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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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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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Goodness-of-Fit Test01:16

Goodness-of-Fit Test

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The goodness-of-fit test is a type of hypothesis test which determines whether the data "fits" a particular distribution. For example, one may suspect that some anonymous data may fit a binomial distribution. A chi-square test (meaning the distribution for the hypothesis test is chi-square) can be used to determine if there is a fit. The null and alternative hypotheses may be written in sentences or stated as equations or inequalities. The test statistic for a goodness-of-fit test is given as...
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Friedman Two-way Analysis of Variance by Ranks01:21

Friedman Two-way Analysis of Variance by Ranks

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Friedman's Two-Way Analysis of Variance by Ranks is a nonparametric test designed to identify differences across multiple test attempts when traditional assumptions of normality and equal variances do not apply. Unlike conventional ANOVA, which requires normally distributed data with equal variances, Friedman's test is ideal for ordinal or non-normally distributed data, making it particularly useful for analyzing dependent samples, such as matched subjects over time or repeated measures...
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Experimental Designs01:16

Experimental Designs

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An experimental design is a systematic process that allows researchers to evaluate the relationship between dependent and independent variables. There are three widely used types of experimental design - pre-experimental design, true experimental design, and quasi-experimental design. In pre-experimental design, the researcher compares the data before and after some interventions or treatments. The true-experimental design has more than one purposefully created group, a commonly measured...
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One-Compartment Open Model: Wagner-Nelson and Loo Riegelman Method for ka Estimation01:24

One-Compartment Open Model: Wagner-Nelson and Loo Riegelman Method for ka Estimation

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This lesson introduces two critical methods in pharmacokinetics, the Wagner-Nelson and Loo-Riegelman methods, used for estimating the absorption rate constant (ka) for drugs administered via non-intravenous routes. The Wagner-Nelson method relates ka to the plasma concentration derived from the slope of a semilog percent unabsorbed time plot. However, it is limited to drugs with one-compartment kinetics and can be impacted by factors like gastrointestinal motility or enzymatic degradation.
On...
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Related Experiment Video

Updated: Dec 28, 2025

The Innovation Arena: A Method for Comparing Innovative Problem-Solving Across Groups
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Bootstrap Model-Based Constrained Optimization Tests of Indirect Effects.

Davood Tofighi1

  • 1University of New Mexico, Albuquerque, NM, United States.

Frontiers in Psychology
|February 11, 2020
PubMed
Summary

Commonly used mediation analysis tests struggle with accuracy in small samples and non-normal data. New bootstrap Model-based Constrained Optimization (MBCO) likelihood ratio tests (LRT) offer more reliable Type I error rates for indirect effects.

Keywords:
confidence intervalconstrained optimizationindirect effectlikelihood ratio testmediation analysis

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Area of Science:

  • Statistics
  • Psychometrics
  • Quantitative Psychology

Background:

  • Mediation analysis is crucial for understanding indirect effects in statistical models.
  • Commonly used confidence interval (CI)-based tests and asymptotic likelihood ratio tests (LRT) often fail to maintain accurate Type I error rates with finite sample sizes and non-normally distributed residuals.
  • Testing null hypotheses for indirect effects presents inherent complexities that challenge existing statistical methods.

Purpose of the Study:

  • To address the limitations of existing methods for testing indirect effects in mediation analysis.
  • To propose and evaluate two novel bootstrap extensions of the Model-based Constrained Optimization (MBCO) likelihood ratio test (LRT).
  • To enhance the accuracy of Type I error rates in mediation analysis, particularly under conditions of small sample sizes and non-normal model residuals.

Main Methods:

  • Developed semi-parametric and parametric bootstrap versions of the Model-based Constrained Optimization (MBCO) likelihood ratio test (LRT).
  • Compared the performance of the proposed bootstrap MBCO LRTs against the asymptotic MBCO LRT and traditional CI-based tests.
  • Evaluated Type I error rates under various conditions, including small sample sizes and different degrees of residual non-normality.

Main Results:

  • The proposed semi-parametric and parametric bootstrap MBCO LRTs demonstrated more accurate Type I error rates compared to the asymptotic MBCO LRT.
  • Bootstrap MBCO LRTs outperformed CI-based methods in maintaining accurate Type I error rates, especially in smaller sample sizes and with non-normal residuals.
  • The study provides R scripts for implementing all three MBCO LRT methods.

Conclusions:

  • Bootstrap extensions of the MBCO LRT offer a more robust approach to testing indirect effects in mediation analysis.
  • These novel methods provide improved accuracy for Type I error rates, addressing critical limitations of existing techniques.
  • The findings suggest that practitioners should consider using bootstrap MBCO LRTs for more reliable mediation analysis, particularly when standard assumptions are violated.