Conference Agenda

In various scientific fields, researchers are interested in exploring the relationship between some response variable $Y$ and a vector of covariates $X$. In order to make use of a specific model for the dependence structure, it first has to be checked whether the conditional density function of $Y$ given $X$ fits into a given parametric family. We propose a consistent bootstrap-based goodness-of-fit test for this purpose. The test statistic traces the difference between a nonparametric and a semi-parametric estimate of the marginal distribution function of $Y$. As its asymptotic null distribution is not distribution-free, a parametric bootstrap method is used to determine the critical value. A simulation study shows that, in some cases, the new method is more sensitive to deviations from the parametric model than other tests found in the literature. We also apply our method to real-world datasets.

Most statistical tests customarily taught in introductory mathematical courses on probability and statistics can be equivalently expressed, perhaps after a rank transformation, in terms of the test statistic of a classical multivariate test for correlation, which is itself an application of Rao’s score test. This talk draws attention to that apparently little-known fact, discusses existing knowledge and presents proofs in a slightly generalized framework to close some gaps. Tests covered include all F-tests (and t-tests) and their multivariate generalizations based on Pillai’s statistic, Wilcoxon rank-sum and signed rank tests, Kruskal-Wallis tests, chi-squared tests of independence and also particular score tests in binomial and multinomial logit models, among others. Recognizing all of those as variations of the same test of independence of two sets of variables given a third one can be of considerable help in teaching. Indeed, the only differences are in the potential prior elementary transformation of the variables, in the level of conditioning on which the statistical model is presented and in the choice of the approximation of the null distribution of the test statistic.

The Erdös‐Rényi graph is a popular choice to model network data as it is parsimoniously parameterized, straightforward to interpret and easy to estimate. However, it has limited suitability in practice, since it often fails to capture crucial characteristics of real-world networks. To check its adequacy, we propose a novel class of goodness-of-fit tests for homogeneous Erdös‐Rényi models against heterogeneous alternatives that permit non-constant edge probabilities. We allow for both asymptotically dense and sparse networks. The tests are based on graph functionals that cover a broad class of network statistics for which we derive limiting distributions in a unified manner. The resulting class of asymptotic tests includes several existing tests as special cases. Further, we propose a parametric bootstrap and prove its consistency, which avoids the often tedious variance estimation for asymptotic tests and enables performance improvements for small network sizes. Moreover, under certain fixed and local alternatives, we provide a power analysis for some popular choices of subgraph counts as goodness-of-fit test statistics. We evaluate the proposed class of tests and illustrate our theoretical findings by simulations.

Generalized estimating equations (GEE) are a popular method to model the effects of covariates on various estimands, which only rely on the specification of a functional relationship without the need of restrictive distributional assumptions. However, if the response variable is not fully observable, e.g. in the case of time-to-event data, the GEE approach is not directly applicable. Andersen et al. (2003) proposed to replace the partially unobservable response variables by jackknife pseudo-observations, and Overgaard et al. (2017) showed that the resulting parameter estimates are consistent and asymptotically normal under very general conditions. For further inference about the parameter vector an estimator of the asymptotic covariance matrix is necessary. But due to the dependence of the pseudo-observations, the limiting covariance matrix is highly complicated and the usual sandwich estimator seems to be inconsistent (Jacobsen and Martinussen (2016), Overgaard et al. (2018)). Overgaard et al. (2017) proposed an alternative estimator which incorporates the dependence of the pseudo-observations and performs well in medium to large samples. \\

These results would in principle allow for the construction of tests for general linear hypotheses about the parameters. However, mainly confidence intervals for individual parameters or simple contrasts, e.g. risk differences, have been considered in the past. \\

In this talk we aim to bridge this gap by introducing different test statistics for general linear hypotheses in pseudo-value regression models. To improve the small sample performance of these tests we discuss different bootstrap methods for pseudo-observations as well as possible extensions to multiple testing problems and simultaneous confidence intervals for contrasts.

\section*{Acknowledgements} We would like to thank Marc Ditzhaus for his invaluable collaboration and guidance in the early phase of this work. Sadly, he has deceased and he could not complete this work together with us. \section*{References} Per Kragh Andersen, John P. Klein, and Susanne Rosthøj. "Generalised linear models for correlated pseudo‐observations, with applications to multi‐state models." Biometrika 90.1 (2003): 15-27. \\

\noindent Morten Overgaard, Erik Thorlund Parner, and Jan Pedersen. "Asymptotic theory of generalized estimating equations based on jack-knife pseudo-observations." The Annals of Statistics 45.5 (2017): 1988-2015. \\

\noindent Martin Jacobsen, and Torben Martinussen. "A note on the large sample properties of estimators based on generalized linear models for correlated pseudo‐observations." Scandinavian Journal of Statistics 43.3 (2016): 845-862. \\

\noindent Morten Overgaard, Erik Thorlund Parner, and Jan Pedersen. "Estimating the variance in a pseudo‐observation scheme with competing risks." Scandinavian Journal of Statistics 45.4 (2018): 923-940.