SPRT.eval: Evaluation of Sequential probability ratio tests

In sequential.pops: Sequential Analysis of Biological Population Sizes

Evaluation of Sequential probability ratio tests

Description

Obtains the average number of sampling bouts and the rate of acceptance for H_{1}, from a sequential test of H_{0}:\mu = \psi_{0} vs. H_{1}:\mu < \psi_{1} across a range of true population densities, based on simulations. Sometimes called "operating characteristics".

Usage

SPRT.eval(obj, eval.range, overdispersion.sim = NA, N, seed = NULL)

Arguments

obj	An object of class "SPRT".
eval.range	A vector with a sequence of true population densities to evaluate.
overdispersion.sim	A character string (if a function) or a non-negative number specifying the overdispersion parameter used to generate simulated counts. Only required when using "negative binomial" or "beta-binomial" as kernel densities. See details.
N	Number of simulations per true population density being evaluated.
seed	Optional seed for random count generation.

Details

The kernel probability density function to evaluate the test is that specified in the argument density_func to create the "SPRT" object, but overdispersion can be different to generate simulated counts. If "negative binomial" is used as kernel density for the test and overdispersion.sim is not specified (NA), then the same specification of the test is used to generate the counts. Ideally, overdispersion for simulations should include uncertainty about the parameter to produce more robust test evaluations. For example, if using a negative binomial kernel and the Taylor's Power Law approach to obtain overdispersion, then overdispersion for simulations should be specified as:

k = \frac{\mu^2}{a \mu^b e^z - \mu}

where k is the overdispersion parameter of the negative binomial distribution, a and b are parameters of the Taylor's Power Law and z is a normally distributed variable with mean 0 and standard deviation \sigma_{e}, which is the root of the mean square error for the regression used to estimate a and b. See examples.

Value

A list with the average number of sampling bouts required to reach a decision ($AvgSamples), and the rate of acceptance for H_{1} across the provided range of population densities ($AcceptRate).

References

Binns, M.R., Nyrop, J.P. & Werf, W.v.d. (2000) Sampling and monitoring in crop protection: the theoretical basis for developing practical decision guides. CABI Pub., Wallingford, Oxon, UK; New York, N.Y.

Examples

# Function that describes negative binomial overdisperion from the mean
# and Taylor's Power Law parameters, a and b:

estimate_k <- function(mean) {
                        a = 1.830012
                        b = 1.218041 # a and b are Taylor's Power Law parameters
                        (mean^2) / ((a * mean^(b)) - mean)
                       }

# Generate a SPRT object with negative binomial and varying overdispersion

counts <- c(2, 5, 6, 2, 7)

test11 <- sprt(data = counts,
                mu0 = 8,
                mu1 = 10,
                density_func = "negative binomial",
                overdispersion = estimate_k(9),
                alpha = 0.1,
                beta = 0.1)

# Stochastic version of 'estimate k', where sd here is the the root of the
# mean square error for the regression used to estimate a and b.

estimate_k_stoch <- function(mean) {
                              a <- 1.830012
                              b <- 1.218041
                              (mean^2) /
                                ((a * mean^(b) *
                                  exp(truncdist::rtrunc(1,
                                                        "norm",
                                                        a = log(1 / (a * mean^(b - 1))),
                                                        b = Inf,
                                                        mean = 0,
                                                        sd = 0.3222354)))
                                         - mean)
                                   }

# Run model evaluation for test11 with varying overdispersion and
# added stochasticity.

eval4 <- SPRT.eval(test11, eval.range = seq(4, 13),
                    overdispersion.sim = "estimate_k_stoch", N = 30)

plot(seq(4, 13), eval4$AvgSamples, type = "o", xlab = "True population size",
      ylab = "Average number of bouts")
plot(seq(4, 13), eval4$AcceptRate, type = "o", xlab = "True population size",
      ylab = "Acceptance rate of H1")

## End (Not run)

sequential.pops documentation built on June 8, 2025, 1:08 p.m.