# prob_accept: Probability of acceptance for grab sampling scheme In grabsampling: Probability of Detection for Grab Sample Selection

## Description

This function calculates the overall probability of acceptance for given microbiological distribution such as lognormal.

## Usage

 `1` ```prob_accept(c, r, t, mu, distribution, K, m, sd) ```

## Arguments

 `c` acceptance number `r` number of primary increments in a grab sample or grab sample size `t` number of grab samples `mu` location parameter (mean log) of the Lognormal and Poisson-lognormal distributions on the log10 scale `distribution` what suitable microbiological distribution we have used such as `'Poisson gamma'` or `'Lognormal'`or `'Poisson lognormal'` `K` dispersion parameter of the Poisson gamma distribution (default value 0.25) `m` microbiological limit with default value zero, generally expressed as number of microorganisms in specific sample weight `sd` standard deviation of the lognormal and Poisson-lognormal distributions on the log10 scale (default value 0.8)

## Details

Based on the food safety literature, for given values of `c`, `r` and `t`, the probability of detection in a primary increment is given by, p_d=P(X > m)=1-P_{distribution}(X ≤ m|μ ,σ) and acceptance probability in `t` selected sample is given by P_a=P_{binomial}(X ≤ c|t,p_d).

If Y be the sum of correlated and identically distributed lognormal random variables X, then the approximate distribution of Y is lognormal distribution with mean μ_y, standard deviation σ_y (see Mehta et al (2006)) where E(Y)=mE(X) and V(Y)=mV(X)+cov(X_i,X_j) for all i \ne j =1 \cdots r.

The parameters μ_y and σ_y of the grab sample unit Y is given by,

μ_y =\log_{10}{(E[Y])} - {{σ_y}^2}/2 \log_e(10)

(see Mussida et al (2013)). For this package development, we have used fixed σ_y value with default value 0.8.

## Value

Probability of acceptance

## References

• Mussida, A., Vose, D. & Butler, F. Efficiency of the sampling plan for Cronobacter spp. assuming a Poisson lognormal distribution of the bacteria in powder infant formula and the implications of assuming a fixed within and between-lot variability, Food Control, Elsevier, 2013 , 33 , 174-185.

• Mehta, N.B, Molisch, A.F, Wu, J, & Zhang, J., 'Approximating the Sum of Correlated Lognormal or, Lognormal-Rice Random Variables,' 2006 IEEE International Conference on Communications, Istanbul, 2006, pp. 1605-1610.

## Examples

 ```1 2 3 4 5 6``` ``` c <- 0 r <- 25 t <- 30 mu <- -3 distribution <- 'Poisson lognormal' prob_accept(c, r, t, mu, distribution) ```

grabsampling documentation built on March 13, 2020, 5:07 p.m.