Transcript Cosinor analysis of accident risk using SPSS’s regression

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)
Mean Stanford Sleepiness Score
1
Time (hrs)
13:00
01:00
13:00
3
01:00
13:00
01:00
13:00
01:00
SSS variation over four days
7
5
1
Time (hrs)
13:00
01:00
13:00
01:00
13:00
01:00
13:00
01:00
Mean Visual Analogue Sleepiness Score
VAS variation over four days
10
8
6
4
2
0
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?
• SSS
• VAS
• Yes!
F(2,157)=73.41, p<0.001; R2=48%
F(2,13)=86.67, p<0.001; R2 =53%
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:
• Acrophase:
A = AMP Cos()
B = -AMP Sin()
A = AMP Cos()
B = -AMP Sin()
Hence
AMP =
Hence
 = ArcTan(-B/A)
A2  B2
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)
9
Sleepiness Score
8
7
6
5
4
3
2
1
0
1:00
AM
7:00
AM
1:00
PM
Time of Day
7:00
PM
1:00
AM
% 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.) = |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, 0.025 standard error of prediction
SSS - 95% Confidence Intervals
9
8
7
6
5
4
3
lower
upper
predicted
mean
2
1
0
D A Y 1 D A Y 2 D A Y 3 D A Y 4
VAS 95% Confidence Intervals
14
12
10
Mean
Lower
Upper
Predicted
8
6
4
2
0
D A Y 1 D A Y 2 D A Y 3 D A Y 4
Rules of Thumb for Fit
• De Prins J, Waldura J (1993)
• Acceptable Fit
95% CI phase range < 30 degrees
SSS
VAS
19 degrees (from NLR)
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 score
Vigilance over the four days
0.5
0.4
0.3
0.2
0.1
0
D A Y 1 D A Y 2 D A Y 3 D A Y 4
Time of day
Results of vigilance analysis
• Linear regression
F(2,13)=1.02, p>0.35,
R2 = 1%
• NLR
Peak : 3-05am
95% CI of peak
(9-58pm , 8-03am)
Phase Range 151 degrees
No evidence of curviture
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