Nonparametric predictive comparison of proportions

Abstract

We use the lower and upper predictive probabilities from Coolen [5] to compare future numbers of successes in Bernoulli trials for different groups. We consider both pairwise and multiple comparisons. These inferences are in terms of lower and upper probabilities that the number of successes in m future trials from one group exceeds the number of successes in m future trials from another group, or such numbers from all other groups. We analyse these lower and upper probabilities via application to two data sets from the literature, and discuss the imprecision in relation to m.

Three-group ROC analysis: A nonparametric predictive approach

“…NPI for real-valued observations is also available for multiple future observations (Arts et al, 2004;Coolen, 2011), where the inter-dependence of these future observations is explicitly taken into account. Development of NPI-based methods for diagnostic accuracy with explicit focus on m ≥ 2 future observations is an interesting topic for future research, where particularly the strength of the inferences as function of m should be studied carefully, see Coolen and Coolen-Schrijner (2007) for a similar study with focus on the role of m for comparison of groups of Bernoulli data. Typically, for increasing m the imprecision in inferences increases, which is likely to imply that, on the basis of the limited information in available data, a specific choice of diagnostic method including the important decision thresholds can be inferred to be good for a number of future patients up to a specific value of m, but for larger values of m the evidence in the data would be too weak to make decisions that are strongly supported by the data without further modelling assumptions.…”

Nonparametric Predictive Inference for Binary Diagnostic Tests

“…We have restricted attention to comparison of two diagnostic tests. This can easily be generalized to comparison of more such tests, using the NPI approach for multiple comparisons of groups of Bernoulli data (Coolen and Coolen-Schrijner 2006;2007).…”

“…These lower and upper probabilities follow from an assumed underlying latent variable representation together with Hill's assumption A (n) (Hill 1968), which has an explicitly predictive nature, and fit in the framework of nonparametric predictive inference (NPI) (Augustin and Coolen 2004;Coolen 2006). Several inferential problems involving Bernoulli data have been addressed using this NPI approach, for example comparisons of groups of Bernoulli data (Coolen and Coolen-Schrijner 2006;2007), acceptance sampling (Coolen and Elsaeiti 2009); and system reliability (Coolen-Schrijner et al 2008). We briefly summarize this approach; for more details and justification we refer to Coolen (1998) and Coolen-Schrijner (2006, 2007).…”

“…Coolen and Coolen-Schrijner (2007) introduced both pairwise and multiple comparisons to compare numbers of successes in m future Bernoulli trials from different groups, in this paper we restrict attention to pairwise comparisons. The NPI upper probabilities for such predictive comparison of two groups, A and B, with related notation using index a and b, respectively, are (Coolen and Coolen-Schrijner 2007)…”

“…With the overview of the results of Coolen (1998) and Coolen and Coolen-Schrijner (2006;2007) and the generalization involving a difference of at least d successes between the two groups, as presented in section 3, we can now introduce NPI for binary diagnostic tests. Downloaded by [University of California, San Diego] at 02:22 03 January 2015…”