Maximin -optimal designs for binary longitudinal responses

Optimal designs for discrete-time survival models with random effects

This paper considers the optimal design for the frailty model with discrete-time survival endpoints in longitudinal studies. We introduce the random effects into the discrete hazard models to account for the heterogeneity between experimental subjects, which causes the observations of the same subject at the sequential time points being correlated. We propose a general design method to collect the survival endpoints as inexpensively and efficiently as possible. A cost-based generalized D ([Formula: see text])-optimal design criterion is proposed to derive the optimal designs for estimating the fixed effects with cost constraint. Different computation strategies based on grid search or particle swarm optimization (PSO) algorithm are provided to obtain generalized D ([Formula: see text])-optimal designs. The equivalence theorem for the cost-based D ([Formula: see text])-optimal design criterion is given to verify the optimality of the designs. Our numerical results indicate that the presence of the random effects has a great influence on the optimal designs. Some useful suggestions are also put forward for future designing longitudinal studies.

Optimal two-stage sampling for mean estimation in multilevel populations when cluster size is informative

To estimate the mean of a quantitative variable in a hierarchical population, it is logistically convenient to sample in two stages (two-stage sampling), i.e. selecting first clusters, and then individuals from the sampled clusters. Allowing cluster size to vary in the population and to be related to the mean of the outcome variable of interest (informative cluster size), the following competing sampling designs are considered: sampling clusters with probability proportional to cluster size, and then the same number of individuals per cluster; drawing clusters with equal probability, and then the same percentage of individuals per cluster; and selecting clusters with equal probability, and then the same number of individuals per cluster. For each design, optimal sample sizes are derived under a budget constraint. The three optimal two-stage sampling designs are compared, in terms of efficiency, with each other and with simple random sampling of individuals. Sampling clusters with probability proportional to size is recommended. To overcome the dependency of the optimal design on unknown nuisance parameters, maximin designs are derived. The results are illustrated, assuming probability proportional to size sampling of clusters, with the planning of a hypothetical survey to compare adolescent alcohol consumption between France and Italy.

Maximin optimal designs for cluster randomized trials

We consider design issues for cluster randomized trials (CRTs) with a binary outcome where both unit costs and intraclass correlation coefficients (ICCs) in the two arms may be unequal. We first propose a design that maximizes cost efficiency (CE), defined as the ratio of the precision of the efficacy measure to the study cost. Because such designs can be highly sensitive to the unknown ICCs and the anticipated success rates in the two arms, a local strategy based on a single set of best guesses for the ICCs and success rates can be risky. To mitigate this issue, we propose a maximin optimal design that permits ranges of values to be specified for the success rate and the ICC in each arm. We derive maximin optimal designs for three common measures of the efficacy of the intervention, risk difference, relative risk and odds ratio, and study their properties. Using a real cancer control and prevention trial example, we ascertain the efficiency of the widely used balanced design relative to the maximin optimal design and show that the former can be quite inefficient and less robust to mis-specifications of the ICCs and the success rates in the two arms.

Optimal designs in longitudinal trials with varying treatment effects and discrete-time survival endpoints

It is plausible to assume that the treatment effect in a longitudinal study will vary over time. It can become either stronger or weaker as time goes on. Here, we extend previous work on optimal designs for discrete-time survival analysis to trials with the treatment effect varying over time. In discrete-time survival analysis, subjects are measured in discrete time intervals, while they may experience the event at any point in time. We focus on studies where the width of time intervals is fixed beforehand, meaning that subjects are measured more often when the study duration increases. The optimal design is defined as the optimal combination of the number of subjects, the number of measurements for each subject, and the optimal proportion of subjects assigned to the experimental condition. We study optimal designs for different optimality criteria and linear cost functions. We illustrate the methodology of finding optimal designs using a clinical trial that studies the effect of an outpatient mental health program on reducing substance abuse among patients with severe mental illness. We observe that optimal designs depend to some extent on the rate at which group differences vary across time intervals and the direction of these changes over time. We conclude that an optimal design based on the assumption of a constant treatment effect is not likely to be efficient if the treatment effect varies across time.

Bayesian design for dichotomous repeated measurements with autocorrelation

In medicine and health sciences, binary outcomes are often measured repeatedly to study their change over time. A problem for such studies is that designs with an optimal efficiency for some parameter values may not be efficient for other values. To handle this problem, we propose Bayesian designs which formally account for the uncertainty in the parameter values for a mixed logistic model which allows quadratic changes over time. Bayesian D-optimal allocations of time points are computed for different priors, costs, covariance structures and values of the autocorrelation. Our results show that the optimal number of time points increases with the subject-to-measurement cost ratio, and that neither the optimal number of time points nor the optimal allocations of time points appear to depend strongly on the prior, the covariance structure or on the size of the autocorrelation. It also appears that for subject-to-measurement cost ratios up to five, four equidistant time points, and for larger cost ratios, five or six equidistant time points are highly efficient. Our results are compared with the actual design of a respiratory infection study in Indonesia and it is shown that, selection of a Bayesian optimal design will increase efficiency, especially for small cost ratios.

Web-based tools for finding optimal designs in biomedical studies

Experimental costs are rising and applications of optimal design ideas are increasingly applied in many disciplines. However, the theory for constructing optimal designs can be esoteric and its implementation can be difficult. To help practitioners have easier access to optimal designs and better appreciate design issues, we present a web site at http://optimal-design.biostat.ucla.edu/optimal/ capable of generating different types of tailor-made optimal designs for popular models in the biological sciences. This site also evaluates various efficiencies of a user-specified design and so enables practitioners to appreciate robustness properties of the design before implementation.

Optimal Experimental Design With Nesting of Persons in Organizations

This paper introduces optimal design of randomized experiments where individuals are nested within organizations, such as schools, health centers, or companies. The focus is on nested designs with two levels (organization, individual) and two treatment conditions (treated, control), with treatment assignment to organizations, or to individuals within organizations. For each type of assignment, a multilevel model is first presented for the analysis of a quantitative dependent variable or outcome. Simple equations are then given for the optimal sample size per level (number of organizations, number of individuals) as a function of the sampling cost and outcome variance at each level, with realistic examples. Next, it is explained how the equations can be applied if the dependent variable is dichotomous, or if there are covariates in the model, or if the effects of two treatment factors are studied in a factorial nested design, or if the dependent variable is repeatedly measured. Designs with three levels of nesting and the optimal number of repeated measures are briefly discussed, and the paper ends with a short discussion of robust design.

Accrual by groups in trials with discrete-time survival endpoints

BACKGROUND

In many fields of science, event status is often recorded in intervals or at discrete points in time and can be investigated in experimental settings. Conducting such trials requires thorough planning before they are actually performed.

PURPOSE

To investigate accrual by groups in a trial with discrete-time survival endpoints and to describe how to choose the number of accrual groups, the size of the accrual groups, and the duration of the trial to achieve a sufficient power level.

METHODS

In trials with multiple time periods, the event status is recorded at the end of each period, but the event may occur at any time between the time points the measurements are taken. Therefore, time is recorded discretely, but the underlying process is continuous. To find the risk of event occurrence in each time interval, a continuous-time survival function is used and the generalized linear model is applied.

RESULTS

It is observed that the combination of the number of accrual groups, the size of the accrual groups, and the duration of the trial that gives a sufficient power level depends on the shape of the continuous-time survival function, the proportion of subjects who have experienced the event after a fixed number of time periods, and the size of the treatment effect.

LIMITATIONS

The results of the study are only presented graphically, because there is no simple closed-form expression for finding the variance of the treatment effect. The authors provide MATLAB software to perform the power calculations.

CONCLUSIONS

More subjects should be recruited in each accrual group or more accrual groups should be included if the effect size or the proportion of the subjects who have experienced the event after a fixed number of time periods decreases, or the probability of the event occurrence is concentrated toward the end of the study duration.

Optimal designs for epidemiologic longitudinal studies with binary outcomes

Alternating presence and absence of a medical condition in human subjects is often modelled as an outcome of underlying process dynamics. Longitudinal studies provide important insights into research questions involving such dynamics. This article concerns optimal designs for studies in which the dynamics are modelled as a binary continuous-time Markov process. Either one or both the transition rate parameters in the model are to be estimated with maximum precision from a sequence of observations made at discrete times on a number of subjects. The design questions concern the choice of time interval between observations, the initial state of each subject and the choice between number of subjects versus repeated observations per subject. Sequential designs are considered due to dependence of the designs on the model parameters. The optimal time spacing can be approximated by the reciprocal of the sum of the two rates. The initial distribution of the study subjects should be taken into account when relatively few repeated samples per subject are to be collected. A study with a reasonably large size should be designed in more than one phase because there are then enough observations to be spent in the first phase to revise the time spacing for the subsequent phases.