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Related Concept Videos

Statistical Hypothesis Testing01:16

Statistical Hypothesis Testing

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Hypothesis testing is a critical statistical procedure facilitating informed, evidence-based decisions. It begins with a hypothesis, which is a tentative explanation, or a prediction about a population parameter. This hypothesis can be either a null hypothesis (H0), indicating no effect or difference, or an alternative hypothesis (Ha), suggesting an effect or difference.
Statistical significance measures the probability that an observed result occurred by chance. If this probability, known as...
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Types of Hypothesis Testing01:11

Types of Hypothesis Testing

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There are three types of hypothesis tests: right-tailed, left-tailed, and two-tailed.
When the null and alternative hypotheses are stated, it is observed that the null hypothesis is a neutral statement against which the alternative hypothesis is tested. The alternative hypothesis is a claim that instead has a certain direction. If the null hypothesis claims that p = 0.5, the alternative hypothesis would be an opposing statement to this and can be put either p > 0.5, p < 0.5, or p...
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Hypothesis Test for Test of Independence01:16

Hypothesis Test for Test of Independence

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The test of independence is a chi-square-based test used to determine whether two variables or factors are independent or dependent. This hypothesis test is used to examine the independence of the variables. One can construct two qualitative survey questions or experiments based on the variables in a contingency table. The goal is to see if the two variables are unrelated (independent) or related (dependent). The null and alternative hypotheses for this test are:
H0: The two variables (factors)...
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Errors In Hypothesis Tests01:14

Errors In Hypothesis Tests

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When performing a hypothesis test, there are four possible outcomes depending on the actual truth (or falseness) of the null hypothesis and the decision to reject or not.
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Statistical Inference Techniques in Hypothesis Testing: Parametric Versus Nonparametric Data01:16

Statistical Inference Techniques in Hypothesis Testing: Parametric Versus Nonparametric Data

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Statistical inference techniques, paramount in hypothesis testing, differentiate into two broad categories: parametric and nonparametric statistics.
Parametric statistics, as the name suggests, assumes that data follow a specific distribution, often a normal distribution. This assumption enables robust hypothesis testing and estimation. Parametric methods, like the Student's t-test or Goodness-of-fit test, are frequently employed in biostatistics due to their robustness. For instance,...
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Null and Alternative Hypotheses01:16

Null and Alternative Hypotheses

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The actual hypothesis testing begins by considering two hypotheses. They are termed  the null hypothesis and the alternative hypothesis. These hypotheses contain opposing viewpoints.
The null hypothesis, denoted by H0 is a statement of no difference between the variables—they are not related. This can often be considered the status quo. As  a result if you cannot accept the null, it requires some action.
The alternative hypothesis, denoted by H1 or Ha, is a claim about the...
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Hypothesis testing in multivariate normal models with block circular covariance structures.

Yuli Liang1, Carlos A Coelho2, Tatjana von Rosen3

  • 1Department of Statistics, Örebro University School of Business, Örebro, Sweden.

Biometrical Journal. Biometrische Zeitschrift
|March 14, 2022
PubMed
Summary

This study develops statistical tests for repeated measures data with patterned mean vectors and covariance matrices. We provide methods for hypothesis testing under specific covariance structures, aiding data analysis.

Keywords:
Toeplitz matrixbeta random variablescanonical reductionexchangeabilitylikelihood ratio testnear-exact distributions

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Area of Science:

  • Statistics
  • Biostatistics
  • Psychometrics

Background:

  • Repeated measures data analysis requires understanding both mean and covariance structures.
  • Patterned covariance matrices (e.g., block circular, doubly exchangeable) are common in longitudinal studies.
  • Simultaneous hypothesis testing on mean and covariance is complex for such data.

Purpose of the Study:

  • To develop methods for simultaneous hypothesis testing on the mean vector and covariance matrix.
  • To address patterned covariance structures in repeated measures.
  • To provide exact or near-exact null distributions for likelihood ratio test statistics.

Main Methods:

  • Likelihood ratio test statistics were derived for patterned mean and covariance matrices.
  • Null distributions for test statistics were established.
  • Exact or near-exact probability density and cumulative distribution functions were obtained.

Main Results:

  • The study established null distributions for likelihood ratio test statistics under block circular and doubly exchangeable covariance structures.
  • Expressions for probability density and cumulative distribution functions were derived.
  • The methodology was validated through simulation and a real-life data example.

Conclusions:

  • The developed methods provide a robust framework for hypothesis testing in repeated measures data with patterned structures.
  • The findings offer practical tools for analyzing complex longitudinal data.
  • The study contributes to statistical methodology for mean and covariance matrix inference.