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Measuring Entrainment: Some Methods and Issues J. Devin McAuley Center for Neuroscience, Mind & Behavior Department of Psychology Bowling Green State University.

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Presentation on theme: "Measuring Entrainment: Some Methods and Issues J. Devin McAuley Center for Neuroscience, Mind & Behavior Department of Psychology Bowling Green State University."— Presentation transcript:

1 Measuring Entrainment: Some Methods and Issues J. Devin McAuley Center for Neuroscience, Mind & Behavior Department of Psychology Bowling Green State University Email: Email: mcauley@bgnet.bgsu.edumcauley@bgnet.bgsu.edu Entrainment Network III, Milton Keynes & Cambridge, UK, December 9 th – 12 th 2005

2 J. Devin McAuley2 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 Can’t I Do?What Can’t I Do?

3 J. Devin McAuley3 A Simple Example Target T (A) Stimulus Sequence Produced Interval (P) (B) Tapping Sequence...

4 J. Devin McAuley4 A More Complex Example …

5 J. Devin McAuley5 A Mystery Example …

6 J. Devin McAuley6 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.

7 J. Devin McAuley7 Polar versus Rectangular Coordinates r (x, y)  180  0 , 360  270  90 

8 J. Devin McAuley8 (1,0) (0, 1) (-1, 0)  00 90  180  270  (x, y) R = 1 x = cos  y = sin 

9 J. Devin McAuley9 A Simple Example Target T (A) Stimulus Sequence Produced Interval (P) (B) Tapping Sequence...

10 J. Devin McAuley10 A Tale of Two Oscillators  r  r Driven OscillatorDriving Oscillator

11 J. Devin McAuley11 Case 1: Perfect Synchrony  = 0   = 0  Driven OscillatorDriving Oscillator Each Produced Tap

12 J. Devin McAuley12 Case 2: Taps Lag Tones  = 0   = 45  Driven OscillatorDriving Oscillator Each Produced Tap

13 J. Devin McAuley13 Case 3: Taps Ahead of Tones  = 0   = 315  Driven OscillatorDriving Oscillator Each Produced Tap

14 J. Devin McAuley14 Case 4: Entrainment  = 0   → , as n ↑ Driven OscillatorDriving Oscillator Each Produced Tap

15 J. Devin McAuley15 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.

16 J. Devin McAuley16 Statistics for Circular Data Descriptive Statistics –Mean Direction,  –Mean Resultant Length, R –Circular Variance, V Inferential Statistical Tests

17 J. Devin McAuley17 90  00 180  270 

18 J. Devin McAuley18 (1,0) (0, 1) (-1, 0)  00 90  180  270  (x, y) R = 1 x = cos  y = sin 

19 J. Devin McAuley19 Calculating a Mean (x 1, y 1 ) (x 2, y 2 )

20 J. Devin McAuley20 Calculating a Mean X = x 1 + x 2 Y = y 1 + y 2 (X, Y)

21 J. Devin McAuley21 Mean Direction, 

22 J. Devin McAuley22 Mean Resultant Length, R (Pythagorean Theorem)

23 J. Devin McAuley23 Circular Variance, V V = 1 – R

24 J. Devin McAuley24 90  00 180  270 

25 J. Devin McAuley25 90  00 180  270   = 50  R = 0.34

26 J. Devin McAuley26 90  00 180  270 

27 J. Devin McAuley27 90  00 180  270   = 344  R = 0.88

28 J. Devin McAuley28 Statistics for Circular Data Descriptive Statistics –Mean Direction,  –Mean Resultant Length, R –Circular Variance, V Inferential Statistical Tests

29 J. Devin McAuley29 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

30 J. Devin McAuley30 Inferential Statistics Test for uniformity Test for unspecified mean direction Test for specified mean direction

31 J. Devin McAuley31 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

32 J. Devin McAuley32 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

33 J. Devin McAuley33 What can’t I do with circular stats? Circular statistics do not address sequential dependencies.

34 J. Devin McAuley34 Stability Across Lifespan

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