Reliability and Risk Analysis Analysis of Model Tasks from Selected Application Areas Using a Computer (software R, Excel)
Task The aim of this study is to assess the wastewater treatment plant (WWTP) with given the presence of synthetic musk compounds in water passed this WWTP. The real sample consists of 60 fish from the carp family, specifically of the European chub (Leuciscus cephalus), which were caught in the Svratka River. Half of them were caught in front of (Group 1), and half of them behind (Group 2) the WWTP. Fish tissue samples (specifically in muscle) were analyzed, and two nitromusk compounds (musk ambrette (AMB), musk tibetene (TIB)), and two polycyclic musk compounds (phantolide (PH), traseolide (TR)) were explored. Fish of approximately the same age were chosen for the analysis. The measuring instrument had two limitations: the limit of detection LOD =x D and the limit of quantification LOQ=x Q, x D < x Q.
Task Data was left doubly time censored (the limit of detection and the limit quantification are in the role of censor time) and for each group we know the number of censored observations, ie the number observation n D prior the detection, the number of observations n Q in the interval (x D, x Q and the number of non-censored observations n 0 and their current values. Data can be found in the article Fusek, Michalek, Zouhar, Vavrova: Statistical Analysis of Musk Compounds Concentrations in Fish Tissue Based on Doubly Left-Censored Samples. DEED 2013.
Suppose that X 1,..., X n is the random sample from the exponential distribution with the parameter λ and the density f (x, λ) and the distribution function F (x, λ). X (1),..., X (n) is the ordered random sample. Double censoring wit the censors x D and x Q : n D is a frequency of censored in 0, x D, n Q is a frequency of censored in (x D, x Q, n O is a frequency of non-censored, bigger than x Q, X (n no +1),..., X (n). The vector (n D, n Q, n O ) Mu 3(n, θ D, θ Q, θ O ), n = n D + n Q + n O (the multinomial distribution), the marginal distributions are n D Bi(n, θ D ), n Q Bi(n, θ Q ), n O Bi(n, θ O ), where θ D = F (x D, λ), θ Q = F (x Q, λ) F (x D, λ), θ O = 1 F (x Q, λ).
The likelihood function is L(λ, n D, n Q, x (n no +1),..., x (n) ) = n! n D!n Q! [F (x D, λ)] n D [F (x Q, λ) F (x D, λ)] n Q The log-likelihood function is of the form ( ) n! l(λ, n D, n Q, x (n no +1),..., x (n) ) = log + n D log [F (x D, λ)] n D!n Q! n + n Q log [F (x Q, λ) F (x D, λ)] + j=n n O +1 n j=n n O +1 f (x j ). log [ f (x (j) ) ].
l λ = n D F λ(x D, λ) F (x D, λ) } {{ } H D (x D,λ) +n Q F λ(x Q, λ) F λ(x D, λ) F (x Q, λ) F (x D, λ) } {{ } H Q (x Q,λ) 2 l H D (x D, λ) H Q (x Q, λ) = n λ 2 D + n Q + λ λ λ + n j=n n O +1 n j=n n O +1 f λ(x (j) ) f (x (j) ) f λ(x (j) ) f (x (j) )
The exponential distribution with the distribution function F (x) = 1 exp ( λx) and the density f (x) = λ exp ( λx). The likelihood function is ( ) n! l(λ, n D, n Q, x (n no +1),..., x (n) ) = log + n D log [1 exp ( λx D )] n D!n Q! + n Q log [exp ( λx D ) exp ( λx Q )] + n O log (λ) λ n i=n n O +1 x (i) l λ x D exp ( λx D ) = n D 1 exp ( λx D ) + n Q n i=n n O +1 x (i) x Q exp ( λx Q ) x D exp ( λx D ) exp ( λx D ) exp ( λx Q ) + n O λ (1)
{ 2 l λ = n 2 D x D 2 exp ( λx D ) 1 exp ( λx D ) x } D 2 exp ( 2λx D ) [1 exp ( λx D )] 2 + n Q { x 2 D exp ( λx D ) x 2 Q exp ( λx Q ) exp ( λx D ) exp ( λx Q ) The empirical Fisher information is of the form [x Q exp ( λx Q ) x D exp ( λx D )] 2 [exp ( λx D ) exp ( λx Q )] 2 { J empir = 2 l x 2 λ = n D exp ( λx D ) 2 D 1 exp ( λx D ) + x } D 2 exp ( 2λx D ) [1 exp ( λx D )] 2 n Q { x 2 D exp ( λx D ) x 2 Q exp ( λx Q ) exp ( λx D ) exp ( λx Q ) [x Q exp ( λx Q ) x D exp ( λx D )] 2 [exp ( λx D ) exp ( λx Q )] 2 } n O λ 2 } + n O λ 2.
The theoretical Fisher information is E (n D ) = nθ D = n [1 exp ( λx D )], E (n Q ) = nθ Q = n [exp ( λx D ) exp ( λx Q )], E (n O ) = nθ O = n exp ( λx Q ). Jteor = E 2 l λ 2 = n x 2 D exp ( 2λx D ) [1 exp ( λx D )] + nx 2 Q exp ( λx Q ) + n [x Q exp ( λx Q ) x D exp ( λx D )] 2 [exp ( λx D ) exp ( λx Q )] + n λ 2 exp ( λx Q).
We will test the null hypothesish 0 : λ 1 λ 2 = 0 against the alternative H 1 : λ 1 λ 2 0, where λ 1 and λ 2 are parameters of the exponential distribution describing data in two locations. The test statistics T is T = λ 1 λ 2, (2) 2 σ 1 + σ 2 2 where σ 2 1 and σ 2 2 are estimated variances of λ 1 and λ 2.
The variance (asympt.) σ 2 k is σ 2 k = J 1 k, k = 1, 2, (3) where J k is the Fisher information and we can calculate is by the second derivatives of the log-likelihood function.
It was discovered that there is no significant difference between Group 1 and Group 2 in expected concentrations of Phantolide, Traseolide and Musk Ambrette. However, there is a difference between Group 1 and Group 2 in expected concentrations of Musk Tibeten. Compound H p-value PH 0 0.47 TR 0 0.58 AMB 0 1.00 TIB 1 0.00 Table: Comparison of the expected concentrations of musk compounds between Group 1 and Group 2. H = 0 (H = 1) denotes that the null hypothesis is not rejected (is rejected) on the significance level 0.05.