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Measuring Entrainment: Some Methods and Issues

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 9 th – 12 th 2005.

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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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