Generalized Confidence Intervals for Ratios of Regression Coefficients with Applications to Bioassays

Abstract

The problem of constructing a confidence interval for the ratio of two regression coefficients is addressed in the context of multiple regression. The concept of a Generalized Confidence Interval is used, and the resulting confidence interval is shown to perform well in terms of coverage probability. The proposed methodology always results in an interval, unlike the confidence region generated from Fieller's theorem. The procedure can easily be implemented for parallel-line assays, slope-ratio assays, and quantal assays under a probit model. Furthermore, this approach can also be extended to compute confidence intervals based on data from multiple bioassays. The results are illustrated using several examples.

The Computational Challenges of Pursuing Multiple Goals: Network Structure of Goal Systems Predicts Human Performance

“…We found that, while had a significant negative effect on solution quality ( , ), had a significant positive effect on the average solution quality ( ) and the additional variance explained by above and beyond the variance explained by was highly significant ( ). To test whether the non-linear effect of was U-shaped, we estimated the location of the minimum point (Lind and Mehlum, 2010;Bebu, Seillier, Moiseiwitsch, & Mathew, 2009): If the minimum point lies at an intermediate value of , then the effect can be called U-shaped. But if the minimum point lies outside the range of feasible then the effect could be still monotonic.…”

“…had a significant positive effect on the average solution quality ( ) and the additional variance explained by above and beyond the variance explained by was highly significant ( ). To test whether the non-linear effect of was U-shaped, we estimated the location of the minimum point (Lind & Mehlum, 2010;Bebu, Seillier-Moiseiwitsch, & Mathew, 2009): According to our data we are 99% confident that the minimum point lies between 6.4 and 9.2 ( , 99% CI:…”

Parametric cost-effectiveness inference with skewed data

“…Since its introduction (Weerahandi, 1993), the GPQ methodology has been successfully applied by a number of researchers in order to obtain satisfactory confidence intervals for several parametric problems for which traditional approaches are unavailable, or are unsatisfactory. In particular, the GPQ methodology has been applied in the context of ratio parameters (Bebu et al, 2009), where the methodology has been used successfully for a ratio of regression coefficients. The asymptotic accuracy of the GPQ methodology is established in Hannig et al (2006); however, satisfactory small sample performance has been noted in many scenarios, for example, in Bebu et al (2009).…”

“…In particular, the GPQ methodology has been applied in the context of ratio parameters (Bebu et al, 2009), where the methodology has been used successfully for a ratio of regression coefficients. The asymptotic accuracy of the GPQ methodology is established in Hannig et al (2006); however, satisfactory small sample performance has been noted in many scenarios, for example, in Bebu et al (2009). An added appeal of the GPQ methodology is that if a vector-GPQ is available for a vector parameter, then a GPQ can be constructed for any function of the parameter, by simply obtaining the corresponding function of the vector-GPQ.…”

Generalized Confidence Intervals and Fiducial Intervals for Some Epidemiological Measures

“…The generalized confidence interval methodology is due to Weerahandi [ 5 ], and it has found numerous applications in interval estimation problems, resulting in confidence intervals that exhibit satisfactory performance in small samples; see also the books by Weerahandi [ 6 , 7 ]. In the context of binary data, the methodology was adopted to obtain satisfactory confidence intervals in a quantal assay problem [ 8 ] and in surrogate endpoint validation [ 9 ]. Recently, the fiducial approach has seen a revival; in fact, some of the generalized confidence intervals are indeed fiducial intervals.…”