented are the means (±SD (standard deviations)) of the off-diagonal entries, 𝑁₁₂, 𝑁₂₁, and their difference 𝑁₁₂−𝑁₂₁ and the rejection rate estimates under Case I patterns in (33) with 𝑁_mc=10000, 𝑛₁=𝑛₂=40 at 𝛼=.05 (a) and rejection rate estimates based on Monte Carlo randomization with 𝑛₁=𝑛₂=40 and 1000 Monte Carlo randomization steps are performed for each of 𝑁_mc=1000 generated samples under Case I patterns (b). 𝛽^𝑃_I and 𝛽^𝑃′_I stand for the empirical rejection rate estimates for Pielou's first type of symmetry test using 𝜒² approximation with and without Yates' continuity correction, respectively; 𝛽^𝑃_bin stands for the exact binomial version of Pielou's first type of symmetry test conditional on 𝑁₁₂+𝑁₂₁=𝑛_𝑡; 𝛽^𝐷_𝑆 stands for Dixon's symmetry test; 𝛽^𝑃_II is for Pielou's second type of symmetry test; 𝛽^𝐹_II stands for the exact test on the 𝑄-symmetry contingency table.

(a)

	Mean ± SD			Rejection rate estimates for Case I patterns
	𝑁12	𝑁21	𝑁12−𝑁21	𝛽^𝑃I	𝛽^𝑃′I	𝛽^𝑃bin	𝛽^𝐷𝑆	𝛽^𝑃II	𝛽^𝐹II
I-(i)	8.5±2.5	6.8±2.7	1.66±2.26	.0018	.0060	.0721	.0092	.0379	.0359
I-(ii)	4.1±1.7	2.2±1.7	1.94±1.58	.0118	.0511	.3109	.0017	.0345	.0324
I-(iii)	2.9±1.4	1.0±1.2	1.87±1.34	.0338	.0919	.5345	.0002	.0324	.0296

(b)

	Rejection rate estimates for Case I patterns based on Monte Carlo randomization
	𝛽^𝑃I	𝛽^𝑃′I	𝛽^𝐷𝑆	𝛽^𝑃II	𝛽^𝐹II
I-(i)	.641219	.480581	.409682	.546873	.546491
I-(ii)	.670268	.402391	.271784	.569909	.569605
I-(iii)	.696817	.397304	.196840	.577398	.576899