Lecture 6 - Health Economics and the QALY
1 Overview
In this lecture, we introduce some key ideas in Health Economics. In particular we will cover
- the Quality-adjusted life-year (QALY)
- Cost-effectiveness
- probabilistic sensitivity analysis (PSA)
This will feed into lectures 7 & 8 and Workshop 4.
2 Health Economics
Put simply, health economics is about maximizing the amount of health that can be achieved with the money that is available. Economic evaluation has been in use in healthcare systems around the world since the 1990s - for example, in the UK, the National Institute for Clinical Excellence (NICE) was formed in 1999. NICE uses economic evaluation to gain information about pharmaceuticals, surgical procedures, diagnostic technologies and medical devices, in order to best allocate the finite resources available. This means that decision making is an important part of their remit.
By economic evaluation in this context, we mean a comparison of different options (by which we could mean treatments, surgical procedures, medicines, screening programmes etc.) in terms of their cost and their consequences.
2.1 Comparing treatments
In a healthcare context, cost-effectiveness analysis (CEA) involves a health-related objective function (ie. the outcome you are trying to optimise) and is constrained by a healthcare budget. The two key questions asked in CEA are (Hill 2012):
- How much more does this treatment/option cost than the current version?
- Is it more effective (and if so, how much more)?
Within a particular disease/condition, this might be relatively simple to answer (or it may be extremely complex!).
For example, let’s imagine we are investigating a new asthma drug, drug B. Drug A (which is the current standard) costs £1000 per patient per year, while drug B costs £1200 per patient per year. The evidence from a clinical trial suggested that patients on drug B had approximately 30% more episode-free days than those on drug A. We might think that spending the extra £200 per person per year for this increase in episode-free days for asthma patients is obviously the right thing to do. For a 20% increase in cost, we achieve a 30% improvement for patients.
However, what if another team simultaneously find a new drug that improves symptoms of sufferers of something else, say Alzheimer’s? How do we compare these new drugs? If we have very limited funds, how do we decide which to spend our resources on? Which spending decision will achieve the ‘most health’? To decide this, we need a measure of health that applies to any possible health-spending decision.
2.2 Measuring Health
The discussion about measuring health and quality of life has been ongoing since the 1920s. Following the formation of the NHS in 1948, as life expectancy increased, the demand for healthcare grew. As well as this, progressively more sophisticated (and expensive) treatments became available. The growing range of treatments needed, coupled with a growing population, many of whom were living with one or more conditions, meant that it was impossible to deliver every treatment.
2.2.1 The Rosser Index
One early example of a scheme to measure the efficacy of a treatment (in this case a hospital stay) is the Rosser index (Rosser and Watts 1972), shown in Figure 2.1.
Figure 2.1: The Rosser index. From Rosser and Watts (1972)
These two scales (for disability and distress) were developed in conjunction with many medical professionals, and the possible states were then ranked, to create a univariate index.
This attracted the attention of Prof. Alan Williams, a health economist who was seconded to the Treasury from 1966-1968. You can watch Alan Williams give a lecture on QALYs here, or read more about the background in Williams (2005). We won’t go into much detail about how QALY values are elicited, but both of these sources do. Williams’ idea was to focus not on people’s clinical symptoms, but on how people live. He was inspired by a measure of crime introduced in the US that ranked crimes not according to Judges’ or the police’s views, but according to the views of the general public. Williams’ aim for the QALY was that it should reflect the valuations of ‘ordinary people’.
In order to meet the needs of cost-effectiveness studies, the QALY should be a single number that encompasses both the value of additional life expectancy and the value of improved quality of life. Rowen et al. (2016) reviews the use of questionnaires to elicit QALY values for many different health states and situations, and discusses some of the issues with the measure. Whitehead and Ali (2010) gives an excellent introduction to the QALY, linking it to the concept of ‘utility’, which is key to decision theory.
3 The Quality-adjusted life-year
The quality-adjusted life-year (QALY) measures health as a combination of the duration of life and the health-related quality of life (HRQoL). We measure HRQoL on a scale where 0 represents the state of death, and 1 is a state of perfect or best-imagineable health. Note that it is theoretically possible to have negative values, and indeed on most related questionnaires some people assign at least one negative value (see Bernfort et al. (2018) for some discussion and analysis around negative QALY values).
The QALY value is found by multiplying the duration by the HRQoL. So, a QALY value of 1 represents one year of perfect health, but it could also represent two years with a HRQoL of 0.5, or four years with a HRQoL of 0.25. If the HRQoL varies over time, then the QALY value is the integral of HRQoL over time.
3.1 Some examples
Williams (1985) gives the example shown in Figure 3.1, where two different treatment pathways are considered for a certain group of patients with angina. Profile B shows the expected pathway associated with medical management (a course of drugs), and profile A shows the expected outcome of a surgical procedure. The expected life expectancy gained by the surgical procedure is around 6 years, and the expected gain in QALYs is represented by the shaded area.
Figure 3.1: Expected value of health-related quality of life for patients with severe angina and left main vessel disease, taken from Williams (1985).
By comparison, Figure 3.2 shows the expected gain in QALYs following a hip replacement. In this situation, there is no increase to life expectancy, but use of the QALY allows us to quantify the improvement in HRQoL so that this treatment can be compared with others.
Figure 3.2: Expected value of health-related quality of life for patients following a hip replacement (from Fordham et al. (2012)).
Exercises
- Suppose a patient has a chronic condition that reduces his health-related quality of life to 60% of that of a fully healthy person, and that he has a life expectancy of five years. A new medicine becomes available which relieves his symptoms somewhat, increasing his HRQoL to 0.8, but not affecting his life expectancy. What is the increase in QALYs?
2. Now suppose that the drug has the same effect on HRQoL, and also increases the patient’s life expectancy by 2 years - what is the increase in QALYs?
3.2 Measuring health-related quality of life
This raises the question of how we measure HRQoL; it cannot be measured in the same way as weight or blood pressure, for example. There is subjectivity involved, both in how people define concepts, and in how people react to different experiences.
Three of the main methods used for eliciting preferences over health states are visual analogue scale (VAS), the time trade-off (TTO) and the standard gamble (SG). The descriptions here are largely based on those in Whitehead and Ali (2010).
3.2.1 Visual Analogue scale
In this method, people are simply asked to place different health states on a scale from 0 (worst imaginable health state) to 1 (best imaginable health state).
Figure 3.3: The visual analogue scale. Taken from Brazier et al. (2003)
The VAS is the simplest direct method, but is considered to be the least accurate / effective of the three, because rating preferences is harder than making simple choices. For example, scaling bias means that people are very unlikely to place states at either extreme end of the scale.
3.2.2 Time trade-off
In the time trade-off method, people are asked to say which they prefer of two possible scenarios. They would be presented with a choice of, for example,
- Living the remainder of their life \(\left(t_{rem}\right)\) in an imperfect health state, OR
- Living in full health for a shorter time \(\left(t_{full}\right)\).
The idea is to vary the amount of time in the full health state \(\left(t_{full}\right)\) until the patient cannot choose between the two states. We call this \(t^*_{full}\). The worse the impaired health state in option 1, the shorter the time period \(t^*_{full}\), will be. The HRQoL can then be inferred, since by construction the QALY values for the two options are equal:
\[ 1 \times {t^*_{full}} = HRQoL\times{t_{rem}} \] and our estimate of HRQoL is \[HRQoL \approx \frac{t^*_{full}}{t_{rem}}.\]
For example, in Figure 3.4 the time period that makes option 2 equivalent to 10 years with reduced HRQoL (in this case due to diabetes) is 8 years. This implies that the impaired health state has a HRQoL of approximately \(8/10 = 0.8\).
Figure 3.4: Two equivalent TTO options.
3.2.3 Standard gamble
The standard gamble approach is similar to TTO, as there is a trade-off involved. However this time, the trade-off involves risk. The choice is between:
- Remaining in a particular health state with certainty OR
- Taking a gamble in which you will either die (probability \(p\)), or be in full health (probability \(1-p\)).
The probability \(p\) is varied until the two options are considered equally desirable, and we call the resulting probability \(p^*\). Again, the HRQoL can be inferred because by construction the QALY values of the two options are equal:
\[ p^*\times{0} + \left(1-p^*\right)\times1 = 1\times{HRQoL} \] and our estimate of HRQoL is \[HRQoL \approx 1-p^*.\]
Figure 3.5: The standard gamble approach. We will see more trees like this in Lectures 7 & 8.
In Figure 3.5, the probability of death that makes the gamble equivalent to option 1 (remaining in a reduced HRQoL state) is \(p^*=0.3\). This means we estimate the HRQoL as \(1-p^*=0.7\).
3.2.4 EQ-5D
This measure, developed by the EuroQol group in the 1980s, is widely used by clinical researchers across the world, as well as several leading pharmaceutical companies, and has been translated into more than 200 languages.
The principal aims of the EuroQol group were “the development of a standardized, non-disease-specific instrument for describing and valuing health-related quality of life that would complement, rather than replace, other health-related quality of life measures.” You can read more about the history (and future) of the EuroQol group and the development of the EQ-5D in Devlin and Brooks (2017).
The EQ-5D is a questionnaire, and the requirements placed on it were that it be:
- An easy ‘add-on’ to studies using existing instruments
- Able to function as a postal questionnaire for self-completion
- Not too demanding (it takes a few minutes to complete)
- Relevant to respondents of all health states
- Capable of producing a single index value
- Consistent with health states judged worse than death (ie. \(HRQoL < 0\))
The EQ-5D defines health in terms of five dimensions:
- mobility
- self-care
- usual activities
- pain/discomfort
- anxiety/depression.
There are two versions. In the EQ-5D-5l, each dimension is divided into five levels:
- No problem
- Slight problem
- Moderate problem
- Severe problem
- Unable to / extreme problems
From this, there are 3125 possible health states. The initial method is now named the EQ-5D-3L, and had only three levels per dimension.
A particular health state can be coded by the numbered responses to each of the five dimensions. So, a health state of 11111 indicates no problems on any dimension, whereas 31121 would indicate moderate problems with mobility and slight problems with pain/discomfort. An example is shown in Figure 3.6.
Figure 3.6: The five dimensions of the EQ-5D-5L.
The respondent also places themselves on the visual analogue scale (from 0 to 100) according to how they would describe their health state on that day.
The 5-digit health state summary can be converted into a single summary number reflecting the respondent’s HRQoL. This uses a formula, and requires a value set, which is specific to each country, to apply weights in the formula that reflect the preferences of the general population of that country/region.
These value sets have been obtained using an exercise in which a representative sample of the general population of each country/region is asked to place a value on various EQ-5D health states, using time-trade off techniques. Golicki et al. (2015) report a study using the EQ-5D-5L to monitor changes in stroke patients, focussing particularly on its responsiveness as a metric.
3.3 Weaknesses with QALYs
There are many arguments against QALYs, and Whitehead and Ali (2010) give a good summary.
Some of the main ones are:
- this works with the ‘average’ patient, allowing for no variability between patients (age, sex, severity of disease, level of deprivation etc.)
- A QALY gain from 0.1 to 0.3 is equivalent to one from 0.7 to 0.9.
- Takes no account of productivity or activity
- A year of perfect health gained by a 6 year old has the same value as one gained by a 30 year old or a 90 year old.
- the aim is to maximise the total health of the population - in theory this could be to the detriment of some.
In this module we will only cosider cohort models, those which think in terms of the ‘average’ patient from a population sharing similar characteristics, but there are also methods that take into account the situation of the individual.
4 Cost-effectiveness
Now that we have defined the QALY, and have explored some ways to measure HRQoL, we are in a better position to think about comparing the cost-effectiveness of treatments.
Two key quantities in health economics are cost per QALY and the willingness-to-pay.
Definition 4.1 The cost per QALY of a treatment is the monetary cost of that treatment per QALY gained. So, if a treatment costs £30,000 per person and generates an expected gain of 2.5 QALYs, the expected cost per QALY is
\[ \frac{30000}{2.5} = £12000.\]
Definition 4.2 The willingness-to-pay threshold is the amount of money a health funder or healthcare provide is willing (or able) to pay for an increase of one QALY. In England, NICE has approximately a £20,000 cost per QALY threshold, which means that treatments costing more than £20,000 per QALY are less likely to be available on the NHS (although there are some caveats to this).
Many analyses are interested in comparisons between treatments, and in this case we will often see the incremental cost-effectiveness ratio (ICER).
Definition 4.3 The incremental cost-effectiveness ratio (ICER) for two options \(A\) and \(B\) is defined as \[ICER = \frac{Cost_A - Cost_B}{Outcome_A - Outcome_B}.\]
The ICER is often one of the main outputs of a health economic evaluation comparing two options (\(A\) and \(B\)). It tells us the extra cost per extra unit of health effect (eg. QALY), and therefore it can be directly compared to the willingness-to-pay threshold, so long as cost and health effect are in the same units. Here, treatment \(A\) is the new / intervention treatment and \(B\) is the standard / control treatment. So, as the cost of treatment \(A\) increases (all other things being equal), the ICER increases. As the outcome of treatment \(A\) increases, the ICER decreases.
Notice that if \(Cost_A > Cost_B\) and \(Outcome_A < Outcome_B\) (ie. treatment \(A\) costs more than treatment \(B\) for worse results), then \(ICER < 0\). The same is true if \(Cost_A < Cost_B\) and \(Outcome_A > Outcome_B\) (ie. if treatment \(A\) costs less and has a better effect). In both of these cases the outcome is obvious, but the ICER is sometimes criticized because of this confusion.
Definition 4.4 The incremental net benefit is \[Z = \lambda\left(Outcome_A - Outcome_B\right)-\left(Cost_A - Cost_B\right)\] where \(\lambda\) is the willingness-to-pay threshold, and \(Z\) is measured in monetary units. The larger \(Z\) is, the more cost-effective our new treatment \(A\) is judged to be. If \(Z<0\) then \(A\) is judged to be less cost-effective than \(B\).
4.1 Visualising cost-effectiveness
It is common to plot the results from cost-effectiveness analyses on the cost-effectiveness plane, as shown in Figure 4.1. The horizontal axis shows the incremental effectiveness and the vertical axis shows the incremental cost. The willingness-to-pay threshold is shown by a dashed line, with its gradient given by the WTP threshold.
Figure 4.1: The cost-effectiveness plane
Treatments whose ICER falls in the top left (blue/purple) are clearly entirely worse than the control treatment, and treatments that fall in the bottom are right are better. In the top right quadrant, treatments cost more and are more effective, so the decision depends on which side of the WTP threshold line they sit. Treatment A is preferable to the control, whereas treatment B is not. Similarly, treatments whose ICER falls in the bottom left quadrant cost less and are less effective than the control treatment, so whether they are preferable depends on which side of the WTP threshold they sit. Treatment C is not preferred, but treatment D is preferred.
There are ethical discussions to be had around this, particularly around choosing a treatment in the bottom left quadrant, where the effect is worse than the current standard. We will not go into this here.
5 Example: health behaviour intervention
Kruger et al. (2014) perform a cost-effectiveness analysis of an online health behaviour intervention aimed at young people, called “U@Uni”. The intervention focussed on aspects of life such as smoking, binge-drinking, eating fruit and vegetables and physical exercise.
A 6 month trial was performed, and this was used to estimate the resulting improvement (if any) to HRQoL. Evidence from the trial was also combined with general population data on health behaviours, and published evidence about the effect of these health behaviours on mortality, to extrapolate the effect of this intervention and estimate the long term cost effectiveness, in terms of cost-per-QALY.
The analysis had three steps:
- Costing analysis to estimate the cost of U@Uni.
- Within-trial analysis to estimate the short-term (6 month) cost-effectiveness of U@Uni
- Economic modelling analysis to estimate the long-term (lifelong) cost-effectiveness.
This involved combining individual follow-up data with cross-sectional data on health behaviours, and using this to estimate the long-term effect. This also involved using published data linking all four health behaviours to mortality risk.
5.1 Estimating the cost
The main cost of the trial was staff time. This was estimated by sending a questionnaire around all staff who had had any involvement, to ask them how much time had been spent on the development and implementation of the intervention. This was then combined with their respective salaries to estimate the total cost of the development and implementation, per student in the trial.
\[\text{Within trial cost} = \frac{\text{Total development and implementation cost}}{\text{Number of students in trial}}\]
To roll the intervention out to other universities would have a different cost, since the intervention would only need to be implemented. The intervention would also be applied to all new students, rather than just a trial cohort. Therefore the rollout cost per student was estimated using
\[\text{Rollout cost} = \frac{\text{Cost of any local developement + implementation + monitoring cost}}{\text{Average number of new students}}\]
5.2 Within trial cost-effectiveness
The starting characteristics of the trial participants are shown in Figure 5.1.
Figure 5.1: A summary from an initial survey of the trial participants. Taken from Kruger et al. (2014).
These same characteristics were surveyed at 1 month and 6 month follow-up periods, as well as cost estimates of the healthcare accessed by each participant in that time (ambulance call outs, GP appointments etc.). The QALYs accrued by each participant were estimated using HRQoL data collected at the start, 1 month and 6 months.
Linear regression was used with the health behaviours (smoking, physical activity, alcohol consumption, fruit and vegetable portions) as covariates to predict the cost per person and QALYs per person within the trial. This in turn enabled them to estimate the cost per QALY, and the probability that U@Uni would be cost effective within the 6 month trial period for various willingness-to-pay thresholds.
5.3 Long term (lifetime) economic modelling
The long-term cost-effectiveness was estimated using a model that translated changes in health behaviours into long-term QALYs (ie. the total QALYs generated over that participant’s life time).
This model first had to predict the expected behaviour of each participant over their lifetime, dependent on whether they had received the U@Uni intervention or not, and then use that expected behaviour to model their probability of death each year. All of this took in results from many other research projects and data sources. These fed into specific hazard rates for Markov modelling (which we will see later this week).
5.4 Results
5.4.1 Costs
The cost of development and implementation per person in the trial was estimated to be £283.29. The cost of rolling out U@Uni to other universities was estimated to be £19.16 per person.
5.4.2 Cost-effectiveness
The estimate of the within trial QALYs gained (ie. over 6 months) was 0.0013 QALYs per person. The additional costs for the trial group were around £326.27 per person (this includes some extra healthcare), resulting in an estimate of around £250,000 per QALY. This is clearly not cost effective, as shown on the cost-effectiveness plane in Figure 5.2.
Figure 5.2: The within-trial estimates of cost per QALY, with the threshold of £20,000 per QALY shown. Taken from Kruger et al. (2014).
Long term modelling showed that the expected gain in life expectancy was 0.29 days, and that the gains in HRQoL would be around 0.0128 QALYs per person compared to the control condition, with an estimated cost per QALY of £22,844. This is clearly much closer to the NICE willingness to pay threshold.
For the roll-out to other universities the cost is much lower, and the mean cost per QALY was estimated to be £1,545, making it very cost-effective. Figure 5.3 shows samples of long-term cost-per-QALY for the initial intervention (left) and roll out to another university (right).
Figure 5.3: The long-term estimates of cost per QALY, with the threshold of £20,000 per QALY. Taken from Kruger et al. (2014).
If you’d like to read more about the details of the models used, see Kruger et al. (2014).
6 Probabilistic sensitivity analysis for CEA
In this lecture so far we have treated most quantities as though they are known, or have used the average without accounting for uncertainty. However, important values for CEA, such as the QALYs accrued from receiving (or not receiving) treatment, the total costs of interventions and so on are not known for certain. In reality, there is uncertainty around the decisions people make, how they respond to treatments and the costs incurred by different courses of action.
A common strategy in CEA for health is to construct a probabilistic model of the intervention scenario. Quantities in the model are described using probability distributions, and the parameters for these distributions are themselves modelled by probability distributions, to capture the uncertainty around their values. By sampling from the parameter distributions, many many repeats of a particular scenario can be simulated, so that the output captures our uncertainty. This process is known as probabilistic sensitivity analysis (PSA), because it enables us to investigate the sensitivity of our model (and therefore our health economic decisions) to the model parameters.
It important to note that PSA only captures uncertainty relating to the parameter values, it doesn’t reflect our uncertainty about the structure of the model itself. If features are entirely missing from our model (which they almost certainly will be) or are not represented realistically, this is not captured by our PSA output. For a discussion of structural uncertainty in health economic models, see Strong, Oakley, and Chilcott (2012).
6.1 PSA: a simple example
Suppose part of our model involves people catching a disease. This is often modelled using a Binomial distribution, where we have a population of \(N\) people and they each have probability \(p\) of catching a disease. Figure 6.1 shows the probability of \(x\) out of 100 people catching a disease if \(p=0.15\), given a binomial distribution. The central 95% of the probability mass is shown by the dashed lines.
Figure 6.1: The binomial distribution for N=100, p=0.15.
However, we are unlikely to know the exact value of \(p\). We know it is definitely between 0 and 1, and we may well have some prior information that leads us to favour some regions over others. Instead of fixing a value for \(p\), we can capture our uncertainty around \(p\) by modelling it with a probability distribution. A common choice for probabilities is the Beta distribution, since it is over \(\left[0,1\right]\). An example, where \(p\sim{Beta\left(\alpha, \beta\right)}\), is shown in Figure 6.2.
Figure 6.2: The Beta distribution with alpha=6, beta=34. The median is shown by the solid vertical lines and the 95% CI by the dashed lines.
We see that rather than being certain that \(p=0.15\), we are now uncertain. Data from studies (for example clinical trials) is often used to infer a confidence interval for the value of a model parameter, from which a distribution can be specified.
To see how this uncertainty affects the infection part of our model, we will use a Monte Carlo approach. Our steps are:
- Sample a value for \(p\) from \(Beta\left(6,34\right)\).
- Sample a value for \(x\) from \(Bin\left(N, p\right)\)
We repeat these steps a large number of times to generate a sample of \(x\) that reflects our uncertainty about \(p\). In Figure 6.3 we have repeated steps 1 and 2 a thousand times, for a population of 100 and \(p\sim{Beta\left(6,32\right)}\).
Figure 6.3: 1000 samples of the number of people becoming infected in 100, sampling \(p\) from a Beta(6, 34) distribution.
We can see that the mean, shown by the solid vertical line, is approximately 15 as before, but the 95% CI is wider. Now, this sampled distribution of people becoming infected reflects our uncertainty about the probability with which people catch the disease. Probability distributions are also used to model quantities like QALYs gained / lost, or the cost of resources. We will see this in our example.
6.2 Example: Flu vaccine
This example is a simplified version of the model in Turner et al. (2006) and Baio and Dawid (2015). In this scenario, we build a probabilistic model to perform a cost effectiveness analysis of a flu vaccination programme. Because we have omitted parts of the real model, our conclusions don’t hold any clinical weight!
For each simulation, we will have two cohorts of the same size \(N\), one in which the vaccine is offered (in which everything will have the superscript ‘V’) and one in which it isn’t (in which everything will have the superscript ‘0’). A subscript will refer to whether or not someone has actually been vaccinated. We will then compare the cost of each scenario and the effectiveness (in terms of QALYs).
6.2.1 Vaccination
For our V cohort, the vaccine is offered, and \(N^V_V \sim Bin\left(N, \phi\right)\) people choose to be vaccinated, with \(\phi \sim {Beta\left(11.31, 14.44\right)}\), the distribution shown in Figure 6.4.
Figure 6.4: Beta(11.31, 14.44)
This means that \(N^V_0 = N - N^V_V\) people choose not to be vaccinated. In cohort 0, the vaccination isn’t offered, and so \(N^0_0 = N\).
The vaccine is associated with a cost \[\psi_V \sim logN\left(1.95, 0.0606\right).\]
(#fig:psi_V)logN(1.95, 0.0606)
We can calculate the total cost due to vaccination as \(C_V = \psi_V {N^V_V}\).
There are also sometimes adverse effects from the vaccination, which we model using
\[\begin{align*} AE & \sim Bin\left(N^V_V, \beta_{AE}\right) \\ \beta_{AE} & \sim Beta\left(3.5, 31.5\right) \end{align*}\]
and the loss of QALYs per person experiencing adverse effects is \[\begin{align*} \omega_{AE} & \sim{logN\left(-0.634, 0.0717\right)}. \end{align*}\] These distributions are shown in Figure 6.5.
Figure 6.5: Distributions for model parameters for adverse effects
The total loss of QALYs due to adverse effects from the vaccine is therefore \[ Q_{AE} = \omega_{AE}AE.\]
6.2.2 Infection
Unvaccinated people are infected with the flu with probability \(\beta_I\), so we have
\[\begin{align*} I^V_0 & \sim{Bin\left(N^V_0, \beta_I\right)}\\ I^0_0 & \sim{Bin\left(N^0_0, \beta_I\right)}. \end{align*}\]
For people who have been vaccinated, the probability of being infected is damped by \(\rho_v\), so we have
\[I^V_V \sim Bin\left(N^V_V, \beta_I\left(1-\rho_v\right)\right).\] From this we can calculate the total number of people infected with the flu from the \(V\) cohort as \(I^V_{tot} = I^V_V + I^V_0\).
The parameters \(\beta_I\) and \(\rho_v\) are modelled by a Beta and a log-normal distribution respectively, as in Figure 6.6.
Figure 6.6: Distributions for model parameters
Some proportion of infected people go to their GP. We model this by
\[\begin{align*} GP^V & \sim Bin\left(I^V_{tot}, \beta_{GP}\right)\\ GP^0 & \sim Bin\left(I^0, \beta_{GP}\right), \end{align*}\]
where \(\beta_{GP}\sim{Beta\left(5.8, 13.80\right)}\). We also have a parameter \(\psi_{GP}\sim{logN\left(3, 0.0606\right)}\) to model the cost per GP visit. These distributions are shown in Figure 6.7.
Figure 6.7: Distributions for model parameters
Using this, we can model the total cost of GP visits in each cohort as
\[\begin{align*} C^V_{GP} & = \psi_{GP}GP^V\\ C^0_{GP} & = \psi_{GP}GP^0. \end{align*}\]
6.2.3 Further complications
Of those who visit the GP, some proportion (each modelled with binomial distributions in a similar way to that described above)
- \(P\) people develop pneumonia
- loss of \(\omega_P\) QALYs per person (pp)
- \(H\) people are hospitalized
- loss of \(\omega_H\) QALYs pp
- cost of \(\psi_H\) pp
- \(D\) people die
- loss of \(\omega_D\) QALYs pp
These parameters, as well as the probabilities for the binomial distributions, are modelled in a similar way to those described above.
For each run of the simulation, we can therefore calculate the cost per person and QALYs lost per person for each cohort:
Cohort \(V\)
- \(\text{Cost} = \frac{1}{N}\left(\psi_VN^V_V + \psi_{GP}GP^V + \psi_HH^V\right)\)
- \(\text{QALYs lost} = \frac{1}{N}\left(\omega_{AE}AE^V + \omega_II_{tot}^V + \omega_PP^V + \omega_HH^V + \omega_DD^V\right)\)
Cohort 0
- \(\text{Cost} = \frac{1}{N}\left(\psi_{GP}GP^0 + \psi_HH^0\right)\)
- \(\text{QALYs lost} = \frac{1}{N}\left(\omega_II_{tot}^0 + \omega_PP^0 + \omega_HH^0 + \omega_DD^0\right)\)
And from this we can find the difference in cost and effect for each simulation to perform our CEA.
6.2.4 Results
We can now repeat the simulation many many times, following a cohort 0 and a cohort V through the same process each time, and directly comparing the overall costs and QALYs for each simulation run. In the following results, we have performed 1000 simulations and assumed each cohort is of size \(N=100000\).
First of all, we can look at the difference in how many people are infected, shown in Figure 6.8.
Figure 6.8: Simulated numbers of infected people from cohort V (left) and cohort 0 (right). The median and central 95% are shown by vertical lines.
We can see (as we would expect) that fewer people are infected, on average, in the group offered the vaccine, although there is significant overlap.
By summing the costs and QALYs for each cohort and finding the difference, we can plot our simulation results in cost-effectiveness space.
Figure 6.9: Flu vaccine simulation results in cost-effectiveness space. The red line shows a willingness-to-pay threshold of £20,000 per QALY, the blue line shows £100 per QALY.
We see that for both values of willingness-to-pay threshold, most simulations conclude that the vaccination programme is cost-effective, though the proportion is higher for £20,000 per QALY than for £100 per QALY.
Using the incremental net benefit
\[Z = \lambda\left(Outcome_A - Outcome_B\right)-\left(Cost_A - Cost_B\right)\]
We can plot the proportion of simulations that conclude the flu vaccine programme to be cost-effective (ie. the proportion for which \(Z>0\)) against \(\lambda\). This is shown in Figure 6.10, and is known as the cost-effectiveness acceptability curve.
Figure 6.10: Proportion of simulations with incremental net benefit > 0 against willingess-to-pay-threshold. The red line is at 0.95.
Because the programme appears to be very cost-effective, the plot is only shown up to \(\lambda=500\), as there is very little change after this. The few simulations which produced a negative value of \(Z\) will never be included in the proportion judged cost-effective.
6.2.5 Simplifications
In our simple example, we have assumed
- The vaccination has no effect on the course of influenza once infected
- People who don’t visit the GP don’t develop complications
and we have greatly simplified the possible complications, and ignored factors such as age or pre-existing medical conditions. This all comes under the umbrella term of ‘structural uncertainty’, which you can read about in Strong, Oakley, and Chilcott (2012).
We have also only considered costs to the healthcare provider in our model, but one could, and Turner et al. (2006) do, include costs from other perspectives, such as from missing time at work, or from purchasing over-the-counter medication.
If you’d like to read more about this type of modelling, Claxton et al. (2005) give an overview of the use of probabilistic sensitivity analysis by NICE, and Gray et al. (2017) work through a cost-effectiveness evaluation with PSA of four candidate national breast screening programmes.
7 Summary
In this lecture we have introduced the QALY (quality adjusted life year)
- The QALY is a measure of health that combines duration and health-related quality of life (HRQoL)
- The QALY allows us to make comparisons of the ‘amount of health’ gained (or lost) by a particular procedure.
- We have looked at several ways QALY values are estimated.
- We can compare treatments using a cost-effectiveness analysis.
- We can account for parameter uncertainty in our cost-effectiveness model using probabilistic sensitivity analysis.