# Logistic regression with mutually exclusive predictor variables

I am working on a logistic regression approach to predict the clinical status of patients (No disease vs Disease). I already have quite strong evidence indicating that the number of genes hit by a certain kind of variation is significant in predicting disease. Among the genes that I am considering, one of them (gene A) apparently plays a central role: I would like to investigate if the effect caused by the number of genes involved is driven by "gene A + others" combinations or "any multiple genes combination". I developed a model with the 3 independent dummy variables geneA.alone (T if a sample has variation only in gene A), geneA.plus.others (T if a sample has variation in gene A and other gene(s)), and only.others (T if a sample has variation in one or more genes, but not in gene A).
I performed two nested model comparisons using the anova() function in R:

anova(glm(Status~geneA.alone,...),glm(Status~geneA.alone+geneA.plus.others,...),
test="Chisq")

anova(glm(Status~geneA.alone+geneA.plus.others,...),
glm(Status~geneA.alone+geneA.plus.others+only.others,...),test="Chisq")


In both cases I observe an improvement of the model fit, with a significant p-value (in the order of e-5 / e-6). So, my first conclusion was that, when multiple genes are involved, it is not only the combinations "geneA+other(s)" that are important in determining disease, but also those not involving gene A. However, I am really doubting that this is the right way to investigate it:

• Does it make sense to have three mutually exclusive variables as predictors? Can they be considered as independent variables? My feeling is that the significant p-values are a natural consequence of the fact that, by using mutually exclusive predictors, I deliberately subtract part of the information from a variable, and shift it to another...hance the significance.
• Would it be more appropriate to use only the variables number.of.genes (numeric) and geneA.involved (T/F), and compare the model Status~n.of.genes+geneA.involved with Status~n.of.genes*geneA.involved? My thought, in this case, is that if the second model results in a significantly better fit, it may be a demonstration that the n.of.genes is important only if geneA is involved...does it make sense?

Any help would be greatly appreciated! Thank you!

• Could you clarify whether you're re-defining the predictors in a given data-set based on the observed relationship to the response of an original set of predictors, or just asking whether the model makes a priori sense? Feb 5 '14 at 12:25
• So far, I know that n.of.genes is a better predictor than the simple presence or absence of variation (has_variation). I also know that, among the 32 genes I am analysing, gene A is the only one significant on its own (following a burden testing approach performed in cases vs controls). Starting from this two observations, now I would like to investigate whether the effect of n.of.genes is driven by the cases geneA.plus.othergenes or it does not matter which genes are involved. So, I thought that maybe comparing the two models in the second bullet-point could answer the question... Feb 5 '14 at 13:18
• The problem seems to be that you have all 3 levels of a dummy variable. Instead, why not a categorical variable called "genes" (or whatever) with three levels: Only A, A + others and Not A. Then the problem is pretty standard. The number of genes is a different issue, I think. Feb 5 '14 at 13:38
• @Peter: Is there not a "no genes hit by a certain kind of variation category", & hence a categorical variable with four levels? Feb 5 '14 at 16:03
• I'm not sure @scortchi, but the idea is the same Feb 5 '14 at 16:05

## 1 Answer

A key issue here is not making the interpretation of the results more difficult than necessary. It would be for dummy coded variables and for categorical variables (not A, A, A + others). As you say in your second bullet point, you could have the number of genes as one predictor and the presence or absence of A as another predictor. This would answer two questions 1) "Is the number of genes, controlling for the presence of A, a significant predictor?" and 2) "Is the presence of A, controlling for the number of genes, a significant predictor?" You also said you could use the interaction between these variables as a predictor. That would answer questions like "Is the presence of A significant depending on the number of genes?". Then, as you say, you compare the model without the interaction to the one with the interaction (also, you would of course see if the interaction term itself is significant). This seems like the best way to go about your analysis.