Calculating 10 year probability using hazard ratios? I am working on some medical research. Given 3 values (BMI, age, gender) I have a list of baseline risks for the probability of an adverse event occurring in the next 10 years for an individual with a given BMI, age, gender.
I also have a list of additional risk factors, given as Hazard Ratios. How can I properly compute the 10-year probability of an adverse event, by modifying the baseline risk if say, 1 or more risk factors apply to an individual.
So, for example: given that an individual with age 30, gender Male, BMI 25 has a 5% risk of adverse event in next 10-years, assume some additional risk factors X and Y, which have respective HRs of 1.3 and 1.6, how can I modify the 5% risk to encompass the risk factors? I know that the HRs were computed using multivariable Cox proportional hazards for a 10 year period, with death considered as a competing hazard.
Any additional statistical wisdom would be appreciated.
 A: I'm going to assume that by "a 5% risk of adverse event in next 10-years " you mean that $S_0(10y) = 95\%$. You can find $S_1(10y)$ given $S_0(10y)$ and a global hazard ratio $R$. 
Consider the following equations:


*

*The cumulative hazard function $H(t)$, with $h(t)$ being the hazard:
$$H(t) = \int_0^t h(\tau)d\tau.$$

*The relationship between survival and cumulative hazard:
$$S(t) = \exp(-H(t)).$$

*The proportional hazards assumption, with $R$ the (global) hazard ratio:
$$\frac{h_1(t)}{h_0(t)} = R.$$


Note: notation may differ between texts. You may encounter $\lambda(t)$ instead of $h(t)$ and $\Lambda(t)$ instead of $H(t)$. Using the equations above, you can write $S_1(t)$ in terms of $S_0(t)$ and the hazard ratio:
$$
\begin{align}
S_1(t) &= \exp(-H_1(t)), \\
&= \exp(-\int_0^t h_1(\tau)d\tau), \\
&= \exp(-\int_0^t R \times h_0(\tau)d\tau), \\
&= \exp(-R\times H_0(t)), \\
&= S_0(t)^{R} = \exp(R\times \log S_0(t)).
\end{align}
$$

What is left now is to determine the global hazard ratio $R$. In Cox PH models, the hazard is modelled as follows:
$$
h_1(t) = h_0(t) \times \exp(\beta' X)
$$
In models you will typically find both the coefficients per covariate ($\beta$) and the hazard ratio associated with a unit change in a given covariate $\exp(\beta_i)$. 
To compute the global hazard ratio $R$, one more piece of the puzzle is needed: the baseline values of the covariates $X_0$. In my experience, most packages use the mean for covariates used in building the model as baseline by default. When you know the baseline covariate values $X_0$, the global hazard ratio $R$ can be obtained using the following equation:
$$
R = \exp(\beta(X-X_0)).
$$
