I am fitting a Cox Proportional Hazards model to clinical data.
Suppose I have two variables, x and y,
and the resulted coefficients are a for x and b for y.
I am curious that will the estimated value of a affect the estimated value of b?
I only know that when we try to estimate the coefficient, we set the partial derivative of the score or loss function to zero, and solve the equation.
In linear regression, when we take the partial derivative of a, the b term disappears and is not included in the results, For example,
$\frac{\partial }{\partial a}\;ax+by = x$
But in exponential regression, taking the derivative of exponential will keep the both terms. For example,
$\frac{\partial }{\partial a}\;e^{ax+by} = xe^{ax+by}$
Because the partial maximum likelihood function contains many natural exponents, will the coefficients affect each other during estimation?