Spectral Analysis of Replicated Biomedical Time Series

SUMMARY Standard methods of spectral analysis are adapted to the interpretation of biomedical time series data with replication across subjects. The methodology is applied to two sets of data consisting of concentrations of lutei nizing hormone in serial blood samples. For such data, the between-subject variability in periodogram ordinates at a given frequency is typically larger than would be implied by the usual asymptotic distribution theory for single series. We interpret this to mean that the underlying spectrum of the stochastic process representing the time variation in hormone concentration varies randomly between subjects. We describe simple random effects models to account for this extra variability and develop likelihood-based methods of inference, using a Monte Carlo integration method to evaluate the likelihood function. For our first data set, which comprises hormone concentrations in blood samples taken from eight subjects at 1 min intervals for 1 h, our model captures the qualitative behaviour of the between-subject variation in the spectrum. We conclude that there is a genuine high frequency component of variation in hormone concentrations and that the amplitude and frequency of this high frequency component vary between subjects. Our second data set relates to a similar sampling protocol, except that each subject is sampled before and after hormone replacement therapy. We conclude that this intervention has a significant effect on the spectrum.

[1]  P. Diggle.  An approach to the analysis of repeated measurements. , 1988, Biometrics.

[2]  J. Naylor,et al.  Applications of a Method for the Efficient Computation of Posterior Distributions , 1982 .

[3]  P. Bloomfield.  An exponential model for the spectrum of a scalar time series , 1973 .

[4]  M. B. Priestley,et al.  A Test for Non‐Stationarity of Time‐Series , 1969 .

[5]  Peter J. Diggle.  Comparing Estimated Spectral Densities Using GLIM , 1985 .

[6]  Marvin H. J. Guber.  Bayesian Spectrum Analysis and Parameter Estimation , 1988 .

[7]  T. Gasser,et al.  Statistical Tools to Analyze Data Representing a Sample of Curves , 1992 .

[8]  P. McCullagh,et al.  Generalized Linear Models, 2nd Edn. , 1990 .

[9]  P. Diggle,et al.  Monte Carlo Methods of Inference for Implicit Statistical Models , 1984 .

[10]  N Lange.  Statistical approaches to human brain mapping by functional magnetic resonance imaging. , 1996, Statistics in medicine.

[11]  S. Zeger,et al.  Longitudinal data analysis using generalized linear models , 1986 .

[12]  T. Anderson.  Statistical analysis of time series , 1974 .

[13]  Peter J. Diggle,et al.  A Non-Gaussian Model for Time Series with Pulses , 1989 .

[14]  R. Tibshirani,et al.  Generalized Additive Models , 1991 .

[15]  P. A. P. Moran,et al.  An introduction to probability theory , 1968 .

[16]  J. Royston,et al.  THE STATISTICAL ANALYSIS OF PULSATILE HORMONE SECRETION DATA , 1989, Clinical endocrinology.

[17]  Murray A. Cameron,et al.  Fitting Models to Spectra Using Regression Packages , 1987 .

[18]  Scott L. Zeger,et al.  Non‐linear Fourier Time Series Analysis for Human Brain Mapping by Functional Magnetic Resonance Imaging , 1997 .

[19]  G. Wahba.  Automatic Smoothing of the Log Periodogram , 1980 .

[20]  P. Whittle.  Curve and Periodogram Smoothing , 1957 .

[21]  Adrian F. M. Smith,et al.  Bayesian Analysis of Linear and Non‐Linear Population Models by Using the Gibbs Sampler , 1994 .

[22]  J. Rice.  On the estimation of the parameters of a power spectrum , 1979 .

[23]  P. J. Diggle,et al.  TESTS FOR COMPARING TWO ESTIMATED SPECTRAL DENSITIES , 1986 .

[24]  J. D. de Koning,et al.  Patterns of LH and FSH in men during high-frequency blood sampling. , 1987, The Journal of endocrinology.

[25]  D. B. Preston.  Spectral Analysis and Time Series , 1983 .

[26]  P. Bloomfield,et al.  A frequency domain analysis of trends in Dobson total ozone records , 1983 .

[27]  Donald B. Percival,et al.  Spectral Analysis for Physical Applications , 1993 .

[28]  J. Doob.  Stochastic processes , 1953 .

[29]  David R. Cox,et al.  A Note on the Efficiency of Least-squares Estimates , 1968 .

[30]  M. Pitt,et al.  Likelihood analysis of non-Gaussian measurement time series , 1997 .

[31]  D. Cox,et al.  On testing for serial correlation in large numbers of small samples , 1988 .

[32]  M. West,et al.  Bayesian forecasting and dynamic models , 1989 .

[33]  Comparison of EEGs before and after stunning of cattle taking account of animal-to-animal variation , 1992 .

[34]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[35]  Sylvia Frühwirth-Schnatter,et al.  Approximate Predictive Integrals for Dynamic Generalized Linear Models , 1992 .

[36]  C. Chatfield,et al.  Fourier Analysis of Time Series: An Introduction , 1977, IEEE Transactions on Systems, Man, and Cybernetics.

[37]  Bradley P. Carlin,et al.  Markov Chain Monte Carlo conver-gence diagnostics: a comparative review , 1996 .

[38]  C C Spicer,et al.  Statistical analysis of heart rates in subsequent victims of sudden infant death syndrome. , 1987, Statistics in medicine.

[39]  P. Diggle,et al.  Analysis of Longitudinal Data , 2003 .

[40]  Brian D. Ripley,et al.  Stochastic Simulation , 2005 .

[41]  J. Beran.  Fitting long-memory models by generalized linear regression , 1993 .

[42]  Martin Crowder,et al.  Gaussian Estimation for Correlated Binomial Data , 1985 .

[43]  C. Geyer,et al.  Constrained Monte Carlo Maximum Likelihood for Dependent Data , 1992 .

[44]  B. Imthurn,et al.  Evidence for in vivo high-frequency periodicity of plasma beta-endorphin in humans. , 1995, Hormone and metabolic research = Hormon- und Stoffwechselforschung = Hormones et metabolisme.

[45]  K. Chaloner,et al.  Bayesian Experimental Design: A Review , 1995 .