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Measuring Entrainment: Some Methods and Issues

This presentation by J. Devin McAuley from Bowling Green State University explores the concept of entrainment, particularly in relation to circular data. Key topics include examples of entrainment behavior, the unique challenges of circular statistics, and the applications of these statistical methods. The discussion covers calculating mean direction and resultant length, the logic of hypothesis testing, and the limitations of circular statistics, emphasizing their role in understanding rhythmic behaviors and synchronization phenomena.

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Measuring Entrainment: Some Methods and Issues

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  1. 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

  2. Outline of Talk • 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?

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

  4. A More Complex Example …

  5. A Mystery Example …

  6. 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. Polar versus Rectangular Coordinates 90 (x, y) r  0, 360 180 270

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

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

  10. A Tale of Two Oscillators Driven Oscillator Driving Oscillator r r  

  11. Case 1: Perfect Synchrony Driven Oscillator Driving Oscillator  = 0  = 0 Each Produced Tap

  12. Case 2: Taps Lag Tones Driven Oscillator Driving Oscillator  = 45  = 0 Each Produced Tap

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

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

  15. 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. Statistics for Circular Data • Descriptive Statistics • Mean Direction,  • Mean Resultant Length, R • Circular Variance, V • Inferential Statistical Tests

  17. 90 180 0 270

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

  19. Calculating a Mean (x1, y1) (x2, y2)

  20. Calculating a Mean (X, Y) X = x1 + x2 Y = y1 + y2

  21. Mean Direction, 

  22. (Pythagorean Theorem) Mean Resultant Length, R

  23. Circular Variance, V V = 1 – R

  24. 90 180 0 270

  25. = 50 R = 0.34 90 180 0 270

  26. 90 180 0 270

  27. = 344 R = 0.88 90 180 0 270

  28. Statistics for Circular Data • Descriptive Statistics • Mean Direction,  • Mean Resultant Length, R • Circular Variance, V • Inferential Statistical Tests

  29. 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. Inferential Statistics • Test for uniformity • Test for unspecified mean direction • Test for specified mean direction

  31. 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. 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. What can’t I do with circular stats? • Circular statistics do not address sequential dependencies.

  34. Stability Across Lifespan

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