# What regression model to use when independent variables are percentages to predict % outcome?

"Independent" variables: time spent (% at work, % sleeping, % exercising), body mass composition (% fat, % muscle, % bone)

Dependent variable: Smoker (1) or Non-Smoker (0)

What kind of regression model should I use when subsets of the "independent" variables are percentages and are therefore not completely independent of each other?

• You could include 2 of the 3 percentages. Eg. include % at work, % sleeping. Similarly for bmc. Obviously you need to use a logistic/probit type model that is suitable for binary outcomes.
– ved
Jun 29 '14 at 23:33
• Thank you so much ved, I see how dropping one of the variables (preferably with the lowest explanatory power) will make the other varibles in the same category independent.
– jc11
Jun 29 '14 at 23:43
• Unless the percentages must add to 100, you might want to include them all anyway. (In the two examples given, the percentages ought to total less than 100 and the totals would likely vary.)
– whuber
Jun 30 '14 at 1:28
• Such variables (with a total of 100%) are known as compositional variables. You could look into compositional data analysis, where transformations of such variables are studied. Some othet questions touching into this area: stats.stackexchange.com/questions/35265/… stats.stackexchange.com/questions/95867/… stats.stackexchange.com/questions/89717/… Jun 30 '14 at 19:14

Your response is binary and so you probably want to look at something like a binomial GLM for that, such as logistic regression.

Having a group of $k$ predictors that add to 1 (e.g. the $k=3$ body proportion predictors) would imply that at most you can have $k-1$ of them in the model because of the multicollinearity issue.

However, I'm going to suggest that you may also want to transform those percentages; they're unlikely to enter the model linearly; indeed with a logit link my first thought would be that you might want to try something like the logit of the proportions instead.

I'd also go for logistic regression since you don't mention that you have a time variable specifying the time from inclusion in the study until start of smoking or censoring (end of study); in that case a Cox regression would be better.

I doubt that there is any difference in using percentages as predictors as compared to other continuous variables. For example, BMI (Body mass Index) is neither a directly measured predictors, as it is derived from a division of two units.

As Glen_b mentions, these predictors might not be truly linearly associated with the dependent variable. But transforming them might make predictors more difficult to interpret and journals typically don't like transformed variables if they are the predictors of main interest.

• "journals typically don't like transformed variables if they are the predictors of main interest": I don't see any evidence for this assertion. As an Editor of a statistical journal, I don't think it is especially accurate or helpful. Reviewers and editors in my experience just want there to be good reasons for any transformation. Oct 6 '14 at 7:04
• I agree partly. Let's say that one examines the effect of blood pressure on risk of alzheimers disease. Blood pressure being the predictor of main interest and adjustments are made for various covariates. If one discovers that entering blood pressure linearly (without any transformation) will provide an adequate fit but less good than entering the log of blood pressure, I believe that many would prefer the linear version, due to simplicity of interpretation. However, I understand your point of view. This often makes me wonder why referees don't ask for regression diagnostics more frequently.. Oct 6 '14 at 7:09
• You've given an example in which a transformation has mixed benefits and reviewers might fairly wonder whether it is worthwhile. That's fine by me and consistent with my position. I don't think one example, or even the implication of many similar, substantiates your much broader claim. Indeed, there are many other examples in which transforming predictors is utterly standard, as when fitting power functions. Oct 6 '14 at 7:34