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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.

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  • $\begingroup$ 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... $\endgroup$ Feb 21, 2015 at 11:09

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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.

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