Solution. A full-credit answer connects the likelihood to the linear predictor through a chain of named components and assumptions. Using \(\tilde X = (L, H, A, P, B, C)\):
Joint likelihood of the data set: \[\mathscr{L}\stackrel{\text{def}}{=}\operatorname{p}(\tilde Y = \tilde y, \tilde D = \tilde d \mid \mathbf X = \mathbf x)\]
Marginal likelihood contribution of observation \(i\): \[\mathscr{L}_i \stackrel{\text{def}}{=}\operatorname{p}(Y_i = y_i, D_i = d_i \mid \tilde X_i = \tilde x_i)\]
Independent-observations assumption (used to factor the likelihood): \[\mathscr{L}= \prod_{i=1}^n \mathscr{L}_i\]
Non-informative censoring assumption (used so each contribution reduces to survival and hazard terms): \(T_i \perp\!\!\!\perp C_i \mid \tilde X_i\), giving \[\mathscr{L}_i \propto \left[\operatorname{f}(y_i \mid \tilde x_i)\right]^{d_i}
\left[\operatorname{S}(y_i \mid \tilde x_i)\right]^{1 - d_i}
= \operatorname{S}(y_i \mid \tilde x_i)\cdot\left[{\lambda}(y_i \mid \tilde x_i)\right]^{d_i}\]
Distribution functions (define each one):
\[\operatorname{S}(t \mid \tilde x) \stackrel{\text{def}}{=}\operatorname{P}(T > t \mid \tilde X = \tilde x) = \operatorname{exp}\mathopen{}\left\{-{\Lambda}(t \mid \tilde x)\right\}\mathclose{}\]
\[\operatorname{f}(t \mid \tilde x) \stackrel{\text{def}}{=}{\lambda}(t \mid \tilde x)\,\operatorname{S}(t \mid \tilde x)\]
\[{\lambda}(t \mid \tilde x) \stackrel{\text{def}}{=}\operatorname{p}(T = t \mid T \ge t, \tilde X = \tilde x) = \frac{\operatorname{f}(t \mid \tilde x)}{\operatorname{S}(t \mid \tilde x)}\]
\[{\Lambda}(t \mid \tilde x) \stackrel{\text{def}}{=}\int_0^t {\lambda}(u \mid \tilde x)\,du = -\log\operatorname{S}(t \mid \tilde x)\]
\[\eta(t \mid \tilde x) \stackrel{\text{def}}{=}\log{\lambda}(t \mid \tilde x)\]
Proportional-hazards assumption (used to split the hazard into a baseline that depends only on time and a factor that depends only on covariates): \[{\lambda}(t \mid \tilde x) = \lambda_0(t)\cdot\theta(\tilde x)\] where \(\lambda_0(t)\) is the unspecified baseline hazard.
Logarithmic-link assumption (used to make the covariate factor a function of a linear predictor): \[\eta(t \mid \tilde x) = \eta_0(t) + \Delta\eta(\tilde x),
\qquad
\theta(\tilde x) = \operatorname{exp}\mathopen{}\left\{\Delta\eta(\tilde x)\right\}\mathclose{}\]
Linear functional-form assumption (used to write the covariate term as a linear combination): \[\Delta\eta(\tilde x) = \tilde x \cdot \tilde\beta
= \beta_L\, l + \beta_H\, h + \beta_A\, a + \beta_P\, p + \beta_B\, b + \beta_C\, c\]
Notice that the baseline hazard \(\lambda_0(t)\) is carried along symbolically the whole time — we never assume a shape for it.