This vignette is an example of an elementary semi-Markov model using the `rdecision`

package. It is based on the example given by Briggs *et al*^{1} (Exercise 2.5) which itself is based on a model described by Chancellor *et al*.^{2} The model compares a combination therapy of Lamivudine/Zidovudine versus Zidovudine monotherapy in people with HIV infection.

The variables used in the model are all numerical constants, and are defined as follows.

```
# transition counts
<- 1251
nAA <- 350
nAB <- 116
nAC <- 17
nAD <- 731
nBB <- 512
nBC <- 15
nBD <- 1312
nCC <- 437
nCD # Healthcare system costs
<- 1701 # direct medical costs associated with state A
dmca <- 1774 # direct medical costs associated with state B
dmcb <- 6948 # direct medical costs associated with state C
dmcc <- 1055 # Community care costs associated with state A
ccca <- 1278 # Community care costs associated with state B
cccb <- 2059 # Community care costs associated with state C
cccc # Drug costs
<- 2278 # zidovudine drug cost
cAZT <- 2087 # lamivudine drug cost
cLam # Treatment effect
<- 0.509
RR # Discount rates
<- 6 # annual discount rate, costs (%)
cDR <- 0 # annual discount rate, benefits (%) oDR
```

The model is constructed by forming a graph, with each state as a node and each transition as an edge. Nodes (of class `MarkovState`

) and edges (class `Transition`

) may have various properties whose values reflect the variables of the model (costs, rates etc.). Because the model is intended to evaluate survival, the utility of states A, B and C are set to 1 (by default) and state D to zero. Thus the incremental quality adjusted life years gained per cycle is equivalent to the survival function.

```
# create Markov states for monotherapy (zidovudine only)
<- MarkovState$new("A", cost=dmca+ccca+cAZT)
sAm <- MarkovState$new("B", cost=dmcb+cccb+cAZT)
sBm <- MarkovState$new("C", cost=dmcc+cccc+cAZT)
sCm <- MarkovState$new("D", cost=0, utility=0)
sDm # create transitions
<- Transition$new(sAm, sAm)
tAAm <- Transition$new(sAm, sBm)
tABm <- Transition$new(sAm, sCm)
tACm <- Transition$new(sAm, sDm)
tADm <- Transition$new(sBm, sBm)
tBBm <- Transition$new(sBm, sCm)
tBCm <- Transition$new(sBm, sDm)
tBDm <- Transition$new(sCm, sCm)
tCCm <- Transition$new(sCm, sDm)
tCDm <- Transition$new(sDm, sDm)
tDDm # construct the model
<- SemiMarkovModel$new(
m.mono V = list(sAm, sBm, sCm, sDm),
E = list(tAAm, tABm, tACm, tADm, tBBm, tBCm, tBDm, tCCm, tCDm, tDDm),
discount.cost = cDR/100,
discount.utility = oDR/100
)
```

Briggs *et al*^{1} interpreted the observed transition counts in 1 year as transition probabilities by dividing counts by the total transitions observed from each state. With this assumption, the annual (per-cycle) transition probabilities are calculated as follows and applied to the model via the `set_probabilities`

function.

```
<- nAA + nAB + nAC + nAD
nA <- nBB + nBC + nBD
nB <- nCC + nCD
nC <- matrix(
Pt c(nAA/nA, nAB/nA, nAC/nA, nAD/nA,
0, nBB/nB, nBC/nB, nBD/nB,
0, 0, nCC/nC, nCD/nC,
0, 0, 0, 1),
nrow=4, byrow=TRUE,
dimnames=list(source=c("A","B","C","D"), target=c("A","B","C","D"))
)$set_probabilities(Pt) m.mono
```

More usually, fully observed transition counts are converted into transition rates (rather than probabilities), as described by Welton and Ades.^{3} This is because counting events and measuring total time at risk includes individuals who make more than one transition during the observation time, and can lead to rates with values which exceed 1. In contrast, the difference between a census of the number of individuals in each state at the start of the interval and another at the end is directly related to the per-cycle probability. As Miller and Homan,^{4} Welton and Ades,^{3} Jones *et al*^{5} and others note, conversion between rates and probabilities for multi-state Markov models is non-trivial^{5} and care is needed when modellers calculate probabilities from published rates for use in `SemiMarkoModel`

s.

A representation of the model in DOT format (Graphviz) can be created using the `as_DOT`

function of `SemiMarkovModel`

. The function returns a character vector which can be saved in a file (`.gv`

extension) for visualization with the `dot`

tool of Graphviz, or plotted directly in R via the `DiagrammeR`

package. The Markov model for monotherapy is shown in Figure 1.

The states in the model can be tabulated with the function `tabulate_states`

. For the monotherapy model, the states are tabulated below. The cost of each state includes the annual cost of AZT (Zidovudine).

Name | Cost |
---|---|

A | 5034 |

B | 5330 |

C | 11285 |

D | 0 |

The per-cycle transition probabilities, which are the cells of the Markov transition matrix, can be extracted from the model via the function `transition_probabilities`

. For the monotherapy model, the transition matrix is shown below. This is consistent with the Table 1 of Chancellor *et al*.^{2}

A | B | C | D | |
---|---|---|---|---|

A | 0.7215 | 0.2018 | 0.0669 | 0.009804 |

B | 0 | 0.5811 | 0.407 | 0.01192 |

C | 0 | 0 | 0.7501 | 0.2499 |

D | 0 | 0 | 0 | 1 |

Model function `cycle`

applies one cycle of a Markov model to a defined starting population in each state. It returns a table with one row per state, and each row containing several columns, including the population at the end of the state and the cost of occupancy of states, normalized by the number of patients in the cohort, with discounting applied.

Multiple cycles are run by feeding the state populations at the end of one cycle into the next. Function `cycles`

does this and returns a data frame with one row per cycle, and each row containing the state populations and the aggregated cost of occupancy for all states, with discounting applied. This is done below for the first 20 cycles of the model for monotherapy, without half cycle correction, with discount. In addition, the proportion of patients alive at each cycle (the Markov trace) is added to the table. The populations and discounted costs are consistent with Briggs *et al*, Table 2.3,^{1} and the QALY column is consistent with Table 2.4 (without half cycle correction). No discount was applied to the utilities.

```
# create starting populations
<- 1000
N <- c(A = N, B = 0, C = 0, D = 0)
populations $reset(populations)
m.mono# run 20 cycles
<- m.mono$cycles(ncycles=20, hcc.pop=FALSE, hcc.cost=FALSE) MT.mono
```

Years | A | B | C | D | Cost | QALY |
---|---|---|---|---|---|---|

0 | 1000 | 0 | 0 | 0 | 0 | 0 |

1 | 721 | 202 | 67 | 10 | 5153 | 0.99 |

2 | 520 | 263 | 181 | 36 | 5393 | 0.964 |

3 | 376 | 258 | 277 | 89 | 5368 | 0.911 |

4 | 271 | 226 | 338 | 165 | 5055 | 0.835 |

5 | 195 | 186 | 364 | 255 | 4541 | 0.745 |

6 | 141 | 147 | 361 | 350 | 3929 | 0.65 |

7 | 102 | 114 | 341 | 444 | 3301 | 0.556 |

8 | 73 | 87 | 309 | 531 | 2708 | 0.469 |

9 | 53 | 65 | 272 | 610 | 2179 | 0.39 |

10 | 38 | 49 | 234 | 679 | 1727 | 0.321 |

11 | 28 | 36 | 198 | 739 | 1350 | 0.261 |

12 | 20 | 26 | 165 | 789 | 1045 | 0.211 |

13 | 14 | 19 | 136 | 830 | 801 | 0.17 |

14 | 10 | 14 | 111 | 865 | 609 | 0.135 |

15 | 7 | 10 | 90 | 893 | 460 | 0.107 |

16 | 5 | 8 | 72 | 915 | 346 | 0.085 |

17 | 4 | 5 | 57 | 933 | 258 | 0.067 |

18 | 3 | 4 | 45 | 948 | 192 | 0.052 |

19 | 2 | 3 | 36 | 959 | 142 | 0.041 |

20 | 1 | 2 | 28 | 968 | 105 | 0.032 |

The estimated life years is approximated by summing the proportions of patients left alive at each cycle (Briggs *et al*,^{1} Exercise 2.5). This is an approximation because it ignores the population who remain alive after 21 years, and assumes all deaths occurred at the start of each cycle. For monotherapy the expected life gained is 7.991 years at a cost of 44663 GBP.

For combination therapy, a similar model was created, with adjusted costs and transition rates. Following Briggs *et al*^{1} the treatment effect was applied to the probabilities, and these were used as inputs to the model. More usually, treatment effects are applied to rates, rather than probabilities.

```
# annual probabilities modified by treatment effect
<- RR*nAB/nA
pAB <- RR*nAC/nC
pAC <- RR*nAD/nA
pAD <- RR*nBC/nB
pBC <- RR*nBD/nB
pBD <- RR*nCD/nC
pCD # annual transition probability matrix
<- matrix(
Ptc c(1-pAB-pAC-pAD, pAB, pAC, pAD,
0, (1-pBC-pBD), pBC, pBD,
0, 0, (1-pCD), pCD,
0, 0, 0, 1),
nrow=4, byrow=TRUE,
dimnames=list(source=c("A","B","C","D"), target=c("A","B","C","D"))
)# create Markov states for combination therapy
<- MarkovState$new("A", cost=dmca+ccca+cAZT+cLam)
sAc <- MarkovState$new("B", cost=dmcb+cccb+cAZT+cLam)
sBc <- MarkovState$new("C", cost=dmcc+cccc+cAZT+cLam)
sCc <- MarkovState$new("D", cost=0, utility=0)
sDc # create transitions
<- Transition$new(sAc, sAc)
tAAc <- Transition$new(sAc, sBc)
tABc <- Transition$new(sAc, sCc)
tACc <- Transition$new(sAc, sDc)
tADc <- Transition$new(sBc, sBc)
tBBc <- Transition$new(sBc, sCc)
tBCc <- Transition$new(sBc, sDc)
tBDc <- Transition$new(sCc, sCc)
tCCc <- Transition$new(sCc, sDc)
tCDc <- Transition$new(sDc, sDc)
tDDc # construct the model
<- SemiMarkovModel$new(
m.comb V = list(sAc, sBc, sCc, sDc),
E = list(tAAc, tABc, tACc, tADc, tBBc, tBCc, tBDc, tCCc, tCDc, tDDc),
discount.cost = cDR/100,
discount.utility = oDR/100
)# set the probabilities
$set_probabilities(Ptc) m.comb
```

The per-cycle transition matrix for the combination therapy is as follows:

A | B | C | D | |
---|---|---|---|---|

A | 0.8585 | 0.1027 | 0.03376 | 0.00499 |

B | 0 | 0.7868 | 0.2072 | 0.006069 |

C | 0 | 0 | 0.8728 | 0.1272 |

D | 0 | 0 | 0 | 1 |

In this model, lamivudine is given for the first 2 years, with the treatment effect assumed to persist for the same period. The state populations and cycle numbers are retained by the model between calls to `cycle`

or `cycles`

and can be retrieved by calling `get_populations`

. In this example, the combination therapy model is run for 2 cycles, then the population is used to continue with the monotherapy model for the remaining 8 years. The `reset`

function is used to set the cycle number and elapsed time of the new run of the mono model.

```
# run combination therapy model for 2 years
<- c('A'=N, 'B'=0, 'C'=0, 'D'=0)
populations $reset(populations)
m.comb# run 2 cycles
<- m.comb$cycles(2, hcc.pop=FALSE, hcc.cost=FALSE)
MT.comb # feed populations into mono model & reset cycle counter and time
<- m.comb$get_populations()
populations $reset(
m.mono
populations, icycle=as.integer(2),
elapsed=as.difftime(365.25*2, units="days")
)# and run model for next 18 years
<- rbind(
MT.comb $cycles(ncycles=18, hcc.pop=FALSE, hcc.cost=FALSE)
MT.comb, m.mono )
```

The Markov trace for combination therapy is as follows:

Years | A | B | C | D | Cost | QALY |
---|---|---|---|---|---|---|

0 | 1000 | 0 | 0 | 0 | 0 | 0 |

1 | 859 | 103 | 34 | 5 | 6912 | 0.995 |

2 | 737 | 169 | 80 | 14 | 6736 | 0.986 |

3 | 532 | 247 | 178 | 43 | 5039 | 0.957 |

4 | 384 | 251 | 270 | 96 | 4998 | 0.904 |

5 | 277 | 223 | 330 | 170 | 4713 | 0.83 |

6 | 200 | 186 | 357 | 258 | 4245 | 0.742 |

7 | 144 | 148 | 357 | 351 | 3684 | 0.649 |

8 | 104 | 115 | 337 | 443 | 3102 | 0.557 |

9 | 75 | 88 | 307 | 530 | 2551 | 0.47 |

10 | 54 | 66 | 271 | 609 | 2057 | 0.391 |

11 | 39 | 49 | 234 | 678 | 1633 | 0.322 |

12 | 28 | 37 | 198 | 737 | 1279 | 0.263 |

13 | 20 | 27 | 165 | 787 | 990 | 0.213 |

14 | 15 | 20 | 136 | 829 | 760 | 0.171 |

15 | 11 | 14 | 111 | 864 | 579 | 0.136 |

16 | 8 | 11 | 90 | 892 | 437 | 0.108 |

17 | 6 | 8 | 72 | 914 | 329 | 0.086 |

18 | 4 | 6 | 58 | 933 | 246 | 0.067 |

19 | 3 | 4 | 46 | 947 | 183 | 0.053 |

20 | 2 | 3 | 36 | 959 | 136 | 0.041 |

The ICER is calculated by running both models and calculating the incremental cost per life year gained. Over the 20 year time horizon, the expected life years gained for monotherapy was 7.991 years at a total cost per patient of 44,663 GBP. The expected life years gained with combination therapy for two years was 8.94 at a total cost per patient of 50,607 GBP. The incremental change in life years was 0.949 years at an incremental cost of 5,944 GBP, giving an ICER of 6264 GBP/QALY. This is consistent with the result obtained by Briggs *et al*^{1} (6276 GBP/QALY), within rounding error.

1.

Briggs, A., Claxton, K. & Sculpher, M. *Decision modelling for health economic evaluation*. (Oxford University Press, 2006).

2.

Chancellor, J. V., Hill, A. M., Sabin, C. A., Simpson, K. N. & Youle, M. Modelling the cost effectiveness of Lamivudine/Zidovudine combination therapy in HIV infection. *Pharmacoeconomics* **12,** 54–66 (1997).

3.

Welton, N. J. & Ades, A. E. Estimation of Markov Chain Transition Probabilities and Rates from Fully and Partially Observed Data: Uncertainty Propagation, Evidence Synthesis, and Model Calibration. *Med Decis Making* **25,** 633–645 (2005).

4.

Miller, D. K. & Homan, S. M. Determining Transition Probabilities: Confusion and Suggestions. *Med Decis Making* **14,** 52–58 (1994).

5.

Jones, E., Epstein, D. & García-Mochón, L. A Procedure for Deriving Formulas to Convert Transition Rates to Probabilities for Multistate Markov Models. *Med Decis Making* **37,** 779–789 (2017).