Inferences about a linear combination of proportions

Abstract

SUMMARYStatistical methods for carrying out asymptotic inferences (tests or confidence intervals) relative to one or two independent binomial proportions are very frequent. However inferences about a linear combination of K independent proportions L=β i p i (in which the first two are special cases) have had very little attention paid to them (focused exclusively on the classic Wald method). In this paper the authors approach the problem from the more efficient viewpoint of the score method, which can be solved using a free program which is available from the webpage quoted in the article. In addition the paper offers approximate formulas that are easy to calculate, gives a general proof of Agresti's heuristic method (consisting of adding a certain number of successes and failures to the original results before applying Wald's method) and, finally, it proves that the score method (which verifies the desirable properties of spatial and parametric convexity) is the best option in comparison with other methods.

The optimal method to make inferences about a linear combination of proportions

“…Historically, literature in this field has paid special attention to the case of K2 (which contains the cases with one proportion and the difference or ratio for two proportions), but there is increasing interest in the case of K>2 (Newcombe, 2001;Price and Bonett, 2004;Schaarschmidt et al 2008;Tebbs and Roths, 2008;Agresti et al, 2008;Zou et al, 2009 andMartín et al, 2010). The linear combination L may be a contrast ( i =0), in which case it is usually interesting to carry out the test for H: L=0 or to determine a confidence interval for L, or may not be ( i 0), in which case it is usually interesting to determine a CI for L; therefore this article has concentrated on the diverse procedures to carry out the test H: L= vs. K: L or to obtain a CI for L through inversion of the previous test.…”

“…In order to make the previous inferences, there are various procedures that have not been compared with each other: the S0 score method and the W3 Wald adjusted method defined by Martín et al (2010) on the one hand, and the N0 method defined by Zou et al (2009) on the other. The article has defined the new P0 method, based on the Peskun method (1993), which is given by expressions (5).…”

“…When K=2, there may be several objectives: the difference between the two proportions if  1 =1 and  2 =+1 (as in Agresti and Caffo, 2000); the sum of two proportions if  1 =+1 and  2 =+1 (as in Pham-Gia and Turkkan, 1994); the ratio  of two proportions if  1 = and  2 =+1 (as in Agresti, 2003); or a linear combination of two proportions with  1 <0 (as in Phillips, 2003). Cases with K>2 are historically rather less frequent, although in recent years they have received more and more attention due to their great practical interest (Newcombe, 2001;Price and Bonett, 2004;Schaarschmidt et al 2008;Tebbs and Roths, 2008;Agresti et al, 2008;Zou et al, 2009 andMartín et al, 2010).…”

“…These procedures originate in similar proposals made by Agresti and Coull (1998) in the case of one proportion and by Agresti and Caffo (2000) in the case of the difference between two proportions. Martín et al (2010) solved the problem through the score method, propose new adjusted Wald methods, justified that all of them are an approach to the score method and, finally, compared the methods obtained. These authors concluded that the score method was the best, closely followed by a modification and generalization of the method proposed by Price and Bonett.…”

A Simple Graphical Way of Evaluating Coverage and Directional Non-coverages

“…Several approximate methods have been proposed in the literature for constructing confidence intervals (CIs) for the difference of two independent binomial proportions ( [1,2,3,4,5,6]). However, few authors have been discussing approximate methods for obtaining CIs for any linear combination of two ( [7,8]) and more than two ( [9,10,11,12,13]) independent binomial populations. Within the context of investigating the properties of each of the different approaches to construct CIs, the performance of each method is commonly evaluated through simulation studies.…”