P values, confidence intervals, or confidence levels for hypotheses?

Null hypothesis significance tests and p values are widely used despite very strong arguments against their use in many contexts. Confidence intervals are often recommended as an alternative, but these do not achieve the objective of assessing the credibility of a hypothesis, and the distinction between confidence and probability is an unnecessary confusion. This paper proposes a more straightforward (probabilistic) definition of confidence, and suggests how the idea can be applied to whatever hypotheses are of interest to researchers. The relative merits of the different approaches are discussed using a series of illustrative examples: usually confidence based approaches seem more transparent and useful, but there are some contexts in which p values may be appropriate. I also suggest some methods for converting results from one format to another. (The attractiveness of the idea of confidence is demonstrated by the widespread persistence of the completely incorrect idea that p=5% is equivalent to 95% confidence in the alternative hypothesis. In this paper I show how p values can be used to derive meaningful confidence statements, and the assumptions underlying the derivation).