# Compare odds ratios that are above and below 1

I am testing a logistic regression model that has two continuous IVs in SPSS. The DV is of course binary. IV-1 has an odds ratio above 1 and IV-2 has an odds ratio below 1. I understand what these mean, but how can I statistically compare the strengths of the influence of the two IVs? I understand that if both of the odds ratios are on the same side of 1, I can use the CIs to make the comparison. But I'm not sure what to do with my current situation.

Thank you very much!

In this case, just take the reciprocal and reverse the meaning. For example, an odds ratio of 0.5 means for every unit increase in the predictor, the odds of a success change by a multiple of 0.5. That's difficult to interpret, so take the reciprocal, and we can then say that for every unit decrease in the predictor, the odds of a success increase by a multiple of 1/0.5 = 2.

Here is a quick example which you can use to understand the odd's ratio of any variable.

lets say $\beta_1$ = 0.7 and $\beta_2$ = -0.3 and let $\beta_o$ = C

So your logistic regression model will be

$$\hat y = \hat \pi = \frac {1}{1 + e^{-(C + 0.7x_1 -0.3x_2)}}$$

you see that inside part of the exponential constant is the same as a linear regression function. But now when we extract odd ratio's

The odd ratio of $\beta1$

$$\beta_1 = e^{\beta_1} = e^{0.7} = 2.01$$

The odd ratio of $\beta2$

$$\beta_2 = e^{\beta_2} = e^{-0.3} = 0.74$$

Now for $\beta1$, when every thing else is constant, 1 unit increase of $X_1$ causes a $(2.01-1)\% = 101\%$ increase in probability $\hat y$.

Now for $\beta2$, when every thing else is constant, 1 unit increase of $X_2$ causes a $(1- 0.74)\% = 26\%$ decrease in probability $\hat y$.

It can also be interpreted as 1 unit decreasae in $X_2$ causes a $\frac {1}{e^{-0.3}} = 1.35$, or the a increase of 35% in probability $\hat y$.

Finaly if odd's ratio is 1, it means there is no change in probability.