Measuring Entrainment:Some Methods and Issues

J. Devin McAuleyJ. Devin McAuley

Center for Neuroscience, Mind & BehaviorCenter for Neuroscience, Mind & Behavior

Department of PsychologyDepartment of Psychology

Bowling Green State UniversityBowling Green State University

Email: Email: mcauley@bgnet.bgsu.edumcauley@bgnet.bgsu.edu

Entrainment Network III, Milton Keynes & Cambridge, UK, December 9 th – 12th 2005

J. Devin McAuley 2

Outline of Talk

• A Few Examples of EntrainmentA Few Examples of Entrainment

• Entrainment Involves Circular DataEntrainment Involves Circular Data

• Statistics for Circular DataStatistics for Circular Data

• What Can I Do With Circular Statistics?What Can I Do With Circular Statistics?

• What What Can’tCan’t I Do? I Do?

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A Simple Example

Target TTarget T

(A) Stimulus Sequence(A) Stimulus Sequence

Produced Interval (P)Produced Interval (P)

(B) Tapping Sequence(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

r

(x, y)

180 0, 360

270

90

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(1,0)

(1, 0)

(0, 1)

(-1, 0)

0

90

180

270

(x, y)

R =

1

x = cos

y = sin

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A Simple Example

Target TTarget T

(A) Stimulus Sequence(A) Stimulus Sequence

Produced Interval (P)Produced Interval (P)

(B) Tapping Sequence(B) Tapping Sequence

...

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A Tale of Two Oscillators

r

r

Driven Oscillator Driving Oscillator

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Case 1: Perfect Synchrony

= 0 = 0

Driven Oscillator Driving Oscillator

Each Produced Tap

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Case 2: Taps Lag Tones

= 0

= 45

Driven Oscillator Driving Oscillator

Each Produced Tap

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Case 3: Taps Ahead of Tones

= 0

= 315

Driven Oscillator Driving Oscillator

Each Produced Tap

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Case 4: Entrainment

= 0

→ ,

as n ↑

Driven Oscillator Driving Oscillator

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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-1.0

-0.5

0.0

0.5

1.0

-1.0 -0.5 0.0 0.5 1.0

90

0180

270

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(1,0)

(1, 0)

(0, 1)

(-1, 0)

0

90

180

270

(x, y)

R =

1

x = cos

y = sin

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Calculating a Mean

(x1, y1)(x2, y2)

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Calculating a Mean

X = x1 + x2

Y = y1 + y2

(X, Y)

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Mean Direction,

0,0

0,02tan

0tan

0,02

0,0tan

1

1

1

YXifundefined

YXifYX

XifYX

YXif

YXifYX

n

jjY

1

sin

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Mean Resultant Length, R

nRR

n

jjC

1

cos

n

jjC

1

cos

n

jjC

1

cos

n

jjY

1

sin

22 YXR (Pythagorean Theorem)

R R

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Circular Variance, V

V = 1 – R

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-1.0

-0.5

0.0

0.5

1.0

-1.0 -0.5 0.0 0.5 1.0

90

0180

270

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-1.0

-0.5

0.0

0.5

1.0

-1.0 -0.5 0.0 0.5 1.0

90

0180

270

= 50

R = 0.34

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90

0180

270

-1.0

-0.5

0.0

0.5

1.0

-1.0 -0.5 0.0 0.5 1.0

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90

0180

270

-1.0

-0.5

0.0

0.5

1.0

-1.0 -0.5 0.0 0.5 1.0

= 344

R = 0.88

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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

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

4-5 6-7 8-9 10-12 18-38 39-59 60-74 75+Age Range (years)

Mea

n R

(p

has

e co

up

lin

g)