Measuring Entrainment:
Some Methods and Issues
J. Devin McAuley
Center for Neuroscience, Mind & Behavior
Department of Psychology
Bowling Green State University
Email: mcauley@bgnet.bgsu.edu
Entrainment Network III, Milton Keynes & Cambridge, UK, December 9th – 12th 2005
Outline of Talk
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A Few Examples of Entrainment
Entrainment Involves Circular Data
Statistics for Circular Data
What Can I Do With Circular Statistics?
What Can’t I Do?
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A Simple Example
Target T
...
(A) Stimulus Sequence
Produced Interval (P)
(B) Tapping Sequence
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A More Complex Example …
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A Mystery Example …
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Entrainment Involves Circular Data
• A simple way to describe any rhythmic
behavior is using a circle.
• Each point on the circle represents a position
in relative time (a phase angle).
• The start point is arbitrary.
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Polar versus Rectangular Coordinates
90
(x, y)
r

180
0, 360
270
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(0, 1)
90
(-1, 0)
(x, y)

180
0
(1,0)
x = cos
y = sin
270
(1, 0)
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A Simple Example
Target T
...
(A) Stimulus Sequence
Produced Interval (P)
(B) Tapping Sequence
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A Tale of Two Oscillators
Driven Oscillator
r
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
Driving Oscillator
r

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Case 1: Perfect Synchrony
Driven Oscillator
Driving Oscillator
 = 0
 = 0
Each Produced Tap
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Case 2: Taps Lag Tones
Driven Oscillator
Driving Oscillator
 = 45
 = 0
Each Produced Tap
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Case 3: Taps Ahead of Tones
Driven Oscillator
Driving Oscillator
 = 0
 = 315
Each Produced Tap
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Case 4: Entrainment
Driven Oscillator
Driving Oscillator
 = 0
 → ,
as n ↑
Each Produced Tap
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Why won’t linear statistics work?
• With circular data there is a cross-over
problem.
• For example, measured in degrees, the
linear mean of 359 and 1 is 180, not 0
• This problem arises no matter what the start
point is, and is independent of unit of
measurement.
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Statistics for Circular Data
• Descriptive Statistics
– Mean Direction, 
– Mean Resultant Length, R
– Circular Variance, V
• Inferential Statistical Tests
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90
1.0
0.5
180
0
0.0
-0.5
-1.0
-1.0
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-0.5
0.0
270
0.5
1.0
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(0, 1)
90
(-1, 0)
(x, y)

180
0
(1,0)
x = cos
y = sin
270
(1, 0)
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Calculating a Mean
(x1, y1)
(x2, y2)
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Calculating a Mean
(X, Y)
X = x1 + x2
Y = y1 + y2
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Mean Direction, 
n
Y   sin  j
j 1
 
tan 1 X
if X  0, Y  0
Y


if X  0, Y  0
2

 1 X
if X  0

  tan
Y
 1
tan X Y  2 if X  0, Y  0

if X  0, Y  0
undefined

 
 
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Mean Resultant Length, R
n
Y   sin  j
j 1
n
C   cos  j
R
j 1
R
X
RR
2
Y2

R
(Pythagorean Theorem)
n
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Circular Variance, V
V=1–R
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90
1.0
0.5
180
0
0.0
-0.5
-1.0
-1.0
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-0.5
0.0
270
0.5
1.0
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90
1.0
0.5
 = 50
180
R = 0.34
0
0.0
-0.5
-1.0
-1.0
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-0.5
0.0
270
0.5
1.0
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90
1.0
0.5
180
0
0.0
-0.5
-1.0
-1.0
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-0.5
0.0
270
0.5
1.0
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90
1.0
0.5
180
0
0.0
 = 344
R = 0.88
-0.5
-1.0
-1.0
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-0.5
0.0
270
0.5
1.0
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Statistics for Circular Data
• Descriptive Statistics
– Mean Direction, 
– Mean Resultant Length, R
– Circular Variance, V
• Inferential Statistical Tests
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Logic of Hypothesis Testing
• State Null & Alternative Hypotheses
• Determine Critical Value
– for pre-selected alpha level (e.g.,  = 0.05)
• Calculate Test Statistic
• If Test Statistic > Critical Value
– then Reject Null (e.g., p < 0.05)
– otherwise Retain Null
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Inferential Statistics
• Test for uniformity
• Test for unspecified mean direction
• Test for specified mean direction
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Logic of Hypothesis Testing
• State Null & Alternative Hypotheses
• Determine Critical Value
– for pre-selected alpha level (e.g.,  = 0.05)
• Calculate Test Statistic
• If Test Statistic > Critical Value
– then Reject Null (e.g., p < 0.05)
– otherwise Retain Null
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What can I do with circular stats?
(not an exhaustive list)
• Descriptive statistics
– Mean direction and length
– Variance, Standard Deviation
– Skewness, Kurtosis
• Inferential statistics
–
–
–
–
Uniformity, symmetry
Unspecified and specified mean direction
Comparison of two or more samples
Confidence intervals
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What can’t I do with circular stats?
• Circular statistics do not address sequential
dependencies.
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Stability Across Lifespan
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