Working with priors

Internally, adoptr is built around the joint distribution of a test statistic and the unknown location parameter of interest given a sample size, i.e. $\mathcal{L}\big[(X_i, \theta)\,|\,n_i\big]$ where $$X_i$$ is the stage-$$i$$ test statistic and $$n_i$$ the corresponding sample size. The distribution class for $$X_i$$ is defined by specifying a DataDistribution object, e.g., a normal distribution

To completely specify the marginal distribution of $$X_i$$, the distribution of $$\theta$$ must also be specified. The classical case where $$\theta$$ is considered fixed, emerges as special case when a single parameter value has probability mass 1.

Discrete priors

The simplest supported prior class are discrete PointMassPrior priors. To specify a discrete prior, one simply specifies the vector of pivot points with positive mass and the vector of corresponding probability masses. E.g., consider an example where the point $$\delta = 0.1$$ has probability mass $$0.4$$ and the point $$\delta = 0.25$$ has mass $$1 - 0.4 = 0.6$$.

disc_prior <- PointMassPrior(c(0.1, 0.25), c(0.4, 0.6))

For details on the provided methods, see ?DiscretePrior.

Continuous priors

adoptr also supports arbitrary continuous priors with support on compact intervals. For instance, we could consider a prior based on a truncated normal via:

cont_prior <- ContinuousPrior(
pdf     = function(x) dnorm(x, mean = 0.3, sd = 0.2),
support = c(-2, 3)
)

For details on the provided methods, see ?ContinuousPrior.

Conditioning

In practice, the most important operation will be conditioning. This is important to implement type one and type two error rate constraints. Consider, e.g., the case of power. Typically, a power constraint is imposed on a single point in the alternative, e.g. using the constraint

Power(Normal(), PointMassPrior(.4, 1)) >= 0.8
#> -E[Pr[x2>=c2(x1)]]<Normal<two-armed>;PointMass<0.40>>  <= -0.8

If uncertainty about the true response rate should be incorporated in the design, it makes sense to assume a continuous prior on $$\theta$$. In this case, the prior should be conditioned for the power constraint to avoid integrating over the null hypothesis:

Power(Normal(), condition(cont_prior, c(0, 3))) >= 0.8
#> -E[Pr[x2>=c2(x1)]]<Normal<two-armed>;ContinuousPrior<[0,3]>>  <= -0.8