Nonparametric inference for functional‐on‐scalar linear models applied to knee kinematic hop data after injury of the anterior cruciate ligament

Abstract

Motivated by the analysis of the dependence of knee movement patterns during functional tasks on subject-specific covariates, we introduce a distribution-free procedure for testing a functional-on-scalar linear model with fixed effects. The procedure does not only test the global hypothesis on the entire domain but also selects the intervals where statistically significant effects are detected. We prove that the proposed tests are provided with an asymptotic control of the intervalwise error rate, that is, the probability of falsely rejecting any interval of true null hypotheses. The procedure is applied to one-leg hop data from a study on anterior cruciate ligament injury. We compare knee kinematics of three groups of individuals (two injured groups with different treatments and one group of healthy controls), taking individual-specific covariates into account.

A permutation approach to the analysis of spatiotemporal geochemical data in the presence of heteroscedasticity

“…When the variance–covariance matrix is of the form Σ = σ 2 I , one can use the test statistic where A decision about hypothesis can be formulated by means of a permutation test, with the global p value of the T OLS test being computed via the adaptation of the Freedman and Lane () scheme to functional data (Abramowicz et al, ). In this setting, the T OLS distribution under permutations is estimated with a Monte Carlo technique, by evaluating it over a high number of permuted data sets, obtained by randomly permuting the residuals estimated from the model under the null hypothesis.…”

“…For this purpose, we propose as a test statistic the absolute value of the log‐proportion of variances, as follows: To perform the test, we consider a permutation scheme similar to the one discussed in Section 3.2. Under H 0 , residuals are approximately exchangeable, and the permutation procedure of Abramowicz et al () can be applied. The global p value of test is computed as the proportion of permutations leading to a value of T Var higher than the one observed on the data.…”

“…The nonparametric approach appears particularly promising, as it allows for minimal assumptions on the data generative model. In this context, permutation tests have been recently successfully developed, leading to flexible approaches for testing in FDA (Abramowicz et al, ; Pini & Vantini, ).…”

“…From a theoretical point of view, since the estimated residuals of the null model are asymptotically exchangeable, the obtained test is asymptotically exact. Winkler, Ridgway, Webster, Smith, and Nichols () have applied this approach into the multivariate case of testing for differences between two sets of images; Abramowicz et al () have adapted the aforementioned permutation scheme to testing in a functional regression model for knee movement.…”

“…In this work, we extend the approach of Abramowicz et al () to the space–time setting. The research is motivated by a real‐world case of interest in geology and chemistry.…”