_sec-understand-coxph.qmd

{{< include latex-macros/macros.qmd >}}

<!-- {{< include _sec_cox-bio.qmd >}} -->

---

::: notes
Let's make two generalizations. 
First, we let the hazard depend on some
covariates $x_1,x_2, \dots, x_p$; 
we will indicate this dependence by extending our notation for hazard:
:::

{{< include _sec-surv-conditional-hazards.qmd >}}

---

{{< include _def-ph-model.qmd >}}

---


:::{#lem-diffloghaz-ph}

If $\loghaz(t|\vx) = \loghaz_0(t) + \reglincomb$, then: 

$$
\ba
\diffloghaz(t|\vx: \vxs)
&= (\vx - \vxs) \cdot \beta
\ea
$$

:::

---

:::{#thm-hazard-ratio-ph}

If $\loghaz(t|\vx) = \loghaz_0(t) + \reglincomb$, then: 

$$
\ba
\theta(t|\vx: \vxs)
&= \expf{\diffloghaz(t|\vx: \vxs)}
\\
&= \expf{(\vx - \vxs) \cdot \beta}
\ea
$$

:::

So for proportional hazards models, we can write the hazard ratio using a shorthand notation:

$$\theta(t| \vx : \vxs)  = \theta(\vx : \vxs)$$

---

:::{#lem-ph-diffloghaz-0}
$$\diffloghaz(t|\vx)= \reglincomb$${#eq-diffloghaz-0-ph}
:::

---

:::{#thm-hazard-ratio-vs-baseline-ph}

If $\loghaz(t|\vx) = \loghaz_0(t) + \reglincomb$, then: 

$$\theta(t|\vx) = \expf{\reglincomb}$$

:::

---

::: proof

$$
\ba
\theta(t|\vx)
&\eqdef \theta(t| \vx : \vec{0})
\\
&= \expf{\diffloghaz(t|\vx)}
\\
&= \expf{\reglincomb}
\ea
$$

:::

---

:::{#thm-ph-haz-decomp}
$$\haz(t|x) = \haz_0(t)\theta(x)$$
:::

---

Also:

:::{#thm-ph-also}

$$
\begin{aligned}
\theta(x) &= \exp{\diffloghaz(x)}
\\
\log{\haz(t|x)} 
&= \log{\haz_0(t)} + \diffloghaz(x)
\\
&= \loghaz_0(t) + \diffloghaz(x)
\\
\diffloghaz(x) &= \reglincomb \\
&\eqdef \beta_1x_1+\cdots+\beta_px_p
\end{aligned}
$$

:::

::: notes
This model is **semi-parametric**,
because the linear predictor depends on estimated
parameters but the base hazard function is unspecified.
There is no constant term in $\eta(x)$,
because it is absorbed in the base hazard.

:::

---

Alternatively, we could define $\beta_0(t) = \log{\haz_0(t)}$, and then:

$$\eta(x,t) = \beta_0(t) + \beta_1x_1+\cdots+\beta_px_p$$

---

::: notes
For two different individuals 
with covariate patterns $\vec x_1$ and $\vec x_2$, 
the ratio of the hazard functions 
(a.k.a. **hazard ratio**, a.k.a. **relative hazard**) is:
:::

$$
\begin{aligned}
\frac{\haz(t|\vec x_1)}{\haz(t|\vec x_2)}
&=\frac{\haz_0(t)\theta(\vec x_1)}{\haz_0(t)\theta(\vec x_2)}\\
&=\frac{\theta(\vec x_1)}{\theta(\vec x_2)}\\
\end{aligned}
$$

::: notes
Under the proportional hazards model,
this ratio (a.k.a. proportion) does not depend on $t$.
This property is a structural limitation of the model;
it is called the **proportional hazards assumption**.
:::

---

::: {#def-pha}
### proportional hazards

A conditional probability distribution $p(T|X)$
has **proportional hazards** if the hazard ratio $\haz(t|\vec x_1)/\haz(t|\vec x_2)$ does not depend on $t$. Mathematically, it can be written as:

$$
\frac{\haz(t|\vec x_1)}{\haz(t|\vec x_2)}
= \theta(\vec x_1,\vec x_2)
$$

:::

::: notes
As we saw above, Cox's proportional hazards model has this property, with $\theta(\vec x_1,\vec x_2) = \frac{\theta(\vec x_1)}{\theta(\vec x_2)}$.
:::

---

:::{#thm-haz-ratio-notations}

::: notes
We are using two similar notations, 
$\theta(\vx,\vxs)$ and $\theta(\vec x)$. 
We can link these notations: 
$$\theta(\vx) \eqdef \theta(\vx, \vec 0)$$

Then:
:::

$$\theta(\vx, \vxs) = \frac{\theta(\vx)}{\theta(\vxs)}$$
$$\theta(\vec 0) = \theta(\vec 0, \vec 0) = 1$$

:::

---

The proportional hazards model also has additional notable properties:

$$
\begin{aligned}
\frac{\haz(t|\vec x_1)}{\haz(t|\vec x_2)}
&=\frac{\theta(\vec x_1)}{\theta(\vec x_2)}\\
&=\frac{\exp{\eta(\vec x_1)}}{\exp{\eta(\vec x_2)}}\\
&=\exp{\eta(\vec x_1)-\eta(\vec x_2)}\\
&=\exp{\vec x_1'\beta-\vec x_2'\beta}\\
&=\exp{(\vec x_1 - \vec x_2)'\beta}\\
\end{aligned}
$$

---

Hence on the log scale, we have:

:::{#thm-diff-loghaz-lincom}
$$
\ba
\log{\frac{\haz(t|\vx)}{\haz(t|\vxs)}}
&= \diffloghaz(t|\vx : \vxs) 
\\
&\eqdef \loghaz(t|\vx) - \loghaz(t|\vxs)
\\
&=\eta(\vec x_1)-\eta(\vec x_2)\\
&= \vec x_1'\beta-\vec x_2'\beta\\
&= (\vec x_1 - \vec x_2)'\beta
\end{aligned}
$$
:::

---

If only one covariate $x_j$ is changing, then:

$$
\begin{aligned}
\log{\frac{\haz(t|\vec x_1)}{\haz(t|\vec x_2)}}
&=  (x_{1j} - x_{2j}) \cdot \beta_j\\
&\propto (x_{1j} - x_{2j})
\end{aligned}
$$

::: notes
That is, under Cox's model $\haz(t|\vec x) = \haz_0(t)\exp{\vec x'\beta}$, the log of the hazard ratio is proportional to the difference in $x_j$, with the proportionality coefficient equal to $\beta_j$.
:::

---

Further,

$$
\begin{aligned}
\log{\haz(t|\vec x)}
&=\log{\haz_0(t)}  + x'\beta
\end{aligned}
$$

::: notes
That is, the covariate effects are additive on the log-hazard scale; hazard functions for different covariate patterns should be vertical shifts of each other.

See also:

<https://en.wikipedia.org/wiki/Proportional_hazards_model#Why_it_is_called_%22proportional%22>

:::

### Additional properties of the proportional hazards model

If $\haz(t|x)= \haz_0(t)\theta(x)$, then:

:::{#thm-ph-cuhaz}

#### Cumulative hazards are also proportional to $\cuhaz_0(t)$

$$
\begin{aligned}
\cuhaz(t|x)
&\eqdef \int_{u=0}^t \haz(u)du\\
&= \int_{u=0}^t \haz_0(u)\theta(x)du\\
&= \theta(x)\int_{u=0}^t \haz_0(u)du\\
&= \theta(x)\cuhaz_0(t)
\end{aligned}
$$

where $\cuhaz_0(t) \eqdef \cuhaz(t|0) = \int_{u=0}^t \haz_0(u)du$.

:::

---

:::{#thm-log-cuhaz-parallel}

#### The logarithms of cumulative hazard should be parallel

$$
\logf{\cuhaz(t|\vx)} =\logf{\cuhaz_0(t)}  + \reglincomb
$$
:::

---

:::{#cor-log-nlog-surv}
### linear model for log-negative-log survival
$$
\logf{-\logf{\surv(t|\vx)}} = 
\logf{-\logf{\surv_0(t)}}  + \reglincomb
$$

:::

---

:::{#thm-ph-surv}
#### Survival functions are exponential multiples of $\surv_0(t)$

$$\surv(t|x) = \sb{\surv_0(t)}^{\theta(x)}$$

:::

---

::: proof
$$
\begin{aligned}
\surv(t|x)
&= \expf{-\cuhaz(t|x)}\\
&= \expf{-\theta(x)\cdot \cuhaz_0(t)}\\
&= \paren{\expf{- \cuhaz_0(t)}}^{\theta(x)}\\
&= \sb{\surv_0(t)}^{\theta(x)}\\
\end{aligned}
$$

:::

### Summary of proportional hazards model structure and assumptions

{{< include _coxph-model-structure.qmd >}}