# 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