I received an email question about developing a logistic regression model with some interaction terms. One of the interaction terms was statistically significant but one or both of the main effects associated with the interaction was not.<U+FFFD> So is it okay, I was asked to include the interaction in the final model but not the non-significant main effects?

First, I need to comment on the “slash and burn”<U+FFFD> model building practice that this person is using. A recent posting to the MedStats email discussion group outlines problems with this approach (although it does not use the term “slash and burn”). The person who adopts a “slash and burn” approach to models has a parsimonious intent. He/she wants to use as few degrees of freedom as possible in the final statistical model and one way to do this is to strip out anything that has an insignificant p-value. The ideal in the “slash and burn” world is a model where every single p-value is smaller than 0.05.

I’ve done my share of “slash and burn” in the past, but as more research is being done, there is increasing evidence that these parsimonious models do not perform as well as expected. The “slash and burn” model is not too dissimilar from automated approaches like stepwise regression and appear to have many of the same flaws. I want to summarize these criticisms (Frank Harrell has some very interesting research in this arena), but I have not had the time to do this well. One reason I wanted to write up a description of the propensity score model (see my other weblog entry for June 26, 2006) is that it offers a different approach to “slash and burn.”

Other ideas worth considering are setting up models based on your initial understanding of the problem and if something that you expected to be highly significant turns out not so, maybe it isn’t so terrible to still include it in the final model. You should also consider including or excluding terms from the model based on considerations other than the p-value. If a covariate has a very limited range, for example, or a large percentage of missing values, it may be best to leave that variable off of the menu entirely. If a covariate has been demonstrated to be important in most previous studies, give it the benefit of the doubt and include it in your model regardless of what your p-value might say.

There are entirely new methods for model fitting, such as the lasso, Bayesian Model Averaging, and smoothing splines that offer intriguing alternatives to “slash and burn.” These new approaches are rather technical but appear to work well in practice. A revolutionary book that describes how these approaches are changing how we practice statistics is

**Regression Modeling Strategies: With Applications to Linear Models, Logistic Regression, and Survival Analysis.**Frank E. Harrell (2001) New York, NY: Springer. [BookFinder4U link]**(Model, Regression)**

But while Frank Harrell and others are making intelligent challenges and offering intriguing alternatives, it is worth noting that “slash and burn” is still quite popular and there are generally accepted protocols that you follow.

First, it is very important to remove terms one at a time. Two terms in a model might both be strongly correlated and the effect of A holding B constant and the effect of B holding A constant might both be statistically insignificant because both A and B are effectively measuring the same thing. Eliminate one of the factors or the other, but not both because things can change in surprising ways each time a factor is removed from your model.

Second, it is very important to respect the hierarchy of terms in a model. If a model includes a cubic term, then you need to include the quadratic and linear terms in the model, even if they are not statistically significant. The reason for this is that simple linear transformations of a cubic term (such as a change in units) will often convert some of those non-significant lower order terms back into significant ones on the transformed scale.

For a similar reason, it is a very bad idea to include an interaction term in your model without the main effects. Simple things like changing your reference level will profoundly change the nature of the interaction unless you are careful to include the main effects.

This is very important because most people who use tools like logistic regression are unaware of the internal mechanics of the programs. Different software has different internal workings so something that produces one set of results in one program might have radically different results in another program.

So how do you investigate an interaction in the “slash and burn” world? There are several choices that are worth looking at, but a lot depends on the context of your problem.

Sometimes a significant interaction is caused by an outlying cell. This is a combination of factors that has a response probability that is far out of line relative to other combinations of factors. You might look to identify this unique combination and estimate its response probability separately with a special indicator variable.

If your factors have more than two levels, you might see if the interaction disappears when a single level of one of your factors is intentionally left out of the model. Is the interaction isolated in a particular subgroup of your data?

If the interaction is one that has no rational explanation based on your knowledge of science, then perhaps you should reconsider whether your original choice to investigate interactions was a good one. Every research study places implicit or explicit limitations on the types of models and the complexity levels that will be considered. Perhaps a troublesome interaction is evidence that your initial plan was simply casting your net too widely.

There are lots of other things you might do, and it depends a lot on the context of your problem. Many of the choices you take will be controversial and there is no generally accepted process that everyone agrees with for building a good statistical model. I’m sure that some of the stuff I have described here will be found to be totally appalling by some.

**Related reading on these web pages:**

- Stats: Exploring interactions in a linear regression model
- Stats: The concepts behind the logistic regression model
- FAQ-12 What are some of the problems with stepwise regression?
- Stats: Stepwise regression to screen for covariates (November 25, 2005, Model, Linear regression)
- Stats: What is the best statistical model? (September 17, 2004, Model, Logistic regression)
- Stats: Interactions in logistic regression (April 8, 2004, Model, Logistic regression)