The analysis of sequential measurements of bones

Analysing sequential bone measurements obtained in experiments in which the duration of observation varies between individuals is a common problem. Two solutions are presented which estimate the population mean rate of change in different ways. The first solution provides a least squares estimate and involves calculating a weighting factor for each individual observation which increases nonlinearly with the observation period. Computation of this solution requires knowledge of the measurement error and also the true variability in the rate of change between individuals, which is rarely if ever known. The second solution, based on cumulative sums, is much simpler, involving a weighting factor which is equal to the observation period. No prior knowledge of the measurement error or variability in the rate of change is required. An expression for the standard error of the computed mean rate of change is also derived. Monte Carlo simulations of a large number of experiments have confirmed that, within the limits of the model parameters tested, the second solution produces valid estimates of the mean rate of change and its standard error.