1 Overview

hesim provides functions for using simulated costs and quality-adjusted life-years (QALYs) from a probabilistic sensitivity analysis (PSA) for decision-analysis in a cost-effectiveness framework, including representing decision uncertainty and conducting value of information analysis. Common metrics that can be produced include the following:

• incremental cost-effectiveness ratios (ICERs)
• cost-effectiveness planes planes
• cost-effectiveness acceptability curves (CEACs)
• cost-effectiveness acceptability frontiers (CEAFs)
• the expected value of perfect information (EVPI)

Cost-effectiveness analysis (CEA) can be performed for a single target population or for different subgroups. The latter is sometimes referred to as individualized CEA (iCEA).

The rest of this article provides an overview of CEA and how it can be conducted using hesim. The perspective is Bayesian in nature in that it is concerned with estimating the distribution of outcomes rather than just expected values (Baio 2012; Baio and Dawid 2015). It also shows how an iCEA can be performed so that both optimal treatments and the cost-effectiveness of those treatments vary across individuals (Basu and Meltzer 2007; Espinoza et al. 2014).

2 Framework

Decision theory provides a formal framework for making treatment decisions based on the utility that a therapy provides to a patient population. Decisions are typically made using a net benefit approach grounded in expected utility theory. The optimal treatment strategy is the one that maximizes expected NMBs where expected NMBs are calculated by averaging over the patient population and uncertain parameters \(\theta\). For a given subgroup \(g\) and parameter set \(\theta\), NMBs are computed as the difference between the monetized health gains from an intervention less costs, or,

\[ \begin{aligned} NMB_g(j,\theta) = e_{gj}\cdot k- c_{gj}, \end{aligned} \]

where \(e_{gj}\) and \(c_{gj}\) are measures of clinical effectiveness (e.g. QALYs) and costs in subgroup \(g\) using treatment \(j\) respectively, and \(k\) is a decision makers willingness to pay (WTP) per unit of clinical effectiveness. The optimal treatment for a given subgroup is the one that maximizes expected NMBs,

\[ \begin{aligned} j^{*}_g = \text{argmax}_j E_{\theta} \left[NMB_g(j,\theta)\right]. \end{aligned} \]

In practice, new interventions are usually compared to a standard treatment often referred to as the comparator. In these cases, a new treatment in a given subgroup is preferred to the comparator if the expected incremental net monetary benefit (INMB) of the new treatment is positive; that is, treatment 1 is preferred to treatment 0 in subgroup \(g\) if \(E_\theta \left[INMB_g\right] > 0\) where the INMB in a particular subgroup is given by

\[ \begin{aligned} INMB_g(\theta) = NMB_g(j = 1, \theta) - NMB_g(j = 0, \theta). \end{aligned} \]

Treatments can be compared in an equivalent manner using the ICER. The most common case occurs when a new treatment is more effective and more costly so that treatment \(1\) is preferred to treatment \(0\) in subgroup \(g\) if the ICER is less than the WTP threshold \(k\),

\[ \begin{aligned} k > \frac{E_\theta [c_{g1} - c_{g0}]}{E_\theta [e_{g1} - e_{g0}]} = ICER_g. \end{aligned} \] There are three additional cases:

• Treatment \(1\) dominates treatment \(0\) if it is more effective and less costly.
  
• Treatment \(1\) is dominated by treatment \(0\) if it is less effective and more costly.
  
• Treatment \(1\) is preferred to treatment \(0\) if it is less effective and less costly when \(k < ICER_g\).

3 Probabilistic sensitivity analysis

Expected NMBs are expected values, which implies that NMBs are uncertain and that optimal treatment strategies may be selected incorrectly. This uncertainty can be quantified using PSA, which uses Bayesian and quasi-Bayesian techniques to estimate the distribution of NMBs given the distribution of the parameters for each treatment strategy.

Since the joint distribution of the model parameters cannot be derived analytically (except in the simplest cases), the distribution of \(\theta\) is typically approximated by simulating the parameters from suitable probability distribution and calculating relevant quantities of interest as a function of the simulated parameters. For each treatment strategy and subgroup, PSA therefore produces \(n\) random draws from the distribution of clinical effectiveness and costs,

\[ \begin{aligned} e_{gj} &= [e_{gj}^1, e_{gj}^2, \dots, e_{gj}^n] \\ c_{gj} &= [c_{gj}^1, c_{gj}^2, \dots, c_{gj}^n]. \end{aligned} \]

Below we simulate costs and QALYs for three treatment strategies and two subgroups (in a real world analysis, this output would be derived from economic models like those supported by hesim). Strategy 1 is the current standard of care; it is the cheapest therapy, but also the least efficacious. Strategies 2 and 3 are equally costly, but Strategy 2 is more effective in subgroup 1 while Strategy 3 is more effective in subgroup 2.

set.seed(131)
n_samples <- 1000

# cost
c <- vector(mode = "list", length = 6)
names(c) <- c("Strategy 1, Grp 1", "Strategy 1, Grp 2", "Strategy 2, Grp 1",
              "Strategy 2, Grp 2", "Strategy 3, Grp 1", "Strategy 3, Grp 2")
c[[1]] <- rlnorm(n_samples, 2, .1)
c[[2]] <- rlnorm(n_samples, 2, .1)
c[[3]] <- rlnorm(n_samples, 11, .15)
c[[4]] <- rlnorm(n_samples, 11, .15)
c[[5]] <- rlnorm(n_samples, 11, .15)
c[[6]] <- rlnorm(n_samples, 11, .15)

# effectiveness
e <- c
e[[1]] <- rnorm(n_samples, 8, .2)
e[[2]] <- rnorm(n_samples, 8, .2)
e[[3]] <- rnorm(n_samples, 10, .8)
e[[4]] <- rnorm(n_samples, 10.5, .8)
e[[5]] <- rnorm(n_samples, 8.5, .6)
e[[6]] <- rnorm(n_samples, 11, .6)

# cost and effectiveness by strategy and simulation
library("data.table")
ce <- data.table(sample = rep(seq(n_samples), length(e)),
                 strategy = rep(paste0("Strategy ", seq(1, 3)), 
                                each = n_samples * 2),
                 grp = rep(rep(c("Group 1", "Group 2"),
                               each = n_samples), 3),
                 cost = do.call("c", c), qalys = do.call("c", e))
head(ce)

##    sample   strategy     grp     cost    qalys
##     <int>     <char>  <char>    <num>    <num>
## 1:      1 Strategy 1 Group 1 6.808115 7.984456
## 2:      2 Strategy 1 Group 1 6.997936 7.984085
## 3:      3 Strategy 1 Group 1 8.183247 8.221359
## 4:      4 Strategy 1 Group 1 7.423647 8.155250
## 5:      5 Strategy 1 Group 1 7.063846 8.292916
## 6:      6 Strategy 1 Group 1 7.234418 8.066187

4 Decision analysis

Decision analysis is performed within a CEA framework using the cea() and cea_pw() functions. The former summarizes results by simultaneously accounting for each treatment strategy in the analysis, while the latter summarizes “pairwise” results in which each treatment is compared to a comparator.

Both are generic functions that can be used to summarize results from a data.table containing simulated costs and QALYs or from a hesim::ce object produced from the $summarize() method of an economic model. In this example we use a data.table object.

library("hesim")
ktop <- 200000
cea_out <-  cea(ce, k = seq(0, ktop, 500), sample = "sample", strategy = "strategy",
                grp = "grp", e = "qalys", c = "cost")

The first argument is a data.table that contains columns for the parameter sample (sample), treatment strategy (strategy), subgroup (grp), clinical effectiveness (e), and costs (c). Users specify the names of the relevant columns in their output table as strings. The other relevant parameter is \(k\), which is a range of WTP values to use for estimating NMBs.

Likewise, we can use cea_pw() to summarize the PSA when directly comparing the two treatment strategies (Strategy 2 and Strategy 3) to the comparator (Strategy 1). The same inputs are used as in cea() except users must specify the name of the comparator strategy.

cea_pw_out <-  cea_pw(ce,  k = seq(0, ktop, 500), comparator = "Strategy 1",
                      sample = "sample", strategy = "strategy", grp = "grp",
                       e = "qalys", c = "cost")

The primary metrics are the ICER and representations of decision uncertainty including cost-effectiveness planes, CEACs, CEAFs, and the EVPI. Plots using ggplot2 graphics can be quickly generated with the plot_ceplane(), plot_ceac(), plot_ceaf(), and plot_evpi() functions, respectively, although output is readily available so users may create custom plots as well.

library("magrittr") # Use pipes
icer(cea_pw_out, k = 50000) %>%
  format()

## Key: <Group, Outcome>
##      Group           Outcome                Strategy 2
##     <char>            <fctr>                    <char>
## 1: Group 1 Incremental QALYs         2.05 (0.44, 3.68)
## 2: Group 1 Incremental costs   60,579 (45,170, 80,640)
## 3: Group 1   Incremental NMB 41,980 (-38,916, 122,227)
## 4: Group 1              ICER                    29,534
## 5: Group 2 Incremental QALYs         2.47 (0.83, 4.06)
## 6: Group 2 Incremental costs   60,689 (44,088, 80,040)
## 7: Group 2   Incremental NMB 62,588 (-23,058, 143,866)
## 8: Group 2              ICER                    24,615
##                    Strategy 3
##                        <char>
## 1:         0.51 (-0.75, 1.72)
## 2:    60,750 (44,408, 79,245)
## 3: -35,460 (-100,074, 32,748)
## 4:                    120,106
## 5:          3.00 (1.74, 4.19)
## 6:    60,726 (44,978, 82,943)
## 7:   89,198 (25,693, 151,890)
## 8:                     20,252

head(cea_pw_out$delta)

##    sample   strategy     grp       ie       ic
##     <int>     <char>  <char>    <num>    <num>
## 1:      1 Strategy 2 Group 1 1.497467 57687.97
## 2:      2 Strategy 2 Group 1 1.428877 65857.23
## 3:      3 Strategy 2 Group 1 3.184584 44830.68
## 4:      4 Strategy 2 Group 1 1.475885 64228.60
## 5:      5 Strategy 2 Group 1 1.045039 58613.55
## 6:      6 Strategy 2 Group 1 1.303221 45173.35

library("ggplot2")
theme_set(theme_bw())
plot_ceplane(cea_pw_out, k = 50000)

ce <- ce[, nmb := 50000 * qalys - cost]
random_rows <- sample(1:n_samples, 10)
nmb <- dcast(ce[sample %in% random_rows & grp == "Group 2"], 
                sample ~ strategy, value.var = "nmb")
setnames(nmb, colnames(nmb), c("sample", "nmb1", "nmb2", "nmb3"))
nmb <- nmb[, maxj := apply(nmb[, .(nmb1, nmb2, nmb3)], 1, which.max)]
nmb <- nmb[, maxj := factor(maxj, levels = c(1, 2, 3))]

mce <- prop.table(table(nmb$maxj))
print(mce)

## 
##   1   2   3 
## 0.0 0.4 0.6

plot_ceac(cea_out)

plot_ceac(cea_pw_out)

plot_ceaf(cea_out)

# Expected net monetary benefit
enmb <- ce[, .(enmb = mean(nmb)), by = c("strategy", "grp")]
enmb <- dcast(enmb, strategy ~ grp, value.var = "enmb")
strategymax_g2 <- which.max(enmb[[3]]) # Optimal strategy group 2

# Net monetary benefit (with perfect and current information)
nmb <- nmb[, nmbpi := apply(nmb[, .(nmb1, nmb2, nmb3)], 1, max)]
nmb <- nmb[, nmbci := nmb[[strategymax_g2 + 1]]]
kable(nmb, digits = 0, format = "html")

enmbpi <- mean(nmb$nmbpi)
enmbci <- mean(nmb$nmbci)
print(enmbpi)

## [1] 481362.9

print(enmbci)

## [1] 470922.7

print(enmbpi - enmbci)

## [1] 10440.18

plot_evpi(cea_out)

w_dt <- data.table(grp = paste0("Group ", seq(1, 2)), w = c(0.25, .75))
evpi <- cea_out$evpi
evpi <- merge(evpi, w_dt, by = "grp")
totevpi <- evpi[,lapply(.SD, weighted.mean, w = w),
                by = "k", .SDcols = c("evpi")]
ggplot(totevpi, aes(x = k, y = evpi)) +
  geom_line() + xlab("Willingness to pay") +
  ylab("Total EVPI") +
  scale_x_continuous(breaks = seq(0, ktop, 100000), 
                     label = scales::dollar) +
  scale_y_continuous(label = scales::dollar) +
  theme(legend.position = "bottom") 

5 Value of individualized care

The previous analyses allow NMBs and optimal treatment decisions to vary by subgroup. In contrast, most CEAs estimate the treatment, \(j^{*}\), that is optimal when averaging NMBs over the entire population. In particular, if the population is broken up into \(G\) distinct subgroups, the optimal treatment is given by,

\[ \begin{aligned} j^{*} = \text{argmax}_j \sum_{g=1}^{G} w_g E_{\theta}\left[NMB_g(j,\theta)\right]. \end{aligned} \]

Basu and Meltzer (2007) have shown that selecting subgroup specific treatments increases expected net benefits relative to this one-size fits all approach. They refer to additional net benefit as the expected value of individualized care (EPIC), which can be computed in terms of NMBs using the subgroup approach illustrated here as,

\[ \begin{aligned} \sum_{g=1}^G w_g E_{\theta}\left[NMB_g(j^{*}_s,\theta)\right] - \sum_{g=1}^G w_g E_{\theta}\left[NMB_g(j^{*},\theta)\right]. \end{aligned} \]

We can estimate the value of individualized care as follows:

# Compute total expected NMB with one-size fits all treatment
ce <- merge(ce, w_dt, by = "grp")
totenmb <- ce[, .(totenmb = weighted.mean(nmb, w = w)), by = c("strategy")]
totenmb_max <- max(totenmb$totenmb)

# Compute total expected NMB with individualized treatment
itotenmb_grp_max <- apply(as.matrix(enmb[, -1]), 2, max)
itotenmb_max <- sum(itotenmb_grp_max * w_dt$w)

# Compute EVIC
totnmb_scenarios <- c(itotenmb_max, totenmb_max)
names(totnmb_scenarios) <- c("Individualized total expected NMB",
                              "One-size fits all total expected NMB")
evic <- totnmb_scenarios[1] - totnmb_scenarios[2]
names(evic) <- "EVIC"
print(evic)

##     EVIC 
## 19359.84

print(evic/150000)

##      EVIC 
## 0.1290656

Our estimate of the EVIC is $19,359, or in terms of net health benefits, 0.129 QALYs.