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Consider a situation with two covariates X and Z. Suppose the data generating mechanism is generated by the Cox model when conditioning on both X and Z:

$$\lambda(t|X,Z)  = \lambda_0(t)e^{\beta_xX +\beta_zZ}$$

a) How would I go about deriving the Hazard function when only conditioning on \(X\) ?

b) Furthermore, what would happen if \(X\) and \(Y\) are independent?

c) Suppose we observe \(n\) iid replicates from the above model and allow for (independent) right-censoring. Suppose also that we now fit a Cox-model to the data using only the first covariate \(X\). Show that the Cox-score function, \(U(β)\), in this case does not have mean zero and give an expression for the bias term.

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