,The bearing analysis (Falsetti and Sokal 1993) is a method of determining the direction of greatest correlation between data distance and geographic distance. The procedure starts with two matrices: a data distance matrix V and a geographic distance matrix D. D is transformed into a new matrix, Gθ, by multiplying each entry of D by the squared cosine of the angle between a fixed bearing (θ) and that of each pair of points,
,
where Gij is the ijth element of matrix G, Dij is the ijth element of matrix D, and αij is the angular bearing of points i and j. If the two bearings point in the same direction (i.e., θ – αij = 0), the cos2 will equal one; if the bearings are at right angles to each other, the cos2 will equal zero. This transformation essentially weights each geographic distance by its alignment with a test direction.
Illustration of the angle between a fixed bearing θ and the bearing for a pair of points (αij ).
The correlation between V and Gθ is calculated via a Mantel test and repeated for a set of θ. The maximal correlation occurs when large distances in the data are correlated with large geographic distances between points that lie roughly in the same direction as the fixed bearing. The direction of maximal correlation indicates the most likely direction of a gradient. The significance of each Mantel correlation can be determined through the asymptotic t-test or by a standard Mantel permutation test.
While not as informative as some of the other anisotropy methods, this method can be applied to data with small sample sizes.
| Menu: | Analysis 2→Scattered Data→Anisotropy→Bearing |
| Button: |  |
| Batch: | Bearing |
To run an analysis, you must specify the data and geographic distance matrices, an angle matrix, and the number of vectors you wish to test. The default is 36 vectors, leading to 5° separation among each vector (since we only need to test through 180°).
Bearing window.
The output consists of a list of each bearing tested, the Mantel correlation between the two matrices, and the asymptotic significance of the correlation. If permutation tests were performed, this significance is reported as well.
The bearing analysis was used to examine anisotropy in mortality due to prostate cancer. A data distance matrix was constructed as the Euclidean distance among mortality rates at different locations (we could have examined patterns in multiple cancers as a combined distance measure, but stuck with a single one for simplicity and to allow us to compare the results directly to other methods). The strongest positive correlation is found at 120° (roughly NW-SE) and, unsurprisingly, at approximately a right angle to that we find the strongest negative correlation at 35° (roughly NE-SW).
Bearing Analysis
Data Distance Matrix: Prostate Distances
Geographic Distance Matrix: Distance Matrix
Angle Matrix : Angle Matrix
Tested 36 vectors
Permutation test based on 99 permutations.
Probability
Bearing Correlation Asymp Perm
0.00000 0.13243 0.00000 0.01000
5.00000 0.08676 0.00000 0.01000
10.00000 0.03695 0.04757 0.04000
15.00000 -0.01399 0.46876 0.56000
20.00000 -0.06131 0.00332 0.01000
25.00000 -0.09957 0.00002 0.01000
30.00000 -0.12443 0.00000 0.01000
35.00000 -0.13381 0.00000 0.01000
40.00000 -0.12783 0.00001 0.01000
45.00000 -0.10793 0.00023 0.01000
50.00000 -0.07605 0.00975 0.03000
55.00000 -0.03444 0.23479 0.20000
60.00000 0.01433 0.60981 0.55000
65.00000 0.06724 0.01222 0.02000
70.00000 0.12093 0.00000 0.01000
75.00000 0.17210 0.00000 0.01000
80.00000 0.21809 0.00000 0.01000
85.00000 0.25725 0.00000 0.01000
90.00000 0.28895 0.00000 0.01000
95.00000 0.31342 0.00000 0.01000
100.00000 0.33136 0.00000 0.01000
105.00000 0.34367 0.00000 0.01000
110.00000 0.35122 0.00000 0.01000
115.00000 0.35479 0.00000 0.01000
120.00000 0.35499 0.00000 0.01000
125.00000 0.35228 0.00000 0.01000
130.00000 0.34697 0.00000 0.01000
135.00000 0.33923 0.00000 0.01000
140.00000 0.32913 0.00000 0.01000
145.00000 0.31661 0.00000 0.01000
150.00000 0.30148 0.00000 0.01000
155.00000 0.28346 0.00000 0.01000
160.00000 0.26215 0.00000 0.01000
165.00000 0.23700 0.00000 0.01000
170.00000 0.20739 0.00000 0.01000
175.00000 0.17268 0.00000 0.01000
Results of a bearing analysis.
Plot from a bearing analysis. Circles indicate significant values, crosses nonsignificant values.