# How to calculate overall odds ratios from two 2X2 contingency tables with correlated variables?

I have analyzed the odds ratio (and risk ratio) from a 2x2 contingency table - I have calculated the odds of Y is true (i.e. =1) given that X1 is true (i.e. =1) versus X1 is false (i.e. =0). The result is a statistically significant odds ratio (and risk ratio) greater than 1.

I now have a second variable (X2), and in the same way (2x2 contingency table) have calculated the odds ratio (and risk ratio) to be greater than 1.

But ... I have an expectation (from domain knowledge in my field) that when X2=0 the influence of X1 on Y may be reduced (i.e. there is an interaction between independent variables). I want to test this hypothesis - and am looking for advice on how best to do so.

An approach I have done so far (but I am looking for suggestions as to why this might not be appropriate) is that I have formed a third contingency table counting the number of times our outcome is true for (X1=1 while X2=0) versus the number of times outcome is true for (X1=1 while X2=1). In doing so I observe that the case (X1=1 while X2=0) results in lower odds of the outcome compared to (X1=1 while X2=1). Is it appropriate to form contingency tables this way? If not can you please explain why not and suggest why other alternate approaches may be better (i.e. logistic regression?).

Thanks in advanced, I am familiar with R if that may guide suggestions.

Probably not appropriate unless I misunderstood what your analysis meant. A 2x2 interaction cannot be examined just by looking at 2 of the 4 categories.

Let's cast this into an ANOVA looking at mean rather than odds. Let's say the mean of $X_{11}$ is 10 and $X_{10}$ is 5. These alone cannot prove interaction, because $X_{01}$ can be 17 and $X_{00}$ can be 12. In that case there is no interaction. Having the first subscript as 1 lowers the mean by 7, having the second subscript as 1 increases the mean by 5.

A better way is to model the four odds in logistic regression:

$logit(Y) = \beta_0 + \beta_1 X1 + \beta_2 X2 + \beta_3 (X1\times X2)$

Four means can be retrieved here:

$\beta_0$: when both are 0

$\beta_0 + \beta_1$: when X1 = 1 and X2 = 0

$\beta_0 + \beta_2$: when X1 = 0 and X2 = 1

$\beta_0 + \beta_1 + \beta_2 + \beta_3$: when both are 1

$\beta_3$ captures the extra adjustment due to interaction. If there is no interaction it should be 0. Hence, the p-value of $\beta_3$ tests the interaction.

• Thank you - that is exactly what I was looking for. My understanding is that this could be coded in R using: Commented Oct 16, 2016 at 21:34
• mylogit <- glm(Y ~ X1 + X2+ (X1*X2), data = mydata2, family = "binomial") Commented Oct 16, 2016 at 21:36