Multistate Models

class: center, middle, inverse, title-slide

.title[
# Multistate Models
]
.author[
### Yue Jiang
]
.date[
### STA 490/690
]

---

### Quick announcements

Looks like learning about stratification next Tuesday won out. Please come by my
office if you have questions on exam material, which will be everything through November 5 (aka nothing from competing risks onward).

Once we cover stratification, I'll provide R code for recurrent events and multi-state models for your future reference (there's no homework on these topics anyway).

Given that we *are* Duke after all, I'll chance some of the special topics to cover Bayesian methodology. Survival is a huge field and there's only so much we can get to in a single introductory semester.

---

### A time to event model

---

### A more realistic model

.question[
What does this model represent/mean in real life?

Draw another type of multi-state model, clearly labeling what each piece means. Must graphs be acyclic for these types of analyses?
]

---

### A more realistic model

.question[
- For `\(\lambda_{TD}(t)\)`, what information would someone who actually experienced graft failure contribute to this pathway?
- Who might be in the risk set when evaluating `\(\lambda_{FD}(t)\)`?
- In general, how would you model the various .vocab[transition hazards]?
]

---

### A more realistic model

.question[
In general, how would you model the various .vocab[transition hazards], and how might this tie into our previous concept of competing risks?
]

---

### A general data formulation

Let `\(\boldsymbol\beta_{jk}\)` be the set of coefficients corresponding to a model which examines hazard of transitioning from state `\(j\)` to state `\(k\)`. Let the corresponding baseline hazard be `\(\lambda_{0jk}(t)\)`. Then for individual `\(i\)`, we can use a proportional hazard model:

`\begin{align*}
\lambda_{ijk} = \lambda_{0jk}\exp(\mathbf{x}_i\boldsymbol\beta_{jk}).
\end{align*}`

As always, be careful with the risk set - only those individuals who are *at risk of transitioning to state* `\(k\)` at time `\(t\)` are allowed to be included.

---

### A general data formulation

In this model, we can estimate separate baseline hazards for each transition (stratifying by which transition it is), and additionally incorporate time-varying covariates.

When modeling the data of individual `\(i\)` going from state `\(j\)` to `\(k\)`, the data must include

- Time of entry into state `\(j\)`
- Time of exit from state `\(j\)`
- Event indicator (e.g., of whether that individual entered state `\(k\)` or was otherwise censored)

We treat all non-entries into state `\(k\)` as censoring events, be they "real" censoring or transitions into other states (this directly relates to Q4/Q5 of the homework due this week, by the way!).

The counting process data structure (assuming no time-varying covariates) must have one row **for each possible transition** for each observation.

---

### Data representation

---

### Data representation

With these data, we might fit a set of Cox models using type of leukemia, age, whether the patient's gender matched their donor's, and whether a patient had T-cell depletion, stratified by which transition we care about.

Functions exist in the `survival` and `mstate` packages to analyze these data.

---

### Intermittently-observed data

.question[
What if we don't observe exact transition times, but rather, follow-up with patients at certain times and observe what state they are in?
]

The likelihood is way scarier (interval censoring tends to do this...). The `msm` package in `R` contains techniques for analyzing such data using the theory of continuous time Markov chains and various assumptions on the underlying processes.