## 18.4 Model-I versus Model-II regression

### 18.4.1 Definitions

Model-I regression is the type we just learned - ordinary least-squares regression. We fit a regression line that minimizes the squared residuals with respect to the “Y” variable.

Model-II regression or “reduced major axis regression” fits a line that minimizes the squared residuals in both the “Y” and “X” direction.

### 18.4.2 Which one do I use?

The biomass experiment example that we’ve used in this tutorial involved direct manipulation of species richness (number of species), and the response variable that was measured was biomass stability.

For two reasons, this experimental study lends itself to Model-I regression. First, there is good theoretical reasons to expect that manipulation of plant richness can cause changes in plant biomass stability. Thus, there is a causal direction that makes $$X$$ a natural “independent” variable and $$Y$$ a “dependent” variable. Second, the number of plant species was directly controlled in the experiment, and thus had zero uncertainty associated with its values (biomass stability would have had at least some uncertainty in its estimation). Under either of these conditions, Model-I regression is most appropriate.

Now imagine an observational study in which we randomly sample 40 vegetation plots, and in each, measure plant species richness (number of plant species) and insect species richness. We are interested in predicting insect richness from plant richness, but is Model-I regression appropriate?

This study arguably lends itself more to Model-II regression. Why? First, because it is conceivable that either plant species richness OR insect species richness could be a “dependent” variable or “independent” variable… the causal direction is not clear. Second, both variables were not directly controlled, and therefore there is necessarily some uncertainty involved.

There is some debate about which method should be used in what contexts. If you’re interested in learning more, see this publication and this one.

For this course, you can assume Model-I regression is appropriate for quiz/exam questions, unless otherwise indicated.