James K. Lindsey

On the appropriateness of marginal models for repeated measurements in clinical trials

Although models developed directly to describe marginal distributions have become widespread in the analysis of repeated measurements, some of their disadvantages are not well enough known. These include producing profile curves that correspond to no possible individual, possibly showing that a treatment is superior on average when it is poorer for each individual subject, implicitly generating complex and implausible physiological explanations, including underdispersion in subgroups, and sometimes corresponding to no possible probabilistic data generating mechanism. We conclude that such marginal models may sometimes be appropriate for descriptive observational studies, such as sample surveys in epidemiology, but should only be used with great care in causal experimental settings, such as clinical trials.

On the use of corrections for overdispersion

In studying fluctuations in the size of a blackgrouse (Tetrao tetrix) population, an autoregressive model using climatic conditions appears to follow the change quite well. However, the deviance of the model is considerably larger than its number of degrees of freedom. A widely used statistical rule of thumb holds that overdispersion is present in such situations, but model selection based on a direct likelihood approach can produce opposing results. Two further examples, of binomial and of Poisson data, have models with deviances that are almost twice the degrees of freedom and yet various overdispersion models do not fit better than the standard model for independent data. This can arise because the rule of thumb only considers a point estimate of dispersion, without regard for any measure of its precision. A reasonable criterion for detecting overdispersion is that the deviance be at least twice the number of degrees of freedom, the familiar Akaike information criterion, but the actual presence of overdispersion should then be checked by some appropriate modelling procedure.

Analysing financial returns by using regression models based on non-symmetric stable distributions

The daily evolution of the price of Abbey National shares over a 10-week period is analysed by using regression models based on possibly non-symmetric stable distributions. These distributions, which are only known through their characteristic function, can be used in practice for interactive modelling of heavy-tailed processes. A regression model for the location parameter is proposed and shown to induce a similar model for the mode. Finally, regression models for the other three parameters of the stable distribution are introduced. The model found to fit best allows the skewness of the distribution, rather than the location or scale parameters, to vary over time. The most likely share return is thus changing over time although the region where most returns are observed is stationary.

Dynamic generalized linear models and repeated measurements

The dynamic generalized linear model for non-normal data is extended for use in repeated measurements, when series of observations are available for more than one individual. Examples are given for count and duration data.

A family of models for uniform and serial dependence in repeated measurements studies

Data arising from a randomized double-masked clinical trial for multiple sclerosis have provided particularly variable longitudinal repeated measurements responses. Specific models for such data, other than those based on the multivariate normal distribution, would be a valuable addition to the applied statisticians toolbox. A useful family of multivariate distributions can be generated by substituting the integrated intensity of one distribution into a second (outer) distribution. The parameters in the second distribution are then used to create a dependence structure among observations on a unit. These may either be a form of serial dependence for longitudinal data or of uniform dependence within clusters. These are respectively analogous to the Kalman filter of state space models and to copulas, but they have the major advantage that they do not require any explicit integration. One useful outer distribution for constructing such multivariate distributions is the Pareto distribution. Certain special models based on it have previously been used in event history analysis, but those considered here have much wider application.

Physical Properties of Particles of Ipratropium and Clenbuterol Generated by Equipment Suitable for the Inhalation of Drugs by Calves

When solutions of ipratropium and clenbuterol were atomised at 300 kPa and 450 kPa in equipment suitable for the inhalation of drugs by calves, the numbers, velocities and diameters of the particles produced were similar. When the pressure was increased to 600 kPa more of the particles were less than 2 microns in diameter and fewer were more than 7 microns in diameter, the fractions of the total mass of the solution generated in these size ranges were similarly increased and decreased, and the velocities of the particles were increased. At any given pressure, the numbers of particles of different sizes, and the proportions of the total mass generated, were similar for the solutions of ipratropium and clenbuterol, but a solution of saline produced more particles with a diameter less than 3 microns. Particles from the solution of ipratropium had the highest velocity and particles from the solution of clenbuterol had the lowest velocity.

Deposition in the distal parts of the bovine respiratory tract: assessment of equipment suitable for drug inhalation

The efficiency of equipment suitable for the inhalation of drugs by calves was assessed in six animals which inhaled radioisotopically labelled particles while suffering from reversible diffuse bronchoconstriction induced experimentally with 5-hydroxytryptamine and while they were breathing normally. Respiratory rates and data from pulmonary function tests and scintiscans were recorded during both investigations. After the first investigation, a mean (se) wash-out period of 9.8 (3.2) days was allowed. Under diffuse bronchoconstriction, the respiratory rate, the oscillatory resistance and the compliance of the respiratory system reached 282.1 (22.0), 161.1 (10.8) and 68.8 (2.7) per cent of their respective baseline values. When the calves were breathing normally these parameters did not change over time. The ratios (Cp/Ct) of the counts of γ-disintegrations in the peripheral part (Cp) of the lungs and in the total lung area (Ct) were not significantly different when comparing the results from the two investigations. The ratios of Cp/Ct in the left lungs did not differ significantly from those in the right lungs.

Normal Theory Models and Some Extensions

One of the most widely used tools in all of statistics is linear regression. This is often misnamed least squares regression, but a least squares estimation refers to a deterministic process, whereby the best straight line is fitted through a series of points. In statistical analysis, the interpretation is much different although the technical calculations remain the same. Normal theory linear regression carries the assumption that the response variable has a normal or Gaussian distribution:

Time Series: The Time Domain

f(y;\mu ,{{\sigma }^{2}}) = \exp [{{(y - \mu )}^{2}}/(2{{\sigma }^{2}})]/\sqrt {{2\pi {{\sigma }^{2}}}}