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Factorial Design02:01

Factorial Design

Factorial Analysis is an experimental design that applies Analysis of Variance (ANOVA) statistical procedures to examine a change in a dependent variable due to more than one independent variable, also known as factors. Changes in worker productivity can be reasoned, for example, to be influenced by salary and other conditions, such as skill level. One way to test this hypothesis is by categorizing salary into three levels (low, moderate, and high) and skills sets into two levels (entry level...
One-Way ANOVA: Unequal Sample Sizes01:15

One-Way ANOVA: Unequal Sample Sizes

One-way ANOVA can be performed on three or more samples of unequal sizes. However, calculations get complicated when sample sizes are not always the same. So, while performing ANOVA with unequal samples size, the following equation is used:
One-Way ANOVA: Equal Sample Sizes01:15

One-Way ANOVA: Equal Sample Sizes

One-Way ANOVA can be performed on three or more samples with equal or unequal sample sizes. When one-way ANOVA is performed on two datasets with samples of equal sizes, it can be easily observed that the computed F statistic is highly sensitive to the sample mean.
Different sample means can result in different values for the variance estimate: variance between samples. This is because the variance between samples is calculated as the product of the sample size and the variance between the...
Two-Way ANOVA01:17

Two-Way ANOVA

The two-way ANOVA is an extension of the one-way ANOVA. It is a statistical test performed on three or more samples categorized by two factors - a row factor and a column factor. Ronald Fischer mentioned it in 1925 in his book 'Statistical Methods for Researchers.'
The two-way ANOVA analysis initially begins by stating the null hypothesis that there is an interaction effect between the two factors of a dataset. This effect can be visualized using line segments formed by joining the means for...
One-Way ANOVA01:18

One-Way ANOVA

One-way ANOVA analyzes more than three samples categorized by one factor. For example, it can compare the average mileage of sports bikes. Here, the data is categorized by one factor - the company. However, one-way ANOVA cannot be used to simultaneously compare the sample mean of three or more samples categorized by two factors. An example of two factors would be sports bikes from different companies driven in different terrains, such as a desert or snowy landscape. Here, two-way ANOVA is used...
Statistical Methods to Analyze Parametric Data: ANOVA01:12

Statistical Methods to Analyze Parametric Data: ANOVA

Analysis of Variance, or ANOVA, is a powerful statistical technique used to analyze parametric data, primarily in research and experimental studies. It's designed to compare the means of two or more groups, assisting researchers in identifying any significant differences between these group means. There are two main types of ANOVA based on the complexity of the analysis: one-way and two-way.
One-way ANOVA is applied when a single independent variable or factor is scrutinized. It compares the...

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Related Experiment Video

Updated: Jul 2, 2026

Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach
04:35

Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach

Published on: July 3, 2020

A practical method for analyzing factorial designs with heteroscedastic data.

Guiillermo Vallejo1, Manuel Ato, M Paula Fernández

  • 1Deprtment of Psychology, University of Oviedo, Spain. gvallejo@uniovi.es

Psychological Reports
|September 4, 2008
PubMed
Summary
This summary is machine-generated.

This study evaluated statistical tests for nonorthogonal factorial designs. The modified Brown-Forsythe procedure offered better Type I error control with asymmetric data, while the SAS MIXED procedure provided higher statistical power.

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Area of Science:

  • Statistics
  • Statistical Methods
  • Experimental Design

Background:

  • Analyzing nonorthogonal factorial designs requires robust statistical methods.
  • Departures from normality and homogeneity assumptions can impact Type I error rates and statistical power.
  • Evaluating the performance of different statistical procedures under these conditions is crucial for accurate data analysis.

Purpose of the Study:

  • To compare the Type I error rates and powers of three statistical tests for nonorthogonal factorial designs.
  • To assess the robustness and sensitivity of these tests under violations of normality and homogeneity assumptions.
  • To identify the most reliable procedure for analyzing complex factorial designs.

Main Methods:

  • Monte Carlo simulation was employed to evaluate test performance.
  • Three procedures were compared: modified Brown-Forsythe, Brunner-Dette-Munk generalization of Box's method, and SAS MIXED procedure with Kenward-Roger adjustment.
  • Data were generated from both symmetric and asymmetric distributions to simulate departures from assumptions.

Main Results:

  • All three methods controlled Type I error adequately with symmetric distributions.
  • The modified Brown-Forsythe procedure demonstrated slightly better Type I error control with asymmetric distributions.
  • The SAS MIXED procedure yielded the highest statistical power across tested conditions.
  • Using the generalization of Box's method resulted in minimal power loss compared to the modified Brown-Forsythe procedure with symmetric data.

Conclusions:

  • The modified Brown-Forsythe procedure is a robust choice for Type I error control in nonorthogonal designs, especially with asymmetric data.
  • The SAS MIXED procedure offers superior statistical power, making it preferable for detecting effects.
  • The Brunner-Dette-Munk generalization of Box's method provides a reasonable balance between Type I error control and power, particularly with symmetric data.