BioCCP

Intro

BioCCP.jl applies the Coupon Collector Problem to combinatorial biotechnology, in particular to aid (expected) minimum sample size determination for screening experiments.

Modular designs are considered, created by randomly combining r modules from a set of n available modules (sampling with replacement). The module probabilities during the generation of the designs are specified by a probability/abundance vector p. Depending on how many complete sets of modules one wants to observe, parameter m can be increased from its default value of 1 to a higher value.

For a specific combinatorial design set-up of interest, a report with results regarding (expected) minimum sample sizes can be easily retrieved by using the provided Interactive Pluto notebook. Additionally, a Pluto notebook with case studies is provided to illustrate the usage of BioCCP onto real biological examples.

References:

Boneh, A., & Hofri, M. (1997). The coupon-collector problem revisited—a survey of engineering problems and computational methods. Stochastic Models, 13(1), 39-66.

Doumas, A. V., & Papanicolaou, V. G. (2016). The coupon collector’s problem revisited: generalizing the double Dixie cup problem of Newman and Shepp. ESAIM: Probability and Statistics, 20, 367-399.

Functions

BioCCP.expectation_minsamplesize — Function

expectation_minsamplesize(n::Integer; p=ones(n)/n, m::Integer=1, r=1, normalize=true)

Calculates the expected minimum number of designs E[T] to observe each module at least m times.

n: number of modules in the design space
p: vector with the probabilities or abundances of the different modules
m: number of times each module has to be observed in the sampled set of designs
r: number of modules per design
normalize: if true, normalize p

julia> n = 100
julia> expectation_minsamplesize(n; p = ones(n)/n, m = 1, r = 1, normalize = true)
518

BioCCP.std_minsamplesize — Function

std_minsamplesize(n::Integer; p=ones(n)/n, m::Integer=1, r=1, normalize=true)

Calculates the standard deviation on the minimum number of designs to observe each module at least m times.

n: number of modules in the design space
p: vector with the probabilities or abundances of the different modules
m: number of times each module has to be observed in the sampled set of designs
r: number of modules per design
normalize: if true, normalize p

julia> n = 100
julia> std_minsamplesize(n; p = ones(n)/n, m = 1, r = 1, normalize = true)
126

BioCCP.success_probability — Function

success_probability(n::Integer, t::Integer; p=ones(n)/n, m::Integer=1, r=1, normalize=true)

Calculates the success probability F(t) = P(T ≤ t) or the probability that the minimum number of designs T to see each module at least m times is smaller than or equal to t.

n: number of modules in design space
t: sample size or number of designs for which to calculate the success probability
p: vector with the probabilities or abundances of the different modules
m: number of times each module has to be observed in the sampled set of designs
r: number of modules per design
normalize: if true, normalize p

julia> n = 100
julia> t = 600
julia> success_probability(n, t; p = ones(n)/n, m = 1, r = 1, normalize = true)
0.7802171997092149

BioCCP.expectation_fraction_collected — Function

expectation_fraction_collected(n::Integer, t::Integer; p=ones(n)/n, r=1, normalize=true)

Calculates the fraction of the total number of modules that is expected to be observed after collecting t designs.

n: number of modules in design space
t: sample size or number of designs
p: vector with the probabilities or abundances of the different modules
r: number of modules per design
normalize: if true, normalize p

julia> n = 100
julia> t = 200
julia> expectation_fraction_collected(n, t; p = ones(n)/n, r = 1, normalize=true)
0.8660203251420364

BioCCP.prob_occurrence_module — Function

prob_occurrence_module(p_i, t::Integer, r, k::Integer)

Calculates the probability that specific module with module probability p_i has occurred k times after collecting t designs.

Sampling processes of individual modules are assumed to be independent Poisson processes.

p_i: module probability
t: sample size or number of designs
r: number of modules per design
k: the number of occurrences

julia> p_i = 0.005
julia> t = 500
julia> r = 1
julia> k = 2
julia> prob_occurrence_module(p_i, t, r, k)
0.25651562069968376