# Interpretation of continuous variable in an odds ratio for logistic regression

I have an odds ratio of 1.02 for x variable (Age, a continuous variable measured in units "1 year"). My response variable is Y. I would interpret this as for every increase in Age by 1 year the odds of Y happening increase by 2%.

My question is can I say for every 10 year increase in Age, the odds of Y happening increase by 20% or would I have to group my continuous variable into 10 year categories to interpret this correctly?

Kind regards,

• If it seems reasonable to extrapolate 10 years ahead and age has only a main linear effect without interactions with other predictors, then yes: since the logistic regresssion predicts the odds increase by 1.02 in 1 year, the increase is exp(log(1.02)*10) = 1.22 in 10 years. Obviously this doesn't say anything about the base odds. And odds are hard to interpret anyway, so you should consider converting the odds to probabilities and reporting those. Apr 25, 2022 at 5:46
• A note about terminology: logistic regression is expressed in terms of the log odds (not a log odds ratio). Apr 25, 2022 at 7:48

## 2 Answers

If it is reasonable to extrapolate age by 10 years and if the effect of age is linear and without interactions, we can compute the odds at 10 years given the odds at 1 year.

If the logistic regression is linear in age, we can write: $$\text{odds} = \exp\{b_{\text{age}}age + \mathbf{b}_{-age}\mathbf{x}_{-age}\} = \exp\{b_{\text{age}}age\}\exp\{\mathbf{b}_{-age}\mathbf{x}_{-age}\}$$ where $$b_{\text{age}}$$ is the age coefficient, $$\mathbf{x}_{-age}$$ are any other predictors incl. the intercept and $$\mathbf{b}_{-age}$$ are the rest of the regression coefficients.

You specify that age is measured in units of 1 year and that $$\exp\{b_{\text{age}}\cdot1\} = 1.02$$. From this we can estimate that at age = 10 years:

$$\text{odds} = \exp\{\log(1.02)10\}\exp\{\mathbf{b}_{-age}\mathbf{-x}_{age}\} = 1.22\exp\{\mathbf{b}_{-age}\mathbf{x}_{-age}\}$$

The odds are a function of both age and the other predictors (there is at least an intercept). And since odds are hard to interpret anyway, you should consider converting the odds to probabilities and reporting both.

More importantly — and this is where I went wrong in my comment as well — your statement seems to imply causality: that a subject will have 22% higher odds event Y in 10 years time. However, the regression results cannot be interpreted causally unless a lot of assumptions are made and/or a lot of requirements met.

The correct, if wordier way, to explain the odds is: Say subject A is 10 years older than subject B but otherwise matches B in all other predictors (eg. has the same education if education is also a covariate). Then subject A has 22% higher odds of event Y.

• If you found this answer helpful, then please consider upvoting and/or accepting it. Apr 25, 2022 at 13:10

Let's say your model, expressed in terms of odds,

$$\dfrac{p}{1-p} = \exp(\beta_0 + \beta_1x) = B\exp(\beta_1x)$$

Here $$B = \exp(\beta_0)$$, and $$\exp(\beta_1)=1.02$$ as per your question. A ten year increase in age would increase the odds by a factor of

$$\exp(\beta_110) = \Big(\exp(\beta_1)\Big)^{10} \approx 1.22$$

• If you found this answer helpful, then please consider upvoting and/or accepting it. Apr 25, 2022 at 13:11