On the use of comparison regions in visualizing stochastic uncertainty in some two-parameter estimation problems.

When considering simultaneous inference for two parameters, it is very common to visualize stochastic uncertainty by plotting two-dimensional confidence regions. This allows us to test post hoc null hypotheses about a single point in a simple manner. However, in some applications the interest is not in rejecting hypotheses on single points, but in demonstrating evidence for the two parameters to be in a convex subset of the parameter space. The specific convex subset to be considered may vary from one post hoc analysis to another. Then it is of interest to have a visualization allowing to perform corresponding analyses. We suggest comparison regions as a simple tool for this task.

Constructing treatment selection rules based on an estimated treatment effect function: different approaches to take stochastic uncertainty into account have a substantial effect on performance.

BACKGROUND: Today we are often interested in the predictive value of a continuous marker with respect to the expected difference in outcome between a new treatment and a standard treatment. We can investigate this in a randomized control trial, allowing us to assess interactions between treatment and marker and to construct a treatment selection rule. A first step is often to estimate the treatment effect as a function of the marker value. A variety of approaches have been suggested for the second step to define explicitly the rule to select the treatment, varying in the way to take uncertainty into account. Little is known about the merits of the different approaches. METHODS: Four construction principles for the second step are compared. They are based on the root of the estimated function, on confidence intervals for the root, or on pointwise or simultaneous confidence bands. All of them have been used implicitly or explicitly in the literature. As performance characteristics we consider the probability to select at least some patients, the probability to classify patients with and without a benefit correctly, and the gain in expected outcome at the population level. These characteristics are investigated in a simulation study. RESULTS: As to be expected confidence interval/band based approaches reduce the risk to select patients who do not benefit from the new treatment, but they tend to overlook patients who can benefit. Simply using positivity of the estimated treatment effect function for selection implies often a larger gain in expected outcome. CONCLUSIONS: The use of 95% confidence intervals/bands in constructing treatment selection rules is a rather conservative approach. There is a need for better construction principles for treatment selection rules aiming to maximize the gain in expected outcome at the population level. Choosing a confidence level of 80% may be a first step in this direction.