This package estimates an ensemble of parametric survival models with different parametric families and uses Bayesian model averaging to combine them. The RoBSA ensemble uses Bayes factors to test for the presence or absence of effects of the individual predictors and evaluates the support for each parametric family. The resulting model-averaged parameter estimates are based on posterior model probabilities. The user can define a wide range of prior distributions for the effect size, intercepts, and auxiliary parameters. The package provides convenient functions for summary, visualizations, and fit diagnostics.

See our pre-print Bartoš et al. (2021) (https://arxiv.org/abs/2112.08311) introducing the methodology.

The package requires JAGS 4.3.0 to be installed. The development version of the package can be installed from GitHub:

`::install_github("fbartos/RoBSA") devtools`

To illustrate the package’s functionality, we use the
`veteran`

data set from the `survival`

package
containing 137 survival times of patients from a randomized trial of two
treatment regimens for lung cancer. We start by loading the package and
data set.

```
library("RoBSA")
#> Loading required namespace: runjags
#> module RoBSA loaded
data(cancer, package = "survival")
head(veteran)
#> trt celltype time status karno diagtime age prior
#> 1 1 squamous 72 1 60 7 69 0
#> 2 1 squamous 411 1 70 5 64 10
#> 3 1 squamous 228 1 60 3 38 0
#> 4 1 squamous 126 1 60 9 63 10
#> 5 1 squamous 118 1 70 11 65 10
#> 6 1 squamous 10 1 20 5 49 0
```

Before we fit the RoBSA ensemble with five accelerated failure times
parametric families (exponential, Weibull, log-normal, log-logistic, and
gamma), we must specify prior distributions for the intercepts and
auxiliary parameters (governing the scales and shapes) of the competing
parametric families. Here, we assume that we would expect the median
survival type in the standard treatment group to be 5 years with an
interquartile range of 10 years. We set the standard deviation of the
prior distributions to 0.5, which provides us with a satisfactory degree
of uncertainty about the parameter values. We obtain the corresponding
prior distributions by using the `calibrate_quartiles()`

function. Subsequently, we print the list containing priors for the
intercepts and auxiliary parameters.

```
<- calibrate_quartiles(median_t = 5, iq_range_t = 10, prior_sd = 0.5)
priors
priors#> $intercept
#> $intercept$`exp-aft`
#> Normal(1.98, 0.5)
#> $intercept$`weibull-aft`
#> Normal(2.06, 0.5)
#> $intercept$`lnorm-aft`
#> Normal(1.61, 0.5)
#> $intercept$`llogis-aft`
#> Normal(1.61, 0.5)
#> $intercept$`gamma-aft`
#> Normal(2.46, 0.5)
#>
#> $aux
#> $aux$`weibull-aft`
#> Lognormal(-0.36, 0.57)
#> $aux$`lnorm-aft`
#> Lognormal(0.2, 0.37)
#> $aux$`llogis-aft`
#> Lognormal(0.15, 0.39)
#> $aux$`gamma-aft`
#> Lognormal(-0.53, 0.63)
```

We create a new data.frame object containing a fit ready data set. We
(1) transform the survival times to years, (2) code the treatment
variable as a factor and set `"standard"`

as the default
level (to test for effect of the `"new"`

treatment effect; so
the intercept of the model corresponds to the standard treatment and the
treatment estimate to the improvement in the test treatment), and (3)
scale the Karnofsky performance score (karno) to range from 0-1 (i.e.,
the coefficient estimate corresponds to the biggest possible
difference).

```
<- data.frame(
df time = veteran$time / 12,
status = veteran$status,
treatment = factor(ifelse(veteran$trt == 1, "standard", "new"), levels = c("standard", "new")),
karno_scaled = veteran$karno / 100
)head(df)
#> time status treatment karno_scaled
#> 1 6.0000000 1 standard 0.6
#> 2 34.2500000 1 standard 0.7
#> 3 19.0000000 1 standard 0.6
#> 4 10.5000000 1 standard 0.6
#> 5 9.8333333 1 standard 0.7
#> 6 0.8333333 1 standard 0.2
```

We proceed by specifying a RoBSA model intended to test an informed
hypothesis of the presence of the treatment effect centered around the
log(AF) of 0.3 with the sd of 0.1,5 via the informed prior distribution
Normal_{+}(0.30,0.15) on the treatment effect. To set a prior
distribution on factor, we use the `prior_factor()`

function.
This allows us to specify the type of contrast we want to use. Here, we
use `"treatment"`

contrast to estimate differences from the
default level (we could also use `orthonormal`

to test for a
difference of the individual levels from the grand mean). Furthermore,
we adjust for the Karnofsky performance score by setting a wider
centered standard normal prior distribution Normal(0,1). To test only
for the presence of the treatment effect and include the covariate in
all models, we set `test_predictors = "treatment"`

. Finally,
we pass the appropriate prior distributions to the `priors`

,
`prior_intercept`

, `prior_aux`

arguments, set
`seed = 1`

for reproducibility, and use
`parallel = TRUE`

for speeding up the computation.

```
<- RoBSA(
fit.test Surv(time, status) ~ treatment + karno_scaled,
data = df,
priors = list(
treatment = prior_factor("normal", parameters = list(mean = 0.30, sd = 0.15),
truncation = list(0, Inf), contrast = "treatment"),
karno_scaled = prior("normal", parameters = list(mean = 0, sd = 1))
),test_predictors = "treatment",
prior_intercept = priors[["intercept"]],
prior_aux = priors[["aux"]],
parallel = TRUE, seed = 1
)
```

The `summary()`

functions provides the main summary of the
fitted model.

```
summary(fit.test)
#> Call:
#> RoBSA(formula = Surv(time, status) ~ treatment + karno_scaled,
#> data = df, priors = list(treatment = prior_factor("normal",
#> parameters = list(mean = 0.3, sd = 0.15), truncation = list(0,
#> Inf), contrast = "treatment"), karno_scaled = prior("normal",
#> parameters = list(mean = 0, sd = 1))), test_predictors = "treatment",
#> prior_intercept = priors[["intercept"]], prior_aux = priors[["aux"]],
#> parallel = TRUE, seed = 1)
#>
#> Robust Bayesian survival analysis
#> Distributions summary:
#> Models Prior prob. Post. prob. Inclusion BF
#> exp-aft 2/10 0.200 0.321 1.894
#> weibull-aft 2/10 0.200 0.026 0.105
#> lnorm-aft 2/10 0.200 0.149 0.699
#> llogis-aft 2/10 0.200 0.499 3.991
#> gamma-aft 2/10 0.200 0.005 0.019
#>
#> Components summary:
#> Models Prior prob. Post. prob. Inclusion BF
#> treatment 5/10 0.500 0.129 0.148
#>
#> Model-averaged estimates:
#> Mean Median 0.025 0.975
#> treatment[new] 0.019 0.000 0.000 0.224
#> karno_scaled 2.499 2.508 1.647 3.336
```

In the first table, we see that most posterior model probability is retained by the log-logistic (0.499), exponential (0.321), and log-normal (0.149) family, with the inclusion Bayes factors quantifying the change from prior to posterior model probabilities.

The second table then summarizes information about hypothesis tests of the model components. Here, in the RoBSA ensemble intended for testing, we are interested in the inclusion Bayes factor for the treatment effect. We find Bayes factor 0.148, indicating that there is moderate evidence in favor of no treatment effect (1/0.148 = 6.76) in comparison to our informed hypothesis of a positive treatment effect.

Now we try to estimate the model-averaged estimate of the difference
between the two treatments. We change the prior distribution from the
informed positive treatment effect to a neutral standard normal prior
distribution allowing for both positive and negative treatment effect.
We further set `test_predictors`

to `""`

in order
to omit models assuming zero treatment effect.

```
<- RoBSA(
fit.est Surv(time, status) ~ treatment + karno_scaled,
data = df,
priors = list(
treatment = prior_factor("normal", parameters = list(mean = 0, sd = 1),
contrast = "treatment"),
karno_scaled = prior("normal", parameters = list(mean = 0, sd = 1))
),test_predictors = "",
prior_intercept = priors[["intercept"]],
prior_aux = priors[["aux"]],
parallel = TRUE, seed = 1
)
```

```
summary(fit.est)
#> Call:
#> RoBSA(formula = Surv(time, status) ~ treatment + karno_scaled,
#> data = df, priors = list(treatment = prior_factor("normal",
#> parameters = list(mean = 0, sd = 1), contrast = "treatment"),
#> karno_scaled = prior("normal", parameters = list(mean = 0,
#> sd = 1))), test_predictors = "", prior_intercept = priors[["intercept"]],
#> prior_aux = priors[["aux"]], parallel = TRUE, seed = 1)
#>
#> Robust Bayesian survival analysis
#> Distributions summary:
#> Models Prior prob. Post. prob. Inclusion BF
#> exp-aft 1/5 0.200 0.254 1.360
#> weibull-aft 1/5 0.200 0.021 0.086
#> lnorm-aft 1/5 0.200 0.206 1.036
#> llogis-aft 1/5 0.200 0.515 4.252
#> gamma-aft 1/5 0.200 0.004 0.016
#>
#> Model-averaged estimates:
#> Mean Median 0.025 0.975
#> treatment[new] -0.179 -0.176 -0.561 0.193
#> karno_scaled 2.539 2.544 1.715 3.336
```

We again use the `summary()`

function to obtain
information about the fitted model. We find out that the experimental
treatment led to notably shorter survival times with the mean
model-averaged log(AF) = -0.19, 95% CI[-0.55, -0.19]. Furthermore, we
observe an enormous effect of the scaled Karnofsky performance score,
showing that moving from 0 to 1 increases the survival times with the
mean model-averaged log(AF) = 2.53, 95% CI [1.72, 3.35].

To assess the convergence of the MCMC model, we can use the
`diagnostics = TRUE`

argument in the `summary()`

function. The resulting table shows the maximum MCMC error, min
effective sample size and maximum R-hat for each of the models.

```
summary(fit.est, type = "diagnostics")
#> Call:
#> RoBSA(formula = Surv(time, status) ~ treatment + karno_scaled,
#> data = df, priors = list(treatment = prior_factor("normal",
#> parameters = list(mean = 0, sd = 1), contrast = "treatment"),
#> karno_scaled = prior("normal", parameters = list(mean = 0,
#> sd = 1))), test_predictors = "", prior_intercept = priors[["intercept"]],
#> prior_aux = priors[["aux"]], parallel = TRUE, seed = 1)
#>
#> Robust Bayesian survival analysis
#> Diagnostics overview:
#> Model Distribution Prior treatment Prior karno_scaled
#> 1 exp-aft treatment contrast: Normal(0, 1) Normal(0, 1)
#> 2 weibull-aft treatment contrast: Normal(0, 1) Normal(0, 1)
#> 3 lnorm-aft treatment contrast: Normal(0, 1) Normal(0, 1)
#> 4 llogis-aft treatment contrast: Normal(0, 1) Normal(0, 1)
#> 5 gamma-aft treatment contrast: Normal(0, 1) Normal(0, 1)
#> max[error(MCMC)] max[error(MCMC)/SD] min(ESS) max(R-hat)
#> 0.01473 0.037 712 1.002
#> 0.01706 0.040 625 1.002
#> 0.01486 0.037 745 1.001
#> 0.01486 0.038 694 1.003
#> 0.01500 0.041 589 1.008
```

We find that while the R-hat and max MCMC error is satisfactory for
all models from the RoBSA estimation ensemble, we might wish for a
larger ESS. To achieve that, we would simply increase the number of
sampling MCMC iterations by using the `iter`

argument in the
`RoBSA()`

function. Furthermore detailed diagnostics such as
trace plot, autocorrelation plots, and density plot for each
parameter/model are provided via the `diagnostics_trace()`

,
`diagnostics_autocorrelation()`

, and
`diagnostics_density()`

functions.

We can also visualize the model ensemble predictions for survival or
hazard. We visualize the model-averaged survival for each treatment
group via the `plot_survival()`

and using the
`predictor = "treatment"`

argument. By default, the remaining
predictors are set to their mean/default level, but predictions for
different values can be obtained by specifying the
`covariates_data`

argument.

`plot_survival(fit.est, predictor = "treatment")`

In alignment with the previous summary, we see that the
model-averaged survival for the new treatment group is below the
estimated treatment of the standard treatment group. Similarly, we could
also visualize the model-averaged hazard `plot_hazard()`

or
obtain numerical estimates for the survival, hazard, density, mean
survival, and standard deviation of the survival at different levels of
detail via the `predict()`

function.

Bartoš, F., Aust, F., & Haaf, J. M. (2021). *Informed Bayesian
survival analysis*. arXiv. https://arxiv.org/abs/2112.08311