How do I interpret this interaction between a continuous variable and a binary variable from a log binomial model? I'm new to modelling and not sure how to interpret this interaction for a results section. The model output tells me that the interaction between cancer (binary) and age (continuous - one year increments) has a p-value of 0.0073 and a relative risk of 1.08 (95%CI of 1.03-1.34). 
Does this mean that for every one year increase in age, the risk of experiencing the outcome increases by 8.0% for people with cancer?
Thank you  
 A: Assuming that the regression model is:
$$
\mathrm{logit}(p)=\beta_{0}+\beta_{1}\mathrm{age} + \beta_{2}\mathrm{cancer} + \beta_{3}\mathrm{age}\times\mathrm{cancer}
$$
where $\mathrm{cancer}$ is a dummy variable that is 1 for people who have cancer and 0 for people who don't. For people who don't have cancer, the model simplifies to
$$
\mathrm{logit}(p)=\beta_{0}+\beta_{1}\mathrm{age} + \beta_{2}\times 0 + \beta_{3}\mathrm{age}\times 0 = \beta_{0}+\beta_{1}\mathrm{age}
$$
so that for those, the odds of developing severe influenza increases by a factor of $\exp(\beta_{1})$ (odds ratio) per additional year of age. For people who do have cancer, the model is
$$
\mathrm{logit}(p)=\beta_{0}+\beta_{1}\mathrm{age} + \beta_{2}\times 1 + \beta_{3}\mathrm{age}\times 1 = (\beta_{0} + \beta_{2}) + (\beta_{1} + \beta_{3})\mathrm{age}
$$
so that the odds of developing severe influenza increases by a factor of $\exp(\beta_{1} + \beta_{3})$ (odds ratio) per additional year of age.
Another way of looking at it is that the regression lines for age on the log-odds scale are not parallel for the two groups (cancer and not cancer). Therefore, $\beta_{3}$ is the difference of the slope of the regression line for age on the log-odds scale for people who have cancer compared to the slope of the line for those who don't have cancer.
Here is a graph that illustrates the situation. The corresponding R code used to generate the graph is at the end of this answer.

The plot on top displays the relationship on the log-odds scale whereas the lower plot is on the response scale (i.e. probability). You can see that the lines on the log-odds scale (upper graph) are not parallel, which indicates an interaction between cancer status and age.
set.seed(142857)

library(visreg)

n <- 500

age <- runif(n, 20, 60)  

cancer <- rbinom(n, 1, 0.25)

linpred <- log(0.2) + log(1.03)*age + log(1.2)*cancer + log(1.08)*age*cancer

pr <- 1/(1 + exp(-linpred))   

y <- rbinom(n, 1, pr)  

mod <- glm(y~age*cancer, family = binomial)

par(cex = 1.5, mar = c(4, 4, 2, 0.2), mfrow = c(2, 1))
visreg(mod
       , xvar = "age"
       , by = "cancer"
       , partial = FALSE
       , rug = FALSE
       , overlay = TRUE
       , ylab = "log-odds"
       , scale = "linear"
)
visreg(mod
       , xvar = "age"
       , by = "cancer"
       , partial = FALSE
       , rug = FALSE
       , overlay = TRUE
       , ylab = "Probability of severe influenza"
       , scale = "response"
)

