## Cost-effectiveness analysis in health care (III)

This will be my last post about the book. Yesterday I finished reading Darwin’s Origin of Species, which was my 100th book this year (here’s the list), but I can’t face blogging that book at the moment so coverage of that one will have to wait a bit.

In my second post about this book I had originally planned to cover chapter 7 – ‘Analysing costs’ – but as I didn’t like to spend too much time on the post I ended up cutting it short. This omission of coverage in the last post means that some themes to be discussed below are closely related to stuff covered in the second post, whereas on the other hand most of the remaining material, more specifically the material from chapters 8, 9 and 10, deal with decision analytic modelling, a quite different topic; in other words the coverage will be slightly more fragmented and less structured than I’d have liked it to be, but there’s not really much to do about that (it doesn’t help in this respect that I decided to not cover chapter 8, but doing that as well was out of the question).

I’ll start with coverage of some of the things they talk about in chapter 7, which as mentioned deals with how to analyze costs in a cost-effectiveness analysis context. They observe in the chapter that health cost data are often skewed to the right, for several reasons (costs incurred by an individual cannot be negative; for many patients the costs may be zero; some study participants may require much more care than the rest, creating a long tail). One way to address skewness is to use the median instead of the mean as the variable of interest, but a problem with this approach is that the median will not be as useful to policy-makers as will be the mean; as the mean times the population of interest will give a good estimate of the total costs of an intervention, whereas the median is not a very useful variable in the context of arriving at an estimate of the total costs. Doing data transformations and analyzing transformed data is another way to deal with skewness, but their use in cost effectiveness analysis have been questioned for a variety of reasons discussed in the chapter (to give a couple of examples, data transformation methods perform badly if inappropriate transformations are used, and many transformations cannot be used if there are data points with zero costs in the data, which is very common). Of the non-parametric methods aimed at dealing with skewness they discuss a variety of tests which are rarely used, as well as the bootstrap, the latter being one approach which has gained widespread use. They observe in the context of the bootstrap that “it has increasingly been recognized that the conditions the bootstrap requires to produce reliable parameter estimates are not fundamentally different from the conditions required by parametric methods” and note in a later chapter (chapter 11) that: “it is not clear that boostrap results in the presence of severe skewness are likely to be any more or less valid than parametric results […] bootstrap and parametric methods both rely on sufficient sample sizes and are likely to be valid or invalid in similar circumstances. Instead, interest in the bootstrap has increasingly focused on its usefulness in dealing simultaneously with issues such as censoring, missing data, multiple statistics of interest such as costs and effects, and non-normality.” Going back to the coverage in chapter 7, in the context of skewness they also briefly touch upon the potential use of a GLM framework to address this problem.

Data is often missing in cost datasets. Some parts of their coverage of these topics was to me but a review of stuff already covered in Bartholomew. Data can be missing for different reasons and through different mechanisms; one distinction is among data missing completely at random (MCAR), missing at random (MAR) (“missing data are correlated in an observable way with the mechanism that generates the cost, i.e. after adjusting the data for observable differences between complete and missing cases, the cost for those with missing data is the same, except for random variation, as for those with complete data”), and not missing at random (NMAR); the last type is also called non-ignorably missing data, and if you have that sort of data the implication is that the costs of those in the observed and unobserved groups differ in unpredictable ways, and if you ignore the process that drives these differences you’ll probably end up with a biased estimator. Another way to distinguish between different types of missing data is to look at patterns within the dataset, where you have:

“***univariate missingness** – a single variable in a dataset is causing a problem through missing values, while the remaining variables contain complete information

***unit non-response** – no data are recorded for any of the variables for some patients

***monotone missing** – caused, for example, by drop-out in panel or longitudinal studies, resulting in variables observed up to a certain time point or wave but not beyond that

***multivariate missing** – also called item non-response or general missingness, where some but not all of the variables are missing for some of the subjects.”

The authors note that the most common types of missingness in cost information analyses are the latter two. They discuss some techniques for dealing with missing data, such as complete-case analysis, available-case analysis, and imputation, but I won’t go into the details here. In the last parts of the chapter they talk a little bit about censoring, which can be viewed as a specific type of missing data, and ways to deal with it. Censoring happens when follow-up information on some subjects is not available for the full duration of interest, which may be caused e.g. by attrition (people dropping out of the trial), or insufficient follow up (the final date of follow-up might be set before all patients reach the endpoint of interest, e.g. death). The two most common methods for dealing with censored cost data are the Kaplan-Meier sample average (-KMSA) estimator and the inverse probability weighting (-IPW) estimator, both of which are non-parametric interval methods. “Comparisons of the IPW and KMSA estimators have shown that they both perform well over different levels of censoring […], and both are considered reasonable approaches for dealing with censoring.” One difference between the two is that the KMSA, unlike the IPW, is not appropriate for dealing with censoring due to attrition unless the attrition is MCAR (and it almost never is), because the KM estimator, and by extension the KMSA estimator, assumes that censoring is independent of the event of interest.

The focus in chapter 8 is on decision tree models, and I decided to skip that chapter as most of it is known stuff which I felt no need to review here (do remember that I to a large extent use this blog as an extended memory, so I’m not only(/mainly?) writing this stuff for other people..). Chapter 9 deals with Markov models, and I’ll talk a little bit about those in the following.

“Markov models analyse uncertain processes over time. They are suited to decisions where the timing of events is important and when events may happen more than once, and therefore they are appropriate where the strategies being evaluated are of a sequential or repetitive nature. Whereas decision trees model uncertain events at chance nodes, Markov models differ in modelling uncertain events as transitions between health states. In particular, Markov models are suited to modelling long-term outcomes, where costs and effects are spread over a long period of time. Therefore Markov models are particularly suited to chronic diseases or situations where events are likely to recur over time […] Over the last decade there has been an increase in the use of Markov models for conducting economic evaluations in a health-care setting […]

A Markov model comprises a finite set of health states in which an individual can be found. The states are such that in any given time interval, the individual will be in only one health state. All individuals in a particular health state have identical characteristics. The number and nature of the states are governed by the decisions problem. […] Markov models are concerned with transitions during a series of cycles consisting of short time intervals. The model is run for several cycles, and patients move between states or remain in the same state between cycles […] Movements between states are defined by transition probabilities which can be time dependent or constant over time. All individuals within a given health state are assumed to be identical, and this leads to a limitation of Markov models in that the transition probabilities only depend on the current health state and not on past health states […the process is memoryless…] – this is known as the Markovian assumption”.

The note that in order to build and analyze a Markov model, you need to do the following: *define states and allowable transitions [for example from ‘non-dead’ to ‘dead’ is okay, but going the other way is, well… For a Markov process to end, you need at least one state that cannot be left after it has been reached, and those states are termed ‘absorbing states’], *specify initial conditions in terms of starting probabilities/initial distribution of patients, *specify transition probabilities, *specify a cycle length, *set a stopping rule, *determine rewards, *implement discounting if required, *analysis and evaluation of the model, and *exploration of uncertainties. They talk about each step in more detail in the book, but I won’t go too much into this.

Markov models may be governed by transitions that are either constant over time or time-dependent. In a Markov *chain* transition probabilities are constant over time, whereas in a Markov *process *transition probabilities vary over time (/from cycle to cycle). In a simple Markov model the baseline assumption is that transitions only occur once in each cycle and usually the transition is modelled as taking place either at the beginning or the end of cycles, but in reality transitions can take place at any point in time during the cycle. One way to deal with the problem of misidentification (people assumed to be in one health state throughout the cycle even though they’ve transfered to another health state during the cycle) is to use half-cycle corrections, in which an assumption is made that on average state transitions occur halfway through the cycle, instead of at the beginning or the end of a cycle. They note that: “the important principle with the half-cycle correction is not when the transitions occur, but when state membership (i.e. the proportion of the cohort in that state) is counted. The longer the cycle length, the more important it may be to use half-cycle corrections.” When state transitions are assumed to take place may influence factors such as cost discounting (if the cycle is long, it can be important to get the state transition timing reasonably right).

When time dependency is introduced into the model, there are in general two types of time dependencies that impact on transition probabilities in the models. One is time dependency depending on the number of cycles since the start of the model (this is e.g. dealing with how transition probabilities depend on factors like age), whereas the other, which is more difficult to implement, deals with state dependence (curiously they don’t use these two words, but I’ve worked with state dependence models before in labour economics and this is what we’re dealing with here); i.e. here the transition probability will depend upon how long you’ve been in a given state.

Below I mostly discuss stuff covered in chapter 10, however I also include a few observations from the final chapter, chapter 11 (on ‘Presenting cost-effectiveness results’). Chapter 10 deals with how to represent uncertainty in decision analytic models. This is an important topic because as noted later in the book, “The primary objective of economic evaluation should not be hypothesis testing, but rather the estimation of the central parameter of interest—the incremental cost-effectiveness ratio—along with appropriate representation of the uncertainty surrounding that estimate.” In chapter 10 a distinction is made between variability, heterogeneity, and uncertainty. Variability has also been termed first-order uncertainty or stochastic uncertainty, and pertains to variation observed when recording information on resource use or outcomes within a homogenous sample of individuals. Heterogeneity relates to differences between patients which can be explained, at least in part. They distinguish between two types of uncertainty, structural uncertainty – dealing with decisions and assumptions made about the structure of the model – and parameter uncertainty, which of course relates to the precision of the parameters estimated. After briefly talking about ways to deal with these, they talk about sensitivity analysis.

“Sensitivity analysis involves varying parameter estimates across a range and seeing how this impacts on he model’s results. […] The simplest form is a one-way analysis where each parameter estimate is varied independently and singly to observe the impact on the model results. […] One-way sensitivity analysis can give some insight into the factors influencing the results, and may provide a validity check to assess what happens when particular variables take extreme values. However, it is likely to grossly underestimate overall uncertainty, and ignores correlation between parameters.”

Multi-way sensitivity analysis is a more refined approach, in which more than one parameter estimate is varied – this is sometimes termed scenario analysis. A different approach is threshold analysis, where one attempts to identify the critical value of one or more variables so that the conclusion/decision changes. All of these approaches are deterministic approaches, and they are not without problems. “They fail to take account of the joint parameter uncertainty and correlation between parameters, and rather than providing the decision-maker with a useful indication of the likelihood of a result, they simply provide a range of results associated with varying one or more input estimates.” So of course an alternative has been developed, namely probabilistic sensitivity analysis (-PSA), which already in the mid-80es started to be used in health economic decision analyses.

“PSA permits the joint uncertainty across all the parameters in the model to be addressed at the same time. It involves sampling model parameter values from distributions imposed on variables in the model. […] The types of distribution imposed are dependent on the nature of the input parameters [but] decision analytic models for the purpose of economic evaluation tend to use homogenous types of input parameters, namely costs, life-years, QALYs, probabilities, and relative treatment effects, and consequently the number of distributions that are frequently used, such as the beta, gamma, and log-normal distributions, is relatively small. […] Uncertainty is then propagated through the model by randomly selecting values from these distributions for each model parameter using Monte Carlo simulation“.