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 luteinizing 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.