Balanced and strongly balanced Pk-designs

AbstractGiven a graph G, a G-decomposition of the complete graph Kv is a set of graphs, all isomorphic to G, whose edge sets partition the edge set of Kv. A G-decomposition of Kv is also called a G-design and the graphs of the partition are said to be the blocks. A G-design is said to be balanced if the number of blocks containing any given vertex of Kv is a constant.In this paper the concept of strongly balanced G-design is introduced and strongly balanced path-designs are studied. Furthermore, we determine the spectrum of those path-designs which are balanced, but not strongly balanced

Handling Covariates in the Design of Clinical Trials

There has been a split in the statistics community about the need for taking covariates into account in the design phase of a clinical trial. There are many advocates of using stratification and covariate-adaptive randomization to promote balance on certain known covariates. However, balance does not always promote efficiency or ensure more patients are assigned to the better treatment. We describe these procedures, including model-based procedures, for incorporating covariates into the design of clinical trials, and give examples where balance, efficiency and ethical considerations may be in conflict. We advocate a new class of procedures, covariate-adjusted response-adaptive (CARA) randomization procedures that attempt to optimize both efficiency and ethical considerations, while maintaining randomization. We review all these procedures, present a few new simulation studies, and conclude with our philosophy.Comment: Published in at http://dx.doi.org/10.1214/08-STS269 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Rank-order conjoint experiments: efficiency and design.

In a rank-order conjoint experiment, the respondent is asked to rank a number of alternatives instead of choosing the preferred one, as is the standard procedure in conjoint choice experiments. In this paper, we study the efficiency of those experiments and propose a D-optimality criterion for rank-order conjoint experiments to find designs yielding the most precise parameter estimators. For that purpose, an expression of the Fisher information matrix for the rank-ordered multinomial logit model is derived which clearly shows how much additional information is provided by each extra ranking step made by the respondent. A simulation study shows that Bayesian D-optimal ranking designs are slightly better than Bayesian D-optimal choice designs and (near-)orthogonal designs and perform considerably better than other commonly used designs in marketing in terms of estimation and prediction accuracy. Finally, it is shown that improvements of about 50% to 60% in estimation and prediction accuracy can be obtained by ranking a second alternative. If the respondent ranks a third alternative, a further improvement of 30%in estimation and prediction accuracy is obtained.

Simple models and concepts as tools for the study of sustained soil productivity in long-term experiments. II. Crop nutrient equivalents, balanced supplies of available nutrients, and NPK triangles

Two NPK factorial trials, one in Vietnam and one in The Netherlands were (re-)analyzed to find causes of success or failure with regard to sustained soil productivity, using the concept of crop nutrient equivalents (CNE). A (k)CNE is the quantity of a nutrient that, under conditions of balanced nutrition, has the same effect on yield as 1 (k)g of nitrogen. The percentages the nutrients take in the (k)CNE sum of N, P and K are plotted along the sides of a triangle. Soil, crop and input NPK are indicated in the triangle. Balanced crop NPK is found in the centre of the triangle, and required NPK inputs are on a straight line in the extension of the line trough the point of soil NPK and the centre. Experimental inputs were compared with inputs required for balanced NPK. In Vietnam, responses to P and soil available N:P:K pointed to severe shortage of P. Rice yields increased over time in dry but not in wet seasons. The lower yields in wet seasons were ascribed to insufficiently long periods between the dry and the next wet seasons for replenishment of labile soil P. In the Netherlands, four crops were grown in rotation on a former sea bottom. Only N had a strong effect on yield. Soil available N:P:K revealed low N, very high K and medium P. Recovery of fertilizer N was high because of capillary rise of groundwater and absence of leaching. In both trials, first-season chemical crop analysis would directly have detected disproportions of soil available N, P and K. This knowledge could have improved the experimental designs, optimized nutrient use efficiency and minimized losses of N and K to the environmen