How should I interpret the following interaction term of 2 continuous predictors in the output of a Cox proportional hazards model?

The Hazard ratio for the interaction of X and Y is >1, which means its log (the original coefficient) is 0-1 (~0.16). The individual items have a HR less than one, and coefficients of X= -0.18 and Y=-0.11.

    |   Variable                   | HR (s.e.)     | p value  
    1 A (5 points)                 |0.756 (0.088)  |    0.001 |      
    2 B (5 points)                 |1.379 (0.11)   |    0.001 |      
    3 X  (10 points)               |0.837 (0.033)  |    0.0011|      
    4 Y  (1 point)                 |0.895 (0.03)   |     0.001|      
    5 X (10 points)x Y (1 point)   |1.016 (0.006)  |    0.011 | 

The effect of a 10 point increase in X, with Y=0 is to decrease the "death" rate by 16%. The effect of a 1 point increase in Y, with X=0, is to decrease the death rate by 10.5%.

What is the effect of a one-point increase in Y on the effect of a 10-point increase in X on the death rate?

X has a range of 0 to 90. Y has a range of 0 to 10.

With a one-point increase in Y, does the effect of a 10-point increase in X increase from 16% to (16% + 1.6%) = 17.6%, or does it decrease by 1.6% to 14.4%?

Thought I had it straight but now very stuck here.


For models beyond the simplest (and an interaction makes it non-simple) I like to look at predictions instead of trying to interpret the coefficients directly. Does the software that you used to fit the model also do predictions for a given set of x and y? (many if not all do). You can then make predictions for patients with the following (x,y): (0,0), (0,1), (10,0), and (10,1) and see how they compare (or maybe use values more meaningful, such as start at the mean or median and then go 1, 10 units either direction). A simple prediction is mean or median survival, but if possible it is really nice for a survival analysis to plot the 4 (or more) predicted survival curves (different colors). These plots/comparisons often make the direction and magnitude of the effects clear.


Did you find out the answers? I would like to know that, too. I think the interpretation is like this: With one-point increase in $Y$ and 10 point increase in $X$, the risk of death increases with 1.6% and this is significant. With keeping $X$ constant, increase in $Y$ decrease the risk (by 16.3%) and with keeping $Y$ constant, increase in $X$ decrease the risk (by 10.5%) but when both of them are working together, they increase the risk of death. We can also check this if we have coefficient value for baseline hazard ($\beta_0$), $X$ ($\beta_1$), $Y$ ($\beta_2$) and $X\times Y$ ($\beta_3$). If there is no interaction then $\exp(\beta_3) = \exp(\beta_1+\beta_2-\beta_0)$. I am not statistician. Please correct me if I am wrong.


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