Knowledge Base

Circular Distributions

Vector Length and Mean Direction

Imagine adding a few more dots between two existing ones on a circle. The more dots you add, the more the average coordinates X,Y will get closer to the circle. This means that the vector length R will increase. The greater R, the better the direction of the vector is defined, because the data points agree with each other.

However, if you place points at other locations far from each other, the length of the vector is reduced and R is low. Consequently, the value of the mean is a poor measure for the average Phase Dispersion because it represents a group of very different locations on the percentages circle. In other words, R helps you quantify the credibility of the average.

When contrasting Phase Dispersions between two experimental groups, you may find a significant difference in R, not in CStat Mean. This means that the two groups do not differ in the average timing of one paw (target, T) relative to the other (anchor, A). However, the higher value of R in one of the two groups indicates lower between-subject variability in the temporal position of T relative to A than in the other group.

Subjects in the group with higher R walk more uniformly than those in the group with a low R, although the average position of T relative to A is not different between the two groups.

Measures of Spread Around the Mean

Angular Deviation

The vector length R is used to compute the Angular Deviation (CStat AD in CatWalk XT). The Angular Deviation ranges from a minimum of 0 when R = 1, to a maximum of 22.508 (100/2π) when R = 0.

Note: The Angular Deviation CStat AD is the same as the CStat Standard Deviation in CatWalk XT 10.6 and earlier versions. Make sure this does not generate confusion when examining old experiments.

Circular Standard Deviation

An alternative measure, the Circular Standard Deviation (CStat SD in CatWalk XT), ranges from 0 to infinity and is defined by formulas E.8 and E.9. The Circular Standard Deviation s0 more closely resembles the standard deviation on a linear scale. However, it is reasonable to assume that circular data have a limited spread; in that case Angular Deviation may be a better option.

Note: The Circular Standard Deviation CStat SD is not the same as the CStat Standard Deviation in CatWalk XT 10.6 and earlier versions. Make sure this does not generate confusion when examining old experiments.

Note that Angular Deviation and Circular Standard Deviation tend to converge as the data cluster around the mean.

The Circular Group Mean

When calculating group statistics (Analyze > View Group Statistics and Charts), the group mean CStat Mean is calculated with the same formulas as for the mean of single values of Phase Dispersions and Couplings.

Example: A group includes three trials. The per-trial averages of Couplings are 5%, 95%, and 100%. The circular mean provided by CatWalk XT is 0% (which is the same as 100%). Using the arithmetic mean would result in a value (66.67%) far from the original ones.

The Excel Sheet "Calculate Circular Statistics"

If you are not familiar with circular statistics, you can practice with the Excel sheet Calculate Circular Statistics for Phase Dispersions and Couplings.xlsx, which you can find in the same folder as this manual:

C:\Program Files\Noldus\CatWalk XT 10\Documentation

1. Open the Excel file.
2. Enter the values of Phase Dispersions or Couplings, for example those displayed in CatWalk XT for a specific run.
3. Explore the formulas that lead to the main results: CStat Mean (%), CStat Angular Deviation (%), and CStat Standard Deviation (%).
4. Enter one or more values in the Values (%) column to see the effect on the statistics and on the orientation of the vector in the chart shown on the right.

Notes:

• If CStat Mean is negative and you prefer positive values, locate the Complementary Mean (%). Do this especially when entering values of Couplings. For example, if the CStat Mean (%) is -25%, the complementary mean valid in the range for Couplings (0–100%) is 75%.
• The formulas for calculating circular statistics are protected and cannot be changed.
• There may be small differences between the statistics given by CatWalk XT and those in the Excel sheet. In general they do not exceed 0.01% and are the effect of rounding of cell values. You can increase the number of decimals in CatWalk XT (File > Preferences > Number of Decimals) to better locate those differences.

Confidence Limits for the Population Mean

CatWalk XT does not calculate the confidence limits for the mean. You can calculate the confidence limits as ā ± d (E.10), where d is defined as follows (if n > 8).

For r ≤ 0.9 (E.11), and for r ≥ 0.9 (E.12), where R = nr and α is the significance level.

Critical values for the χ² distribution with 1 degree of freedom are given in the table below.

Critical Values for the χ² Distribution with v=1 (Zar 1998, Table B.1)