Harmonic Regression in the Biological Setting Michael Gaffney, - - PowerPoint PPT Presentation

Harmonic Regression in the Biological Setting

Michael Gaffney, Ph.D., Pfizer Inc

Two primary aims of harmonic regression

• 1. To describe the timing (phase) or degree of the diurnal variation

(amplitude) of a physiological variable, e.g., blood pressure

the amplitude of mean data underestimates the true amplitude unless all subjects are in phase

• 2. To analyze the treatment effect over a specific time period such

as 24 hours in the presence of diurnal variability.

harmonic regression analysis yields specific hypotheses that relate directly to the treatment effect over 24 hours.

Two Harmonic Model

Rit = Mi + A1i Cos (2π t / 24 – T1i) + A2i Cos (2π t / 12 – T2i) Rit is the response variable for the subject i at hour t t = 1,…,24 Mi is the 24 hour mean rate for subject i A1i and T1i are the amplitude and peak time of the 24 hour harmonic A2i and T2i are the amplitude and peak time of the 12 hour harmonic

Fourier Coefficients

• 1. Rit = Mi + A1i Cos (2π t / 24 – T1i) + A2i Cos (2π t / 12 – T2i)

An alternative representation is

• 2. Rit = Mi + a1i Cos (2π t / 24 ) + b1i Sin (2π t / 24 )

+ a2i Cos (2π t / 12 ) + b2i Sin (2π t / 12 ) Where, a1i = A1i Cos T1i and b1i = A1i Sin T1i Consequently, A1i = ( a1i

2 + b1i 2 ) ½ and T1i = arc tan (b1i / a1i )

Diurnal Rhythm of Mean Systolic BP in 41 Subjects

Distribution of Peak Times Hour Systolic BP N % 0-4 2 5 4-8 2 5 8-12 12 29% 12-16 8 20% 16-20 13 32% 20-24 4 10% 41

Diurnal Rhythm of Systolic BP in 6 Individual Subjects

Peak-trough difference Subject Estimated* Observed** 1 39.1 38.2 2 35.6 32.7 3 22.8 25.7 4 38.1 37.8 5 33.3 33.1 6 25.0 27.1 Avg 32.3 32.4

The estimated peak-trough difference of the hourly means is 20 mmHg.

Mean of individual subject amplitudes compared to amplitudes of mean hourly measurements Peak-trough difference A1 A2 Estimated Observed S M S M S M Systolic BP 10.5 7.8 6.1 3.3 28.6 16.8 26.0

Conclusions

• Mean of the Fourier coefficients = Fourier coefficients of the mean data

but Mean of the transformed Fourier coefficients ≠ transformed Fourier coefficients

• f the mean data
• Diurnal rhythm of mean data does not adequately describe the diurnal rhythm
• f individual patients because individual patient harmonics are not in phase.
• Amplitude of mean data underestimates the mean of the amplitudes
• Timing of the diurnal rhythm of the mean data does not represent the timing
• f the diurnal rhythm of individual subjects

There is no “average” diurnal curve when subjects are not in phase

Treatment Effect over 24 Hours

• The UUI rates at hour t for 3 baseline days were averaged to obtain the

baseline UUI rate at hour t

• The UUI rates at hour t for 3 on-treatment days were averaged to obtain the

treatment UUI rate at hour t

• The 5-hour moving average UUI rate at hour t was obtained by

( Ut-2 + Ut-1 + Ut + Ut+1 + Ut+2 ) / 5

Properties of Moving Average

• The moving average transformation improves the harmonic fit of the

data.

• The moving average is a high frequency filter that will filter out noise

and result in a better estimation of low-frequency amplitudes.

• The amplitude is reduced after applying a moving average. The

peak time is not affected by the moving average transformation.

• With respect to inferences regarding treatment effects, the moving

average transformation has no effect because the standard deviation of the amplitude is reduced proportionally.

Example: 3-hour Moving Average

Xt = ( Xt-1 + Xt+ Xt+1 ) / 3 is the subject’s transformed measurement at hour t Xt = a cos 2π t / 24 + b sin 2π t / 24 Order time from the peak time for each subject, i.e., peak time is 0 for each subject. The harmonic regression on the 3-hour moving average is identical to the average of the

• riginal harmonic regression and the harmonic regressions with a time shift of + - 1 hour

Time shift 0: a= A b = 0 Time shift 1: Tan 15o = b* / a* therefore, [ (a*)2 + (a*Tan 15o)2 ] ½ = a a* = a / (1 + Tan2 15o ) ½ Therefore the amplitude of the 3-hour moving average data is: A [ 1 + 2 / (1 + Tan2 15o )½ ] / 3

Mean Baseline, Treatment and Change in Amplitudes for UUI rate - 5 Hour Moving Average

Placebo Active (N=416) (N=443) Variable Mean SD Mean SD Mean (B) 0.151 0.132 0.156 0.132 Mean (T) 0.105 0.125 0.065 0.089 C0

• 0.046 0.107
• 0.091 0.095

(p<0.0001) A1 (B) 0.089 0.064 0.091 0.069 A1 (T) 0.058 0.057 0.039 0.049 C1

• 0.031 0.065 -0.052 0.062

(p<0.0001) A2 (B) 0.056 0.039 0.057 0.041 A2 (T) 0.032 0.029 0.024 0.026 C2

• 0.024 0.037 -0.033 0.039

(p=0.001)

Change from baseline in UUI Rate by Peak time

Placebo High Dose Peak Time

• 0.095
• 0.150

(peak hour + - 4 hours) (p<0.0001) Off-Peak

• 0.017
• 0.055

(p<0.0001)