Cosinor analysis of accident risk using SPSS

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Cosinor analysis of accident risk using SPSSs
regression procedures

• Peter Watson
• 31st October 1997
• MRC Cognition Brain Sciences Unit

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Aims Objectives

• To help understand accident risk we investigate 3
  alertness measures over time
• Two self-reported measures of sleep Stanford
  Sleepiness Score (SSS) and Visual Analogue Score
  (VAS)
• Attention measure Sustained Attention to
  Response Task (SART)

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Study

• 10 healthy Peterhouse college undergrads
• (5 male)
• Studied at 1am, 7am, 1pm and 7pm for four
  consecutive days
• How do vigilance (SART) and perceived vigilance
  (SSS, VAS) behave over time?

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Characteristics of Sleepiness

• Most subjects most sleepy early in morning or
  late at night
• Theoretical evidence of cyclic behaviour
• (ie repeated behaviour over a period of 24 hours)

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SSS variation over four days
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VAS variation over four days
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Aspects of cyclic behaviour

• Features considered
• Length of a cycle (period)
• Overall value of response (mesor)
• Location of peak and nadir (acrophase)
• Half the difference between peak and nadir scores
  (amplitude)

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Cosinor Model - cyclic behaviour

• f(t) M AMP.Cos(2?t ?) ?t
• T
• Parameters of Interest
• f(t) sleepiness score
• M intercept (Mesor)
• AMP amplitude ?phase Ttrial period (in
  hours) under study 24 ?t Residual

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Period, T

• May be estimated
• Previous experience (as in our example)
• Constrained so that Peak and Nadir are T/2
• hours apart (12 hours in our sleep example)

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Periodicity

• 24 hour Periodicity upheld via absence of Time by
  Day interactions
• SSS F(9,81)0.57, pgt0.8
• VAS F(9,81)0.63, pgt0.7

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Fitting using SPSS linear regression

• For g(t)2?t/24 and since
• Cos(g(t)?) Cos(?)Cos(g(t))-Sin(?)Sin(g(t))
• it follows the linear regression
• f(t) M A.Cos(2?t/24) B.Sin(2?t/24)
• is equivalent to the above single cosine function
  - now fittable in SPSS linear regression
  combining Cos and Sine function

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SPSSRegression Linear

• Look at the combined sine and cosine
• Evidence of curviture about the mean?
• SSS F(2,157)73.41, plt0.001 R248
• VAS F(2,13)86.67, plt0.001 R2 53
• Yes!

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Fitting via SPSS NLR

• Estimates f, AMP and M
• SSS Peak at 5-11am
• VAS Peak at 5-05am
• M not generally of interest
• Can also obtain CIs for AMP and Peak sleepiness
  time

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Equivalence of NLR and Linear regression models

• Amplitude
• A AMP Cos(f)
• B -AMP Sin(f)
• Hence
• AMP

• Acrophase
• A AMP Cos(f)
• B -AMP Sin(f)
• Hence
• f ArcTan(-B/A)

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Model terms

• Amplitude
• 1/2(peak-nadir)
• Mesor M
• Mean Response

• (Acro)Phase ?
• time of peak in 24 hour cycle
• In hours peak -? 24
• 2?
• In degrees
• peak -? 360
• 2?

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Fitted Cosinor Functions (VAS in black SSS in
red)
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Amplitude

• Amplitude 100 x (Peak-Nadir)
• overall mean
• 100 x 2 AMP
• MESOR

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95 Confidence interval for peak

• Use SPSS NLR - estimates acrophase directly
• acrophase t13,0.025 x standard error
• multiply endpoints by -3.82 (-24/2?)
• Ie
• standard error(C.?) C?x standard error(?)

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Levels of Sleepiness

• CIs for peak sleepiness and amplitude
• Stanford Sleepiness Score
• 95 CI (4-33,5-48), amplitude97
• Visual Analogue Score
• 95 CI (4-31,5-40), amplitude129

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95 confidence intervals for predictions

• Using Multiple Linear Regression
• Individual predictions in statistics option
  window
• This corresponds to prediction
• pred t 13, 0.025 standard error of prediction

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SSS - 95 Confidence Intervals
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VAS 95 Confidence Intervals
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Rules of Thumb for Fit

• De Prins J, Waldura J (1993)
• Acceptable Fit
• 95 CI phase range lt 30 degrees
• SSS 19 degrees (from NLR)
• VAS 17 degrees (from NLR)

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Conclusions

• Perceived alertness has a 24 hour cycle
• No Time by Day interaction - alertness consistent
  each day
• We feel most sleepy around early morning

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Unperceived Vigilance

• Vigilance task (same 10 students as sleep
  indices)
• Proportion of correct responses to an attention
  task at 1am, 7am, 1pm and 7pm over 4 days

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Vigilance over the four days
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Results of vigilance analysis

• Linear regression
• F(2,13)1.02, pgt0.35,
• R2 1
• No evidence of curviture

• NLR
• Peak 3-05am
• 95 CI of peak
• (9-58pm , 8-03am)
• Phase Range 151 degrees
• Amplitude 18

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Vigilance - linear over time

• Plot suggests no obvious periodicity
• Acrophase of 151 degrees gt 30 degrees (badly
  inaccurate fit)
• Cyclic terms statistically nonsignificant, low R2
• Flat profile suggested by low amplitude
• Vigilance, itself, may be linear with time

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Polynomial Regression

• An alternative strategy is the fitting of cubic
  polynomials
• Similar results to cosinor functions
• two turning points for perceived sleepiness
• no turning points (linear) for attention measure

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Conclusions

• Cosinor analysis is a natural way of modelling
  cyclic behaviour
• Can be fitted in SPSS using either linear or
  nonlinear regression procedures

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Thanks to helpful colleagues..

• Avijit Datta
• Geraint Lewis
• Tom Manly
• Ian Robertson