Models for estimating the change-point in gas exchange data.

In subjects undertaking an incremental exercise test to exhaustion, the onset of metabolic acidosis can be detected by an increased rate of carbon dioxide output (VCO2) relative to the rate of increase of oxygen uptake (VO2). To locate the change-point (the gas exchange threshold) in such subjects, a two-line regression model relating these two quantities has been used, where the location of the change-point is unknown. We argue that statistical models where the change-point is set on time (rather than VO2) are more appropriate. This is because VO2 is not monotone in time. We use novel statistical methodology of hidden Markov models to demonstrate the existence of the change-point. We use time series models, to estimate the position of the change-point. In these models distributions other than the multivariate normal are considered. For some subjects, the variance of VCO2 increases with time because of increasing ventilation and this is also modelled. The results are illustrated using gas exchange data on three healthy subjects who performed a 20 W min(-1) workrate ramp test.

Robust estimation of the median lethal dose.

Alternatives to M-estimation for robust estimation of the median lethal dose in biological assays are developed. A class of link functions based on the Student-t distribution is proposed, where degrees of freedom are estimated from the data by maximum likelihood. Other alternatives include slash and finite mixture distributions. For bioassays from a pharmaceutical company, these methods extend the standard probit and logistic models, as well as the Huber's M-estimator. They are also applied to several standard examples from the literature.

Dynamic modelling of a challenge-escalation cross-over study of treatment of capsaicin-induced coughing.

In a complex seven-period cross-over trial to study the effects of a drug in attenuating capsaicin-induced coughing, counts of numbers of coughs were recorded 32 times in each period. Subjects were subjected to four escalating levels of capsaicin at each of one and five hours after treatment, with counts of coughs being recorded in four one-minute intervals at each level. Such longitudinal count studies often show considerable individual variability about any regression curve that might be fitted. We develop a non-linear autoregressive model for such count data that also allows for overdispersion.

Some statistical issues in modelling pharmacokinetic data.

A fundamental assumption underlying pharmacokinetic compartment modelling is that each subject has a different individual curve. To some extent this runs counter to the statistical principle that similar individuals will have similar curves, thus making inferences to a wider population possible. In population pharmacokinetics, the compromise is to use random effects. We recommend that such models also be used in data rich situations instead of independently fitting individual curves. However, the additional information available in such studies shows that random effects are often not sufficient; generally, an autoregressive process is also required. This has the added advantage that it provides a means of tracking each individual, yielding predictions for the next observation. The compartment model curve being fitted may also be distorted in other ways. A widely held assumption is that most, if not all, pharmacokinetic concentration data follow a log-normal distribution. By examples, we show that this is not generally true, with the gamma distribution often being more suitable. When extreme individuals are present, a heavy-tailed distribution, such as the log Cauchy, can often provide more robust results. Finally, other assumptions that can distort the results include a direct dependence of the variance, or other dispersion parameter, on the mean and setting non-detectable values to some arbitrarily small value instead of treating them as censored. By pointing out these problems with standard methods of statistical modelling of pharmacokinetic data, we hope that commercial software will soon make more flexible and suitable models available.

A general family of distributions for longitudinal dependence with special reference to event histories.

Event histories play an increasingly important role in medical studies. Examples include times between recurrences of tumours, as with bladder cancer, and between repeated infections, as with chronic granulotomous disease. A general method for generating new distributions is proposed by introducing an intensity function into a density. This procedure yields, as special cases, several distributions already proposed in the literature. The families of distributions based on the Pareto distribution are of particular interest for event history analysis because of their relationship to the Laplace transform of a gamma distribution. They can yield multivariate distributions, with longitudinal (serial) dependence by a procedure similar to updating in the Kalman filter and with uniform dependence in a similar way to copulas. For longitudinal dependence, several such updating procedures are proposed.

Dropouts in longitudinal studies: definitions and models.

The widely used distinction of Little and Rubin (1) about types of randomness for missing data presents difficulties in its application to dropouts in longitudinal repeated measurement studies. In its place, a new typology of randomness for dropouts is proposed that relies on using a survival model for the dropout process. In terms of a stochastic process, dropping out is a change of state. Then, the longitudinal measures and dropout processes can be modeled simultaneously, each conditional on the complete previous history of both repeated measures and states. In this context, Poisson regression is used to fit various proportional hazards models, some of which are new, to the dropout process using the longitudinal measurements responses as time-varying covariates. As examples of longitudinal measurement studies displaying nonrandom dropout processes, a dental study of testosterone production in rats and clinical trials for treatment of gallstones and of depression are analyzed.

Modeling pharmacokinetic data using heavy-tailed multivariate distributions.

Pharmacokinetic studies of drug and metabolite concentrations in the blood are usually conducted as crossover trials, especially in Phases I and II. A longitudinal series of measurements is collected on each subject within each period. Dependence among such observations, within and between periods, will generally be fairly complex, requiring two levels of variance components, for the subjects and for the periods within subjects, and an autocorrelation within periods as well as a time-varying variance. Until now, the standard way in which this has been modeled is using a multivariate normal distribution. Here, we introduce procedures for simultaneously handling these various types of dependence in a wider class of distributions called the multivariate power exponential and Student t families. They can have the heavy tails required for handling the extreme observations that may occur in such contexts. We also consider various forms of serial dependence among the observations and find that they provide more improvement to our models than do the variance components. An integrated Ornstein-Uhlenbeck (IOU) stochastic process fits much better to our data set than the conventional continuous first-order autoregression, CAR(1). We apply these models to a Phase I study of the drug, flosequinan, and its metabolite.

Simultaneous modelling of flosequinan and its metabolite.

DESIGN: A randomised, double-blind, prospective, placebo-controlled four-way study of the pharmacokinetics of single oral doses of flosequinan. We do not report the placebo data in this paper. Flosequinan was given at doses of 50, 100 and 150 mg, with a 2-week wash-out between periods. Blood samples were taken at a series of times up to 96 h after dosing. SETTING: Clinical pharmacology unit in a pharmaceutical company. PARTICIPANTS: Eighteen healthy volunteers of both genders, aged from 18 years to 55 years. MAIN OUTCOME MEASURES: Plasma concentrations of flosequinan and of its metabolite, flosequinoxan. RESULTS: We demonstrate that it is possible to model parent and metabolite concentration time profiles simultaneously and, in doing so, to estimate the first-pass effect using data from an oral administration. In our modelling approach, we propose a reasonably wide class of statistical models, allowing for left censoring. CONCLUSIONS: A parent-metabolite model that ignores the first-pass results in misleading predictions in a case where significant first-pass metabolism occurs. Thus, in phase-I studies, the new approach described in this paper can provide additional knowledge that may be useful in future formal studies.

Generalized nonlinear models for pharmacokinetic data.

Phase I trials to study the pharmacokinetic properties of a new drug generally involve a restricted number of healthy volunteers. Because of the nature of the group involved in such studies, the appropriate distributional assumptions are not always obvious. These model assumptions include the actual distribution but also the ways in which the dispersion of responses is allowed to vary over time and the fact that small concentrations of a substance are not easily detectable and hence are left censored. We propose that a reasonably wide class of generalized nonlinear models allowing for left censoring be considered now that this is feasible with current computer power and sophisticated statistical packages. These modelling strategies are applied to a Phase I study of the drug flosequinan and its metabolite. This drug was developed for the treatment of heart failure. Because the metabolite also exhibits an active pharmacologic effect, study of both the parent drug and the metabolite is of interest.

Obtaining marginal estimates from conditional categorical repeated measurements models with missing data.

The most commonly used models for categorical repeated measurement data are log-linear models. Not only are they easy to fit with standard software but they include such useful models as Markov chains and graphical models. However, these are conditional models and one often also requires the marginal probabilities of responses, for example, at each time point in a longitudinal study. Here a simple method of matrix manipulation is used to derive the maximum likelihood estimates of the marginal probabilities from any such conditional categorical repeated measures model. The technique is applied to the classical Muscatine data set, taking into account the dependence of missingness on previous observed values, as well as serial dependence and a random effect.

Directly modelling matched case-control data.

Matching in case-control studies is a situation in which one wishes to make inferences about a parameter of interest in the presence of nuisance parameters. The usual approach is to apply a conditional likelihood. A bivariate latent class log-linear model for binomial responses is shown to yield a standard likelihood identical to the usual conditional one. This extension of the Rasch model for binary responses gives consistent estimates and a suitable likelihood function for cases matched with any fixed number of controls.

A review of some extensions to generalized linear models.

Although generalized linear models are reasonably well known, they are not as widely used in medical statistics as might be appropriate, with the exception of logistic, log-linear, and some survival models. At the same time, the generalized linear modelling methodology is decidedly outdated in that more powerful methods, involving wider classes of distributions, non-linear regression, censoring and dependence among responses, are required. Limitations of the generalized linear modelling approach include the need for the iterated weighted least squares (IWLS) procedure for estimation and deviances for inferences; these restrict the class of models that can be used and do not allow direct comparisons among models from different distributions. Powerful non-linear optimization routines are now available and comparisons can more fruitfully be made using the complete likelihood function. The link function is an artefact, necessary for IWLS to function with linear models, but that disappears once the class is extended to truly non-linear models. Restricting comparisons of responses under different treatments to differences in means can be extremely misleading if the shape of the distribution is changing. This may involve changes in dispersion, or of other shape-related parameters such as the skewness in a stable distribution, with the treatments or covariates. Any exact likelihood function, defined as the probability of the observed data, takes into account the fact that all observable data are interval censored, thus directly encompassing the various types of censoring possible with duration-type data. In most situations this can now be as easily used as the traditional approximate likelihood based on densities. Finally, methods are required for incorporating dependencies among responses in models including conditioning on previous history and on random effects. One important procedure for constructing such likelihoods is based on Kalman filtering.

Modeling the covariance structure in pharmacokinetic crossover trials.

Pharmacokinetic studies of drug and metabolite concentrations in the blood are usually conducted as crossover trials, especially in phases I and II. A longitudinal series of measurements is collected on each subject within each period. However, much of the dependence among such observations, within and between periods, is generally ignored in analyzing this type of data. Usually, only a random coefficient model is fitted for the parameters in the nonlinear mean function, along with allowing the variance to depend on the mean so that it changes over time. Here, we develop models to allow more fully for the structure of the crossover study. We introduce two levels of variance components, for the subjects and for the periods within subjects, and also an autocorrelation within periods. We also retain the time-varying variance, using a separate variance function for this, different from that for the mean. We apply this model to a phase I study of the drug flosequinan and its metabolite. This drug was developed for the treatment of heart failure. Because the metabolite also exhibits an active pharmacologic effect, study of both the parent drug and the metabolite is of interest. We find that the autocorrelation is the element in the covariance structure that most improves the fit of the model but that two levels of variance components can also be necessary.

Multivariate elliptically contoured distributions for repeated measurements.

The multivariate power exponential distribution, a member of the multivariate elliptically contoured family, provides a useful generalization of the multivariate normal distribution for the modeling of repeated measurements. Both light and heavy tailed distributions are included. The covariance matrix retains its interpretation so that it can easily be structured for serial dependence and several levels of variance components. A crossover trial on insulin applied to rabbits, with a series of repeated measurements within each period, is analyzed by means of this distribution using autocorrelation and two levels of variance components.

Response surfaces for overdispersion in the study of the conditions for fish eggs hatching.

Response surface methodology, originally developed for determining optimal conditions in industrial experiments, was early adapted to experiments in marine ecology. However, these involved studying the shape of the complete response surface, not only detecting the optimum, and often had counts or durations as the response variable. Thus, nonlinear, nonnormal response models were required. For counts, binomial and beta-binomial models have been used, the latter because of substantial overdispersion. In closely controlled experiments, overdispersion among units held under the same conditions might indicate that some mishap has occurred in conducting the study. One possible check is to model the dispersion as a second response surface. This procedure is used to show that overdispersion in fish egg hatching experiments has a biological explanation in that it occurs only under suboptimal hatching conditions.

Counts and times to events.

A common response recorded in medical statistics involves the occurrence of events. Often this is summarized as a count for each patient. Reasons are given for disaggregating the data as much as possible and instead studying the time to each event. Global counts may hide evolution of the state of the patient over time. This will appear as overdispersion which may wrongly be interpreted as differential frailty among patients. Interval censoring is only important if the rate of events is very high as compared to the time unit of observation chosen. Repeated times to events on the same patient will be interrelated and so should be appropriately treated, as for any repeated measure. Both serial dependence among successive times to events on the same patient and frailty among patients may be present.

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.

Choosing among generalized linear models applied to medical data.

When testing for a treatment effect or a difference among groups, the distributional assumptions made about the response variable can have a critical impact on the conclusions drawn. For example, controversy has arisen over transformations of the response (Keene). An alternative approach is to use some member of the family of generalized linear models. However, this raises the issue of selecting the appropriate member, a problem of testing non-nested hypotheses. Standard model selection criteria, such as the Akaike information criterion (AIC), can be used to resolve problems. These procedures for comparing generalized linear models are applied to checking for difference in T4 cell counts between two disease groups. We conclude that appropriate model selection criteria should be specified in the protocol for any study, including clinical trials, in order that optimal inferences can be drawn about treatment differences.

Analysis of the human sex ratio by using overdispersion models.

Data from nineteenth-century Germany are reanalyzed to determine how the probability for having a child of each sex changes with family size. The data, collected by A. Geissler, concern some 1 million birth registrations and 3.7 million births occurring in Saxony between 1876 and 1885. Three models are fitted to the data. "The multiplicative and beta-binomial models provide similar fits, substantially better than that of the double-binomial model. All models show that both the probability that the child is a boy and the dispersion are greater in larger families. There is also some indication that a point probability mass is needed for families containing children uniquely of one sex."

A study of interval censoring in parametric regression models.

Parametric models for interval censored data can now easily be fitted with minimal programming in certain standard statistical software packages. Regression equations can be introduced, both for the location and for the dispersion parameters. Finite mixture models can also be fitted, with a point mass on right (or left) censored observations, to allow for individuals who cannot have the event (or already have it). This mixing probability can also be allowed to follow a regression equation. Here, models based on nine different distributions are compared for three examples of heavily censored data as well as a set of simulated data. We find that, for parametric models, interval censoring can often be ignored and that the density, at centres of intervals, can be used instead in the likelihood function, although the approximation is not always reliable. In the context of heavily interval censored data, the conclusions from parametric models are remarkably robust with changing distributional assumptions and generally more informative than the corresponding non-parametric models.