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

1

3

5

70

1:0

0

13

:00

01

:00

13

:00

01

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13

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13

:00

Time (hrs)

Mea

n S

tan

ford

Sle

epin

ess

Sco

re

1

VAS variation over four days

0

2

4

6

8

100

1:0

0

13

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01

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13

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Time (hrs)

Me

an

Vis

ua

l An

alo

gu

e S

lee

pin

ess

Sco

re

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(2t + ) + 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)=2t/24 and since

Cos(g(t)+) = Cos()Cos(g(t))-Sin()Sin(g(t))

it follows the linear regression:

f(t) = M + A.Cos(2t/24) + B.Sin(2t/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?

• SSS F(2,157)=73.41, p<0.001; R2=48%• VAS F(2,13)=86.67, p<0.001; R2 =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)22 BA

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

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

0

1

2

3

4

5

6

7

8

9

1:00AM

7:00AM

1:00PM

7:00PM

1:00AM

Time of Day

Sle

ep

ine

ss

Sc

ore

% 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 ± t13,0.025 x standard error

• multiply endpoints by -3.82 (=-24/2)

• Ie

standard error(C.) = |Cx 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, 0.025 standard error of prediction

SSS - 95% Confidence Intervals

0

1

2

3

4

5

6

7

8

9

D A Y 1 D A Y 2 D A Y 3 D A Y 4

lowerupperpredictedmean

VAS 95% Confidence Intervals

0

2

4

6

8

10

12

14

D A Y 1 D A Y 2 D A Y 3 D A Y 4

MeanLowerUpperPredicted

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

0

0.1

0.2

0.3

0.4

0.5

D A Y 1 D A Y 2 D A Y 3 D A Y 4

Time of day

Vigi

lanc

e sc

ore

Results of vigilance analysis

• Linear regression

F(2,13)=1.02, p>0.35,

R2 = 1%

No evidence of curviture

• NLR

Peak : 3-05am

95% CI of peak

(9-58pm , 8-03am)

Phase Range 151 degrees

Amplitude 18%

Vigilance - linear over time

• Plot suggests no obvious periodicity

• Acrophase of 151 degrees > 30 degrees (badly inaccurate fit)

• Cyclic terms statistically nonsignificant, low R2

• 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