Equivalence of Regression Curves

Abstract

This paper investigates the problem whether the difference between two parametric models m 1 , m 2 describing the relation between a response variable and several covariates in two different groups is practically irrelevant, such that inference can be performed on the basis of the pooled sample. Statistical methodology is developed to test the hypotheses H 0 : d(m 1 , m 2 ) ≥ ε versus H 1 : d(m 1 , m 2 ) < ε to demonstrate equivalence between the two regression curves m 1 , m 2 for a pre-specified threshold ε, where d denotes a distance measuring the distance between m 1 and m 2 . Our approach is based on the asymptotic properties of a suitable estimator d(m 1 ,m 2 ) of this distance. In order to improve the approximation of the nominal level for small sample sizes a bootstrap test is developed, which addresses the specific form of the interval hypotheses. In particular, data has to be generated under the null hypothesis, which implicitly defines a manifold for the parameter vector. The results are illustrated by means of a simulation study and a data example. It is demonstrated that the new methods substantially improve currently available approaches with respect to power and approximation of the nominal level. 1 arXiv:1505.05266v3 [stat.ME] 7 Jun 2016 particular difficulties arising in the problem of testing (parametric) interval hypotheses. In particular, resampling has to be performed under the null hypothesis H 0 : d(m 1 , m 2 ) ≥ ε, which defines (implicitly) a manifold in the parameter space. We prove consistency of the bootstrap test and demonstrate by means of a simulation study that it yields an improvement of the approximation of the nominal level for small sample sizes. In Section 4 the maximal deviation between the two curves is considered as a measure of similarity, for which corresponding results are substantially harder to derive. For example, we prove weak convergence of a corresponding test statistic, but the limit distribution depends in a complicated way on the extremal points of the difference between the "true" curves. This problem is again solved by developing a bootstrap test. The finite sample properties of the new methodology are illustrated in Section 5, where we also provide a comparison with the method of Gsteiger et al. (2011). In particular, it is demonstrated that the methodology proposed in this paper is more powerful than the test proposed by these authors. The methods are illustrated with an example in Section 6. Technical details and proofs are deferred to an appendix in Section 7. Equivalence of regression curvesWe consider two, possibly different, regression models m 1 , m 2 to describe the relationship between a response variable Y and several covariates for two different groups = 1, 2:Y ,i,j = m (x ,i , β ) + η ,i,j , j = 1, . . . , n ,i , i = 1, . . . , k .(2.1)Here, the covariate region is denoted by X ⊂ R d , x ,i denotes the ith dose level (in group ), n ,i the number of patients treated at dose level x ,i and k the number of different dose levels in group . Further, n ...

Lessons learned from IDeAl — 33 recommendations from the IDeAl-net about design and analysis of small population clinical trials

“…We showed that for small-populations groups, a causal inference framework is especially useful [ 43 – 45 ]. Further, to account for missing data, the use of pseudo-likelihood and inverse probability weighting methods are shown to be advantageous over commonly used full pseudo-likelihood methods while validation of surrogate endpoints [ 46 , 47 ]. Efficient and stable estimation strategies for the validation model which of course could be non-linear as well are developed [ 48 ].…”

“…Additionally, we developed a new statistical test for the hypothesis of similarity of dose response curves. The test decides for equivalence of the curves if an estimate of a distance is smaller than a given threshold, which is obtained by a (non-standard) constrained parametric bootstrap procedure [ 47 ]. A corresponding R package “TestingSimilarity” was developed [ 82 ].…”

“…In summary, we developed a new optimized design and analysis strategy for comparing dose-response profiles to extrapolate clinical trial results from a larger to a smaller population. The comparison of dose response curves should be done by the bootstrap approach [ 47 , 87 ]. If the aim of the study is the extrapolation of efficacy and safety information, we recommend considering and comparing the MEDs of two given populations [ 84 ].…”

Optimal designs for comparing regression models with correlated observations

“…Their approach is based on estimating the regression curves in the different samples and constructing confidence bands for the maximal deviation distance between these estimates. More recently, Dette et al (2015b) propose to directly estimate the maximal deviation distance or an L 2 -distance between the curves and to establish equivalence if the estimator is smaller than a given threshold. On the other hand, the efficient planning of experiments for comparing curves has not been dealt with in the literature although this would substantially improve the accuracy of the conclusions drawn regarding non-superiority or equivalence.…”

Equivalence analyses of dissolution profiles with the Mahalanobis distance: a regulatory perspective and a comparison with a parametric maximum deviation‐based approach

“…Our second contribution to the discussion regards Dr. Hoffelder's comment: “ future work could be to analyze the statistical properties of the approach of ( Bretz, Möllenhoff, Dette, Liu, & Trampisch , ) if it is applied to realistic dissolution profile scenarios where the variances at various time points can be considerably different, especially for so‐called “highly variable” profiles .” We actually improved that method in Dette, Moellenhoff, Volgushev, & Bretz (). We then extended this approach to drug dissolution experiments by integrating the time dependency in the method via the covariance matrix of the profiles and by providing an inferential framework for testing if the maximum deviation is below a certain acceptance limit (Moellenhoff et al., ).…”