multi_trial {adaptDiag} | R Documentation |

## Simulate and analyse multiple trials

### Description

Multiple trials and simulated and analysed up to the final analysis stage, irrespective of whether it would have been stopped for early success or expected futility. The output of the trials is handled elsewhere.

### Usage

```
multi_trial(
sens_true,
spec_true,
prev_true,
endpoint = "both",
sens_pg = 0.8,
spec_pg = 0.8,
prior_sens = c(0.1, 0.1),
prior_spec = c(0.1, 0.1),
prior_prev = c(0.1, 0.1),
succ_sens = 0.95,
succ_spec = 0.95,
n_at_looks,
n_mc = 10000,
n_trials = 1000,
ncores
)
```

### Arguments

`sens_true` |
scalar. True assumed sensitivity (must be between 0 and 1). |

`spec_true` |
scalar. True assumed specificity (must be between 0 and 1). |

`prev_true` |
scalar. True assumed prevalence as measured by the gold-standard reference test (must be between 0 and 1). |

`endpoint` |
character. The endpoint(s) that must meet a performance goal
criterion. The default is |

`sens_pg` |
scalar. Performance goal (PG) for the sensitivity endpoint, such that the the posterior probability that the PG is exceeded is calculated. Must be between 0 and 1. |

`spec_pg` |
scalar. Performance goal (PG) for the specificity endpoint, such that the the posterior probability that the PG is exceeded is calculated. Must be between 0 and 1. |

`prior_sens` |
vector. A vector of length 2 with the prior shape parameters for the sensitivity Beta distribution. |

`prior_spec` |
vector. A vector of length 2 with the prior shape parameters for the specificity Beta distribution. |

`prior_prev` |
vector. A vector of length 2 with the prior shape parameters for the prevalence Beta distribution. |

`succ_sens` |
scalar. Probability threshold for the sensitivity to exceed in order to declare a success. Must be between 0 and 1. |

`succ_spec` |
scalar. Probability threshold for the specificity to exceed in order to declare a success. Must be between 0 and 1. |

`n_at_looks` |
vector. Sample sizes for each interim look. The final value (or only value if no interim looks are planned) is the maximum allowable sample size for the trial. |

`n_mc` |
integer. Number of Monte Carlo draws to use for sampling from the Beta-Binomial distribution. |

`n_trials` |
integer. The number of clinical trials to simulate overall, which will be used to evaluate the operating characteristics. |

`ncores` |
integer. The number of cores to use for parallel processing. If 'ncores' is missing, it defaults to the maximum number of cores available (spare 1). |

### Details

This function simulates multiple trials and analyses each stage of the trial (i.e. at each interim analysis sample size look) irrespective of whether a stopping rule was triggered or not. The operating characteristics are handled by a separate function, which accounts for the stopping rules and any other trial constraints. By enumerating each stage of the trial, additional insights can be gained such as: for a trial that stopped early for futility, what is the probability that it would eventually go on to be successful if the trial had not stopped. The details on how each trial are simulated here are described below.

**Simulating a single trial**

Given true values for the test sensitivity (`sens_true`

), specificity
(`spec_true`

), and the prevalence (`prev_true`

) of disease, along
with a sample size look strategy (`n_at_looks`

), it is straightforward
to simulate a complete dataset using the binomial distribution. That is, a
data frame with true disease status (reference test), and the new diagnostic
test result.

**Posterior probability of exceeding PG at current look**

At a given sample size look, the posterior probability of an endpoint (e.g.
sensitivity) exceeding the pre-specified PG (`sens_pg`

) can be
calculated as follows.

If we let `\theta`

be the test property of interest (e.g. sensitivity),
and if we assume a prior distribution of the form

`\theta ~ Beta(\alpha, \beta),`

then with `X | \theta \sim Bin(n, \theta)`

, where `X`

is the number
of new test positive cases from the reference positive cases, the posterior
distribution of `\theta`

is

`\theta | X=x ~ Beta(\alpha + x, \beta + n - x).`

The posterior probability of exceeding the PG is then calculated as

`P[\theta \ge sens_pg | X = x, n]`

.

A similar calculation can be performed for the specificity, with
corresponding PG, `spec_pg`

.

**Posterior predictive probability of eventual success**

When at an interim sample size that is less the maximum
(i.e. `max(n_at_looks)`

), we can calculate the probability that the trial
will go on to eventually meet the success criteria.

At the `j`

-th look, we have observed `n_j`

tests, with ```
n_j^* =
n_{max} - n_j
```

subjects yet to be enrolled for testing. For the `n_j^*`

subjects remaining, we can simulate the number of reference positive results,
`y_j^*`

, using the posterior predictive distribution for the prevalence
(reference positive tests), which is off the form

`y_j^* | y_j, n_j, n_j^* ~ Beta-Bin(n_j^*, \alpha_0 + y_j, \beta + n_j - y_j),`

where `y_j`

is the observed number of reference positive cases.
Conditional on the number of subjects with a positive reference test in the
remaining sample together with `n_j^*`

, one can simulate the complete 2x2
contingency table by using the posterior predictive distributions for
sensitivity and specificity, each of which has a Beta-Binomial form.
Combining the observed `n_j`

subjects' data with a sample of the
`n_j^*`

subjects' data drawn from the predictive distribution, one can
then calculate the posterior probability of trial success (exceeding a PG)
for a specific endpoint. Repeating this many times and calculating the
proportion of probabilities that exceed the probability success threshold
yields the probability of eventual trial success at the maximum sample size.

As well as calculating the predictive posterior probability of eventual success for sensitivity and specificity, separately, we can also calculate the probability for both endpoints simultaneously.

### Value

A list containing a data frame with rows for each stage of the trial (i.e. each sample size look), irrespective of whether the trial meets the stopping criteria. Multiple trial simulations are stacked longways and indicated by the 'trial' column. The data frame has the following columns:

`stage`

: Trial stage.`pp_sens`

: Posterior probability of exceeding the performance goal for sensitivity.`pp_spec`

: Posterior probability of exceeding the performance goal for specificity.`ppp_succ_sens`

: Posterior predictive probability of eventual success for sensitivity at the maximum sample size.`ppp_succ_spec`

: Posterior predictive probability of eventual success for specificity at the maximum sample size.`ppp_succ_both`

: Posterior predictive probability of eventual success for *both* sensitivity and specificity at the maximum sample size.`tp`

: True positive count.`tn`

: True negative count.`fp`

: False positive count.`fn`

: False negative count.`sens_hat`

: Posterior median estimate of the test sensitivity.`sens_CrI2.5`

: Lower bound of the 95 the test sensitivity.`sens_CrI97.5`

: Upper bound of the 95 the test sensitivity.`spec_hat`

: Posterior median estimate of the test specificity.`spec_CrI2.5`

: Lower bound of the 95 the test specificity.`spec_CrI97.5`

: Upper bound of the 95 the test specificity.`n`

: The sample size at the given look for the row.`trial`

: The trial number, which will range from 1 to 'n_trials'.

The list also contains the arguments used and the call.

### Parallelization

To use multiple cores (where available), the argument `ncores`

can be
increased from the default of 1. On UNIX machines (including macOS),
parallelization is performed using the `mclapply`

function with `ncores`

`>1`

. On Windows machines, parallel
processing is implemented via the `foreach`

function.

### Examples

```
multi_trial(
sens_true = 0.9,
spec_true = 0.95,
prev_true = 0.1,
endpoint = "both",
sens_pg = 0.8,
spec_pg = 0.8,
prior_sens = c(0.1, 0.1),
prior_spec = c(0.1, 0.1),
prior_prev = c(0.1, 0.1),
succ_sens = 0.95,
succ_spec = 0.95,
n_at_looks = c(200, 400, 600, 800, 1000),
n_mc = 10000,
n_trials = 2,
ncores = 1
)
```

*adaptDiag*version 0.1.0 Index]