---
title: "Probabilistic sensitivity analysis"
author: "Sheeja Manchira Krishnan"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Probabilistic sensitivity analysis}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

In this example we will model the cost effectiveness of lamivudine/zidovudine
combination therapy in HIV infection ([Chancellor, 1997] (https://pubmed.ncbi.nlm.nih.gov/10169387/)) and also described in Decision 
Modelling for Health Economic Evaluation, page 32. We have implemented the 
scenario in which the effect of combination therapy of lamivudine/zidovudine 
(as the probability of moving to worse health states and having to accrue 
more cost) is only seen in first 2 years. This means that the transition 
probability and costs are dependent on first two cycles (As the cycle length is years).

This model aims to compare costs and utilities of two treatment
strategies, mono therapy and combined therapy.

Four states are described, from best to worst health-wise:

A: CD4 cells > 200 and < 500 cells/mm3;
B: CD4 < 200 cells/mm3, non-AIDS;
C: AIDS;
D: Death.
```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = ">",
 fig.width = 6, 
 fig.height = 4
)
```

```{r setup}
library(packDAMipd)
```

Define health states for mono therapy - now the costs are just defined, 
but not defined any value.

```{r}
A <- health_state("A", cost = "cost_health_A+ cost_drug ",utility = 1)
B <- health_state("B", cost = "cost_health_B + cost_drug",utility = 1)
C <- health_state("C", cost = "cost_health_C + cost_drug",utility = 1)
D <- health_state("D", cost = 0,utility = 0)
```

Define allowed transition probabilities and number them. The below matrix is 
numbered so that the maximum entry in the matrix gives the total number of 
allowed transitions. Column names and row names are just the names of the 
health states.

```{r}
tmat <- rbind(c(1, 2,3,4), c(NA, 5,6,7),c(NA, NA, 8,9), c(NA, NA, NA,10))
colnames(tmat) <- rownames(tmat) <- c("A", "B" , "C", "D")

```

Define transition matrix now, but parameters are fine now
```{r}
tm <- populate_transition_matrix(4, tmat, c("tpAtoA", "tpAtoB", "tpAtoC", 
                                            "tpAtoD",
                                   "tpBtoB", "tpBtoC", "tpBtoD",
                                   "tpCtoC", "tpCtoD", "tpDtoD" ), 
                                 colnames(tmat) )
```
Combine the health states and define the strategy. The current strategy ie. 
control or intervention - here it is mono therapy


```{r}
health_states <- combine_state(A,B,C,D)
mono_strategy <- strategy(tm, health_states, "mono")

```
Before we run the model, we need to give values to parameters, thus we define 
the parameter list. We need to make sure that the parameters with numerical 
values are given first and derived parameters later in the list. The parameters 
are assigned sequentially as given in the parameter list. So if there is any calculations needed (or using functions), they are defined earlier.

```{r}
mono_param_list = define_parameters(cost_zido = 2278,
cost_direct_med_A = 1701,
cost_comm_care_A = 1055,
cost_direct_med_B = 1774,
cost_comm_care_B = 1278,
cost_direct_med_C = 6948,
cost_comm_care_C = 2059,
tpAtoA = 1251/(1251 + 483),
tpAtoB = 350/(350 + 1384),
tpAtoC = 116/(116 + 1618),
tpAtoD = 17/(17 + 1717),
tpBtoB = 731/(731 + 527),
tpBtoC = 512/(512 + 746),
tpBtoD = 15/(15 + 1243),
tpCtoC = 1312/(1312 + 437),
tpCtoD = 437/(437 + 1312),
tpDtoD = 1,
cost_health_A = "cost_direct_med_A+ cost_comm_care_A",
cost_health_B = "cost_direct_med_B+ cost_comm_care_B",
cost_health_C = "cost_direct_med_C+ cost_comm_care_C",
cost_drug = "cost_zido")
```


We run the model with strategy mono_strategy for 20 cycles with initial state 
values 1,0,0,0 corresponding to A,B, C and D. Costs and quays are discounted 
following the values given in discount. Parameter values will have any parameter defined as health state values or transition probabilities, but not their 
values were assigned.
```{r}
mono_markov <- markov_model(mono_strategy, 20,c(1,0,0,0),discount = c(0.06,0),mono_param_list, method = "half cycle correction",TRUE)
```
Now for combination therapy, the transition probabilities and cost are 
dependent on the years or the cycle number. Thus we need to define functions 
that given the correct output to calculate the probabilities and costs as a 
function of cycle
```{r}
 #Define function to set the cost to be different for first two cycles
define_comb_cost = function(cycle,cost_lami) {
  if (cycle == 2 || cycle == 3 )
    return(cost_lami)
  else
    return(0)
}
 #Define function to set the risk ratio to be different for first two cycles

define_rr = function(cycle,rr) {
  if (cycle == 2 || cycle == 3)
    return(rr)
  else
    return(1)
}
```

 Define health states - now the costs are just defined, but not defined any 
 value.
```{r}
A <-  health_state("A", cost = "cost_health_A + cost_drug", utility = 1)
B <- health_state("B", cost = "cost_health_B + cost_drug", utility = 1)
C <- health_state("C", cost = "cost_health_C + cost_drug", utility = 1)
D <- health_state("D", cost = 0, utility = 0)

```

Define allowed transition probabilities and number them. The below matrix is 
numbered so that the maximum entry in the matrix gives the total number of 
allowed transitions. Column names and row names are just the names of the 
health states.


```{r}
tmat <- rbind(c(1, 2,3,4), c(NA, 5,6,7),c(NA, NA, 8,9), c(NA, NA, NA,10))
colnames(tmat) <- rownames(tmat) <- c("A", "B" , "C", "D")

```


 Define transition matrix now, but parameters are fine now
```{r}
tm <- populate_transition_matrix(4, tmat, c("tpAtoA_rr", "tpAtoB_rr", 
                                            "tpAtoC_rr", "tpAtoD_rr",
                                   "tpBtoB_rr", "tpBtoC_rr", "tpBtoD_rr",
                                   "tpCtoC_rr", "tpCtoD_rr", "tpDtoD_rr" ), colnames(tmat) )

```
Before we run the model, we need to give values to parameters, thus we define 
the parameter list.
```{r}
comb_param_list <- define_parameters(cost_zido = 2278,
                     cost_direct_med_A = 1701,
                     cost_comm_care_A = 1055,
                     cost_direct_med_B = 1774,
                     cost_comm_care_B = 1278,
                     cost_direct_med_C = 6948,
                     cost_comm_care_C = 2059,
                     tpAtoA = 1251/(1251 + 483),
                     tpAtoB = 350/(350 + 1384),
                     tpAtoC = 116/(116 + 1618),
                     tpAtoD = 17/(17 + 1717),
                     tpBtoB = 731/(731 + 527),
                     tpBtoC = 512/(512 + 746),
                     tpBtoD = 15/(15 + 1243),
                     tpCtoC = 1312/(1312 + 437),
                     tpCtoD = 437/(437 + 1312),
                     tpDtoD = 1,
                     rr = 0.509,
                     cost_lami = 2086.50,
                     rr_cycle = "define_rr(markov_cycle,rr)",
                     tpAtoA_rr = "1-tpAtoB*rr_cycle-tpAtoC*rr_cycle-
                     tpAtoD*rr_cycle",
                     tpAtoB_rr = "tpAtoB*rr_cycle",
                     tpAtoC_rr = "tpAtoC*rr_cycle",
                     tpAtoD_rr = "tpAtoD*rr_cycle",
                     tpBtoB_rr = "1-tpBtoC*rr_cycle-tpBtoD*rr_cycle",
                     tpBtoC_rr = "tpBtoC*rr_cycle",
                     tpBtoD_rr = "tpBtoD*rr_cycle",
                     tpCtoC_rr = "1-tpCtoD*rr_cycle",
                     tpCtoD_rr = "tpCtoD*rr_cycle",
                     tpDtoD_rr = 1,
                     cost_health_A = "cost_direct_med_A + cost_comm_care_A",
                     cost_health_B = "cost_direct_med_B + cost_comm_care_B",
                     cost_health_C = "cost_direct_med_C + cost_comm_care_C",
                     cost_lami_cycle = "define_comb_cost(markov_cycle,cost_lami)",
                     cost_drug = "cost_zido + cost_lami_cycle")
```


```{r}
# Combine the health states
health_states <- combine_state(A,B,C,D)

#The current strategy ie. control or intervention - here it is combination 
#therapy
comb_strategy <- strategy(tm, health_states, "comb")

```

We run the model with strategy mono_strategy for 20 cycles with initial state 
values 1,0,0,0 corresponding to A,B, C and D. Costs and quays are discounted
following the values given in discount. Parameter values will have any parameter defined as health state values or transition probabilities, but not their 
values were assigned

```{r}
comb_markov <- markov_model(comb_strategy, 20, c(1, 0,0,0), discount = c(0.06,0.0),comb_param_list, method = "half cycle correction", TRUE)
```

Now we will define the parameters for the sensitivity analysis. The parameters 
are already defined as mono_param_list and comb_param_list. The parameter with sampling distributions are defined using definer_parameters() and create 
parameter table using define_parameters_psa(). Then probabilistic sensitivity 
analysis is done for the Markov model of the choice here the mono_Markov using 
the function do_psa Using the result obtained, a full report or plot (for 
specific variable) can be generated using summary_plot_psa(). To get a report 
or plot of sensitivity analysis on  ICER on NMB, two Markov models need to be 
passed on.

```{r}
sample_list <- define_parameters(rr = "lognormal(mean = 0.509, sd = 0.173)",
                                 cost_comm_care_C = "gamma(mean = 2756, sd = sqrt(2756))")
param_table_mono <- define_parameters_psa(mono_param_list, sample_list)
param_table_combo <- define_parameters_psa(comb_param_list, sample_list)

result_psa_mono = do_psa(mono_markov, param_table_mono, 5)
result_psa_comb = do_psa(comb_markov, param_table_combo, 5)

list_result_psa_mono <- list_paramwise_psa_result(result_psa_mono, NULL, NULL, 
                                                  NULL)
list_result_psa_comb <- list_paramwise_psa_result(result_psa_comb, NULL, NULL, 
                                                  NULL)
list_all <- list_paramwise_psa_result(result_psa_mono,result_psa_comb, 20000, 
                                      "mono")

summary_plot_psa(result_psa_mono, NULL, NULL, NULL)
summary_plot_psa(result_psa_comb, NULL, NULL, NULL)
summary_plot_psa(result_psa_mono,result_psa_comb, 20000, "mono")
```
