On the reduction theory of binary forms

Cremona developed a reduction theory for binary forms of degree 3 and 4 with integer coefficients, the motivation in the case of quartics being to improve 2-descent algorithms for elliptic curves over Q. In this paper we extend some of these results to forms of higher degree. One application of this is to the study of hyperelliptic curves.

RoMEO Studies 8: self-archiving: the logic behind the colour-coding used in the Copyright Knowledge Bank

Purpose – The purpose of this research is to show how the self-archiving of journal papers is a major step towards providing open access to research. However, copyright transfer agreements (CTAs) that are signed by an author prior to publication often indicate whether, and in what form, self-archiving is allowed. The SHERPA/RoMEO database enables easy access to publishers' policies in this area and uses a colour-coding scheme to classify publishers according to their self-archiving status. The database is currently being redeveloped and renamed the Copyright Knowledge Bank. However, it will still assign a colour to individual publishers indicating whether pre-prints can be self-archived (yellow), post-prints can be self-archived (blue), both pre-print and post-print can be archived (green) or neither (white). The nature of CTAs means that these decisions are rarely as straightforward as they may seem, and this paper describes the thinking and considerations that were used in assigning these colours in the light of the underlying principles and definitions of open access. Approach – Detailed analysis of a large number of CTAs led to the development of controlled vocabulary of terms which was carefully analysed to determine how these terms equate to the definition and “spirit” of open access. Findings – The paper reports on how conditions outlined by publishers in their CTAs, such as how or where a paper can be self-archived, affect the assignment of a self-archiving colour to the publisher. Value – The colour assignment is widely used by authors and repository administrators in determining whether academic papers can be self-archived. This paper provides a starting-point for further discussion and development of publisher classification in the open access environment.

Calculation of sample size for stroke trials assessing functional outcome: comparison of binary and ordinal approaches

Background Many acute stroke trials have given neutral results. Sub-optimal statistical analyses may be failing to detect efficacy. Methods which take account of the ordinal nature of functional outcome data are more efficient. We compare sample size calculations for dichotomous and ordinal outcomes for use in stroke trials. Methods Data from stroke trials studying the effects of interventions known to positively or negatively alter functional outcome – Rankin Scale and Barthel Index – were assessed. Sample size was calculated using comparisons of proportions, means, medians (according to Payne), and ordinal data (according to Whitehead). The sample sizes gained from each method were compared using Friedman 2 way ANOVA. Results Fifty-five comparisons (54 173 patients) of active vs. control treatment were assessed. Estimated sample sizes differed significantly depending on the method of calculation (Po00001). The ordering of the methods showed that the ordinal method of Whitehead and comparison of means produced significantly lower sample sizes than the other methods. The ordinal data method on average reduced sample size by 28% (inter-quartile range 14–53%) compared with the comparison of proportions; however, a 22% increase in sample size was seen with the ordinal method for trials assessing thrombolysis. The comparison of medians method of Payne gave the largest sample sizes. Conclusions Choosing an ordinal rather than binary method of analysis allows most trials to be, on average, smaller by approximately 28% for a given statistical power. Smaller trial sample sizes may help by reducing time to completion, complexity, and financial expense. However, ordinal methods may not be optimal for interventions which both improve functional outcome

Use of ordinal outcomes in vascular prevention trials: comparison with binary outcomes in published trials

Background and Purpose—Vascular prevention trials mostly count “yes/no” (binary) outcome events, eg, stroke/no stroke. Analysis of ordered categorical vascular events (eg, fatal stroke/nonfatal stroke/no stroke) is clinically relevant and could be more powerful statistically. Although this is not a novel idea in the statistical community, ordinal outcomes have not been applied to stroke prevention trials in the past. Methods—Summary data on stroke, myocardial infarction, combined vascular events, and bleeding were obtained by treatment group from published vascular prevention trials. Data were analyzed using 10 statistical approaches which allow comparison of 2 ordinal or binary treatment groups. The results for each statistical test for each trial were then compared using Friedman 2-way analysis of variance with multiple comparison procedures. Results—Across 85 trials (335 305 subjects) the test results differed substantially so that approaches which used the ordinal nature of stroke events (fatal/nonfatal/no stroke) were more efficient than those which combined the data to form 2 groups (P0.0001). The most efficient tests were bootstrapping the difference in mean rank, Mann–Whitney U test, and ordinal logistic regression; 4- and 5-level data were more efficient still. Similar findings were obtained for myocardial infarction, combined vascular outcomes, and bleeding. The findings were consistent across different types, designs and sizes of trial, and for the different types of intervention. Conclusions—When analyzing vascular events from prevention trials, statistical tests which use ordered categorical data are more efficient and are more likely to yield reliable results than binary tests. This approach gives additional information on treatment effects by severity of event and will allow trials to be smaller. (Stroke. 2008;39:000-000.)