(2012) used an arcsin
transformation if percent coverage data sets were not normal.
[[??].sup.ref.sub.k] = arcsin
([angle][[PHI].sub.c]].sub.k,k]/4[pi]d/[lambda]), k = 1, ..., K.
Data were transformed to arcsin
[square root of (x/100)] for statistical analysis (Taylor, 1984).
Y = sgn(y) [square root of [x.sup.2] + [y.sup.2]]/[square root of 2[pi]] ([pi]/2 - arcsin
[x.sup.2] - [y.sup.2]/[x.sup.2] + [y.sup.2]).
[alpha] [less than or equal to] 90[degrees] - arcsin
([n.sub.2]/[n.sub.1]) = 90[degrees] - arcsin
(1.46/1.49) = 90[degrees] - 78[degrees] = 12[degrees] (11)
[alpha] = [2/[pi]] arcsin
([7/128] [[square root of 207] + [square root of 15]]) (21)
Dendrogram was performed on the basis of arcsin
transformed percentage relative frequency (%O) and consumed biomass (%B) data of 10 main food types (same food types as listed above, except pheasant and other birds were merged).
[alpha] = arcsin
h/R = arcsin
2gh/[g.sup.2] + [h.sup.2].
(1.3) [q.summation over (j=1)] ([[alpha].sub.k+1] - [[beta].sub.j]) < 4/[sigma] arcsin
[square root of ([delta]/2)],
Therefore, we also transformed the original data using arcsin
, a process typical for percentage data (Sokal and Rolf, 1981), and analyzed them, using a paired t-test in addition to a related-samples Wilcoxon signed-rank test.
c) If L is quasismooth (in the sense of Lavrentiev), that is, for every pair [z.sub.1], [z.sub.2] [member of] L, if s([z.sub.1], [z.sub.2]) represents the smallest of the lengths of the arcs joining [z.sub.1] to [z.sub.2] on L, there exists a constant c > 1 such that s([z.sub.1], [z.sub.2]) [less than or equal to] c [absolute value of ([z.sub.1] - [z.sub.2])], then [PHI] [member of] Lip [alpha] for [alpha] = [1/2][(1 - [1/[pi]] arcsin
[1/c]).sup.-1] and [PSI] [member of] Lip [beta] for [beta] = [2/[(1 + c).sup.2]] , .
Analysis of variance was used for all statistical comparisons (PROC GLM, SAS Institute, 2013) and proportions were arcsin
, square root transformed prior to analysis.