Basics

Tests for nominal variables

Descriptive statistics

Tests for one measurement variable

Tests for multiple measurement variables

Multiple tests

Miscellany

Model I vs. Model II anova


One of the first steps in performing a one-way anova is deciding whether to do a Model I or Model II anova. The test of homogeneity of means is the same for both models, but the choice of models determines what you do if the means are significantly heterogeneous.

Model I anova

In a model I anova (also known as a fixed-effects model anova), the groups are identified by some characteristic that is repeatable and interesting. If there is a difference among the group means and you repeat the experiment, you would expect to see the same pattern of differences among the means, because you could classify the observations into the same groups. The group labels are meaningful (such as "seawater, glucose solution, mannose solution"). You are interested in the relationship between the way the data are grouped (the "treatments") and the group means. Examples of data for which a model I anova would be appropriate are:

If you have significant heterogeneity among the means in a model I anova, the next step (if there are more than two groups) is usually to try to determine which means are significantly different from other means. In the amphipod example, if there were significant heterogeneity in time of death among the treatments, the next question would be "Is that because mannose kills amphipods, while glucose has similar effects to plain seawater? Or does either sugar kill amphipods, compared with plain seawater? Or is it glucose that is deadly?" To answer questions like these, you will do either planned comparisons of means (if you decided, before looking at the data, on a limited number of comparisons) or unplanned comparisons of means (if you just looked at the data and picked out interesting comparisons to do).

Model II anova

In a model II anova (also known as a random-effects model anova), the groups are identified by some characteristic that is not interesting; they are just groups chosen from a larger number of possible groups. If there is heterogeneity among the group means and you repeat the experiment, you would expect to see heterogeneity again, but you would not expect to see the same pattern of differences. The group labels are generally arbitrary (such as "family A, family B, family C"). You are interested in the amount of variation among the means, compared with the amount of variation within groups. Examples of data for which a model II anova would be appropriate are:

If you have significant heterogeneity among the means in a model II anova, the next step is to partition the variance into the proportion due to the treatment effects and the proportion within treatments.

How to tell the difference

If you are going to follow up a significant result with planned or unplanned comparisons of means, it's model I; if you are going to follow up by partitioning the variance, it's model II. Sometimes it's not obvious which model to use; I've seen many examples of researchers partitioning the variance after a model I anova, or doing comparisons of means after a model II anova, just because their software outputs both sets of numbers. I find it helpful to imagine that you've written all the observations for the measurement variable on cards, with one card for each group. At the top of each card you've written the name of the group. For example, imagine you've written the tastiness measurements for 10 peaches from tree A on one card, 10 peaches from tree B on a second card, etc. Then imagine that your scientific arch-enemy sneaks into your lab and erases the tree identification letter (A, B, C, ...) from each card. Now if one of the trees has significantly better tastiness measures than the other trees, you don't know which tree it was. If your experiment is completely ruined, and you have to wait a year until you can go back to the same peach trees and get new tastiness measures, and therefore your scientific arch-enemy is cackling with glee, that's a model I anova. But if your experiment isn't ruined—if your goal was to see how much tree-to-tree variation there was, and you can just write new arbitrary tree names on each card and still answer the question, and your arch enemy is going "Curses! Foiled again!"—that's a model II anova.

Further reading

Sokal and Rohlf, pp. 201-205.

Zar, pp. 184-185.



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This page was last revised August 31, 2009. Its address is http://udel.edu/~mcdonald/statanovamodel.html. It may be cited as pp. 127-129 in: McDonald, J.H. 2009. Handbook of Biological Statistics (2nd ed.). Sparky House Publishing, Baltimore, Maryland.

©2009 by John H. McDonald. You can probably do what you want with this content; see the permissions page for details.