In chjackson/msm: Multi-State Markov and Hidden Markov Models in Continuous Time

Multi-state modelling theory: recap {#theory}

Multi-state models for intermittently-observed data

• Slides from the introductory workshop in May 2021

More information about the msm package in

• the msm package vignette

• the Journal of Statistical Software paper

A more in-depth treatment of this kind of modelling can be found in

• van den Hout (2017)

• applications in life-course / ageing epidemiology

• Cook and Lawless (2018)

• more advanced mathematically, greater variety of models

Multi-state processes

Multi-state models represent processes that can be described as discrete states that change through time.

The msm package can be used for any state and transition structure.

Examples in the course represent stages of a disease.

```{tikz, tikz-header, echo=FALSE, eval=FALSE} \usetikzlibrary{positioning} \usetikzlibrary{calc} \usetikzlibrary{arrows} \definecolor{deepskyblue}{rgb}{0, 0.75, 1} \tikzstyle{state}=[minimum size = 0.8cm, draw, rounded corners, fill=deepskyblue]

For example, a progressive disease with death from any state (CAV after heart transplantation, from [Sharples et al.](https://doi.org/10.1097/01.TP.0000071200.37399.1D) )

```{tikz, cav-model, echo=FALSE, fig.width=5}
<<tikz-header>>
\begin{tikzpicture}[]
\node [state] (well) {1. No CAV};
\node [state, right=of well] (mild) {2. Mild CAV};
\node [state, right=of mild] (severe) {3. Severe CAV};
\node [state, below=of mild] (death) {4. Death};
\draw[->] ($(well.east)+(0cm,0.1cm)$) -- ($(mild.west)+(0cm,0.1cm)$) node [] {};
\draw[->] ($(mild.west)-(0cm,0.1cm)$) -- ($(well.east)-(0cm,0.1cm)$) node [] {};
\draw[->] ($(mild.east)+(0cm,0.1cm)$) -- ($(severe.west)+(0cm,0.1cm)$) node [] {};
\draw[->] ($(severe.west)-(0cm,0.1cm)$) -- ($(mild.east)-(0cm,0.1cm)$) node [] {}; 
\draw[->] (mild) -- (death) node [midway] {};
\draw[->] (well) -- (death) node [midway, left] {};
\draw[->] (severe) -- (death) node [midway, right] {};
\end{tikzpicture}

Progression through stages of an irreversible non-fatal condition (psoriatic arthritis, from Gladman and Farewell)

```{tikz, psor-model, echo=FALSE} <> \begin{tikzpicture}[] \node [state] (state1) {1. 0-1 damaged joints}; \node [state, right=of state1] (state2) {2. 1-4 damaged joints}; \node [state, right=of state2] (state3) {3. 5-9 damaged joints}; \node [state, right=of state3] (state4) {4. 10+ damaged joints}; \draw[->] ($(state1.east)+(0cm,0.1cm)$) -- ($(state2.west)+(0cm,0.1cm)$) node [] {}; \draw[->] ($(state2.east)+(0cm,0.1cm)$) -- ($(state3.west)+(0cm,0.1cm)$) node [] {}; \draw[->] ($(state3.east)-(0cm,0.1cm)$) -- ($(state4.west)-(0cm,0.1cm)$) node [] {}; \end{tikzpicture}

A relapsing-remitting condition with no "final" state (also in psoriatic arthritis, from [Jackson et al.](https://doi.org/10.1002/acr.22687))

```{tikz, echo=FALSE, fig.width=5}
<<tikz-header>>
\begin{tikzpicture}[]
\node [state] (nomda) {1. Disease activity};
\node [state, right=of nomda] (mda) {2. Minimal disease activity};
\draw[->] ($(nomda.east)+(0cm,0.1cm)$) -- ($(mda.west)+(0cm,0.1cm)$) node [] {};
\draw[->] ($(mda.west)-(0cm,0.1cm)$) -- ($(nomda.east)-(0cm,0.1cm)$) node [] {};
\end{tikzpicture}

msm models work in continuous time

Model represents movement between discrete states in continuous time

Transition intensities

Continuous-time Markov models are defined by transition intensities $q_{rs}(t)$ between pairs of states $r,s$.

Transition intensity matrix $Q$ with diagonals $q_{rr}(t) = -\sum_{s!=r} q_{rs}(t)$.

Transition probabilities

If $Q(t) = Q$ is constant over time, we can compute the transition probability matrix over any time unit as $P(t) = Exp(tQ)$ where Exp is the matrix exponential.

Entries $p_{rs}(t)$ define the probability someone in state $r$ now is in state $s$ at a time $t$ years from now.

Discrete-time models (not covered in this course) are defined by transition probabilities $p_{rs}$ over one unit of time. Less common than continuous-time models in medical applications, since data are not generally on a regular discrete-time grid.

Be careful of these distinctions!

Intermittently-observed data {#intermittent}

msm is mainly designed for data that are intermittently observed

changetimes <- c(1,4,6,8,10)
obstimes <- c(0, 1.5, 3.5, 5, 9)
states <- c(1,2,1,3,4)

nst <- 4
maxtime <- 10
par(lwd=2, cex=1.2, mgp=c(2,1,0), mar=c(3,4,1,1))
plot(0,0, xlim=c(0,maxtime), ylim=c(0.7,nst+1.0), type="n", axes=F, xlab="Years after transplant", ylab="")
axis(2, at = c(1,2,3,4), labels = c("State 1","State 2","State 3","State 4"), tick=F, las=1)

axis(1, at=c(0, 1.5, 3.5, 5, 9), lwd=1)

arrows(0,states[1], changetimes[1], states[1], length=0.1, col="red")
for (i in seq(along=changetimes))
  arrows(changetimes[i], states[i+1], changetimes[i+1], states[i+1], length=0.1, col="red")
for (i in seq(along=changetimes))
  lines(c(changetimes[i], changetimes[i]), c(states[i], states[i+1]), col="red")

for (i in seq(along=obstimes))
  segments(obstimes, 0, obstimes, nst, lty=2)

legend("topright", col=c("red","black"), lty=c(1, 2), lwd=c(2,2), c("Underlying process", "Observation times"), bty="n")

and variants of this format.

Time-to-event data (not covered in this course)

In multi-state time-to-event data, we know the exact times of transitions between all states, potentially with right-censoring

| Patient | Months | Event | | ----- | ---- | ---- | | 1 | 0 | Start of follow up | | 1 | 50 | Disease diagnosis | 1 | 60 | Death |

Competing risks analyses are a special case: e.g. survival data with one "alive" state and multiple "death" states for different causes of death.

msm can be used for this kind of data, but relies on strong assumptions:

• intensites are piecewise constant, i.e. constant over a series of time intervals, or a step function of time.
• models are Markov

Other R packages are better designed for multi-state time-to-event data:

• flexsurv for arbitrarily flexible, fully parametric models

• survival and mstate for semiparametric (Cox-type) models

Examples used for illustration

Psoriatic arthritis

```{tikz, fig.cap = "Psoriatic arthritis", fig.ext = 'png', echo=FALSE} <>

Factors governing progression of joint damage, an irreversible condition. 

Available as `psor` in the `msm` package. 


### Cardiac allograft vasculopathy after heart transplantation

```{tikz, fig.cap = "CAV after heart transplantation", fig.ext = 'png', echo=FALSE}
<<cav-model>>

A condition similar to coronary artery disease that occurs in people who have had heart transplants. Clinically irreversible, but measured state (diagnosed by angiography) can go backwards or forwards

Available as cav in the msm package.

chjackson/msm documentation built on March 3, 2024, 1:05 a.m.