CIRCULAR DISTRIBUTION IN BIOSTATISTICAL ANALYSIS

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CIRCULAR DISTRIBUTION IN BIOSTATISTICAL ANALYSIS
DONGMEI LI Department of Statistics, University
of Connecticut MAY 2005
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Circular Distributions
CONTENTS

• Definition Of A Circular Scale
• Descriptive Statistics Of Circular Distributions
• Hypothesis Testing For Circular Distributions

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Circular Distributions
I. Definition Of A Circular Scale

• 1.Type of Biological Data
• Data on a Ratio Scale
• Data on an Interval Scale
• Data on an Ordinal Scale
• Data on a Nominal Scale
• Continuous and Discrete Data

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Circular Distributions
2.Data On a Circular Scale

• An Interval Scale of measurement was defined as
  a scale with
• equal intervals but with no true zero point .

• A circular scale is a special type of an
  interval scale, where not only is there no
  true zero, but any designation of high or low
  values is arbitrary.

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Circular Distributions
2.Data On a Circular Scale (Examples)


Times of day
Compass direction
Months of year
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Circular Distributions
2.Data On a Circular Scale
Time units X ? an angular direction
a, in degrees k is total time units in the full
cycle
Example time of day 0615 hr X6.25 hr,
k24hr, then
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Circular Distributions
II. Descriptive Statistics Of Circular
Distributions

• Graphical Presentation of Circular Data
• Sines and Cosines of Circular Data
• The Mean Angle
• Angle Dispersion
• The Median and Modal of Angles
• Confidence Limits for the Population Mean and
  Median Angle
• Second-Order Analysis The Mean and Mean Angle
• Confidence Limits for the Second-Order Mean Angle

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Circular Distributions
1.Graphical Presentation of Circular Scale --
Scatter Diagram
Example 1 Eight trees leaning compass
directions
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Figure1. A circular scatter diagram for the data
of example 1 (The dashed line is position of
median)
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96
110
117
132
154
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Circular Distributions
1.Graphical Presentation of Circular
ScaleCircular Histogram
Example2 A sample of circular data, presented
as a frequency table
Figure2. (a)Circular histogram
(b) A relative frequency histogram
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Circular Distributions

2.Sines and Cosines of Circular Scale





Figure3. A unit circle, showing four points and
their polar (a,r) and rectangular (X,Y)
coordinates.
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Circular Distributions

3.The Mean Angle

Sample n angles, a1..an, To compute the
sample mean angle, , we first consider the
rectangular coordinates of the mean angle



and, then get
r is the length of the mean vector, the value of
is determined as the angle having the
following cosine and sin
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Circular Distributions

3.The Mean Angle

Example3 Calculating the mean angle for the data
of example1




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Circular Distributions

3.The Mean Angle (for grouped data)

Often circular data are recorded in a frequency
table. For such data, the following computation
are needed for the rectangular coordinates of the
mean angle



In these equations, is the midpoint of
the measurement interval recorded, is the
frequency of occurrence of the data within the
interval . And
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Circular Distributions

3.The Mean Angle (for grouped data)

Example4 Calculating the mean angle



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Circular Distributions

4. Angular Dispersion

• Range is defined as the smallest arc that
  contains all the data in the distribution.
• r is a measure of concentration, the value of r
  varies inversely with the amount of dispersion in
  the data. It has no unit and it may vary from 0
  (when there is so much dispersion that a mean
  angle can not be described) to 1.0 (when all the
  data are concentrated at same direction).
• 1-r is a measure of dispersion. Lack of
  dispersion would be indicated by 1-r0, and
  maximum dispersion by 1-r1.0

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Circular Distributions

4. Angular Dispersion


Mardia(1972a45)? Circular variance
Batschelet(1965,198134) ? Angular
variance Mardia (1972a45) ? Standard
variance unit radian squared


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Circular Distributions

4. Angular Dispersion



Angular deviation Circular standard
deviation Unit degrees

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Circular Distributions

4. Angular Dispersion

• r1.00, s0, s00
• (b) r0.99, s8.10, s08.12
• (c) r0.90,s25.62,s026.30
• (d) r0.60, s51.25s057.91
• (e) r0.30,s67.79,s088.91
• (f) r0.00, s81.03, s08

Figure4. Circular distributions with various
amount of dispersion. The values of r varies
inverse with the amount of dispersion. The mean
angle is 50.
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Circular Distributions
5. The Median and Modal Angles

To find the median angle we first determine
which diameter of the circle divides the data
into two equal sized groups. The median angle is
the angle indicated by that diameters radius
that is near to the majority of the data point.



The mode is defined as is the mode for linear
scale data. Just as with linear data, there may
be more than one mode or there may be no modes.
Figure1.
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Circular Distributions

6.Confidence Limits for the population mean Angles


The confidence limits may be expressed as

For n as small as 8 the following method maybe
used

If
then
then
If
where
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Circular Distributions

6.Confidence Limits for the population mean Angles

Example5 The 95 confidence interval for the
data of Example 3 (slide12)

n 8 r 0.82522 R nr (8)(0.82522)
6.60176 Using the 1st Equation of d




arcos(0.85779)
31º or 329º Here the
number 31 º is a reasonable choice The 95
confidence interval is L1 68 º and L2 130 º


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Circular Distributions

7.The Mean of Mean Angles (Second order Analysis)

The grand mean is determined for each of k groups
of angles, the rectangular coordinates



and
Figure5. The data of Example 6. The mean of 7
angles is indicated by the broken line
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Circular Distributions

7.The Mean of Mean Angles (Second order Analysis)

Example6 The mean of set of mean angles



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Circular Distributions

8. Confidence Limits for the Second order Mean
Angle

Batschelet(1981144) method



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Circular Distributions

8. Confidence Limits for the Second order Mean
Angle

Batschelet(1981144) method (cont)

The quantities b1 and b2 are then examined
separately, each yielding one of the confidence
limits, as follows


After which we determine that the angle having
and
The confidence limit is either the angle thus
determined or that angle 180, whichever is
nearer the sample mean angle (and, if the angle
180 is greater than 360, simply subtract
360). The confidence interval thus computed is a
little conservative, and the confidence limits
are not necessarily symmetrical about the mean.
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Circular Distributions

8. Confidence Limits for the Second order Mean
Angle
Batschelet(1981144) method

Example7 Confidence limits for the mean a set of
angles, given



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Circular Distributions

8. Confidence Limits for the Second order Mean
Angle

Example7(cont)



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8. Confidence Limits for the Second order Mean
Angle

Example7(cont)



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Circular Distributions

8. Confidence Limits for the Second order Mean
Angle

Batschelet(1981144) method (Example)
Example7(cont)



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III. Hypothesis Testing For Circular Distributions

• One Sample Testing
• Multisample Testing
• Second-Order Analysis of Angles
• Paired-Sample Testing (parametric
  nonparametric)
• Angular Correlation and Regression
• Goodness of Fit Testing for Circular
  Distributions

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Circular Distributions
1. Testing of Significance of the Mean angle
Unimodal Distributions
The Rayleigh Test. H0 The Sampled distribution
is uniformly distributed around the circle. Ha
The population is not a uniform circular
distribution. The Rayleigh test asks how large
a sample r must be to indicate confidently a
nonuniform population distribution. A quantity
referred to as Rayleighs R is obtained as
and the so-called Rayleighs z is utilized
for testing the null hypothesis of no population
mean direction

or
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Circular Distributions
1. Testing of Significance of the Mean angle
Unimodal Distributions
The Rayleigh Test.

TableB.34 presents critical values of ,
An excellent approximation P value of Rayleighs
R is
If null hypothesis is rejected, we may conclude
that there is a mean population direction, and if
not, we may conclude that population distribution
to be uniform around the circle
Assumption The population does not have more
than one mode.
Note TableB.34 is in the reference book on page
App188.
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Circular Distributions
1. Testing of Significance of the Mean angle
Unimodal Distributions
Example8Rayleighs test for circular uniformity,
applied to the data of example1


figure1
3
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Circular Distributions
1. Testing of Significance of the Mean angle
Unimodal Distributions


One-Sample Test for Mean Angle If it is desired
to test that whether the population mean angle µa
is equal to a specified value, say µ0 . The
hypothesis test are
and
We determine the 1-a confidence interval under
H0, If µa lies outside of the confidence
interval, then H0 is reject.
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Circular Distributions
1. Testing of Significance of the Mean angle
Unimodal Distributions


Example9 The one sample test for the mean angle
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Circular Distributions
1. Testing of Significance of the Mean angle
Unimodal Distributions
One-Sample Test for Mean Angle
Example9(cont)


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Circular Distributions
2. Two-Sample Testing of Angular Distances

Angular distance is simply the shortest distance,
in angles, between two points on a circle. In
general, we shall refer to the angular distance
between angles a1 and a2 as da1-a2 (Ex, d
95-120 25). Angular distance are useful in
drawing inferences about departures of data from
a specified direction. We may have two samples,
sample1 and sample 2, each of which has
associated with it a specified angle, µ1 and µ2,
respectively. We may ask whether the angle
distances for sample 1 (d a1i- µ1 ) are
significantly different from those for sample 2
(d a2i- µ2 ) . As shown in Example 27.13, we rank
the angular distances of both samples combined
and then perform a Mann-Whitney test (see section
8.10 in reference). This was suggested by
Wallraff(1979).
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2. Two-Sample Testing of Angular Distances

Example10Two-sample testing of angular
distances
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2. Two-Sample Testing of Angular Distances

Example10(cont)
EXAMPLE 27.13 (cont)
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3. Parametric One-Sample Second-Order Analysis
of Angles
Hotelling (1931) procedure


For a second-order sample of k mean angles, we
can obtain general mean and with
equations on slides 20. Assuming that the
second-order sample comes from a bivariate normal
distribution i.e., a population in which both the
Xjs and Yjs are normal distributed.
The sums of squares and crossproducts of the k
means are
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Circular Distributions
3. Parametric One-Sample Second-Order Analysis
of Angles
Hotelling (1931) procedure (cont) Then we
can test the null hypothesis
from which the second-order sample came by using
as a test statistics


with the critical value being the one tailed F
with degrees of freedom of 2 and k-2,
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4. Nonparametric One-Sample Second-Order
Analysis of Angles


Moore(1980) procedure Moore has provided
a nonparametric modification of the Reyleigh
test, which can be used to test a sample mean
angles First rank the k vector lengths
from small to large r1,r2, .rk , then compute
Where is the rank of r.
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Circular Distributions
5. Parametric Two-Sample SecondOrder analysis

Batschelet (1978,1981150-154) explains how the
Hotelling procedure can be extended to consider
the hypothesis of equality of the means of two
populations of means (assuming each population to
be bivariate normal) . We proceed as in slides
20, obtaining and for each of two
samples. Then, we apply equations on slides 26 to
each of the two samples, obtaining the sums of
squares and crossproducts for two samples. Then
we calculate
And the null hypothesis of the two population
mean angles being equal is tested by
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5. Parametric Two-Sample SecondOrder analysis
Batschelet (1978,1981150-154) procedure (cont)

And the null hypothesis of the two population
mean angles being equal is tested by
where Nk1k2 , and F is one-tailed with 2 and
N-3 df.
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Circular Distributions
5. Parametric Two-Sample SecondOrder analysis
Batschelet (1978,1981150-154) procedure (cont)
Example11 Parametric two-sample second-order
analysis for testing the difference between mean
angles

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Circular Distributions
5. Parametric Two-Sample SecondOrder analysis
Batschelet (1978,1981150-154) procedure (cont)

Example11(cont)
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Circular Distributions
5. Parametric Two-Sample SecondOrder analysis

Batschelet (1978,1981150-154) procedure (cont)
Figure6. The data of Example 11. The open
circles indicate the ends of the seven mean
vectors of sample1, with mean indicated by the
broken line vectors. The solid circles
indicated the ten data of sample2, with their
mean shown as a solid-line data
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6. Nonparametric Paired-Sample Testing with
Angles

Circular data in a pairedsample
experimental design may be tested
nonparametrically by forming a single sample of
the paired differences, which can then be
subjected to the Moore procedure (slide 42).
First, calculate rectangular coordinates for each
paired difference
Second, for each of j paired differences, we
compute
Then, rank and perform Moore procedure .
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6. Nonparametric Paired-Sample Testing with
Angles
Example12The Moore test for paired data on a
circular scale of measurement

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6. Nonparametric Paired-Sample Testing with
Angles
Example12(cont)

n 10 Reject Null Hypothesis.
Note The critical value can be find
on page App198, Table B.39, in the reference book.
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7. Parametric Angular Correlation and Regression

• There two kinds of correlation involving angular
  data
• Angular-Angular Correlation both variables are
  measured on a
• circular scale, also called
  spherical correlation.
• Angular-Linear Correlation one variable is on
  a circle scale with
• the other measured on a linear
  scale, also called
• cylindrical correlation.

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7. Parametric Angular Correlation and Regression

Angular-Angular Correlation Fisher and Lee (1983)
presented a correlation coefficient
Fisher (1993151) gives an alternate computation
of
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7. Parametric Angular Correlation and Regression
Angular-Angular Correlation Test the significance
of -- Fingleton(1989303) procedure

The procedure involves computing an
additional n times for the sample, each time
eliminating a different one of the n pairs of a
and b data. Then calculate the mean and
variance of n additional s
And confidence limits for are obtained as
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Circular Distributions
7. Parametric Angular Correlation and
Regression

Example13 Angular-angular correlation
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7. Parametric Angular Correlation and Regression
Example13(cont)

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7. Parametric Angular Correlation and Regression
Angular-Linear Correlation From the work of
Mardia(1976) and Johnson Wehrly(1977) , the
angular-linear correlation
coefficient of an angular variable a and a linear
variable X is

where -- the correlation between X and the
cosine of a, -- the correlation
between X and the sine of a, --
the correlation between the sine and the cosine
of a.
they can be calculated by using equation
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7. Parametric Angular Correlation and Regression

Regression (Fisher, 1993139-140) Linearcircular
regression, in which the dependent variable (Y)
is linear and the independent variable (a)
circular, may be analyzed , by the regression
methods
Where b0 is the Y-intercept and b1 and b2 are
partial regression coefficients
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8. Goodness Fit Testing for Circular Distributions
Chi-square goodness of fit test

This test is used to test the goodness fit of a
theoretical to an observed circular frequency
distribution. The procedure is to determine each
expected frequency, , corresponding to each
observed frequency, , in each category
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8. Goodness Fit Testing for Circular Distributions
Example14 Chi-square goodness of fit for the
circular data

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8. Goodness Fit Testing for Circular Distributions
Watson (1961,1962) one-sample goodness of
fit test

To test the null hypothesis of uniformity, we
first transform each angular measurement, ,
by dividing it by 360
The test statistics called Watsons
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8. Goodness Fit Testing for Circular Distributions
Watson (1961,1962) one-sample goodness of
fit test

Example15 Watsons goodness of fit test
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8. Goodness Fit Testing for Circular Distributions
Watson (1961,1962) one-sample goodness of
fit test

Example15(cont)
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• Reference
• Biostatistical Analysis Jerrold H. Zar
  fourth edition.
• Topics in Circular Statistics S Rao
  Jammalamadaka (University of California, Santa
  Barbara, USA) Ashis SenGupta (Indian
  Statistical Institute, India).
• Statistical Analysis of Circular Data
  Nicholas I. Fisher (Division of Applied Physics,
  CSIRO, Canberra )