# Cox hazard Model: HR of 1 variable greater than 1 in simple regression becomes smaller than 1 in a multiple regression

I'm performing a simple and multiple Cox regression. I have 1 dependent variable $$y$$ and 10 independent variables:

$$X = x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9, x_{10}$$

A particularity of the model is that

$${x_8}_i + {x_9}_i = {x_{10}}_i$$

(i.e. that the vector $$x_{10}$$ is the addition of $$x_8$$ and $$x_9$$)

When running the simple Cox regression model

• $$x_1$$, $$x_2$$, $$x_3$$, $$x_4$$, $$x_5$$, $$x_6$$, $$x_7$$ are not survival predictors (i.e. their HR are cross 1 and are of the type HR: 1.1, 95% CI: [0.8-1.2}, p > 0.05)
• whereas $$x_8$$, $$x_9$$, $$x_{10}$$ are predictor of poorer (i.e. their HR are above 1 and are of the type HR: 1.2, 95% CI: [1.1-2.0}, p < 0.05)

When running the multiple Cox regression model

• $$x_1$$, $$x_2$$, $$x_3$$, $$x_4$$, $$x_5$$, $$x_6$$, $$x_7$$ are still not survival predictors
• $$x_8$$, $$x_9$$, $$x_{10}$$ are still predictor of poorer (i.e. their HR are above 1 and are of the type HR: 1.2, 95% CI: [1.1-2.0}, p < 0.05)
• whereas $$x_{10}$$ are predictor of better (i.e. their HR are under 1 and are of the type HR: 0.8, 95% CI: [0.6-0.9}, p < 0.05)

Here are my questions:

1. Is it ok to use a variable such $$x_{10}$$ which is a linear combination of other 2 explanatory variables (in this case $$x_8$$ and $$x_9$$)
2. How come $$x_{10}$$ which is statistically significant predictor of poorer survival in the simple model becomes a significant predictor of better survival in the multiple model (instead of simply becoming a non significant predictor of poorer survival). Does it have to do with the fact that it is a linear combination of $$x_8$$ and $$x_9$$
• For the point 1, I would not use linearly dependent variables in a linear model. For the second point, it is normal, read about confounding. – marc1s Jan 29 at 13:24
• thank you for your comment @marc1s. Regarding the point 2, do you have a link that talks about that precise behaviour? – ecjb Jan 29 at 17:41
• en.wikipedia.org/wiki/Confounding – marc1s Jan 30 at 13:19