Description Usage Arguments Details Value References Examples

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

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

`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 |

`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) |

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.

Probability of acceptance

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.

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

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