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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 $x_9$
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  • $\begingroup$ 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. $\endgroup$ – marc1s Jan 29 at 13:24
  • $\begingroup$ thank you for your comment @marc1s. Regarding the point 2, do you have a link that talks about that precise behaviour? $\endgroup$ – ecjb Jan 29 at 17:41
  • $\begingroup$ en.wikipedia.org/wiki/Confounding $\endgroup$ – marc1s Jan 30 at 13:19

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