Introducing the {msm.stacked} R package

Today we introduce {msm.stacked}, an R package that can be used to simplify the calculation of state transition probabilities over time and the creation of stacked probability plots from multi-state model fits from the {msm} package. Let me show you some examples of the package functionality in practice.

We start by building upon the example provided within the documentation for the msm::msm() function. Specifically, the data we are using is the heart
transplant data that comes bundled with {msm}:

library(msm)
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

Further details on this example dataset are included in the vignette of the {msm} package.

We start with a matrix of possible transitions:

tm <- rbind(
  c(-0.5, 0.25, 0, 0.25),
  c(0.166, -0.498, 0.166, 0.166),
  c(0, 0.25, -0.5, 0.25),
  c(0, 0, 0, 0)
)
tm
#>        [,1]   [,2]   [,3]  [,4]
#> [1,] -0.500  0.250  0.000 0.250
#> [2,]  0.166 -0.498  0.166 0.166
#> [3,]  0.000  0.250 -0.500 0.250
#> [4,]  0.000  0.000  0.000 0.000

This is then used to provide starting values for the model without any additional covariate:

cav.msm <- msm(
  formula = state ~ years,
  subject = PTNUM,
  data = cav,
  qmatrix = tm,
  deathexact = 4
)
cav.msm
#> 
#> Call:
#> msm(formula = state ~ years, subject = PTNUM, data = cav, qmatrix = tm, deathexact = 4)
#> 
#> Maximum likelihood estimates
#> 
#> Transition intensities
#>                   Baseline                    
#> State 1 - State 1 -0.17037 (-0.19027,-0.15255)
#> State 1 - State 2  0.12787 ( 0.11135, 0.14684)
#> State 1 - State 4  0.04250 ( 0.03412, 0.05294)
#> State 2 - State 1  0.22512 ( 0.16755, 0.30247)
#> State 2 - State 2 -0.60794 (-0.70880,-0.52143)
#> State 2 - State 3  0.34261 ( 0.27317, 0.42970)
#> State 2 - State 4  0.04021 ( 0.01129, 0.14324)
#> State 3 - State 2  0.13062 ( 0.07952, 0.21457)
#> State 3 - State 3 -0.43710 (-0.55292,-0.34554)
#> State 3 - State 4  0.30648 ( 0.23822, 0.39429)
#> 
#> -2 * log-likelihood:  3968.798

We don’t focus to much here on the arguments of the msm() function, check its documentation to learn more about it.

We can use the plot.msm() function to plot survival curves from every transient state to the final, absorbing state (e.g., a state denoting death). This is denoted in the cav dataset by State 4:

plot(cav.msm, from = 1:3, to = 4)

The {msm} package also provides functionality to calculate state transition probabilities at a given point in time. Say we are interested in estimating the probability of being in a given state, from each state, five years after baseline; we can use the pmatrix.msm() function to obtain just that:

pmatrix.msm(x = cav.msm, t = 5)
#>            State 1    State 2    State 3   State 4
#> State 1 0.51965804 0.13851775 0.09119847 0.2506257
#> State 2 0.24386420 0.13881410 0.18090731 0.4364144
#> State 3 0.06121333 0.06897186 0.16909991 0.7007149
#> State 4 0.00000000 0.00000000 0.00000000 1.0000000

This shows that, for instance, study participants in State 1 at time zero have (approximately) a 52% probability of still being in State 1 after years, 14% probability of being in State 2, 9% probability of being in State 3, and 25% probability of being in State 4.

We can repeatedly call the pmatrix.msm() function to obtain probabilities over time, but that’s a bit tedious. This is where the {msm.stacked} package comes in handy. Specifically, we can use the stacked.data.msm() function to calculate
transition probabilities over time, say, at 1 to 5 years:

library(msm.stacked)
sdd <- stacked.data.msm(model = cav.msm, tstart = 0, tforward = 5, tseqn = 6)
str(sdd)
#> 'data.frame':    96 obs. of  5 variables:
#>  $ from  : Factor w/ 4 levels "State 1","State 2",..: 1 2 3 4 1 2 3 4 1 2 ...
#>  $ to    : Factor w/ 4 levels "State 1","State 2",..: 1 1 1 1 2 2 2 2 3 3 ...
#>  $ p     : num  1 0 0 0 0 1 0 0 0 0 ...
#>  $ tstart: num  0 0 0 0 0 0 0 0 0 0 ...
#>  $ t     : num  0 0 0 0 0 0 0 0 0 0 ...

This returns a tidy dataset with all transition probabilities (column p), from and
to every state (columns from and to), over tseqn = 6 equally-spaced time intervals between time zero and time five. Columns tstart and t denotes starting point for the predictions and how many units of time after the starting point predictions are for, respectively. Focussing on transitions from State 1 only:

subset(sdd, sdd$from == "State 1")
#>       from      to          p tstart t
#> 1  State 1 State 1 1.00000000      0 0
#> 5  State 1 State 2 0.00000000      0 0
#> 9  State 1 State 3 0.00000000      0 0
#> 13 State 1 State 4 0.00000000      0 0
#> 17 State 1 State 1 0.85395872      0 1
#> 21 State 1 State 2 0.08836953      0 1
#> 25 State 1 State 3 0.01475543      0 1
#> 29 State 1 State 4 0.04291632      0 1
#> 33 State 1 State 1 0.74313989      0 2
#> 37 State 1 State 2 0.12669585      0 2
#> 41 State 1 State 3 0.04053779      0 2
#> 45 State 1 State 4 0.08962646      0 2
#> 49 State 1 State 1 0.65472323      0 3
#> 53 State 1 State 2 0.14064466      0 3
#> 57 State 1 State 3 0.06380519      0 3
#> 61 State 1 State 4 0.14082692      0 3
#> 65 State 1 State 1 0.58161960      0 4
#> 69 State 1 State 2 0.14256253      0 4
#> 73 State 1 State 3 0.08072247      0 4
#> 77 State 1 State 4 0.19509541      0 4
#> 81 State 1 State 1 0.51965804      0 5
#> 85 State 1 State 2 0.13851775      0 5
#> 89 State 1 State 3 0.09119847      0 5
#> 93 State 1 State 4 0.25062574      0 5

Here we see, for instance, that the probability of still being in State 1, starting from State 1, is (approximately) 85% after one year, 74% after two years, 65% after three years, 58% after four years, and 52% after five years:

subset(sdd, sdd$from == "State 1" & sdd$to == "State 1")
#>       from      to         p tstart t
#> 1  State 1 State 1 1.0000000      0 0
#> 17 State 1 State 1 0.8539587      0 1
#> 33 State 1 State 1 0.7431399      0 2
#> 49 State 1 State 1 0.6547232      0 3
#> 65 State 1 State 1 0.5816196      0 4
#> 81 State 1 State 1 0.5196580      0 5

The package also provides functionality to automatically produce stacked probabilities plots, for transition probabilities from and to every state. This is implemented in the stacked.plot.msm() function:

stacked.plot.msm(model = cav.msm, tstart = 0, tforward = 5)

This relies on {ggplot2} functionality and returns a standard ggplot object, which can of course be further customised beyond the default settings:

library(ggplot2)

stacked.plot.msm(model = cav.msm, tstart = 0, tforward = 5) +
  scale_fill_viridis_d(option = "plasma") +
  theme_minimal() +
  theme(legend.position = "bottom") +
  labs(fill = "To:")

Models with Covariates

We can of course incorporate covariates in a multi-state model and obtain predictions for a specific covariates pattern; let’s demonstrate this by incorporating sex in the model above. First, we fit a second model:

cav.msm.cov <- msm(
  formula = state ~ years,
  subject = PTNUM,
  data = cav,
  covariates = ~sex,
  qmatrix = tm,
  deathexact = 4
)
cav.msm.cov
#> 
#> Call:
#> msm(formula = state ~ years, subject = PTNUM, data = cav, qmatrix = tm, covariates = ~sex, deathexact = 4)
#> 
#> Maximum likelihood estimates
#> Baselines are with covariates set to their means
#> 
#> Transition intensities with hazard ratios for each covariate
#>                   Baseline                        
#> State 1 - State 1 -0.16938 (-1.894e-01,-1.515e-01)
#> State 1 - State 2  0.12745 ( 1.108e-01, 1.466e-01)
#> State 1 - State 4  0.04193 ( 3.354e-02, 5.241e-02)
#> State 2 - State 1  0.22645 ( 1.686e-01, 3.042e-01)
#> State 2 - State 2 -0.58403 (-1.053e+00,-3.238e-01)
#> State 2 - State 3  0.33693 ( 2.697e-01, 4.209e-01)
#> State 2 - State 4  0.02065 ( 2.196e-09, 1.941e+05)
#> State 3 - State 2  0.13050 ( 7.830e-02, 2.175e-01)
#> State 3 - State 3 -0.44178 (-5.582e-01,-3.497e-01)
#> State 3 - State 4  0.31128 ( 2.425e-01, 3.996e-01)
#>                   sex                            
#> State 1 - State 1                                
#> State 1 - State 2 0.5632779 (3.333e-01,9.518e-01)
#> State 1 - State 4 1.1289701 (6.262e-01,2.035e+00)
#> State 2 - State 1 1.2905854 (4.916e-01,3.388e+00)
#> State 2 - State 2                                
#> State 2 - State 3 1.0765518 (5.194e-01,2.231e+00)
#> State 2 - State 4 0.0003805 (7.241e-65,1.999e+57)
#> State 3 - State 2 1.0965531 (1.345e-01,8.937e+00)
#> State 3 - State 3                                
#> State 3 - State 4 2.4135380 (1.176e+00,4.952e+00)
#> 
#> -2 * log-likelihood:  3954.777

Then, we can use the same functionality as before to obtain stacked probabilities plots:

stacked.plot.msm(model = cav.msm.cov, tstart = 0, tforward = 5) +
  labs(title = "Predictions for average covariates")

By default, this will set all covariates to their average value (as in pmatrix.msm()); we can, however, pass specific covariates patterns that we want to predict for:

stacked.plot.msm(model = cav.msm.cov, tstart = 0, tforward = 5, covariates = list(sex = 0)) +
  labs(title = "Predictions for 'sex = 0'")
stacked.plot.msm(model = cav.msm.cov, tstart = 0, tforward = 5, covariates = list(sex = 1)) +
  labs(title = "Predictions for 'sex = 1'")

This way we can provide clinically meaningful predictions that highlight the effect of covariates of interest on state occupancy probabilities over time.

Models with Piecewise-Constant Intensities

By default, the {msm} package assumes constant (i.e., exponential) baseline transition intensities. This means that predictions at t years will be the same, irrespectively of when the starting point is:

stacked.plot.msm(model = cav.msm, tstart = 0, tforward = 3)
stacked.plot.msm(model = cav.msm, tstart = 5, tforward = 3, start0 = FALSE)

We can relax this assumption by allowing piecewise-constant baseline transition rates. This can be done by setting the pci argument of msm():

cav.msm.pw <- msm(
  formula = state ~ years,
  subject = PTNUM,
  data = cav,
  qmatrix = tm,
  deathexact = 4,
  pci = quantile(x = cav$years, probs = c(0.25, 0.50, 0.75))
)
#> Warning in msm(formula = state ~ years, subject = PTNUM, data = cav, qmatrix = tm, : Optimisation has probably not converged to the maximum likelihood -
#> Hessian is not positive definite.
cav.msm.pw
#> 
#> Call:
#> msm(formula = state ~ years, subject = PTNUM, data = cav, qmatrix = tm, deathexact = 4, pci = quantile(x = cav$years, probs = c(0.25, 0.5, 0.75)))
#> 
#> Optimisation probably not converged to the maximum likelihood.
#> optim() reported convergence but estimated Hessian not positive-definite.
#> 
#> -2 * log-likelihood:  3887.911

Specifically, here we set cut-points at quartiles of the observed distribution of (possibly censored) transition times. We ignore the warning about non-convergence for now, as we are only illustrating the package functionality here – in practice, we should investigate this and try a different optimiser or “consider tightening the tolerance criteria for convergence” (according to the documentation
of msm()).

First, we can do a likelihood ratio test to check whether the model with piecewise-constant intensities fits the data better:

lrtest.msm(cav.msm, cav.msm.pw)
#>            -2 log LR df            p
#> cav.msm.pw   80.8872 21 5.736045e-09

The test is statistically significant at any usual level, thus the more flexible model seems appropriate. Predictions of transition probabilities will now depend on the starting point tstart, even though tforward is the same:

stacked.plot.msm(model = cav.msm.pw, tstart = 0, tforward = 3)
stacked.plot.msm(model = cav.msm.pw, tstart = 5, tforward = 3, start0 = FALSE)

As expected, we see that the predicted probabilities between 0 and 3 units of time are now different compared to those between 5 and 8, given the (now) non-constant baseline intensities.

Excluding States

The {msm.stacked} package also include functionality to calculate (and plot) transition probabilities from only certain states of interest. For instance, the models we fit in the previous examples includes an absorbing state, State 4, from which there will be no transitions. For this example, we will be using the model with constant baseline transition rates (cav.msm).

Let’s start with a utility function, included in {msm.stacked}, to determine the names of the state of a {msm} model fit. This is called states.msm():

states.msm(cav.msm)
#> [1] "State 1" "State 2" "State 3" "State 4"

Now we know the correct names used by {msm} to define each state. Let’s calculate transitions probabilities from all states, but excluding the absorbing State 4:

stacked.data.msm(model = cav.msm, tstart = 0, tforward = 1, tseqn = 3, exclude = "State 4")
#>       from      to           p tstart   t
#> 1  State 1 State 1 1.000000000      0 0.0
#> 2  State 2 State 1 0.000000000      0 0.0
#> 3  State 3 State 1 0.000000000      0 0.0
#> 4  State 1 State 2 0.000000000      0 0.0
#> 5  State 2 State 2 1.000000000      0 0.0
#> 6  State 3 State 2 0.000000000      0 0.0
#> 7  State 1 State 3 0.000000000      0 0.0
#> 8  State 2 State 3 0.000000000      0 0.0
#> 9  State 3 State 3 1.000000000      0 0.0
#> 10 State 1 State 4 0.000000000      0 0.0
#> 11 State 2 State 4 0.000000000      0 0.0
#> 12 State 3 State 4 0.000000000      0 0.0
#> 13 State 1 State 1 0.921422669      0 0.5
#> 14 State 2 State 1 0.093120696      0 0.5
#> 15 State 3 State 1 0.003009286      0 0.5
#> 16 State 1 State 2 0.052893658      0 0.5
#> 17 State 2 State 2 0.745001419      0 0.5
#> 18 State 3 State 2 0.050466555      0 0.5
#> 19 State 1 State 3 0.004483375      0 0.5
#> 20 State 2 State 3 0.132369477      0 0.5
#> 21 State 3 State 3 0.808061596      0 0.5
#> 22 State 1 State 4 0.021200298      0 0.5
#> 23 State 2 State 4 0.029508408      0 0.5
#> 24 State 3 State 4 0.138462563      0 0.5
#> 25 State 1 State 1 0.853958721      0 1.0
#> 26 State 2 State 1 0.155576908      0 1.0
#> 27 State 3 State 1 0.009903994      0 1.0
#> 28 State 1 State 2 0.088369526      0 1.0
#> 29 State 2 State 2 0.566632840      0 1.0
#> 30 State 3 State 2 0.078536913      0 1.0
#> 31 State 1 State 3 0.014755432      0 1.0
#> 32 State 2 State 3 0.205995634      0 1.0
#> 33 State 3 State 3 0.659657266      0 1.0
#> 34 State 1 State 4 0.042916321      0 1.0
#> 35 State 2 State 4 0.071794618      0 1.0
#> 36 State 3 State 4 0.251901827      0 1.0

As you can see, transitions from the state passed to exclude are not reported. We can also exclude more than one state, for instance if we want to calculate only transitions from State 1:

stacked.data.msm(model = cav.msm, tstart = 0, tforward = 1, tseqn = 3, exclude = c("State 2", "State 3", "State 4"))
#>       from      to           p tstart   t
#> 1  State 1 State 1 1.000000000      0 0.0
#> 2  State 1 State 2 0.000000000      0 0.0
#> 3  State 1 State 3 0.000000000      0 0.0
#> 4  State 1 State 4 0.000000000      0 0.0
#> 5  State 1 State 1 0.921422669      0 0.5
#> 6  State 1 State 2 0.052893658      0 0.5
#> 7  State 1 State 3 0.004483375      0 0.5
#> 8  State 1 State 4 0.021200298      0 0.5
#> 9  State 1 State 1 0.853958721      0 1.0
#> 10 State 1 State 2 0.088369526      0 1.0
#> 11 State 1 State 3 0.014755432      0 1.0
#> 12 State 1 State 4 0.042916321      0 1.0

Of course, this functionality is also included in the plotting function:

stacked.plot.msm(model = cav.msm, tstart = 0, tforward = 10, exclude = "State 4") +
  theme(legend.position = "bottom")

Wrap-up

Summing up, we have developed and here introduced the {msm.stacked} R package which can simplify some prediction tasks when fitting multi-state models with the {msm} package.

You can install and test the package from GitHub by typing the following code in your R console:

# install.packages("devtools") devtools::install_github("RedDoorAnalytics/msm.stacked")

Make sure to check it out, and please remember to file an issue on GitHub if you spot any bug!


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