statistics.bgwoodruff.com > ANOVA Hub > Here

ANOVA Results: Monte Carlo Simulations


The almost-final step in a research project is to decide if the ANOVA F-ratio is indicating whether the independent variable was or was not effective!

Monte Carlo techniques simulate the results we would get from research if the under the two conditions. Then we can decide what the F-ratio from the research is telling us about the independent variable.

In a Monte Carlo simulation, numbers are randomly generated using the mathematical model upon which ANOVA is based.

Listed below are a series of links that open animations illustrating the results we would expect to get under various conditions.

All of the animations show ANOVAs computed on a completely randomized research design (different subjects are assigned randomly to each of the conditions) where there are four levels of the independent variable.

Monte Carlo Simulations of an Experiment Analyzed with ANOVA.
Simulation Run Condition
1

The Null Hypothesis is true for an experiment of 4 groups of 12 observations each.

2 The Null Hypothesis is true for an experiment of 3 groups with 5 observations each.
3

The Null Hypothesis is false in a particular way--the independent variable has a "large" effect.

This animation shows the sampling distribution of the F-ratio for "Ho is false"

4

The Null Hypothesis is false in a particular way--the independent variable has a "large" effect.

Now contrast the results to when the "Ho is true."

5 The Null Hypothesis is false--the indpendent variable has a "medium" effect. This animation shows the results when the Ho is false and compares it to the distribution of F when Ho is true.
6 The Null Hypothesis is false--the indpendent variable has a "medium" effect. This animation shows the results when the Ho is true in comparison to when the Ho is false.
7

Observe the distribution of F-ratios when the independent variable has a "small" effect (Ho is false).

8 And to re-emphasize the point, here is the "Ho true" distribution added to the "small" effect.
9 Here is a simulation of of the Sampling Distribution of the Variance Ratio (aka "F") for 10,000 replications with 3 and 44 degrees of freedom. This animation used the "Histogrammer" tool in the Statistics Tool application.

© 2006 by Burrton Woodruff. All rights reserved. Last modified 9/26/06