Calculate the minimum number of samples required for a one-sided exact binomial test

Description

Calculate the minimum number of samples required for a one-sided exact binomial test to distinguish between two success probabilities with specified alpha and power.

Usage

binom_sample_size(alpha = 0.05, power = 0.9, p0 = 0.9, p1 = 0.95)

Arguments

Details

This is a one-sided function, such that p_0 < p_1. It determines the minimum sample size to evaluate the hypothesis test:

H_0: \, p_1 \le p_0, \, vs.

H_1: \, p_1 > p_0

Value

A list containing the required sample size and the number of successful trials required.

References

Chow S-C, Shao J, Wang H, Lokhnygina Y. (2017) Sample Size Calculations in Clinical Research, Boca Raton, FL: CRC Press.

Examples

# The minimum number of reference positive cases required to demonstrate
# the true sensitivity is >0.7, assuming that the true value is 0.824, with
# 90% power is

binom_sample_size(alpha = 0.05, power = 0.9, p0 = 0.7, p1 = 0.824)

# With a sample size of n = 104, if the true prevalence is 0.2, we would
# require a sample size of at least n = 520 randomly sampled subjects to
# have adequate power to demonstrate the sensitivity of the new test.

# The minimum number of reference negative cases required to demonstrate
# the true specificity is >0.9, assuming that the true value is 0.963, with
# 90% power is

binom_sample_size(alpha = 0.05, power = 0.9, p0 = 0.9, p1 = 0.963)

# The proposed total sample size of n = 520 would be sufficient to
# demonstrate both endpoint goals are met.

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

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
)

Summarise results of multiple simulated trials to give the operating characteristics

Description

Summarise results of multiple simulated trials to give the operating characteristics

Usage

summarise_trials(data, min_pos = 1, fut = 0)

Arguments

Value

A data frame of row length 1, with the following columns:

• power: Power is defined as the proportion of trials that result in success, irrespective of whether it is an early stop for success or not. Trials that stop for futility, but which subsequently go on to be successful, are not considered as a success. In other words, the futility decision is binding, and in practice, if a trial triggered a futility rule, the sponsor would not see the eventual outcome if the trial were to continue enrolling. When the performance goals are set equal to the respective true values, the power returned is the type I error.

• stop_futility: The proportion of trials that stopped early for expected futility.

• n_avg: The average sample size for trials at the stage they stopped.

• sens: The average sensitivity for trials at the stage they stopped.

• spec: The average specificity for trials at the stage they stopped.

• mean_pos: The average number of reference positive cases for trials at the stage they stopped.

Examples

data <- 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(1, 1),
    prior_spec = c(1, 1),
    prior_prev = c(1, 1),
    succ_sens = 0.95,
    succ_spec = 0.95,
    n_at_looks = c(200, 400, 600, 800, 1000),
    n_mc = 10000,
    n_trials = 20,
    ncores = 1
    )

summarise_trials(data, fut = 0.05, min_pos = 10)