# expert-surv

The goal of `expertsurv` is to incorporate expert opinion into an analysis of time to event data. `expertsurv` uses many of the core functions of the `survHE` package (Baio 2020) and also the `flexsurv` package (Jackson 2016). Technical details of the implementation are detailed in (Cooney and White 2023) and will not be repeated here.

The key function is `fit.models.expert` and operates almost identically to the `fit.models` function of `survHE`.

## Installation

You can install the released version of expertsurv from CRAN with:

    install("expertsurv")

## Expert Opinion on Survival at timepoints

If we have elicited expert opinion of the survival probability at certain timepoint(s) and assigned distributions to these beliefs, we encode that information as follows:

    #A param_expert object; which is a list of
    #length equal to the number of timepoints
    param_expert_example1 <- list()

    #If we have 1 timepoint and 2 experts
    #dist is the names of the distributions
    #wi is the weight assigned to each expert (usually 1)
    #param1, param2, param3 are the parameters of the distribution
    #e.g. for norm, param1 = mean, param2 = sd
    #param3 is only used for the t-distribution and is the degress of freedom.
    #We allow the following distributions:
    #c("normal","t","gamma","lognormal","beta")

    param_expert_example1[[1]] <- data.frame(dist = c("norm","t"),
                                             wi = c(0.5,0.5), # Ensure Weights sum to 1
                                             param1 = c(0.1,0.12),
                                             param2 = c(0.005,0.005),
                                             param3 = c(NA,3))
    param_expert_example1
    #> [[1]]
    #>   dist  wi param1 param2 param3
    #> 1 norm 0.5   0.10  0.005     NA
    #> 2    t 0.5   0.12  0.005      3

    #Naturally we will specify the timepoint for which these probabilities where elicited

    timepoint_expert <- 14

    #In case we wanted a second timepoint -- Just for illustration

    # param_expert_example1[[2]] <- data.frame(dist = c("norm","norm"),
    #                                          wi = c(1,1),
    #                                          param1 = c(0.05,0.045),
    #                                          param2 = c(0.005,0.005),
    #                                          param3 = c(NA,NA))
    #
    # timepoint_expert <- c(timepoint_expert,18)

If we wanted opinions at multiple timepoints we just include append another list (i.e. param_expert_example1[[2]] with the relevant parameters) and specify `timepoint_expert` as a vector of length 2 with the second element being the second timepoint.

For details on assigning distributions to elicited probabilities and quantiles see the `SHELF` package (Oakley 2021) and for an overview on methodological approaches to eliciting expert opinion see (O’Hagan 2019). We can see both the individual and pooled distributions using the following code (note that we could have used the output of the `fitdist` function from `SHELF` if we actually elicited quantiles from an expert):

    plot_opinion1 <- plot_expert_opinion(param_expert_example1[[1]],
                        weights = param_expert_example1[[1]]$wi)
    ggsave("Vignette_Example 1 - Expert Opinion.png")

For the log pool we have a uni-modal distribution (in contrast to the bi-modal linear pool) which has a \(95\%\) credible interval between \(9.0−11.9\%\) calculated with the function below:

    cred_int_val <- cred_int(plot_opinion1,val = "log pool", interval = c(0.025, 0.975))

Expert prior distributions

We load and fit the data as follows (in this example considering just the Weibull and Gompertz models), with `pool_type = "log pool"` specifying that we want to use the logarithmic pooling (rather than default “linear pool”). We do this as we wish to compare the results to the penalized maximum likelihood estimates in the next section.

    data2 <- data %>% rename(status = censored) %>% mutate(time2 = ifelse(time > 10, 10, time),
                                                                  status2 = ifelse(time> 10, 0, status))

    #Set the opinion type to "survival"

    example1  <- fit.models.expert(formula=Surv(time2,status2)~1,data=data2,
                                            distr=c("wph", "gomp"),
                                            method="bayes",
                                            iter = 5000,
                                            pool_type = "log pool",
                                            opinion_type = "survival",
                                            times_expert = timepoint_expert,
                                            param_expert = param_expert_example1)

Both visual fit and model fit statistics highlight that the Weibull model is a poor fit to both the expert opinion and data (black line referring to the \(95\%\) confidence region for the experts prior belief).

    model.fit.plot(example1, type = "dic")

    #N.B. plot.expertsurv (ported directly from survHE) plots the survival function at the posterior mean parameter values
    #     while it is more robust to use the entire posterior sample (make.surv), however, in this case both results are similar.

     plot(example1, add.km = T, t = 0:30)+
      theme_light()+
      scale_x_continuous(expand = c(0, 0), limits = c(0,NA), breaks=seq(0, 30, 2)) +
      scale_y_continuous(expand = c(0, 0), limits = c(0, NA), breaks=seq(0, 1, 0.05))+
      geom_segment(aes(x = 14, y = cred_int_val[1], xend = 14, yend = cred_int_val[2]))

Model Comparison

Survival function with Expert prior

## Expert Opinion using Penalized Maximum Likelihood

We can also fit the model by Penalized Maximum Likelihood approaches based on code taken from the `flexsurv` package (Jackson 2016). All that is required that the `method="bayes"` is changed to `method="mle"` with the `iter` argument now redundant. One argument that maybe of interest is the `method_mle` which is the optimization procedure that `flexsurv` uses. In case the optimization fails, we can sometimes obtain convergence with the `Nelder-Mead` algorithm. If the procedure is still failing, it may relate to the expert opinion being too informative.

It should be noted that the results will be similar to the Bayesian approach when the expert opinion is unimodal and relatively more informative, therefore we use the logarithmic pool which is unimodal.

We find that the AIC values also favour the Gompertz model by a large factor (not shown) and are very similar to the DIC presented for the Bayesian model.

## Expert Opinion on Survival of a comparator arm

In this situation we place an opinion on the comparator arm. In this example `id_St` is used to specify the row number in the data frame of the comparator arm, indicating that the covariate pattern for this row represents the group for which the expert opinion is provided.

    param_expert_example2[[1]] <- data.frame(dist = c("norm"),
                                             wi = c(1),
                                             param1 = c(0.1),
                                             param2 = c(0.005),
                                             param3 = c(NA))

    #Check the coding of the arm variable
    #Comparator is 0, which is our id_St
    unique(data$arm)
    #> [1] 0 1

    survHE.data.model  <- fit.models.expert(formula=Surv(time2,status2)~as.factor(arm),data=data2,
                                            distr=c("wei"),
                                            method="hmc",
                                            iter = 5000,
                                            opinion_type = "survival",
                                            id_St = min(which(data2$arm ==0)), #We want our opinion to refer to the treatment called "0"
                                            times_expert = timepoint_expert,
                                            param_expert = param_expert_example2)

We can remove the impact of expert opinion omitting the arguments relating to `opinion_type,id_St,times_expert,param_expert`, \(\dots\) with the function.

Survival function with Expert prior (left) and Vague prior (right)

The survival function for “arm 1” has been shifted downwards slightly, however the covariate for the accelerated time factor has markedly increased to counteract the lower survival probability for the reference (arm 0).

## Expert Opinion on Survival Difference

This example illustrates an opinion on the survival difference. For illustration we use the Gompertz, noting that a negative shape parameter will lead to a proportion of subjects living forever. Clearly the mean is not defined in these cases so the code automatically constrains the shape to be positive. It should also be noted that there are many potential combinations of survival curves which could produce a difference in the expected survival of 5 months. If this approach is considered, the number of iterations should be quite large to ensure convergence.

    param_expert3 <- list()

    #Prior belief of 5 "months" difference in expected survival
    param_expert3[[1]] <- data.frame(dist = "norm", wi = 1, param1 = 5, param2 = 0.2, param3 = NA)

    survHE.data.model  <- fit.models.expert(formula=Surv(time2,status2)~as.factor(arm),data=data2,
                                                         distr=c("gom"),
                                                         method="hmc",
                                                         iter = 5000,
                                                         opinion_type = "mean",
                                                         id_trt = min(which(data2$arm ==1)), # Survival difference is  Mean_surv[id_trt]- Mean_surv[id_comp]
                                                         id_comp = min(which(data2$arm ==0)),
                                                         param_expert = param_expert3)

Survival difference

## Compatability with underlying packages survHE and flexsurv

As stated in the introduction this package relies on many of the core functions of the `survHE, flexsurv` packages Jackson (2016). Because future versions of `survHE` and `flexsurv` may introduce conflicts with the current implementation, we have directly ported the key functions from these packages into the package so that `expertsurv` no longer imports `survHE,flexsurv` (of course all credit for those functions goes to Jackson (2016)).

If you run in issues, bugs or just features which you feel would be useful, please let me know (phcooney@tcd.ie) and I will investigate and update as required.

## Model Diagnostics

As this is a Bayesian analysis convergence diagnostics should be performed. Poor convergence can be observed for many reasons, however, because of our use of expert opinion it may be a symptom of conflict between the observed data and the expert’s opinion.

Default priors should work in most situations, but still need to be considered. At a minimum the Bayesian results without expert opinion should be compared against the maximum likelihood estimates. If considerable differences are present the prior distributions should be investigated.

Because the analysis is done in JAGS and Stan we can leverage the `ggmcmc` package (ggmcmc.2016?):

    #For Stan Models # Log-Normal, RP, Exponential, Weibull
    ggmcmc(ggs(example1$models$`Exponential`), file = "Exponential.pdf")

    #For JAGS Models # Gamma, Gompertz, Generalized Gamma
    ggmcmc(ggs(as.mcmc(example1$models$`Gamma`)), file = "Gamma.pdf")

## References

Baio, Gianluca. 2020. “survHE: Survival Analysis for Health Economic Evaluation and Cost-Effectiveness Modeling.” Journal of Statistical Software 95 (14): 1–47. https://doi.org/10.18637/jss.v095.i14.
Cooney, Philip, and Arthur White. 2023. “Direct Incorporation of Expert Opinion into Parametric Survival Models to Inform Survival Extrapolation.” Medical Decision Making 1 (1): 0272989X221150212. https://doi.org/10.1177/0272989X221150212.
Jackson, Christopher. 2016. “flexsurv: A Platform for Parametric Survival Modeling in R.” Journal of Statistical Software 70 (8): 1–33. https://doi.org/10.18637/jss.v070.i08.
O’Hagan, Anthony. 2019. “Expert Knowledge Elicitation: Subjective but Scientific.” The American Statistician 73 (sup1): 69–81. https://doi.org/10.1080/00031305.2018.1518265.
Oakley, Jeremy. 2021. SHELF: Tools to Support the Sheffield Elicitation Framework. https://CRAN.R-project.org/package=SHELF.