# Interpretation of Odds Ratio of Zero

In logistic regression, does an odds ratio of zero make sense, and if so, what's the interpretation.

I've only been able to locate one reference that specifically refers to OR of 0: Making Sense of Data: A Self Instruction Manual on the Interpretation of Epidemiological Data (Abramson et al, 2001 ) p113 which just states that an odds ratio of zero indicates one of the odds being compared must be zero, and that it indicates a strong negative association (unless the other odds is close to zero).

If an OR of 1 indicates no association, OR of 0 must indicate infinite association ? In that case, is a predictor with an OR of zero a predictor that results in complete separation?

I'm not sure the background of the study is relevant to the question, but just in case: the DV is the presence or absence of medium scale residential development in a neighbourhood, and the IV with a odds ratio of 0 is the median house / land sales price per square metre, which theory says is likely to be a predictor of the scale of residential development. The variable was significant in the logistic regression.

• You should provide some details on your data; the dependent, independent variables and the aim of your analysis. I think it could help a bit... Feb 21, 2015 at 11:09

It's easiest to illustrate what is going on with a simple example with a single predictor that is dichotomous (e.g., to distinguish two groups). Suppose these are the data (using R for illustration):

y   <- c(0,0,0,1,1,0,0,0,0,0)
grp <- c(0,0,0,0,0,1,1,1,1,1)
cbind(grp, y)


So:

      grp y
[1,]   0 0
[2,]   0 0
[3,]   0 0
[4,]   0 1
[5,]   0 1
[6,]   1 0
[7,]   1 0
[8,]   1 0
[9,]   1 0
[10,]   1 0


There are 5 observations for each group. In group 0 (the reference group), there are 2 events, so the odds of the event are $2/3$. So, the log odds of the event happening are $\ln(2/3) = -0.4055$. In the second group, the are 0 events, so the odds of the event happening are $0/5$. And the log odds of the event are $\ln(0/5) = -\infty$. So, the odds ratio of the event happening in group 1 versus 0 is $(0/5)/(2/3) = 0$. So, the log odds ratio is $\ln((0/5)/(2/3)) = -\infty$ or, equivalently, $\ln(0/5) - \ln(2/3) = -\infty$.

Now let's actually fit the model:

res <- glm(y ~ grp, family=binomial)
summary(res)


This yields:

Call:
glm(formula = y ~ grp, family = binomial)

Deviance Residuals:
Min        1Q    Median        3Q       Max
-1.01077  -0.75810  -0.00008  -0.00008   1.35373

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept)   -0.4055     0.9129  -0.444    0.657
grp          -19.1606  4809.3409  -0.004    0.997

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 10.0080  on 9  degrees of freedom
Residual deviance:  6.7301  on 8  degrees of freedom
AIC: 10.73

Number of Fisher Scoring iterations: 18


So, the estimated intercept is $-0.4055$, which is the log odds in group 0. The coefficient for grp is the log odds ratio, which is estimated to be $-19.1606$. Hmmm, that's not quite $-\infty$. But after exponentiation, we get the odds ratio, which we can round to, let's say, 8 digits:

round(exp(coef(res)[2]), 8)


And that is in essence zero. The coefficient for grp is not $-\infty$ due to numerical issues when fitting the model when there is complete separation in the data (and to answer that part of your question: that is indeed exactly what is going on here). But for all practical purposes, the model implies an odds ratio that is in essence zero.