Decision criterion based on posterior predictive probability

Specifies a Go/NoGo decision criterion based on the posterior predictive probability of the treatment effect \(\tilde\theta\) that will be observed in a future study with m_t and m_c subjects in the treatment and control groups, given the proof-of-concept data \(D\). The future effect is the difference between the future sample proportions (binary endpoints) or sample means (continuous endpoints), or, for a time-to-event endpoint, the difference of the estimated milestone survival probabilities or the ratio of the estimated hazards (see design_tte).

Usage

criterion_predictive(
  theta_null,
  m_t,
  m_c = m_t,
  gamma_go,
  gamma_nogo,
  go_regions = 1L,
  nogo_regions = 4L,
  rule = c("joint", "and", "or"),
  direction = "greater"
)

Arguments

theta_null: Null value of the treatment effect in the future study: a number for one endpoint or a numeric vector of length 2 for two endpoints.
m_t, m_c: Numbers of subjects in the treatment and control groups of the future study (whole numbers of at least 1).
gamma_go: Probability threshold for the Go criterion, strictly between 0 and 1. With rule = "and" or "or", one threshold per endpoint (a single number is used for both).
gamma_nogo: Probability threshold for the NoGo criterion, strictly between 0 and 1, given in the same way as gamma_go.
go_regions: Region numbers (from 1 to 4) forming the Go region for two endpoints with rule = "joint". Ignored for one endpoint. The default, region 1, requires both future effects to exceed their null values.
nogo_regions: Region numbers (from 1 to 4) forming the NoGo region for two endpoints with rule = "joint". Ignored for one endpoint. The default, region 4, requires both future effects to be at most their null values.
rule: For two endpoints, "joint" (the default; probabilities of unions of regions), "and" or "or" (marginal probabilities of each endpoint, combined with AND or OR). Ignored for one endpoint.
direction: "greater" (the default) if larger treatment effects are favorable, or "less" if smaller effects are favorable. For two endpoints, a single value for both or one value per endpoint.

Value

An object of class c("qdm_criterion_predictive", "qdm_criterion").

Details

For one endpoint, the Go region is \(\tilde\theta > \theta_{NULL}\) and the NoGo region is \(\tilde\theta \le \theta_{NULL}\). The Go criterion is \(\Pr(\tilde\theta > \theta_{NULL} \mid D) \ge \gamma_{go}\) and the NoGo criterion is \(\Pr(\tilde\theta \le \theta_{NULL} \mid D) \ge \gamma_{nogo}\). Because the two probabilities add up to 1, a Gray decision is possible only when \(\gamma_{go} + \gamma_{nogo} > 1\), and Miss decisions can occur only when \(\gamma_{go} + \gamma_{nogo} \le 1\).

With direction = "less", for an endpoint on which smaller effects are favorable, the Go region is \(\tilde\theta < \theta_{NULL}\) and the NoGo region is \(\tilde\theta \ge \theta_{NULL}\).

For two endpoints, the null values divide the plane of \((\tilde\theta_1, \tilde\theta_2)\) into four regions (see qdm_regions), and the criteria are based on the predictive probabilities of the unions of go_regions and nogo_regions (Yamaguchi et al., submitted, equations 7 and 8). With rule = "and" or "or", they are based on the marginal predictive probabilities of each endpoint, \(\Pr(\tilde\theta_k > \theta_{NULLk} \mid D)\) for Go and \(\Pr(\tilde\theta_k \le \theta_{NULLk} \mid D)\) for NoGo, combined with AND or OR (equations 9 and 10). With two endpoints, direction can differ between the endpoints, and the regions are defined on the favorable side of each endpoint.

References

Yamaguchi, Y., Hu, H., Huang, Y., Han, C., Homma, G. and Wang, A. (submitted). Bivariate Bayesian quantitative decision-making for two endpoints in rare disease drug development.

Examples