The Statistical Analysis of Circadian Phase and Amplitude in Constant-Routine Core-Temperature Data

Abstract

Accurate estimation of the phases and amplitude of the endogenous circadian pacemaker from constant-routine core-temperature series is crucial for making inferences about the properties of the human biological clock from data collected under this protocol. This paper presents a set of statistical methods based on a harmonic-regression-plus-correlated-noise model for estimating the phases and the amplitude of the endogenous circadian pacemaker from constant-routine core-temperature data. The methods include a Bayesian Monte Carlo procedure for computing the uncertainty in these circadian functions. We illustrate the techniques with a detailed study of a single subject's core-temperature series and describe their relationship to other statistical methods for circadian data analysis. In our laboratory, these methods have been successfully used to analyze more than 300 constant routines and provide a highly reliable means of extracting phase and amplitude information from core-temperature data.

Keywords

Bayesian analysis circadian rhythms correlated noise harmonic regression internal time Kalman filter maximum-likelihood estimation Monte Carlo methods

References

Apostol, T.M. (1974) Mathematical Analysis, 2nd ed. , Addison-Wesley, Reading, MA.

Atkinson, K.E. (1978) An Introduction to Numerical Analysis , Wiley,

New York.

Bingham, C.B. Abrogast, C.G. Guillaume, J. K. Lee , and F. Halberg (1982) Inferential statistical methods for estimating and comparing parameters Chronobiologia 9: 397-439.

Box, G.E.P. , and G.C. Tiao (1973) Bayesian Inference in Statistical Analysis , Addison-Wesley, Reading, MA.

Brown, E.N. (1987a) Estimating Circadian Phase, MD thesis , Harvard Medical School, Boston .

Brown, E.N. (1987b) Identification and Estimation of Differential Equation Models for Circadian Data, PhD dissertation Department of Statistics, Harvard Umversity.

Brown, E.N. (1990) A note on the asymptotic distribution of the parameter estimates for the harmonic regression model Biometrika 77: 653-655.

Brown, E.N. , and C.A. Czeisler (1985) A method for determining the phase position of the deep circadian oscillator and determining a confidence interval Sleep Res. 14: 290.

Czeisler, C.A. , J.S. Allan , S.H. Strogatz , J.M. Ronda , R Sanchez , C.D. Rios , W.F. Freitag , G.S. Richardson , and R.E. Kronauer (1986) Bright light resets the human circadian pacemaker independent of the timing of the sleep- wake cycle Science 233: 667-671.

10.

Czeisler, C.A. , E.N. Brown , J.M. Ronda , G.S. Richardson , R.E. Kronauer , and W.F. Freitag (1985) A clinical method to assess the endogenous circadian phase (ecp) of the deep circadian oscillator in man Sleep Res. 14: 295.

11.

Czeisler, C.A. , M.P. Johnson , J.F. Duffy , E.N. Brown , J.M. Ronda , and R.E. Kronauer (1990) Exposure to bright light and darkness to treat physiologic maladaption to night work. N Eng. J. Med. 322: 1253-1259.

12.

Czeisler, C.A. , R.E. Kronauer , J.S. Allan , J.F. Duffy , M.E. Jewett , E.N. Brown , and J.M. Ronda (1989) Bright light induction of strong (Type O) resetting of the human circadian pacemaker Science 244: 1328-1332.

13.

Degroot, M.H. (1975) Probability and Statistics, Addison-Wesley, Reading, MA.

14.

Deprins, J. , G Cornelissen , D Hillman , F Halberg , and C. Van Dijck (1981) Harmonic interpolation yields paraphases and orthophases for biological rhythms In Proceedings of the XIII International Conference of the International Society of Chronobiology , F. Halberg, L. E. Scheving, E. W. Powell, and D. K. Hayes, eds., pp. 333-343, Il Ponte, Milan.

15.

Fuller, W.A. (1976) Introduction to Statistical Time-Series , Wiley, New York.

16.

Glotzbach, S.F. , and H.C. Heller (1989) Sleep and thermoregulation In Principles and Practice of Sleep Medicine, M. H. Kryger, ed., pp. 300-309, W. B. Saunders, Philadelphia.

17.

Greenhouse, J.B. , R Kass , and R Tsay ( 1987) Fitting nonlinear models with ARMA errors to biological rhythm data Stat. Med. 6: 167-183.

18.

Halberg, F. , E. A. Johnson , W Nelson , W Runge , and R Sothern (1972) Autorhythmometry procedures for physiologic self-measurements and their analysis Physiol. Teacher 1: 1-11.

19.

Halberg, F. , Y. L. Tong , and E. A. Johnson ( 1967) Circadian system phase, an aspect of temporal morphology ; procedures and illustrative examples: Proceedings of an International Congress of Anatomists. In The Cellular Aspect of Biorhythms: Symposium on Biorhythms , pp. 20-48, Springer-Verlag , Berlin.

20.

Hoel, P.G. (1961) Asymptotic efficiency in polynomial estimation Ann. Math. Stat. 32: 1042-1047.

21.

Jones, R.H. (1980) Maximum likelihood fitting of ARMA models to time-series with missing observations Technometrics 22: 389-395.

22.

Kronauer, R.E. (1990) A quantitative model for the effects of light on the amplitude and phase of the deep circadian pacemaker, based on human data In Sleep '90, J. A. Home, ed., pp. 306-309, Pontenagel Press, Bochum, Germany.

23.

Kronauer, R.E. , C.A.Czfisler , and E.N.Brown (1989) Circadian modulation of circadian pacemaker responsiveness to light—an important consideration in the use of bright light Sleep Res. 18: 424.

24.

Mills, J.N. , D.S.Minors , and J.M.Waterhouse (1978) Adaptation to abrupt time shifts of the oscillator controlling human circadian rhythms J. Physiol. 285: 455-470.

25.

Minors, D.S. , and J.M.Waterhouse (1984) The use of the constant routines in unmasking the endogenous component of the human circadian rhythms Chronobiol. Int. 1: 205-216.

26.

Morris, M.D. , and S.F.Ebey (1984) An interesting property of the sample mean under a first-order autoregressive model Am. Stat. 38: 127-129.

27.

Press, S.J. (1989) Bayesian Statistics: Principles, Models and Applications, Wiley, New York.

28.

Priestley, M.B. (1981) Spectral Analysis and Time Series , Academic Press, London.

29.

Ramos, E.R. (1988) Resampling Methods for Time Series, PhD dissertation, Department of Statistics, Harvard University .

30.

Shanahan, T.L. , and C.A. Czeisler (1991) Light exposure induces equivalent phase shifts of the endogenous circadian rhythms of circulating plasma melatonin and core body temperature in men J. Clin. Endocrinol. Metab. 73: 227-235.

31.

Shumway, R.H. (1988) Applied Statistical Time Series Analysis , Prentice-Hall, Englewood Cliffs, NJ .

32.

Tong, Y.L. (1976) Parameter estimation in studying circadian rhythms Biometrics 32: 85-94.

33.

Tong, Y.L. , W.L.Nelson , R.B.Sothern , and F.Halberg (1977) Estimation of the orthophase timing of the high values on a nonsinusoidal rhythm—il— lustrated by the best timing for experimental chemotherapy In Proceedings of the XII International Conference of the International Society of Chronobiology , pp. 765-769, Il Ponte, Milan .

34.

Van Cauter , E (1979) Method for characterization of 24-h temporal variation of blood components Am. J. Physiol. 237: E255-E264.

35.

Wecker, W.E. , and Ansley, C.F. (1983) The signal extraction approach to nonlinear regression and spline smoothing J. Am. Stat. Assoc. 78: 81-89.

36.

Wever, R.A. (1979) The Circadian System of Man, Springer-Verlag, Berlin.