--- title: '3-state cSTM in R' subtitle: "With state-residence time dependency" author: "The DARTH workgroup" output: html_document: default pdf_document: default --- Developed by the Decision Analysis in R for Technologies in Health (DARTH) workgroup. If you use this code as a basis for your own work, please cite our papers (see suggestions below) and in your acknowledgments for both the code and publications please add: **The authors thank the Decision Analysis in R for Technologies in Health (DARTH) workgroup for providing the coding framework in R.** \newpage ```{r setup, include=FALSE} knitr::opts_chunk$set(echo = TRUE, warning = FALSE, message = FALSE, eval = TRUE) ``` Change `eval` to `TRUE` if you want to knit this document. ```{r} rm(list = ls()) # clear memory (removes all the variables from the workspace) ``` # 01 Load packages ```{r} if (!require('pacman')) install.packages('pacman'); library(pacman) # use this package to conveniently install other packages # load (install if required) packages from CRAN p_load("dplyr", "tidyr", "reshape2", "devtools", "scales", "ellipse", "ggplot2", "lazyeval", "igraph", "truncnorm", "ggraph", "reshape2", "knitr", "stringr", "diagram", "dampack") # load (install if required) packages from GitHub # install_github("DARTH-git/darthtools", force = TRUE) #Uncomment if there is a newer version p_load_gh("DARTH-git/darthtools") ``` # 02 Load functions ```{r} # all functions are in the darthtools package ``` # 03 Model input ```{r} ## General setup cycle_length <- 1 # cycle length equal to one year (use 1/12 for monthly) n_cycles <- 60 / cycle_length # number of cycles v_names_cycles <- paste("cycle", 0:n_cycles) # cycle names v_names_states <- c("H", "S", "D") # state names n_states <- length(v_names_states) # number of health states ### Discounting factors d_c <- 0.03 # annual discount rate for costs d_e <- 0.015 # annual discount rate for QALYs ### Strategies v_names_str <- c("Standard of Care", # store the strategy names "Treatment A", "Treatment B") n_str <- length(v_names_str) # number of strategies ## Within-cycle correction (WCC) using Simpson's 1/3 rule v_wcc <- gen_wcc(n_cycles = n_cycles, method = "Simpson1/3") ### Transition probabilities r_HS_SoC <- 0.05 # rate of becoming sick when healthy, under standard of care r_HS_trtA <- 0.04 # rate of becoming sick when healthy, under treatment A r_HS_trtB <- 0.02 # rate of becoming sick when healthy, under treatment B r_SD <- 0.1 # probability of dying when sick # probabilities of dying when healthy (age-dependent) s v_r_HD <- read.csv("data/lifetable_rates.csv")$x # select lifetable mortality rates when healthy #v_r_HD <- v_r_HD[1:n_cycles] # if shorter cycle lengths are preferred, select only values needed if(n_cycles > length(v_r_HD)){paste("Note: the lifetable data only supports as simulation of", length(v_r_HD), "years", sep = " ")} p_HS_SoC <- rate_to_prob(r = r_HS_SoC, time = cycle_length) # probability of becoming sick when healthy, under SoC p_HS_trtA <- rate_to_prob(r = r_HS_trtA, time = cycle_length) # probability of becoming sick when healthy, under treatment A p_HS_trtB <- rate_to_prob(r = r_HS_trtB, time = cycle_length) # probability of becoming sick when healthy, under treatment B p_SD <- rate_to_prob(r = r_SD, time = cycle_length) # probability of dying when sick v_p_HD <- rate_to_prob(r = v_r_HD, time = cycle_length) # probability of dying when healthy (vector) # State-residence-dependent # Transition probability to die in sick state by cycle of being sick v_p_SD <- c(0.1, 0.2, 0.3, 0.4, 0.5, 0.7) # probability ### Tunnels n_tunnel_size <- length(v_p_SD) # Sick state v_Sick_tunnel <- c(paste("S_", seq(1, n_tunnel_size-1), "Yr", sep = ""), paste("S_", n_tunnel_size, "Yr+", sep = "")) # Create variables for time-dependent model v_names_states_tunnels <- c("H", v_Sick_tunnel, "D") # state names n_states_tunnels <- length(v_names_states_tunnels) # number of states ### State rewards #### Costs c_H <- 400 # cost of one cycle in healthy state c_S <- 1000 # cost of one cycle in sick state c_D <- 0 # cost of one cycle in dead state c_trtA <- 800 # cost of treatment A (per cycle) in healthy state c_trtB <- 1500 # cost of treatment B (per cycle) in healthy state #### Utilities u_H <- 1 # utility when healthy u_S <- 0.5 # utility when sick u_D <- 0 # utility when dead d_e <- 0.03 # discount rate per cycle equal discount of costs and QALYs by 3% d_c <- 0.03 # discount rate per cycle equal discount of costs and QAL ### Discount weight for costs and effects v_dwc <- 1 / ((1 + (d_c * cycle_length)) ^ (0:n_cycles)) v_dwe <- 1 / ((1 + (d_e * cycle_length)) ^ (0:n_cycles)) ``` # 04 Construct state-transition models ```{r} m_P_diag <- matrix(0, nrow = n_states, ncol = n_states, dimnames = list(v_names_states, v_names_states)) m_P_diag["H", "H" ] = "" m_P_diag["H", "S" ] = "" m_P_diag["H", "D" ] = "" m_P_diag["S", "S" ] = "" m_P_diag["S", "D" ] = "" m_P_diag["D", "D" ] = "" layout.fig <- c(2, 1) plotmat(t(m_P_diag), t(layout.fig), self.cex = 0.5, curve = 0, arr.pos = 0.8, latex = T, arr.type = "curved", relsize = 0.85, box.prop = 0.8, cex = 0.8, box.cex = 0.7, lwd = 1) ``` ## 04.1 Initial state vector ```{r} # All starting healthy v_m_init_tunnels <- c(1, rep(0, n_tunnel_size), 0) ``` ## 04.2 Initialize cohort traces with tunnels ```{r} ### Initialize cohort trace for state-residence dependent cSTM under SoC m_M_tunnels_SoC <- matrix(0, nrow = (n_cycles + 1), ncol = n_states_tunnels, dimnames = list(0:n_cycles, v_names_states_tunnels)) # Store the initial state vector in the first row of the cohort trace m_M_tunnels_SoC[1, ] <- v_m_init_tunnels ### Initialize cohort trace for strategies A and B # Structure and initial states are the same as for SoC m_M_tunnels_trtA <- m_M_tunnels_SoC # Strategy A m_M_tunnels_trtB <- m_M_tunnels_SoC # Strategy B ``` ## 04.3 Create transition probability arrays ```{r} ## Create transition probability arrays for strategy SoC ### Initialize transition probability array for strategy SoC a_P_tunnels_SoC <- array(0, # Create 3-D array dim = c(n_states_tunnels, n_states_tunnels, n_cycles), dimnames = list(v_names_states_tunnels, v_names_states_tunnels, v_names_cycles[-length(v_names_cycles)])) # name the dimensions of the array ### Fill in array ## from Healthy a_P_tunnels_SoC["H", "H", ] <- 1 - p_HS_SoC - v_p_HD a_P_tunnels_SoC["H", "S_1Yr", ] <- p_HS_SoC a_P_tunnels_SoC["H", "D", ] <- v_p_HD ## from Sick for(i in 1:(n_tunnel_size - 1)){ a_P_tunnels_SoC[v_Sick_tunnel[i], v_Sick_tunnel[i + 1], ] <- 1 - v_p_SD[i] a_P_tunnels_SoC[v_Sick_tunnel[i], "D", ] <- v_p_SD[i] } a_P_tunnels_SoC[v_Sick_tunnel[n_tunnel_size], v_Sick_tunnel[n_tunnel_size], ] <- 1 - v_p_SD[n_tunnel_size] a_P_tunnels_SoC[v_Sick_tunnel[n_tunnel_size], "D", ] <- v_p_SD[n_tunnel_size] # from Dead a_P_tunnels_SoC["D", "D", ] <- 1 ## Treatment A a_P_tunnels_trtA <- a_P_tunnels_SoC # make a copy # Update the probabilities a_P_tunnels_trtA["H", "H", ] <- 1 - p_HS_trtA - v_p_HD a_P_tunnels_trtA["H", "S_1Yr", ] <- p_HS_trtA ## Treatment B a_P_tunnels_trtB <- a_P_tunnels_SoC # make a copy # Update the probabilities a_P_tunnels_trtB["H", "H", ] <- 1 - p_HS_trtB - v_p_HD a_P_tunnels_trtB["H", "S_1Yr", ] <- p_HS_trtB # Check if transition array and probabilities are valid # Check that transition probabilities are in [0, 1] check_transition_probability(a_P_tunnels_SoC, verbose = TRUE, err_stop = TRUE) check_transition_probability(a_P_tunnels_trtA, verbose = TRUE, err_stop = TRUE) check_transition_probability(a_P_tunnels_trtB, verbose = TRUE, err_stop = TRUE) # Check that all rows sum to 1 check_sum_of_transition_array(a_P_tunnels_SoC, n_states = n_states_tunnels, n_cycles = n_cycles, verbose = TRUE, err_stop = TRUE) check_sum_of_transition_array(a_P_tunnels_trtA, n_states = n_states_tunnels, n_cycles = n_cycles, verbose = TRUE, err_stop = TRUE) check_sum_of_transition_array(a_P_tunnels_trtB, n_states = n_states_tunnels, n_cycles = n_cycles, verbose = TRUE, err_stop = TRUE) ``` ## 04.4 Create transition dynamics arrays ```{r} # These arrays will capture transitions from each state to another over time ### Initialize transition dynamics array for strategy SoC a_A_tunnels_SoC <- array(0, dim = c(n_states_tunnels, n_states_tunnels, n_cycles + 1), dimnames = list(v_names_states_tunnels, v_names_states_tunnels, 0:n_cycles)) # Set first slice of A with the initial state vector in its diagonal diag(a_A_tunnels_SoC[, , 1]) <- v_m_init_tunnels ### Initialize transition-dynamics array for strategies A and B # Structure and initial states are the same as for SoC a_A_tunnels_trtA <- a_A_tunnels_SoC # Strategy A a_A_tunnels_trtB <- a_A_tunnels_SoC # Strategy B ``` # 05 Run Markov model ```{r} # Iterative solution of state-residence dependent cSTM for (t in 1:n_cycles){ ## Fill in cohort trace # For SoC m_M_tunnels_SoC [t + 1, ] <- m_M_tunnels_SoC [t, ] %*% a_P_tunnels_SoC [, , t] # For Strategy A m_M_tunnels_trtA[t + 1, ] <- m_M_tunnels_trtA[t, ] %*% a_P_tunnels_trtA[, , t] # For Strategy B m_M_tunnels_trtB[t + 1, ] <- m_M_tunnels_trtB[t, ] %*% a_P_tunnels_trtB[, , t] ## Fill in transition-dynamics array # For SoC a_A_tunnels_SoC[, , t + 1] <- diag(m_M_tunnels_SoC[t, ]) %*% a_P_tunnels_SoC[, , t] # For strategy A a_A_tunnels_trtA[, , t + 1] <- diag(m_M_tunnels_trtA[t, ]) %*% a_P_tunnels_trtA[, , t] # For strategy B a_A_tunnels_trtB[, , t + 1] <- diag(m_M_tunnels_trtB[t, ]) %*% a_P_tunnels_trtB[, , t] } # Create aggregated trace m_M_tunnels_SoC_sum <- cbind(H = m_M_tunnels_SoC[, "H"], S = rowSums(m_M_tunnels_SoC[, 2:(n_tunnel_size + 1)]), D = m_M_tunnels_SoC[, "D"]) m_M_tunnels_trtA_sum <- cbind(H = m_M_tunnels_trtA[, "H"], S = rowSums(m_M_tunnels_trtA[, 2:(n_tunnel_size + 1)]), D = m_M_tunnels_trtA[, "D"]) m_M_tunnels_trtB_sum <- cbind(H = m_M_tunnels_trtB[, "H"], S = rowSums(m_M_tunnels_trtB[, 2:(n_tunnel_size + 1)]), D = m_M_tunnels_trtB[, "D"]) ## Store the cohort traces in a list l_m_M <- list(m_M_tunnels_SoC_sum, m_M_tunnels_trtA_sum, m_M_tunnels_trtB_sum) names(l_m_M) <- v_names_str ## Store the transition dynamics array for each strategy in a list l_a_A <- list(a_A_tunnels_SoC, a_A_tunnels_trtA, a_A_tunnels_trtB) names(l_a_A) <- v_names_str ``` # 06 Plot Outputs ## 06.1 Plot the cohort trace for strategy SoC ```{r} ## Plot the cohort trace for strategy SoC plot_trace(m_M_tunnels_SoC_sum) ``` ## 06.2 Overall Survival (OS) ```{r} v_os <- 1 - m_M_tunnels_SoC_sum[, "D"] # calculate the overall survival (OS) probability v_os <- rowSums(m_M_tunnels_SoC_sum[, 1:2]) # alternative way of calculating the OS probability # create a simple plot showing the OS plot(v_os, type = 'l', ylim = c(0, 1), ylab = "Survival probability", xlab = "Cycle", main = "Overall Survival") # add grid grid(nx = n_cycles, ny = 10, col = "lightgray", lty = "dotted", lwd = par("lwd"), equilogs = TRUE) ``` ## 06.2.1 Life Expectancy (LE) ```{r} v_le <- sum(v_os) # summing probablity of OS over time (i.e. life expectancy) ``` ## 06.3 Disease prevalence ```{r} v_prev <- m_M_tunnels_SoC_sum[, "S"]/v_os plot(v_prev, ylim = c(0, 1), ylab = "Prevalence", xlab = "Cycle", main = "Disease prevalence", type = "line") ``` # 07 State Rewards ```{r} ## Scale by the cycle length # Vector of utilities per cycle for Sick under strategy SoC v_u_S_SoC <- rep(u_S, n_tunnel_size) names(v_u_S_SoC) <- v_Sick_tunnel # Vector of utilities under strategy SoC v_u_SoC <- c(H = u_H, S = v_u_S_SoC, D = u_D) * cycle_length # Vector of costs per cycle for Sick under strategy SoC v_c_S_SoC <- rep(c_S, n_tunnel_size) names(v_c_S_SoC) <- v_Sick_tunnel # Vector of costs per cycle under strategy SoC v_c_SoC <- c(H = c_H, S = v_c_S_SoC, D = c_D) * cycle_length # Vector of utilities per cycle for Sick under strategy A v_u_S_trtA <- rep(u_S, n_tunnel_size) names(v_u_S_trtA) <- v_Sick_tunnel # Vector of utilities under strategy A v_u_trtA <- c(H = u_H, S = v_u_S_trtA, D = u_D) * cycle_length # Vector of costs per cycle for Sick under strategy A v_c_S_trtA <- rep(c_S, n_tunnel_size) names(v_c_S_trtA) <- v_Sick_tunnel # Vector of costs per cycle under strategy A v_c_trtA <- c(H = c_H + c_trtA, S = v_c_S_trtA, D = c_D) * cycle_length # Vector of utilities per cycle for Sick under strategy B v_u_S_trtB <- rep(u_S, n_tunnel_size) names(v_u_S_trtB) <- v_Sick_tunnel # Vector of utilities under strategy B v_u_trtB <- c(H = u_H, S = v_u_S_trtB, D = u_D) * cycle_length # Vector of costs per cycle for Sick under strategy B v_c_S_trtB <- rep(c_S + c_trtB, n_tunnel_size) names(v_c_S_trtB) <- v_Sick_tunnel # Vector of costs per cycle under strategy B v_c_trtB <- c(H = c_H + c_trtB, S = v_c_S_trtB, D = c_D) * cycle_length ## Store state rewards # Store the vectors of state utilities for each strategy in a list l_u <- list(SoC = v_u_SoC, A = v_u_trtA, B = v_u_trtB) # Store the vectors of state cost for each strategy in a list l_c <- list(SoC = v_c_SoC, A = v_c_trtA, B = v_c_trtB) # assign strategy names to matching items in the lists names(l_u) <- names(l_c) <- v_names_str ``` # 08 Compute expected outcomes ```{r} # Create empty vectors to store total utilities and costs v_tot_qaly <- v_tot_cost <- vector(mode = "numeric", length = n_str) names(v_tot_qaly) <- names(v_tot_cost) <- v_names_str ## Loop through each strategy and calculate total utilities and costs for (i in 1:n_str) { # i <- 1 v_u_str <- l_u[[i]] # select the vector of state utilities for the i-th strategy v_c_str <- l_c[[i]] # select the vector of state costs for the i-th strategy a_A_tunnels_str <- l_a_A[[i]] # select the transition array for the i-th strategy ## Array of state rewards # Create transition matrices of state utilities and state costs for the i-th strategy m_u_str <- matrix(v_u_str, nrow = n_states_tunnels, ncol = n_states_tunnels, byrow = T) m_c_str <- matrix(v_c_str, nrow = n_states_tunnels, ncol = n_states_tunnels, byrow = T) # Expand the transition matrix of state utilities across cycles to form a transition array of state utilities a_R_u_str <- array(m_u_str, dim = c(n_states_tunnels, n_states_tunnels, n_cycles + 1), dimnames = list(v_names_states_tunnels, v_names_states_tunnels, 0:n_cycles)) # Expand the transition matrix of state costs across cycles to form a transition array of state costs a_R_c_str <- array(m_c_str, dim = c(n_states_tunnels, n_states_tunnels, n_cycles + 1), dimnames = list(v_names_states_tunnels, v_names_states_tunnels, 0:n_cycles)) ### Expected QALYs and costs for all transitions per cycle # QALYs = life years x QoL # Note: all parameters are annual in our example. In case your own case example is different make sure you correctly apply. a_Y_c_str <- a_A_tunnels_str * a_R_c_str a_Y_u_str <- a_A_tunnels_str * a_R_u_str ### Expected QALYs and costs per cycle ## Vector of QALYs and costs v_qaly_str <- apply(a_Y_u_str, 3, sum) # sum the proportion of the cohort across transitions v_cost_str <- apply(a_Y_c_str, 3, sum) # sum the proportion of the cohort across transitions ## Discounted total expected QALYs and Costs per strategy and apply within-cycle correction if applicable # QALYs v_tot_qaly[i] <- t(v_qaly_str) %*% (v_dwe * v_wcc) # Costs v_tot_cost[i] <- t(v_cost_str) %*% (v_dwc * v_wcc) } ``` # 09 Cost-effectiveness analysis (CEA) ```{r} ## Incremental cost-effectiveness ratios (ICERs) # using the dampack package df_cea <- calculate_icers(cost = v_tot_cost, effect = v_tot_qaly, strategies = v_names_str) df_cea ``` ```{r} ## CEA table in proper format # using the dampack package table_cea <- format_table_cea(df_cea) table_cea ``` ```{r} ## CEA frontier # using the dampack package plot_icers(df_cea, label = "all", txtsize = 16) + expand_limits(x = max(table_cea$QALYs) + 0.1) + theme(legend.position = c(0.8, 0.3)) ``` # Acknowlegdement For this work we made use of the template developed by the Decision Analysis in R for Technologies in Health (DARTH) workgroup: . The notation of our code is based on the following provided framework and coding convention: Alarid-Escudero, F., Krijkamp, E., Pechlivanoglou, P. et al. A Need for Change! A Coding Framework for Improving Transparency in Decision Modeling. PharmacoEconomics 37, 1329–1339 (2019). . - Alarid-Escudero F, Krijkamp EM, Enns EA, Yang A, Hunink MGM Pechlivanoglou P, Jalal H. An Introductory Tutorial on Cohort State-Transition Models in R Using a Cost-Effectiveness Analysis Example. Medical Decision Making, 2023; 43(1). (Epub). - Alarid-Escudero F, Krijkamp EM, Enns EA, Yang A, Hunink MGM Pechlivanoglou P, Jalal H. A Tutorial on Time-Dependent Cohort State-Transition Models in R using a Cost-Effectiveness Analysis Example. Medical Decision Making, 2023; 43(1). Other work from DARTH can be found on the website: # Copyright for assignment work Copyright 2017, THE HOSPITAL FOR SICK CHILDREN AND THE COLLABORATING INSTITUTIONS.All rights reserved in Canada, the United States and worldwide. Copyright, trademarks, trade names and any and all associated intellectual property are exclusively owned by THE HOSPITAL FOR Sick CHILDREN and the collaborating institutions. These materials may be used, reproduced, modified, distributed and adapted with proper attribution.