Solved – Use multi-state modelling (MSM) to predict individual transition probabilities in R

predictive-modelsrtransition matrix

I am using R package msm to model multi-state transitions in the sample dataset cav:

> head(cav)
   PTNUM      age    years dage sex pdiag cumrej state firstobs statemax
1 100002 52.49589 0.000000   21   0   IHD      0     1        1        1
2 100002 53.49863 1.002740   21   0   IHD      2     1        0        1
3 100002 54.49863 2.002740   21   0   IHD      2     2        0        2
4 100002 55.58904 3.093151   21   0   IHD      2     2        0        2
5 100002 56.49589 4.000000   21   0   IHD      3     2        0        2
6 100002 57.49315 4.997260   21   0   IHD      3     3        0        3

I use the following to model:

Q = rbind ( c(0,     0.25, 0,     0.25  ),
            c(0.166, 0   , 0.166, 0.166 ),
            c(0,     0.25, 0,     0.25  ),
            c(0,     0,    0,     0     ) )

cav.msm = msm(state ~ years, subject=PTNUM, data = cav, qmatrix = Q, deathexact = 4)

I can also use pnext.msm(cav.msm) to generate the probability of transition, resulting in:

        State 1             State 2             State 3                 State 4                 
State 1 0                   0.75 (0.70,0.80)    0                       0.25 (0.20,0.30)
State 2 0.37 (0.29,0.45)    0                   0.56 (0.44,0.66)        0.66 (0.02,0.21)
State 3 0                   0.30 (0.20,0.42)    0                       0.70 (0.58,0.80)
State 4 0                   0                   0                       0

My Question

Is it possible to generate the transition probabilities for an INDIVIDUAL (new data) given their current state and other factors (e.g., modeling covariates with msm's covariates= parameter)

Maybe Solution?

If I run the following:

#gender as covariate
cavsex.msm = msm(state ~ years, subject=PTNUM, data=cav, qmatrix=Q, deathexact=4, covariates = ~ sex)

and then generate a probability of transition matrix using a specified value for the covariate:

#probability each state is next (given a specified gender)
covariates = list(sex=0)
pnm = pnext.msm(cavsex.msm,covariates)

Does this generate the predicted transition matrix given covariate values?

> print(pnm$estimates)

          State 1   State 2   State 3    State 4
State 1 0.0000000 0.7670212 0.0000000 0.23297881
State 2 0.3635454 0.0000000 0.5522745 0.08418016
State 3 0.0000000 0.3145707 0.0000000 0.68542935
State 4 0.0000000 0.0000000 0.0000000 0.00000000

Here it is for sex=1:

          State 1   State 2   State 3     State 4
State 1 0.0000000 0.6215845 0.0000000 0.378415538
State 2 0.4410597 0.0000000 0.5589102 0.000030109
State 3 0.0000000 0.1725361 0.0000000 0.827463858
State 4 0.0000000 0.0000000 0.0000000 0.000000000

Can this be generally interpreted as, from State 3, the probability of transitioning to State 2 on the next step is 17% if sex=1 and 32% of sex=0?

Best Answer

it is possible in multiple ways. You can either use pmatrix.msm and pass some covariates to compute the estimated transition probability matrix. The function allows you to define a covariate profile which can be very specific. It also permits the definition of a custom time sequence over which computing the estimates.

If you are interested in computing profiles using survival probabilities, then I suggest you to look also at the package msmtools which has been specifically developed to work together with msm. You can use the function survplot() to visualize a patient specific survival curve with respect to a given set of covariates.

franz

Short Answer

Use the Kolmogorow-Smirnov test to identify a time window such that the eigenvalues of the Transition matrix is not sensitive to the precise choice of window.

Longer Answer

In Markov theory, there are two ways of formulating the dynamics.

Continuous time: We talk about transition rates in units of inverse time. So $Q_{rs}$ is the number of transitions from $r$ to $s$ expected per unit time (e.g. 5 transitions per second).
Discrete time. Here we talk about the probability of transitioning from one state to another in some specified interval of time. So $T_{rs}(\delta t)$ would be the probability to be in state $s$ at time $t+\delta t$ given that we were in state $r$ at time $t$.

One can show that these two formulations are related through a matrix exponential.

\begin{equation} T = e^{Q\delta t} \end{equation}

Continuous Time Being Well Defined

So regarding your first point regarding self transition being well-defined:

The continuous time formulation requires that the diagonal elements $Q_{rr}$ are equal to the negative sum of all the other transitions ($-\sum_{i\neq r}Q_{ri}$). You can read more about this here.
The pnext.msm quantity gives the probability of which state will come next. This is achieved by the equation in your question which just calculates the relative probabilities of transitions.
Notice that since the diagonal is the negative sum, the equation becomes: \begin{equation} \frac{-Q_{rs}}{Q_{rr}}=\frac{-Q_{rs}}{-\sum_{i\neq r}Q_{ri}}=\frac{Q_{rs}}{\sum_{i\neq r}Q_{ri}} \end{equation}
So now it is clearly just a relative likelihood!
Since the continuous time approach asks "Which state comes next?", the answer to that question can never be the current state.

Discrete Time Choice of Time Window

Now, let's say we want to look at discrete time and talk about "Which state will we be in 1 second from now?". Then we can easily find our $T$ from $Q$ if we know what our $\delta t$ is.

How do we choose the time window? Well we want a window such that the process is memoryless (Markovian). Often real world processes are non-Markovian on short time scales.

To check for Markovianity, we compute the matrix at a range of windows and examine that the dominant eigenvalue $\lambda$ is insensitive to the precise choice of window.

\begin{equation} \frac{log(\lambda(\delta t))}{\delta t}\approx \frac{log(\lambda(2\delta t))}{2\delta t} \end{equation}

My Question

Maybe Solution?

Best Answer

Related Solutions

Solved – Multi-State Numerical overflow using MSM package in R

Markov Process – Estimating Transition Probability to the Same State Using MSM Package

Short Answer

Longer Answer

Continuous Time Being Well Defined

Discrete Time Choice of Time Window

Related Question