binom_PRC_h {bayespm}

Derivation of the decision limit for the PRC for Binomial data with probability parameter unknown

Description

binom_PRC_h is used to estimate the decision limit h for the Predictive Ratio CUSUM (PRC) for Binomial data. The process controls the false alarms tolerance, providing a lower bound of ARL_0 or FAP, based on the marginal distribution.

Usage

binom_PRC_h( ARL_0 = 370.4, FAP = NULL, N = NULL, n = NULL,
             historical_data = NULL, historical_n = NULL,
             a0 = NULL, b0 = NULL, alpha_0 = NULL, k = 2,
             it = 1e4, ARL0tol = 10/it )

Arguments

ARL_0

scalar (positive); lower bound of the In Control (IC) Average Run Length (ARL). It is average number of IC data points that we will plot in the PCC before a false alarm occurs. The default value is 370.4.

FAP

scalar (between 0 and 1); lower bound of the False Alarm Probability (FAP). It is the probability of raising at least one false alarm out of a pre-determined number of N hypothesis tests and it can be used instead of ARL_0.

N

scalar (natural); number of N hypothesis tests. It is used for the FAP determination.

n

vector; number of trials for given dataset. It needs to be in a vector form.

historical_data

vector; an optional univariate dataset of historical data. Historical data needs to be in a vector form.

historical_n

vector; number of trials for given historical dataset. It needs to be in a vector form.

a0

scalar (positive); shape hyperparameter of the prior Beta(a, b). It needs to be a number.

b0

scalar (positive); shape hyperparameter of the prior Beta(a, b). It needs to be a number.

alpha_0

scalar (non negative); It is a power prior parameter controlling the influence of the historical data on the posterior distribution. The default is 1/n_0, where n_0 is the size of the historical data.

k

scalar (positive); tuning parameter, which represents deviation between IC and OOC state. Precisely, if k>1 then the OOC state corresponds to a (k-1)100% increase in the expected odds of the success probability p and if k<1 corresponds (1-k)100% decrease. The dafault value is 2.

it

scalar (natural); iterations of the simulation. The default value is 10000.

ARL0tol

scalar (positive); relative tolerance of ARL_0 estimate. It represents the maximum amount of error which is allowed. The default value is 10/it.

Details

binom_PRC_h estimates the desicion limit h regarding the desired upper bound of false alarms tolerance for the Predictive Ratio CUSUM (PRC) process for Binomial data, when the probability parameter is unknown. The estimate concerns an one-sided PRC process and the default setup corresponds to a doubling of the expected odds ratio of the probability of success (k=2). The estimate is based on the use of the marginal distribution (prior predictive) and the incorporation of historical data is possible via the power prior.

The default metric of the false alarms tolerance is ARL_0=370.4, while a FAP option is available. The desicion limit is obtained via the numerical method regula falsi when ARL_0 is selected, while simulated samples of fixed size are used when FAP is of interest.

Examples

binom_PRC_h(ARL_0 = NULL, FAP = 0.05, N = 20, n = 10, a0 = 20, b0 = 180)

# derivation of the decision limit of the third application in
# "Design and properties of the Predictive Ratio Cusum (PRC) control charts"

arl0 <- 400
ap <- 66.5
bp <- 1434.5
kl <- 2


# To replicate results from application set 'ARL0tol = .001' and 'it = 1e4'
binom_PRC_h(ARL_0 = arl0, ARL0tol = .01, k = kl, n = 50, a0 = ap, b0 = bp, it = 1e3)