Estimating a preference-based index for mental health from the Recovering Quality of Life (ReQoL) measure : valuation of ReQoL-UI

Objectives There are increasing concerns about the appropriateness of generic preference-based measures to capture health benefits in the area of mental health. This study estimates preference weights for a new measure, Recovering Quality of Life (ReQoL-10), to better capture the benefits of mental health care. Methods Psychometric analyses of a larger sample of mental health service users (n = 4266) using confirmatory factor analyses and item response theory (IRT) were used to derive a health state classification system and inform the selection of health states for utility assessment. A valuation survey with members of the UK public representative in terms of age, gender and region was conducted using face-to-face interviewer administered time-trade-off (TTO) with props. A series of regression models were fitted to the data and the best performing model selected for the scoring algorithm. Results The ReQoL-UI classification system comprises six mental health items and one physical health (PH) item. Sixty-four health states were valued by 305 participants. The preferred model was a random effects model, with significant and consistent coefficients and best model fit. Estimated utilities modelled for all health states ranged from -0.195 (state worse than dead) to 1 (best possible state). Conclusions The development of the ReQoL-UI is based on a novel application of IRT methods for generating the classification system and selecting health states for valuation. Conventional TTO was used to elicit utility values that are modelled to enable the generation of QALYs for use in cost-utility analysis of mental health interventions

Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams

We consider the problem of finding the number of matrices over a finite field with a certain rank and with support that avoids a subset of the entries. These matrices are a q-analogue of permutations with restricted positions (i.e., rook placements). For general sets of entries these numbers of matrices are not polynomials in q (Stembridge 98); however, when the set of entries is a Young diagram, the numbers, up to a power of q-1, are polynomials with nonnegative coefficients (Haglund 98). In this paper, we give a number of conditions under which these numbers are polynomials in q, or even polynomials with nonnegative integer coefficients. We extend Haglund's result to complements of skew Young diagrams, and we apply this result to the case when the set of entries is the Rothe diagram of a permutation. In particular, we give a necessary and sufficient condition on the permutation for its Rothe diagram to be the complement of a skew Young diagram up to rearrangement of rows and columns. We end by giving conjectures connecting invertible matrices whose support avoids a Rothe diagram and Poincar\'e polynomials of the strong Bruhat order.Comment: 24 pages, 9 figures, 1 tabl

An update on the Hirsch conjecture

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n - d. Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite scarce. Most notably, no polynomial upper bound is known for the diameters that are conjectured to be linear. In contrast, very few polytopes are known where the bound $n-d$ is attained. This paper collects known results and remarks both on the positive and on the negative side of the conjecture. Some proofs are included, but only those that we hope are accessible to a general mathematical audience without introducing too many technicalities.Comment: 28 pages, 6 figures. Many proofs have been taken out from version 2 and put into the appendix arXiv:0912.423

Task swapping networks in distributed systems

In this paper we propose task swapping networks for task reassignments by using task swappings in distributed systems. Some classes of task reassignments are achieved by using iterative local task swappings between software agents in distributed systems. We use group-theoretic methods to find a minimum-length sequence of adjacent task swappings needed from a source task assignment to a target task assignment in a task swapping network of several well-known topologies.Comment: This is a preprint of a paper whose final and definite form is published in: Int. J. Comput. Math. 90 (2013), 2221-2243 (DOI: 10.1080/00207160.2013.772985