# Can you compare Odds Ratio on the same population

I am a newbie in stats, so please bear with me. I am not an epidemiologist, but I have stated my problem is similar terms:

I have the data for calculating the Odds Ratio for Disease D when exposed to Injury I. I use Mantel Haenszel (MH) for this using age as stratas (confounding factor for MH). Say this gives me OR_1. Over that same population, I have similar data for Disease D when exposed to condition C. Again I use the MH and age. This gives me OR_2.

If OR_1 > OR_2, can I conclude that Injury I is more important than condition C for getting disease D ?

• Is it one data set or different data sets? If they are the same, and it is the same individuals that have the disease D in both analyses, you might consider using logistic regression. – JonB Nov 15 '16 at 7:01

The problem with the approach you have described is that those who are exposed to Injury $$I$$ are not mutually exclusive of those who are exposed to Condition $$C$$. So there are presumably individuals who have condition $$C$$ who are also Injured ($$I$$). But you also could have individuals who have condition $$C$$ who are not injured ($$I$$). As a result simply knowing $$OR_1 > OR_2$$ doesn't really tell you anything. Put another way, you can classify individuals into one of four categories:

So, if you look at the the odds ratio of individuals who have Disease $$D$$, when exposed to injury $$I$$, you are capturing $$n_{11}$$ as well as $$n_{12}$$ in the table above. When you then compare that odds-ratio to those who have disease $$D$$ when exposed to condition $$C$$, you are capturing those in $$n_{11}$$ and $$n_{21}$$ in the table above. As you can see, both odds-ratios will involve the same individuals from $$n_{11}$$. The problem here essentially boils down to contamination of the comparison. In other words, you are changing multiple variables at the same time making it impossible to determine the association between an exposure and the disease. To see this more easily, it's helpful to write out the possible categories of disease and exposures into which a person can fall. The tables below will help with this.

Recall that the odds ratio $$OR$$ can be written as:

$$\begin{eqnarray*} OR & = & \frac{{\color{blue}{\color{blue}P(Disease|Exposure)}}}{1-P(Disease|Exposure)}/\frac{P(No\,Disease|Exposure)}{P(No\,Disease|Exposure)} \end{eqnarray*}$$

It's also helpful to focus on the top Numerator $$NOR$$ in each of the odds ratios ($$OR_1$$ and $$OR_2$$) which, more or less, are compared in your analysis. This is the term in blue font above. The can be written as:

$$\begin{eqnarray*} NOR_{1} & = & P(Disease|Injury)\\ & = & P(Disease|Injury\,and\,Condition)+P(Disease|Injury\,and\,No\,Condition) \end{eqnarray*}$$ \ $$\begin{eqnarray*} NOR_{2} & = & P(Disease|Condition)\\ & = & P(Disease|Injury\,and\,Condition)+P(Disease|No\,Injury\,and\,Condition) \end{eqnarray*}$$

The comparison between $$OR_1$$ and $$OR_2$$ can largely be boiled down to comparing these two terms, $$NOR_1$$ and $$NOR_2$$. When you do so, you see the that $$P(Disease|Injury\,and\,Condition)$$ terms cancel each other out, since it appears in both $$NOR_1$$ and $$NOR_2$$, and we are left comparing:

$$P(Disease|Injury\,and\,No\,Condition)$$ from $$NOR_1$$ with $$P(Disease|No\,Injury\,and\,Condition)$$ from $$NOR_2$$.

But you see that this is a comparison of those who have the disease given they were injured and not exposed to condition $$C$$ to those who were not injured and were exposed to condition $$C$$. So both the injury exposure and condition exposure are changing. This makes it impossible to determine if it is the injury exposure or the condition exposure that is associated with the changes in disease state.

What is likely going to be a better analysis for you is to perform a logistic regression where you regress the log of the odds of disease (the logit) onto condition, injury and their interaction (see here for a seminar on this). This will allow you to disentangle and determine which is the more important factor associated with disease state.

Do you mean you are comparing two coefficients (odds ratios) in the same model? If so, I was taught that you can compare standardized coefficients but not unstandardized coefficients. This is because the scale of the unstandardized coefficients may be different, causing one to have a higher odds ratio than the other, but not a stronger effect on D. For example if Injury I is nominal but Condition C is continuous, you may see OR_1 > OR_2 without saying anything about the "usual" effect of Condition C. It may be that patients commonly score a "10" on Condition C, and 10*OR_2 > 1*OR_1.

Another way to compare effect sizes is to calculate marginal predicted probabilities for some "typical" respondents by taking means on all the other variables and calculating the probability of contracting Disease D for respondents with Injury I and Condition C.