Are you an EPFL student looking for a semester project?

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept

Null hypothesis

Summary

In scientific research, the null hypothesis (often denoted H0) is the claim that no relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is due to chance alone, and an underlying causative relationship does not exist, hence the term "null". In addition to the null hypothesis, an alternative hypothesis is also developed, which claims that a relationship does exist between two variables. Basic definitions The null hypothesis and the alternative hypothesis are types of conjectures used in statistical tests, which are formal methods of reaching conclusions or making decisions on the basis of data. The hypotheses are conjectures about a statistical model of the population, which are based on a sample of the population. The tests are core elements of statistical inference, heavily used in the interpretation of scientific experimental data, to separate scientific claims from statistical noise.

Official source

https://en.wikipedia.org/wiki/Null_hypothesis

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (46)

Related publications

Related people (13)

Related people

Related units (11)

Related units

Related concepts (34)

Related concepts

Related courses (43)

Related courses

Related lectures (119)

Related lectures

Related publications (46)

Multiple testing with test statistics following heavy-tailed distributions

Zhiwen Jiang

In multiple testing problems where the components come from a mixture model of noise and true effect, we seek to first test for the existence of the non-zero components, and then identify the true alternatives under a fixed significance level

\alpha

. Two parameters, namely the fraction of the non-null components

\varepsilon

and the size of the effects

\mu

, characterise the two-point mixture model under the global alternative. When the number of hypotheses

m

goes to infinity, we are interested in an asymptotic framework where the fraction of the non-null components is vanishing, and the true effects need to be sizable to be detected. Donoho and Jin give an explicit form of the asymptotic detectable boundary based on the Gaussian mixture model under the classic calibration of the parameters of the mixture model. We prove the analogous results for the Cauchy mixture distribution as an example heavy-tailed case. This requires a different formulation of the parameters, which reflects the added difficulties. We also propose a multiple testing procedure based on a filtering approach that can discover the true alternatives. Benjamini and Hochberg (BH) compare the observed

p

-values to a linear threshold curve and reject the null hypotheses from the minimum up to the last up-crossing, and prove the false discovery rate (FDR) is controlled. However, there is an intrinsic difference in heavy-tailed settings. Were we to use the BH procedure we would get a highly variable positive false discovery rate (pFDR). In our study we analyse the distribution of the

p

-values and devise a new multiple testing procedure to combine the usual case and the heavy-tailed case based on the empirical properties of the

p

-values. The filtering approach is designed to eliminate most

p

-values that are more likely to be uniform, while preserving most of the true alternatives. Based on the filtered

p

-values, we estimate the mode

\vartheta

and define the rejection region

\mathscr{R}(\vartheta, \delta)=\left[ \vartheta -\delta/2, \vartheta +\delta/2 \right]

such that the most informative

p

-values are included. The length

\delta

is chosen by controlling the data-dependent estimation of FDR at a desired level.

EPFL2021

During the last twenty years, Random matrix theory (RMT) has produced numerous results that allow a better understanding of large random matrices. These advances have enabled interesting applications in the domain of communication. Although this theory can contribute to many other domains such as brain imaging or genetic research, its has been rarely applied. The main barrier to the adoption of RMT may be the lack of concrete statistical results from probabilistic Random matrix theory. Indeed, direct generalisation of classical multivariate theory to high dimensional assumptions is often difficult and the proposed procedures often assume strong hypotheses on the data matrix such as normality or overly restrictive independence conditions on the data. This thesis proposes a statistical procedure for testing the equality of two independent estimated covariance matrices when the number of potentially dependent data vectors is large and proportional to the size of the vectors corresponding to the number of observed variables. Although the existing theory builds a very good intuition of the behaviour of these matrices, it does not provide enough results to build a satisfactory test for both the power and the robustness. Hence, inspired by spike models, we define the residual spikes and prove many theorems describing the behaviour of many statistics using eigenvectors and eigenvalues in very general cases. For example in the two central theorems of this thesis, the Invariant Angle Theorem and the Invariant Dot Product Theorem. Using numerous generalisations of the theory, this thesis finally proposes a description of the behaviour of a statistic under a null hypothesis. This statistic allows the user to test the equality of two populations, but also other null hypotheses such as the independence of two sets of variables. Finally, the robustness of the procedure is demonstrated for different classes of models and criteria for evaluating robustness are proposed to the reader. Therefore, the major contribution of this thesis is to propose a methodology both easy to apply and having good properties. Secondly, a large number of theoretical results are demonstrated and could be easily used to build other applications.

EPFL2019

Measures of surprise have been recently studied in statistics. This new concept can be used as the first exploratory tool to verify if a model under the null hypothesis fits appropriately. As no alternative models are necessary, the use of the measures of surprise is considered very simple. At the same time, this new alternative to test the goodness of a model cannot replace the full Bayesian analysis. The aim of this project is threshold selection for threshold models. The estimate of the threshold could be investigated by using the measures of surprise. The reason is that no alternative models are specified for non-extreme data, for which the distribution in extreme value theory is unknown, and thus the surprise would be a credible tool.

2010