2.1 Terminology and assumptions
The Fisher statistical method can be used to estimate descriptive and inferential statistics from a set of directional observations. Just as one-dimensional statistics are based on the Gaussian probability density function with true mean (μ) and variance (σ2), these directional statistics are based on the Fisher probability density function, with true direction (υ) and precision parameter (κ).
The type of directional data that can be interrogated using Fisher statistics are normally distributed sets of unit vectors where each vector has a length of 1 and originates from x=y=z=0 so that each set may be represented by a group of points on the unit sphere. For a set of N vectors, V = (v1, …, vN), a mean direction vector, vm, can be determined and each vector in the set, vi = (xi, yi, zi), can be described by it's angle, αi, relative to vm using the dot product αi,=cos−1(vi˙ vm).
Ultimately, an estimate can be made for the confidence angle, α(1-p), within which (1-p)% of all directional observations can be expected to be found and F-statistic hypothesis testing may be performed to make inferences about group differences in orientation.
2.2 The Fisher distribution
The Fisher probability density function (pdf) (Fisher, 1953
) describes the probability of a point on a sphere of unit radius within an angular area dA (in steradians) centered at an angle α from υ
where κ is a precision parameter that is inversely proportional to the dispersion.
Since it is more useful for this type of statistics to consider the pdf in terms of the spatial variables α and
, the azimuthal angle, and because d
ϕ, the pdf is more often written as:
which gives the probability of finding a direction vector in the circular strip on the unit sphere between the angles α and α+dα from the true direction.
2.3 Descriptive statistics
The Fisher probability distribution is the basis for the estimation of descriptive and inferential statistics presented in this section.
2.3.1 Mean direction and resultant vector
The mean direction vector, vm
, describing the average of a set of N unit vectors can be calculated as follows:
and R is the length of the resultant vector sum of all observations, which is given by:
2.3.2 The precision parameter
The precision parameter, κ, is a population parameter that describes the concentration of the pdf around the true direction. As κ approaches 0, the distribution becomes uniform over the unit sphere and as κ approaches∞
, the distribution becomes singular at the true direction. The sample estimate of κ is k (Fisher, 1953
; McFadden, 1980
) given by:
where N is the number of unit vectors in the sample and R is the length of the resultant vector calculated in (4).
2.3.3 Confidence angles
For a confidence level of (1-p)% (typically 95% or p=0.05), a circle can be defined by all points at an angle of α(1-p)
from the calculated mean direction, vm
is given by:
where N is the number of unit vectors in the sample, R is the resultant vector calculated in (4) and p is defined according to confidence level.
2.5 Hypothesis testing
In order to test the null hypothesis - that sample observations from 2 or more groups are taken from the same population - the mean direction vectors and confidence angles may be compared across groups or inferential statistics may be calculated.
An intuitive comparison of group directional information is to use the circle of confidence defined by α95. The clearest inferences can be made when: 1) α95 circles for the groups do not overlap, then the null hypothesis is unlikely or 2) the mean direction vector from one group lies inside α95 of the other, then the null hypothesis is likely. A less clear case occurs when α95 circles for the groups do overlap, but the mean from each group lies outside the α95-circle of the other.
A more quantitative way to perform hypothesis testing is by calculation of the F statistic. The following equation was derived by Watson (Watson, 1956
), to compare two groups with N1
observed unit vectors respectively and resultant vectors of length R1
and R is the length of the resultant vector for the pooled direction vector observations from both groups. The larger the value of F, the more different the two group mean directions and, a p-value may be obtained using the appropriate degrees of freedom (2 and 2(N-2) respectively). Equation 7
also extends to hypothesis testing of more than 2 groups.
2.5 Special considerations for DTI data
Although Fisher statistics are generalizable to the analysis of many types of directional data sets, it is important to consider certain inherent attributes of DTI data for accurate directional analysis. Here, it is suggested that the potential pitfalls of DTI data be avoided or accounted for prior to statistical analysis so that the input for Fisher statistics are groups of direction vectors where each vector accurately represents the tissue orientation within a given region of interest (ROI) for one brain sample and is collected and calculated consistently across all samples and groups.
2.5.1 Sources of experimental error
Sources of variance that may contribute to erroneous directional measurement include: positioning of the sample within the scanner reference frame, gross brain structure abnormalities (e.g. by uneven fixation) and ROI mask placement variations. To the extent possible, these should be reduced by experimental practices or corrected for by post-processing techniques, for example DTI-appropriate registration to standard space (Jones et al., 2002
2.5.2 DEC ambiguity and lateralization
Most DEC maps report directional coloration based on an absolute value algorithm, which creates maps that are qualitatively intuitive and do not suffer from discontinuity artifacts (Pajevic & Pierpaoli, 1999
), however this approach suffers from ambiguity in that each color represents 4 unique directions. For example, if <R,G,B> = < |x|, |y|, |z|>, then the directions: x,y,z = (0.6667, 0.3333, 0.6667), (−0.6667, 0.3333, 0.6667), (0.6667, −0.3333, 0.6667) and (−0.6667, −0.3333, 0.6667) are all different from one another, but coded for by the same shade of “pink”. This is particularly evident for bilateral structures, such as the fimbria (see ), which are encoded for by the same color in both brain hemispheres when in fact the ε1
vectors for symmetric anatomical locations are oriented differently, albeit symmetrically. Consequently, care should be taken to consider ROIs from different hemispheres separately or carefully transform directional data from anatomically symmetric regions to a single hemisphere.
Figure 1 Determination of DTI directional information. (a) Representative ROIs for the CC and fimbria are shown on a DEC map. The approach for determination of single sample (b) and group (c) directional information for the CC is shown, where red points are included (more ...) 2.5.3 Antipodal symmetry
Perhaps the most substantial issue in the analysis of directional DTI data arises from the arbitrary sign of the DTI eigenvectors (i.e. ε1=−ε1) which implies that the collection of eigenvectors in an ROI may be clustered about two points, known as antipodes, that are exactly opposite one another on the unit sphere (illustrated in ). Uncorrected, this could lead to severe misestimation of the direction vector. To avert this, it is possible to map all vectors to a single antipode before determining the direction vector. Similarly, direction vectors for all observations within and across groups should be taken from antipodes in the same hemisphere, although choice of the hemisphere will depend on the orientation of the structure in question.