class: center, middle, inverse, title-slide .title[ # Competing Risks ] .author[ ### Yue Jiang ] .date[ ### STA 490/690 ] --- ### An old research question .question[ Suppose you are an investigator in the DIG trial, examining time to death (knowing that some patients might be censored. > *What is the predicted probability that, by three years, patients taking digoxin have not yet died due to heart failure?* > *Is there a relationship between treatment and time to death?* > *What is this relationship, perhaps accounting for other variables?* ] --- ### Cumulative incidence .question[ Suppose we estimate the survival probability (let's say using a Kaplan-Meier curve). How might you be able to estimate the cumulative incidence of death in the presence of censoring using the Kaplan-Meier curve? ] --- ### A competing event Now suppose we're specifically interested in death *due to heart failure*. Patients in our study might die due to heart failure, die due to other causes, or be censored. Death due to heart failure (our main event of interest), and death due to other causes are known as .vocab[competing events]. If one has happened, it is impossible for the other to happen. Associated probabilities of these events happening are called .vocab[competing risks]. --- ### Some new quantitites of interest Suppose now that the observed `\(\delta_i \in {0, 1, \cdots, K}\)`, where it is equal to 0 if an observation was censored, or `\(1, 2, 3, \cdots, K\)` corresponding to which of the `\(K\)` competing events occurred. For `\(k = 1, \cdots, K\)`, the .vocab[cause-specific hazard] for cause `\(k\)` is given by: `$$\lambda_k(t) = \lim_{dt \to 0+} \frac{P(t \le T < t + dt, \delta = k | T \ge t)}{dt}$$` and represents the instantaneous failure rate due to cause `\(k\)` given that you've made it until time `\(t\)`. The .vocab[cumulative incidence function] provides the probability of failure due to cause `\(k\)` by time `\(t\)`: `$$F_k(t) = P(T \le t, \delta = k)$$` --- ### Cause-specific hazards and cumulative incidence functions .question[ What is the relationship between cause-specific hazards and cumulative incidence functions? Is there a one-to-one relationship between the cause-specific hazard for cause `\(k\)` and the corresponding cumulative incidence function for cause `\(k\)`? ] --- ### Cause-specific hazards and cumulative incidence functions `$$F_k(t) = \int_0^t S(u)\lambda_k(u)du$$` .question[ What do you notice about this expression? ] --- ### Two regression models A cause-specific proportional hazards model can be used (e.g., a Cox model). Sometimes people combine all events into one and simply think of a composite endpoint instead (e.g., simply "death," without caring for cause). You can model cause-specific hazard by fitting separate proportional hazards model for each of the mutually exclusive events: `$$\lambda_k(t | \mathbf{X}) = \lambda_{k, 0}(t)\exp(\mathbf{X}\boldsymbol\beta_k)$$` which can be estimated as before. .question[ When thinking about the cause-specific partial likelihood, who is in the risk set at time `\(t_j\)` for cause `\(k\)`? ] --- ### Two regression models There is a corresponding hazard to the cumulative incidence function (or subdistribution), known as the .vocab[subdistribution hazard] (notice the tilde below): `$$\tilde\lambda_k(t) = \frac{d}{dt}\log(1 - F_k(t))$$` In the presence of competing risks, the subdistribution hazard is *not* the same as the cause-specific hazard. We can use a proportional subdistribution hazards model as follows: `$$\tilde\lambda_k(t | \mathbf{X}) = \tilde\lambda_{k, 0}(t)\exp(\mathbf{X}\boldsymbol\beta_k)$$` where `\(\tilde\lambda_{k, 0}(t)\)` is the baseline subdistribution hazard for cause `\(k\)`. This model is known as the .vocab[Fine-Gray model], and uses a modified risk set which includes subjects who have not yet been censored *or failed specifically due to cause* `\(k\)`. --- ### A thought experiment Suppose investigators are interested in a hypothetical drug for heart failure, busilomide, are randomly allocate half of the patients to busilomide and half to placebo. You are interested in time to death due to heart failure. Unfortunately, busilomide is not associated at all with death due to heart failure, but has a side effect in 50% of patients of summoning a bus to run them over (this usually happens within a few years). --- ### A thought experiment Two questions of interest: .question[ > *Is there an association between busilomide and time to death due to heart failure?* > *What is the proportion of the population who has died of heart failure at a given time?* Suppose you use a cause-specific Cox model and a Fine-Gray model to model these data. How well would they answer each of these questions? ]