Continuous variable has very large odds ratio in binary logistic regression

Question

Any help would be very much appreciated. I have run a binary logistic regression model. My dependent variable is impaired or not-impaired (following stroke), I have two predictors one is age the other is an EEG variable called theta power. Both predictors are continuous variables. I have checked the assumptions and there is no indication for multicolinearity (both variables VIF=1.154, tolerance =0.866).In addition there is no indication that the model violates the assumption of linearity (I ran a check using linear regression).

My results show that the model is significant (Chi-square=13.779, df=2, sig=0.001). The sample is not large, only 31, however I only included two predictors with this in mind.

One of my predictors has an enormous Exp(B) value and confidence intervals to go along with it.

Age: B=0.084, S.E.=0.046, Wald=3.364, df=1, sig=0.067, Exp(B)=1.087, 95%C.I. for Exp(B)=0.994-1.189

ThetaPower: B=16.259, S.E.=8.019, Wald=4.112, df=1, sig=0.043, Exp(B)=11516574.29, 95%C.I. for Exp(B)=1.721-7.705+E13

There is only a very small variance in theta values (0.012) compared to age (202.034), could this be the reason for the extremely large odds ratio and confidence intervals? Is there anyway I can 'fix' this to keep this variable in my model and report the statistics in a more meaningful way?

My data are included now

Answer 1

After reading Scortchi and Ben's comments, I think I may have found a solution for you.

I think the problem is your scale of predictor. You know a regression coefficient represents the change in Y (outcome variable) relative to a one unit change in the respective independent variable. For your logistic regression your Y is probability of impaired.

For age, it is 1 years change to affect your probability to be impaired.

For your Theta, it is also 1 units change, but attention, your values are 0.1, 0.2,..., so the value "1" might not in a reasonable rage of your measurement. The same as a human can not live for 1000 years, if your unit is 1000 years, then the coefficient will be huge.

I think you may either divide yoru coefficient by 10 directly or may

change your theta power's unit, I don't know what it is.

Such as multiple your theta values by 10 then the results seem more reasonable.

age<-c(77,84,45,47,72,61,78,49,79,77,74,54,65,52,80)
  theta<-10*c(0.117,0.443,0.136,0.285,0.107,0.113,0.263,0.146,0.182,0.299,0.148,0.097,0.091,0.151,0.302)
impaired<-c(0,1,1,1,1,NA,1,0,1,1,1,0,0,0,NA)
mydata<-data.frame(cbind(age,theta,impaired))

logistic <- glm(impaired ~ age+theta, data = mydata, family = "binomial")
summary(logistic)

The results seem much better, but need to be understood according to your new unit.

Answer 2

Your problem is a lack of information. An explanation follows, but your options are to

1. Collect more data
2. Use a bayesian method where you provide information about the variable's beta via a prior
3. Use a penalized objective function, like those provided in the glmnet package

Explanation

Without typing in the data manually, what I believe is happening is that you have a dichotomous predictor where the sample proportion is 1.0 within one of its levels. Note that when the true proportion is exactly 1, the linear link function $\log p / (1 - p)$ must be infinity. In reality however, when the likelihood is being maximized on your computer, that variable's value of beta just keeps getting larger as the optimization routine continues, until it terminates due to some stopping condition. It's not infinity, but it's pretty close ;)

Edit: From comments below, I learned that the situation was not the dichotomous predictor situation described above. So I took a closer look at the data and noticed what I believe is the culprit: whenever ZRE_1 is positive, MOCHA2DICH is 1, and whenever ZRE_1 is negative, MOCHADICH is 0. The same basic reasoning from my earlier explanation would apply to the coefficient of ZRE_1, if MOCA2DICH is indeed the response.

Answer 3

I used the first 15 observation to check your calculations. I think I did not find very very strange results. The followings are my R code.

age<-c(77,84,45,47,72,61,78,49,79,77,74,54,65,52,80)
theta<-c(0.117,0.443,0.136,0.285,0.107,0.113,0.263,0.146,0.182,0.299,0.148,0.097,0.091,0.151,0.302)
impaired<-c(0,1,1,1,1,NA,1,0,1,1,1,0,0,0,NA)
mydata<-data.frame(cbind(age,theta,impaired))

logistic <- glm(impaired ~ age+theta, data = mydata, family = "binomial")
summary(logistic)

I did not see big age coefficient but theta's coefficient might be too big.