Estimation after subpopulation selection in adaptive seamless trials

During the development of new therapies, it is not uncommon to test whether a new treatment works better than the existing treatment for all patients who suffer from a condition (full population) or for a subset of the full population (subpopulation). One approach that may be used for this objective is to have two separate trials, where in the first trial, data are collected to determine if the new treatment benefits the full population or the subpopulation. The second trial is a confirmatory trial to test the new treatment in the population selected in the first trial. In this paper, we consider the more efficient two-stage adaptive seamless designs (ASDs), where in stage 1, data are collected to select the population to test in stage 2. In stage 2, additional data are collected to perform confirmatory analysis for the selected population. Unlike the approach that uses two separate trials, for ASDs, stage 1 data are also used in the confirmatory analysis. Although ASDs are efficient, using stage 1 data both for selection and confirmatory analysis introduces selection bias and consequently statistical challenges in making inference. We will focus on point estimation for such trials. In this paper, we describe the extent of bias for estimators that ignore multiple hypotheses and selecting the population that is most likely to give positive trial results based on observed stage 1 data. We then derive conditionally unbiased estimators and examine their mean squared errors for different scenarios.

Designing a sampling scheme to reveal correlations between weeds and soil properties at multiple spatial scales

Weeds tend to aggregate in patches within fields and there is evidence that this is partly owing to variation in soil properties. Because the processes driving soil heterogeneity operate at different scales, the strength of the relationships between soil properties and weed density would also be expected to be scale-dependent. Quantifying these effects of scale on weed patch dynamics is essential to guide the design of discrete sampling protocols for mapping weed distribution. We have developed a general method that uses novel within-field nested sampling and residual maximum likelihood (REML) estimation to explore scale-dependent relationships between weeds and soil properties. We have validated the method using a case study of Alopecurus myosuroides in winter wheat. Using REML, we partitioned the variance and covariance into scale-specific components and estimated the correlations between the weed counts and soil properties at each scale. We used variograms to quantify the spatial structure in the data and to map variables by kriging. Our methodology successfully captured the effect of scale on a number of edaphic drivers of weed patchiness. The overall Pearson correlations between A. myosuroides and soil organic matter and clay content were weak and masked the stronger correlations at >50 m. Knowing how the variance was partitioned across the spatial scales we optimized the sampling design to focus sampling effort at those scales that contributed most to the total variance. The methods have the potential to guide patch spraying of weeds by identifying areas of the field that are vulnerable to weed establishment.