Cosinor analysis of accident risk using SPSS’s regression procedures Peter Watson 31st October 1997 MRC Cognition & Brain Sciences Unit

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)

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?

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)

SSS variation over four days

VAS variation over four days

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)

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; T=trial period (in hours) under study = 24;  t = Residual

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)

Periodicity 24 hour Periodicity upheld via absence of Time by Day interactions SSS : F(9,81)=0.57, p>0.8 VAS : F(9,81)=0.63, p>0.7

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

SPSS:Regression: “Linear” Look at the combined sine and cosine Evidence of curviture about the mean? SSSF(2,157)=73.41, p<0.001; R 2 =48% VASF(2,13)=86.67, p<0.001; R 2 =53% Yes!

Fitting via SPSS NLR Estimates  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

Equivalence of NLR and “Linear” regression models Amplitude: A = AMP Cos(  ) B = -AMP Sin(  ) Hence AMP = Acrophase: A = AMP Cos(  ) B = -AMP Sin(  ) Hence  ArcTan(-B/A)

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 = -  

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

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

95% Confidence interval for peak Use SPSS NLR - estimates acrophase directly acrophase ± t 13,0.025 x standard error multiply endpoints by (=-24/2  ) Ie standard error(C.  ) = |C  x standard error(  )

Levels of Sleepiness CIs for peak sleepiness and % amplitude Stanford Sleepiness Score: 95% CI = (4-33,5-48), amplitude=97% Visual Analogue Score: 95% CI = (4-31,5-40), amplitude=129%

95% confidence intervals for predictions Using Multiple “Linear” Regression: Individual predictions in “statistics” option window This corresponds to prediction pred ± t 13, standard error of prediction

SSS - 95% Confidence Intervals

VAS 95% Confidence Intervals

Rules of Thumb for Fit De Prins J, Waldura J (1993) Acceptable Fit 95% CI phase range < 30 degrees SSS 19 degrees (from NLR) VAS 17 degrees (from NLR)

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

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

Vigilance over the four days

Results of vigilance analysis Linear regression F(2,13)=1.02, p>0.35, R 2 = 1% No evidence of curviture NLR Peak : 3-05am 95% CI of peak (9-58pm, 8-03am) Phase Range 151 degrees Amplitude 18%

Vigilance - linear over time Plot suggests no obvious periodicity Acrophase of 151 degrees > 30 degrees (badly inaccurate fit) Cyclic terms statistically nonsignificant, low R 2 Flat profile suggested by low % amplitude Vigilance, itself, may be linear with time

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

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

Thanks to helpful colleagues….. Avijit Datta Geraint Lewis Tom Manly Ian Robertson