# voi for Value of Information calculation: package overview

#### Christopher Jackson

#### 2024-09-16

Value of Information methods are a decision-theoretic framework for estimating the expected value of getting more information of particular kinds.

They are used in mathematical and statistical models, where parameters of the models represent quantities that are uncertain, and uncertainty is described by probability distributions.

A comprehensive guide to VoI methods is given in the the book Value of Information for Health Economic Evaluations (eds. Heath, Kunst, Jackson), (Chapman and Hall/CRC, 2024), which this R package accompanies.

This document gives a simple overview of how the `voi` package is used to calculate measures of Value of Information. A simple example model is used, but the same methods work in more complex models.

The example model is a model used for decision making, which has been the most common application of VoI, e.g. in health economic evaluations (see, e.g. Fenwick et al. and Rothery et al.)

A later section describes the use of VoI methods for a model that is used for estimation of uncertain quantities, rather than for explicit decision-making. For more information about the theory behind this, see Jackson et al. 2019 and Jackson et al. 2021.

## Simple example model

Suppose we are making a decision between two treatments. Treatment 1 has no costs or effects. Treatment 2 has a net benefit which describes its average costs and effects for a population. We choose Treatment 2 if its incremental net benefit, relative to treatment 1, is positive. The incremental net benefit in this simple case is identical to the net benefit of treatment 2, since the net benefit of treatment 1 is zero.

Suppose that the net benefit is simply defined as the difference between two uncertain parameters, \(y(p_1,p_2) = p_1 - p_2\), where \(p_1\) gives the effects, and \(p_2\) gives the costs. Our current uncertainty can be described by normal distributions \(p_1 \sim N(1,1)\) and \(p_2 \sim N(0,2)\).

To make a decision under parameter uncertainty, one option is preferred to another if the expectation of its net benefit, with respect to the uncertainty, is greater. In this case, we choose treatment 2, because the net benefit is distributed as \(N(1, \sqrt{1^2+2^2}) = N(1, \sqrt{5})\) which has an expectation of 1, whereas treatment 1 has a known net benefit of zero.

Most of the functions in the `voi` package work with a random sample of model inputs and outputs, generated from “uncertainty analysis”, also known as “probabilistic sensitivity analysis” or “probabilistic analysis”. For the example model, these are simple to generate, as follows.

### Specifying model inputs

The inputs should be a data frame with one column per parameter and one row per random sample.

    set.seed(1)
    nsam <- 10000
    inputs <- data.frame(p1 = rnorm(nsam, 1, 1),
                         p2 = rnorm(nsam, 0, 2))

### Specifying model outputs

The outputs can be supplied in either of two forms.

Net benefit form. A data frame with one column per treatment, and one row per random sample, giving the net benefit of each treatment. In this example, the net benefit of treatment 1 is zero.

    outputs_nb <- data.frame(t1 = 0,
                             t2 = inputs$p1 - inputs$p2)

Cost-effectiveness analysis form. This should be a list that includes the following three named elements (in any order)

- `"c"`: a data frame with one column per treatment and one row per random sample, containing sampled values for the expected costs of the treatment.

- `"e"`: a data frame with one column per treatment and one row per random sample, containing sampled values for the expected effects of the treatment.

- `"k"`: a vector of values giving alternative amounts that a decision-maker is willing to pay for one unit of effectiveness, so that the net (monetary) benefit is \(e \times k - c\).

In this simple example, the parameter \(p_1\) gives the effects, and \(p_2\) the costs of treatment 2, and the net benefit \(y = p_1 - p_2\) defined in `outputs_nb` corresponds to a willingness-to-pay of \(k=1\). The cost-effectiveness format allows us to compare VoI between different willingness-to-pay values, e.g. 1, 2 and 3 say here.

    outputs_cea <- list(
      e = data.frame(t1 = 0, t2 = inputs$p1),
      c = data.frame(t1 = 0, t2 = inputs$p2),
      k = c(1, 2, 3)
    )

Note that objects returned by the `bcea` function in the BCEA package satisfy this “cost-effectiveness analysis” format.

## Expected value of perfect information

The expected value of perfect information is the expected net benefit given perfect information minus the expected net benefit given current information.

#### Computation using random sampling

- Given current information, we decided on treatment 2. In the example we know that the expected net benefit under current information is 1, the mean of the distribution of treatment 2’s net benefit.

- Random sampling can be used to illustrate how to compute the expected net benefit given perfect information. Each sample of parameter values mimics a situation of decision-making given perfect information, where we know the parameters take these values. For each sample, we compare the corresponding treatment 2 net benefit to the threshold of zero, and prefer treatment 1 if the net benefit is negative, and treatment 2 if the net benefit is positive. The net benefit for each sample is the net benefit of the chosen treatment given the “known” sampled parameter values. The expected net benefit given perfect information is then computed as the average of this sample (`nb_perfect`).

    decision_current <- 2
    nb_current <- 1
    decision_perfect <- ifelse(outputs_nb$t2 < 0, 1, 2)
    nb_perfect <- ifelse(decision_perfect == 1, 0, outputs_nb$t2)
    (evpi1 <- mean(nb_perfect) - nb_current)

    ## [1] 0.4778316

In practice we would not usually know the exact expectation (under the current uncertainty distribution) of the net benefit for any treatment, so we must compute it as the mean of the random sample. In this case, `colMeans(outputs_nb)` would give a vector of the expected net benefit for each treatment. The maximum of these is the net benefit of the decision we take under current information, which in this case is 1.0018431. This would become closer to the exact value of 1, the more random samples are drawn.

An alternative view of EVPI is in terms of opportunity loss, which is the net benefit of the better decision we should have made (if we had known the truth), minus the net benefit of the decision we did make. The opportunity loss can be computed at each sample as follows. The EVPI is the mean of the opportunity loss.

    opp_loss <- nb_perfect - nb_current
    mean(opp_loss)

    ## [1] 0.4778316

#### Using the `voi` package to calculate EVPI

The `voi` package contains a simple function `evpi` to compute the EVPI using the above procedure. The function automatically detects whether your outputs are in net benefit or cost-effectiveness format.

    library(voi)
    evpi(outputs_nb)

    ## [1] 0.4759885

    evpi(outputs_cea)

    ##   k      evpi
    ## 1 1 0.4759885
    ## 2 2 0.4028003
    ## 3 3 0.4174207

Note the result is slightly different from `evpi1`, since it uses the sample-based estimate of 1.0018431 of the expected net benefit under current information, rather than the known expectation of 1.

#### Analytic computation

In this simple example, the EVPI can also be calculated “by hand”, because the model just involves normal distributions. The probability that the decision under perfect information agrees with the decision under current information, in this case, is the probability that the true value of a \(N(1, \sqrt{5})\) is actually positive.

    prob_correct <- 1 - pnorm(0, 1, sqrt(5))

The mean of `nb_perfect` can then be calculated as the expected net benefit given a correct decision, multiplied by the probability of a correct decision. The former is the mean of the values of `outputs_nb$t2` which are positive, which is the mean of a \(N(1,\sqrt{5})\) truncated below at zero. The mean of the truncated normal distribution has a known analytic form, represented in the following R function.

    mean_truncnorm <- function(mu, sig, lower=-Inf, upper=Inf){
      a <- (lower-mu)/sig
      b <- (upper-mu)/sig
      mu + sig * (dnorm(a) - dnorm(b)) / (pnorm(b) - pnorm(a))
    }
    enb_correct <- mean_truncnorm(1, sqrt(5), lower=0)
    mean_nb_perfect <- enb_correct * prob_correct
    (evpi_exact <- mean_nb_perfect - nb_current)

    ## [1] 0.4798107

This is the exact value of the EVPI in this example, which differs slightly from the estimate based on Monte Carlo simulation. Unfortunately most realistic decision-analytic models do not have such a nice form, and we must rely on Monte Carlo methods to calculate the expected value of information.

## Expected value of partial perfect information

The expected value of partial perfect information (EVPPI) for a parameter \(\phi\) in a decision-analytic model is the expected value of learning the exact value of that parameter, while the other parameters remain uncertain. \(\phi\) can comprise a single scalar parameter, or multiple parameters. If \(\phi\) refers to multiple parameters then the EVPPI describes the expected value of learning all of these parameters, often referred to as the multiparameter EVPPI.

The EVPPI is defined as the expected net benefit given perfect knowledge of \(\phi\), minus the expected net benefit given current information.

The function `evppi` can be used to compute this.

There are a variety of alternative computational methods implemented in this function. The default methods are based on nonparametric regression, and come from Strong et al. (2013). If there are four or fewer parameters, then a generalized additive model is used (the default spline model in `gam` from the `mgcv` package). With five or more, then Gaussian process regression is used.

### Invoking the `evppi` function.

To call `evppi`, supply a sample of outputs and inputs (in the same form as defined above) in the first two arguments. The parameter or parameters of interest (whose EVPPI is desired) is supplied in the `"pars"` argument. This can be expressed in various ways.

(a) As a vector. The joint EVPPI is computed for all parameters in this vector. If the vector has more than one element, then the function returns the expected value of perfect information on all of these parameters simultaneously (described as the “multiparameter” EVPPI by Strong et al.).

    evppi(outputs_nb, inputs, pars="p1")

    ##   pars      evppi
    ## 1   p1 0.08210074

    evppi(outputs_nb, inputs, pars=c("p1","p2"))

    ##    pars     evppi
    ## 1 p1,p2 0.4759885

(b) As a list. A separate EVPPI is computed for each element of the list. In the second example below, this is the EVPPI of \(p_1\), followed by the multiparameter EVPPI of \((p_1,p_2)\). Note that the multiparameter EVPPI is the theoretically same as the EVPI if, as in this case, the vector includes all of the parameters in the model (though note the difference from EVPI estimates above due to the different computational method).

    evppi(outputs_nb, inputs, pars=list("p1","p2"))

    ##   pars      evppi
    ## 1   p1 0.08210074
    ## 2   p2 0.39029827

    evppi(outputs_nb, inputs, pars=list("p1",c("p1","p2")))

    ##    pars      evppi
    ## 1    p1 0.08210074
    ## 2 p1,p2 0.47598851

The `evppi` function returns a data frame with columns indicating the parameter (or parameters), and the corresponding EVPPI. If the outputs are in cost-effectiveness analysis format, then a separate column is returned indicating the willingness-to-pay.

    evppi(outputs_cea, inputs, pars=list("p1",c("p1","p2")))

    ##    pars k      evppi
    ## 1    p1 1 0.08210074
    ## 2    p1 2 0.17052564
    ## 3    p1 3 0.25895085
    ## 4 p1,p2 1 0.47598851
    ## 5 p1,p2 2 0.40280031
    ## 6 p1,p2 3 0.41742073

### Changing the default calculation method

The method can be changed by supplying the `method` argument to `evppi`. Some methods have additional options to tune them. For a full list of these options, see `help(evppi)`.

#### Gaussian process regression

(from Strong et al. (2013)). The number of random samples to use in this computation can be changed using the `nsim` argument, which can be useful for this method as it can be prohibitive for large samples. Here the sample of 10000 is reduced to 1000.

    evppi(outputs_nb, inputs, pars="p1", method="gp", nsim=1000)

    ##   pars      evppi
    ## 1   p1 0.08834038

#### Multivariate adaptive regression splines

This is a variant of generalized additive models based on linear splines, which uses a package called earth.

    evppi(outputs_nb, inputs, pars="p1", method="earth")

    ##   pars      evppi
    ## 1   p1 0.08810086

While the merits of this method for EVPPI computation have not been systematically investigated, I have found this to be generally faster than comparable `mgcv` spline models for similar levels of accuracy and around 5 or fewer parameters.

#### INLA method

(from Heath et al., Baio et al. ). This needs the following extra packages to be installed, using the following commands.

    install.packages('INLA', repos='https://inla.r-inla-download.org/R/stable')
    install.packages('splancs')

It is only applicable to calculating the multiparameter EVPPI for 2 or more parameters.

In this toy example it is overkill, since the two-parameter EVPPI is simply the EVPI, and the default method needs an esoteric tweak (`pfc_struc`) to work.

However it has been found to be more efficient than the Gaussian process method in many other situations. See Heath et al., Baio et al. for more details about implementing and tuning this method.

    evppi(outputs_nb, inputs, pars=c("p1","p2"), method="inla", pfc_struc="iso")

#### Bayesian additive regression trees (BART)

This is another general nonparametric regression procedure. It is designed for regression with lots of predictors, so it may be particularly efficient for calculating multiparameter EVPPI, as the following demonstration shows.

The `voi` package includes a (fictitious) example health economic model based on a decision tree and Markov model: see the help page `voi::chemo_model`. There are 14 uncertain parameters. Outputs and inputs from probabilistic analysis are stored in the datasets `chemo_nb` (net benefit for willingness-to-pay £20000) and `chemo_pars`. The multiparameter EVPPI for all fourteen of these parameters is by definition equal to the EVPI.

Using the BART estimation method, the EVPPI estimate for all 14 parameters is very close to the estimate of the EVPI, and the computation is quick (about 16 seconds on my laptop. Call as `evppi(...,verbose=TRUE)` to see the progress of the estimation).

    evppi(chemo_nb, chemo_pars, pars=colnames(chemo_pars), method="bart")

    ##                                                                                                                                                                                        pars
    ## 1 p_side_effects_t1,p_side_effects_t2,c_home_care,c_hospital,c_death,u_recovery,u_home_care,u_hospital,logor_side_effects,p_hospitalised_total,p_died,lambda_home,lambda_hosp,rate_longterm
    ##      evppi
    ## 1 368.4953

    evpi(chemo_nb)

    ## [1] 368.6051

The BART estimation is being performed using the `bart()` function from the dbarts package, and in this case the function’s default settings are used.

While the BART method has not been investigated systematically as a way of estimating EVPPI, these are promising results.

#### Tuning the generalized additive model method

The generalized additive model formula can be changed with the `gam_formula` argument. This is supplied to the `gam` function from the `mgcv` package. The default formula uses a tensor product, and if there are more than four parameters, then a basis dimension of 4 terms per parameter is assumed.
A challenge of estimating EVPPI using GAMs is to define a GAM that is sufficiently flexible to represent how the outputs depend on the inputs, but can also be estimated in practice, given the complexity of the GAM and the number of random samples available to fit it to.

    evppi(outputs_nb, inputs, pars=c("p1","p2"), method="gam", gam_formula="s(p1) + s(p2)")

    ##    pars     evppi
    ## 1 p1,p2 0.4759885

Note that if there are spaces in the variable names in `inputs` and `pars`, then for `gam_formula` the spaces should be converted to underscores, or else an `"unexpected symbol"` error will be returned from `gam`.

A standard error for the EVPPI estimates from the GAM method, resulting from uncertainty about the parameters of the GAM approximation, can be obtained by calling `evppi` with `se=TRUE`. This uses \(B\) samples from the distribution of the GAM parameters, thus the standard error can be estimated more accurately by increasing B.

    evppi(outputs_nb, inputs, pars="p1", se=TRUE, B=100)

    ##   pars      evppi          se
    ## 1   p1 0.08210074 0.007093807

#### Single-parameter methods

These are only applicable for computing the EVPPI for a single scalar parameter. They are supplied in the package for academic interest, but for single-parameter EVPPI we have found it to be sufficiently reliable to use the default GAM method, which requires less tuning than these methods.

The method of Strong and Oakley:

    evppi(outputs_nb, inputs, pars="p1", n.blocks=20, method="so")

    ##   pars      evppi
    ## 1   p1 0.07997543

The method of Sadatsafavi et al.:

    evppi(outputs_nb, inputs, pars="p1", method="sal")

    ##   pars      evppi
    ## 1   p1 0.08151919

### Traditional Monte Carlo nested loop method

(see e.g. Brennan et al.)

This is generally too slow to provide reliable EVPPI estimates in realistic models, but is provided in this package for technical completeness.

This method is available in the function `evppi_mc`. It requires the user to supply two functions; one to evaluate the decision-analytic model, and one to generate parameter values.

#### Model evaluation function

This function evaluates the decision-analytic model for specific parameter values. This must have one argument for each parameter. The return value can be in either a “net benefit” form or a “costs and effects” form. The “net benefit” form is a vector giving the net benefit for each decision option.

    model_fn_nb <- function(p1, p2){
      c(0, p1 - p2)
    }

The “costs and effects” form is a matrix with two rows, and one column for each decision option. The rows gives the effects and costs respectively for each decision option. If they have names `"e"` and `"c"` then these are assumed to identify the effects and costs. Otherwise the first row is assumed to contain the effects, and the second the costs.

    model_fn_cea <- function(p1, p2){
      rbind(e = c(0, p1),
            c = c(0, p2))
    }

#### Parameter simulation function

This function generates a random sample of \(n\) values from the current (joint) uncertainty distribution of the model parameters. This returns a data frame with \(n\) rows and one named column for each parameter.

    par_fn <- function(n){
      data.frame(p1 = rnorm(n, 1, 1),
                 p2 = rnorm(n, 0, 2))
    }

#### Invoking `evppi_mc`

These functions are then supplied as arguments to `evppi_mc`, along with the number of samples to draw in the inner and outer loops. 1000 inner samples and 100 outer samples give a reasonable EVPPI estimate in this example, but many more samples may be required for the result to converge to the EVPPI in more complex models.

    evppi_mc(model_fn_nb, par_fn, pars="p1", ninner=1000, nouter=100)

#### Accounting for parameter correlation

We may want the EVPPI for a parameter which is correlated with another parameter. To account for this correlation, `par_fn` requires an extra argument or arguments to enable a sample to be drawn from the appropriate conditional distribution. For example, the function below specifies a bivariate normal distribution for \((p_1,p_2)\) where a correlation is induced by defining \(E(p_2|p_1) = p_1\). To draw a sample from the conditional distribution of \(p_2\) given \(p_1=2\), for example, call `par_fn_corr(1, p1=2)$p2`.

If the argument `p1` is not supplied, then the function should return a sample from the joint distribution marginalised over \(p_1\), as in this case where if we do not supply `p1` then a random `p1` is drawn followed by `p2|p1`.

A function of this form should then be passed to `evppi_mc` if the parameters are correlated. This allows `evppi_mc` to draw from the appropriate distribution in the inner loop.

    par_fn_corr <- function(n, p1=NULL){
      p1_new <- if (is.null(p1)) rnorm(n, 1, 1) else p1
      data.frame(p1 = p1_new,
                 p2 = rnorm(n, p1_new, 2))
    }
    evppi_mc(model_fn_nb, par_fn_corr, pars="p1", ninner=1000, nouter=100)

## Expected value of sample information

The expected value of sample information is the expected value of collecting a specific amount of data from a study designed to give information about some model parameter or parameters. It is defined as the expected net benefit given the study data, minus the expected net benefit with current information.

The function `evsi` can be used to calculate this. The default method is based on nonparametric regression (from Strong et al.). This requires the user to either

1. supply an R function to generate and summarise the study data, or

2. use one of the built-in study designs, and specify which of the model parameters are informed by this study.

To illustrate how to use `evsi`, suppose we want to collect a sample of \(n\) normally-distributed observations in order to get a better estimate of the treatment 2 effectiveness \(p_1\). Under current information, \(p_1\) is distributed as \(N(1,1)\). After collecting the sample, we would expect this distribution to become more precise, hence reduce the chance of making a wrong decision. The EVSI measures the expected improvement in net benefit from this sample.

Denote the study data as \(x_1,\ldots,x_n\), and suppose that they are distributed as \(x_i \sim N(p_1, \sigma)\). Hence the mean of the sample \(\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i\) is a summary statistic containing the information provided by the data about \(p_1\).

The sample mean is distributed as \(\bar{x} \sim N(p_1, \sigma / \sqrt{n})\). Suppose for simplicity that the sampling variance \(\sigma\) of the data is known to equal 1.

To calculate the EVSI using this method, we generate a sample from the predictive distribution of this summary statistic under current information. This is achieved by generating a value of \(p_1\) from its current \(N(1,1)\) distribution, followed by a value of \(\bar{x}\) from \(N(p_1, \sigma / \sqrt{n})\).

### Function to generate study data

The function should generate a sample from the predictive distribution of the summary statistic, given a sample `inputs` from the current uncertainty distribution of the parameters.

`inputs` has the same format as described above, a data frame with one row per sample and one column per parameter.

The function must return a data frame with one row per sample, and one column per parameter that is informed by the study data. Each data frame cell contains a summary statistic for that parameter from a simulated study.

The function `datagen_normal` below does this in a vectorised way for the example. Each row of the returned data frame is based on a different simulated \(p_1\) taken from the first column of `inputs`, and contains a summary statistic \(\bar{x}\) obtained from a dataset generated conditionally on that value of \(p_1\).

The sample size is included as an argument `n` to the data generation function. The names of the returned data frame can be anything (`xbar` was used in this case to be descriptive).

The `evsi` function can then be used to compute the EVSI for a series of different sample sizes from this design. Note how the EVSI converges to the EVPPI as the sample size increases.

    datagen_normal <- function(inputs, n=100, sd=1){
      data.frame(xbar = rnorm(nrow(inputs),
                              mean = inputs[,"p1"],
                              sd = sd / sqrt(n)))
    }

    set.seed(1)
    evsi(outputs_nb, inputs, datagen_fn = datagen_normal, n=c(10,100,1000))

    ##      n       evsi
    ## 1   10 0.07440250
    ## 2  100 0.08106261
    ## 3 1000 0.08189925

### Built-in study designs

The function `datagen_normal` is also included in the `voi` package as a built-in study design. To invoke the `evsi` function for a built-in study design, we have to supply the name of the design (in this case `"normal_known"`) and the name of the parameter or parameters (corresponding to a column of “inputs”) which is estimated by the study data.

    evsi(outputs_nb, inputs, study = "normal_known", n=c(100,1000), pars = "p1")

    ##      n       evsi
    ## 1  100 0.07998166
    ## 2 1000 0.08197188

    evsi(outputs_cea, inputs, study = "normal_known", n=c(100,1000), pars = "p1")

    ##      n k       evsi
    ## 1  100 1 0.07976014
    ## 2  100 2 0.16666306
    ## 3  100 3 0.25356597
    ## 4 1000 1 0.08216435
    ## 5 1000 2 0.17052103
    ## 6 1000 3 0.25887874

The known standard deviation defaults to 1, but can be changed, e.g. to 2, by calling `evsi` with an `aux_pars` argument, e.g. `evsi(..., aux_pars=list(sd=2), ...)`.

Note that the results will be slightly different every time the `evsi` function is invoked with the same arguments, due to Monte Carlo error from generating the data (unless the seed is set with the R function `set_seed` before each invocation).

Other built-in study designs include

`"binary"`: A single sample of observations of a binary outcome. Requires one parameter to be specified in `pars`, that is, the probability of the outcome.

`"trial_binary"`: A two-arm trial with a binary outcome. Requires two parameters to be specified in `pars`: the probability of the outcome in arm 1 and 2 respectively. The sample size is the same in each arm, specifed in the `n` argument to `evsi`, and the binomial outcomes are returned in the first and second column respectively.

### Importance sampling method

An alternative method comes from Menzies (2015) and is based on importance sampling. This can be invoked as `evsi(..., method="is")`.

As well as a data generation function in the above format, this also requires the user to supply a likelihood function for the study data.

This is illustrated here for the simple normal example. The likelihood function acts on one row of the data frame \(Y\) which is produced by the data generation function, and returns a data frame with number of rows matching the rows of `inputs`. Each row of the returned data frame gives the sampling density for that row of \(Y\) given the corresponding parameter values in `inputs`. The corresponding EVSI calculation then involves building a large matrix of likelihoods for combinations of simulated datasets and simulated parameters.

Any user-supplied likelihood function should consistently define the same model for the data as the data generation function (the package does not check this!), and any names of parameters and outputs should match the names defined in `inputs` and the data generation function.

This method is typically slower than the default nonparametric regression method, so it may be worth setting `nsim` to a value lower than the number of samples in `inputs`. Below `nsim=1000` is used so that only 1000 samples are used, instead of the full 10000 samples contained in `inputs`.

    likelihood_normal <- function(Y, inputs, n=100, sig=1){
      mu <- inputs[,"p1"]
      dnorm(Y[,"xbar"], mu, sig/sqrt(n))
    }

    evsi(outputs_nb, inputs, datagen_fn = datagen_normal, likelihood = likelihood_normal,
         n=100, pars = "p1", method="is", nsim=1000)

    ##     n      evsi
    ## 1 100 0.0864487

Again, this study model is available as a built-in study design, so instead of writing a user-defined likelihood and data generation function, `evsi` can also be invoked with `study="normal_known", method="is"`.

    evsi(outputs_nb, inputs, study = "normal_known", n=100, pars = "p1", method="is", nsim=1000)

    ##     n       evsi
    ## 1 100 0.08619358

### Moment matching method

The momemt matching method from Heath et al. is available as `evsi(..., method="mm")`. This includes the extension of this method to efficiently estimate the EVSI for the same design but with many different sample sizes (from Heath et al).

Roughly, this method works as follows (see Heath et al. for full details)

- a small set of values \(\theta_q: q = 1,\ldots, Q\) are simulated for the decision model parameters \(\theta\) (typically \(Q<50\) is sufficient).

- for each \(\theta_q\), future study data \(x_q|\theta_q\) are generated from the sampling distribution (as specified through `study` for built-in study designs, or `datagen_fn` for custom designs).

- a sample is generated from the posterior distribution of \(\theta | x_q\).

- the posterior variance \(v_q\) of the decision model net benefit \(NB(\theta)\) is deduced from this sample.

- the average posterior variance over all samples \(q\) is compared to the prior variance, obtaining an estimate of the proportion of uncertainty explained by the proposed study. This is used as a “shrinkage” factor to modify the regression-based EVPPI computation to produce an estimate of the EVSI.

To use the moment matching method, `evsi` needs to know some information that is not needed by the other EVSI calculation functions. This includes:

- a function `model_fn` to evaluate the decision model, and a function `par_fn` to sample from the distribution of the model parameters under current information. These are supplied in the same form as in the EVPPI Monte Carlo method function, `evppi_mc`.

- information about how data \(x_q\) are analysed to produce the posterior.

  For the built-in study designs, nothing extra is needed from the user here, as this information is built in.

  For custom study designs specified using `datagen_fn`, an extra function should be defined, and supplied as the `analysis_fn` argument to `evsi(..., method="mm")`. This function should take three arguments:

  - `data` a data frame with names matching the output of `datagen_fn`,

  - `args` a list of constants for the analysis that the user might want to vary, e.g. prior parameters or options for posterior computation. The user supplies these in `analysis_args`.

  - `pars` names of the parameters whose posterior is being sampled.

  and return a data frame with each row containing a draw from the posterior distribution of the parameters named in the columns. If specialised Bayesian analysis software such as JAGS or Stan is needed, then this function should wrap around a complete call to this software. An example is given below.

- `analysis_args`: a list of constants that the data analysis function needs to know. This is needed whether or not a built-in design is used. This would typically include parameters that define the prior distributions, and settings to control posterior computation (e.g. number of MCMC iterations). The specific components that are needed in this list depends on the study design, as specified by `analysis_fn` or `study`.

#### Moment matching method: example using a built-in study design

This is the first EVSI example computation shown above, implemented using the moment matching method with a study sample size of 1000.

Recall the sampling distribution for the study data is a normal with known variance, specified through `study="normal_known"`. The Bayesian inference procedure here is a simple conjugate normal analysis, and is built in to the package, so we do not need to supply `analysis_fn`.

However we do need to supply `analysis_args`. The constants required by the conjugate normal analysis are the prior mean and SD of the parameter `p1`, and the sampling SD of an individual-level observation in the study, here \(\sigma=1\).

    evsi(outputs_nb, inputs, study = "normal_known", n=10000, pars = "p1", method="mm", Q=30,
         model_fn = model_fn_nb, par_fn = par_fn,
         analysis_args = list(prior_mean=1, prior_sd=1, sampling_sd=1, niter=1000))

    ##       n       evsi
    ## 1 10000 0.05094176

The estimate of the EVSI is fairly close to the result from the regression method. The moment matching method also computes the regression-based estimate of the EVPPI, so this is returned alongside the EVSI estimate.

A lot of randomness is involved in the computation for this method, thus there is some Monte Carlo error and it is not reproducible unless the seed is set. This error can be controlled somewhat by changing \(Q\) (default 30) in the call to `evsi`, the number of posterior samples (default 1000) through the `niter` component of `analysis_args`, and the size of `outputs` and `inputs` (as in most VoI calculation methods).

#### Moment matching method: example using a custom study design

A more complex decision model is included with the `voi` package, see `help(chemo_model)`. In this model, the key uncertain parameters describe the probabilities of an adverse outcome (risk of side-effects), \(p_1\) and \(p_2\), under the standard of care and some novel treatment respectively. We want to design a study that informs only the relative effect of the new treatment on the risk of that outcome. The baseline risk \(p_1\) would be informed by some other source. This might be because we believe that the decision population has the same relative risk as the population in the designed study, but the baseline risk may be different.

To calculate the EVSI for the proposed study, we define two R functions:

1. a function to simulate study data

2. a function defining a Bayesian model to analyse the potential study data.

Analysis function. First we show the function to analyse the data. We parameterise the relative treatment effect as a log odds ratio \(log(p_1/(1-p_1)) - log(p_2/(1-p_2))\), supposing that there is previous information about the likely size of this quantity that can be expressed as a Normal\((\mu,\sigma^2)\) prior. The baseline risk \(p_1\) is given a Beta\((a_1,b_1)\) prior. These priors match the current uncertainty distributions used in `par_fn` and used to produce the parameter sample `inputs` supplied to `evsi`.

Since this model does not have an analytic posterior distribution, we use JAGS (via the `rjags` R package) to implement MCMC sampling. It is not necessary to use JAGS however - the analysis function may call any Bayesian modelling software.

The following R function encapsulates how the posterior is obtained from the data in this analysis.

    analysis_fn <- function(data, args, pars){
      dat <- list(y=c(data[,"y1"], data[,"y2"]))
      design <- list(n = rep(args$n, 2))
      priors <- list(a1=53, b1=60, mu=log(0.54), sigma=0.3)
      jagsdat <- c(dat, design, priors)
      or_jagsmod <- "
      model {
        y[1] ~ dbinom(p[1], n[1])
        y[2] ~ dbinom(p[2], n[2])
        p[1] <- p1
        p[2] <- odds[2] / (1 + odds[2])
        p1 ~ dbeta(a1, b1)
        odds[1] <- p[1] / (1 - p[1])
        odds[2] <- odds[1] * exp(logor)
        logor ~ dnorm(mu, 1/sigma^2)
      }
      "
      or.jag <- rjags::jags.model(textConnection(or_jagsmod),
                                  data=jagsdat, inits=list(logor=0, p1=0.5), quiet = TRUE)
      update(or.jag, 100, progress.bar="none")
      sam <- rjags::coda.samples(or.jag, c("logor"), 500, progress.bar="none")
      data.frame(logor_side_effects = as.numeric(sam[[1]][,"logor"]))
    }

A full understanding of JAGS model specification and sampling syntax is beyond the scope of this vignette - we just explain what the `voi` package is expecting. The inputs and outputs of `analysis_fn` take the following form.

- The return value is a data frame with a sample from the posterior of the parameters “learnt” from the study to inform the decision model. There should be one column per parameter learnt. Here we will only be using the information gained about `logor_side_effects`, so there is only column. The names of the data frame (here `logor_side_effects`) should match the names of arguments to the decision model function supplied as the `model_fn` argument to `evsi` (here this function is `chemo_model_lor_nb`).

- The `data` input argument is a data frame with the outcome data from the study to be analysed. In this example, the study is a trial with two arms, and the study results are `"y1"` and `"y2"`, the number of people experiencing the outcome in each arm. One row of data is sufficient here, but in other cases (e.g. with individual-level data) we might need multiple rows to define the study data.

  The names (`"y1"` and `"y2"` here) should match the names of the data frame returned by the function to simulate the study data (`datagen_fn` below).

- The `args` input argument is a list of constants that are needed in the analysis. The sample size of the proposed study, supplied as the `n` argument to `evsi()` is automatically added to this list.

  In this example, `n` is the the size of each trial arm, and is available to the analysis code as `args$n`. The sample size could also have been hard-coded into the analysis code, but supplying it through `args` is advisable as it allows the EVSI calculation to be defined transparently in the call to `evsi()`, and allows multiple calculations to be easily done for different sample sizes by supplying a vector as the `n` argument to `evsi()`.

  Any other constants can be supplied through the `analysis_args` argument to `evsi()`. This might be used for constants that define prior distributions. In this example, these are instead hard-coded inside `analysis_fn`.

- The `pars` argument is not used in this example, but this is a more general way of setting the names of the returned data frame. The names supplied in the `pars` argument to the `evsi()` function will be automatically passed to this argument, so they can be used in naming the returned data frame. We might do this if we wanted to write a generic function to fit a particular Bayesian model, that we could use in different decision models or EVSI calculations. Here we simply hard-coded the names as `logor_side_effects` for clarity.

Data generation function As in previous examples, this takes an input data frame `inputs` with parameter values sampled from the current uncertainty distribution, and outputs a data frame with a corresponding sample from the predictive distribution of the data. An additional argument `n` defines the sample size of the study. Note in this example how `inputs` is parameterised in terms of a baseline risk and a log odds ratio, and we combine these to obtain the absolute risk in group 2.

    datagen_fn <- function(inputs, n=100){
      p1 <- inputs[,"p_side_effects_t1"]
      logor <- inputs[,"logor_side_effects"]
      odds1 <- p1 / (1 - p1)
      odds2 <- odds1 * exp(logor)
      p2 <- odds2 / (1 + odds2)
      nsim <- nrow(inputs)
      data.frame(y1 = rbinom(nsim, n, p1),
                 y2 = rbinom(nsim, n, p2))
    }

Finally `evsi` is called to perform the EVSI calculation using the moment matching method.

    ev <- evsi(outputs=chemo_nb, inputs=chemo_pars,
               method="mm",
               pars="logor_side_effects",
               pars_datagen = c("p_side_effects_t1", "logor_side_effects"),
               datagen_fn = datagen_fn, analysis_fn = analysis_fn,
               n = 100, Q = 10,
               model_fn = chemo_model_lor_nb, par_fn = chemo_pars_fn)

One aspect of the syntax of the `evsi` call is new for this example. `pars_datagen` identifies which parameters (columns of `inputs`) are required to generate data from the proposed study, and `pars` identifies which parameters are learned from the study. These are different in this example. We need to know the baseline risk `"p_side_effects_t1"` to be able to generate study data, but we then choose to ignore the data that the study provides about this parameter when measuring the value of the study.

(In theory, these names need not be supplied to `evsi` if they are hard-coded into the analysis and data-generating functions, but it is safer in general to supply them, as they are required when using built-in study designs, and it allows the analysis and data-generating functions to be written in an abstract manner that can be re-used for different analyses).

## Value of Information in models for estimation

Suppose that the aim of the analysis is to get a precise estimate of a quantity, rather than to make an explicit decision between policies. VoI methodology can still be used to determine which uncertain parameters the estimate is most sensitive to (EVPPI) or the improvements in precision expected from new data (EVSI). The expected value of information is the expected reduction in variance of the quantity of interest given the further information.

We illustrate with an example. Suppose we want to estimate the prevalence of an infection. There are two sources of information. We have survey data which we think is biased, so that the true infection rate is higher than the rate of infection observed in the survey. We then have an expert judgement about the extent of bias in the data, and uncertainty associated with this.

Firstly, suppose the survey observed 100 people and 5 of them were infected. Using a vague Beta(0,0) prior (flat on the logit scale), the posterior distribution of the survey infection rate \(p_1\) is a Beta(5,95). We draw a random sample from this.

    p1 <- rbeta(10000, 5, 95)

Secondly, we guess that the true risk of being infected is twice the risk in the survey population, but we are uncertain about this relative risk. We might approximate our belief by placing a normal prior distribution on the log odds ratio \(\beta\), with a mean designed to reflect the doubled relative risk, and a variance that is high but concentrates the relative risk on positive values. Hence the true infection probability \(p_2\) is a function of the two parameters \(p_1\) and \(\beta\):

\[p_2 = expit(logit(p_1) + \beta)\].

We draw random samples from the current belief distributions of \(\beta\) and \(p_2\), and then graphically compare the distributions of the infection probability \(p_1\) in the survey data (black) and the true infection probability \(p_2\) (red).

    beta <- rnorm(10000, 0.8, 0.4)
    p2 <- plogis(qlogis(p1) + beta)
    plot(density(p1), lwd=2, xlim=c(0,1), main="")
    lines(density(p2), col="red", lwd=2)
    legend("topright", col=c("black","red"), lwd=c(2,2),
           legend=c("Surveyed infection probability", "True infection probability"))

The model output quantity is \(p_2\), and the model inputs are \(p_1\) and \(\beta\). We now want to determine the expected value of further information. This is measured in terms of expected reductions in variance of \(p_2\). This has a decision-theoretic interpretation, with “loss” being measured by squared error of a parameter estimate compared to the true parameter value. See Jackson et al. 2019.

### EVPI and EVPPI for estimation

The EVPI is trivially \(var(p_2)\), the variance under current information, which we can compute from the sample as

    var(p2)

    ## [1] 0.003566403

A more interesting quantity is the EVPPI for a parameter. It describes the expected reduction in variance given perfect knowledge of a particular parameter. In this example, we compute the EVPPI for \(p_1\) and \(\beta\), respectively, defined as

\[EVPPI[p_1] = var(p_2) - E(var(p_2| p_1))\] \[EVPPI[\beta] = var(p_2) - E(var(p_2 | \beta))\]

These can be computed using nonparametric regression, as described in Jackson et al. 2019. This is implemented in the function `evppivar` in the `voi` package.

    inputs <- data.frame(p1, beta)
    (evppi_beta <- evppivar(p2, inputs, par="beta"))

    ##   pars       evppi
    ## 1 beta 0.001498406

    (evppi_p1 <- evppivar(p2, inputs, par="p1"))

    ##   pars       evppi
    ## 1   p1 0.001915543

Hence a slightly greater improvement in variance is expected from knowing the true risk in the biased sample, compared to knowing the relative odds of infection.

These EVPPI values are easier to interpret if they are converted to the scale of a standard deviation. If we subtract the EVPPI from the original variance of \(p_2\) we get, for example \(E(var(p_2) | p_1)\). The square root of this is an estimate of what the standard deviation of \(p_2\) would be if we learnt \(p_1\) (note that is not exactly the expected SD remaining, since we cannot swap the order of the square root and expectation operators).

    sqrt(var(p2)) # or sd(p2)

    ## [1] 0.05971937

    sqrt(var(p2) - evppi_beta$evppi)

    ## [1] 0.04547524

    sqrt(var(p2) - evppi_p1$evppi)

    ## [1] 0.04063079

Hence we would expect to reduce the SD of \(p_2\) to around 2/3 of its original value by learning \(p_1\) or \(\beta\).

### How regression-based EVPPI estimation works

EVPPI is estimated here by nonparametric regression of the output on the input. Recall that the method of Strong et al. (2013) was used in `evppi` for models with decisions. Jackson et al. 2019 showed how this method applied in a wider class of problems, including models for estimation.

To estimate the expected reduction in variance in \(p_2\), given knowledge of \(p_1\), the `evppivar` function fits a regression model with `p2` as the outcome, and `p1` as the single predictor. A generalized additive model based on splines is fitted using the `gam` function from the `mgcv` package.

    plot(x=p1, y=p2, pch=".")
    mod <- mgcv::gam(p2 ~ te(p1, bs="cr"))
    p1fit <- fitted(mod)
    lines(sort(p1), p1fit[order(p1)], col="blue")

Taking the variance of the residuals from this regression (observed minus fitted values) produces an estimate of \(E(var(p_2 | p_1 = x))\), intuitively, the expected variance given knowledge of \(p_1\), which, when subtracted from the variance of \(p_2\), gives the EVPPI.

    p1res <- p2 - p1fit
    var(p2) - var(p1res)

    ## [1] 0.001916577

This agrees (up to Monte Carlo error) with the value produced by `evppivar`, which is obtained by a closely-related method, as the variance of the fitted values from this regression. This is equal to the total variance minus the variance of the residuals, through the “law of total variance”:

\[var(Y) - E_{X}\left[var_{Y| X}(Y |X)\right] = var_{X} \left[E_{Y|X}(Y|X)\right]\]

    var(p1fit)

    ## [1] 0.001915543

### EVSI for estimation

Now suppose we planned to collect additional survey data on the prevalence of infection.

First suppose that we can collect more data on the biased population that was used to estimate \(p_1\). The EVSI can be computed to show the expected value of surveying \(n\) individuals, for different sample sizes \(n\).

This is achieved using the function `evsivar` in the `voi` package, as follows.

- As in `evppivar`, samples from the prior distributions of the parameters are supplied in the data frame `inputs`.

- `study="binary"` indicates that the proposed study consists of a binary outcome observed from `n` individuals.

- `pars="p1"` indicates which input parameter is informed by the study, in other words, which parameter is assumed to generate the study data.

    evsivar(p2, inputs, study = "binary", pars="p1", n=c(100,1000,10000))

    ##       n         evsi
    ## 1   100 0.0009579006
    ## 2  1000 0.0017476769
    ## 3 10000 0.0018962755

As the proposed sample sizes increase, the expected value of sample information informing \(p_1\) converges to the EVPPI, the expected value of perfect information about \(p_1\).

Alternatively, suppose we were able to collect information about the unbiased population, whose infection prevalence is \(p_2\). That is, suppose we surveyed \(n\) individuals, each infected with probability \(p_2\). In this case, we can compute the EVSI by using `evsivar` with the model `inputs` defined to equal the model outputs, as follows:

    inputs_p2 = data.frame(p2 = p2)
    evsivar(p2, inputs=inputs_p2, study = "binary", pars="p2", n=c(100, 1000, 10000))

    ##       n        evsi
    ## 1   100 0.002808475
    ## 2  1000 0.003471948
    ## 3 10000 0.003556959

The unbiased data is clearly more valuable than the biased data, with double the variance reductions from biased data of the same sample size. As the sample size increases, the value converges to the EVPI, hence (in the asymptote) we will eliminate all uncertainty about our quantity of interest \(p_2\).

## Expected net benefit of sampling

The `voi` package includes a function `enbs()` to calculate the expected net benefit of sampling for a simple proposed study, given estimates of EVSI, and some other information including study costs and the size of the decision population.

This is described in a separate vignette, which also demonstrates how to plot the results of VoI analyses.