Extraction efficacy of the modified Baermann's Funnel method:
Baermann's Funnel method has been widely used for its low set up input and it's convenience of operation. A critical analysis of different extraction methods was made and concluded that the Baerman method was the most efficient and that there was no significant difference between the number of nematodes extracted by the sedimentation and sieving techniques (Harrison and Green, 1976
). Thus, the extraction efficacy of this method was tested in this study. Aphelenchoides besseyi
was selected for the default test because of its notoriety as a quarantine nematode in many countries. Here, to demonstrate the developed computer program, we used the extraction efficacy of the modified Baermann's Funnel method (Wu et al., 2010
), which obtains nematodes by artificial seeding of soil samples with known numbers of nematodes.
The funnel used in this study was open ended made of plastic and was 14 cm in diameter. For nematode collection, a small glass vial (1.4 cm diameter) was attached to the funnel with a rubber tube (8 cm long). Two tissue papers (Kimberly-Clark ®, Taipei, Taiwan) were placed on a mesh, and 100 g sand, with particle diameters between 0.42-0.84 mm, was placed on the tissue. One thousand all-stage Aphelenchoide besseyi were added evenly into the sand, and the mesh containing the sand and nematodes was placed on top of the funnel. Water was immediately added nearly to the brim of the funnel to cover the sample; no mist was applied during the incubation period. Nematodes from each funnel were collected twice at 24-hr and 48-hr intervals. In the case study, three replicates were used in the experiment. The entire experiment was repeated five times to determine the repeatability of treatment in the laboratory. We also investigated whether the extraction efficacies were consistent across experiments.
Statistical models: In general, in order to characterize units as infested or not, the probability of any combination of infested and non-infested units in a sample can be determined from the binomial or from the hypergeometric distribution, when the distribution of the pest among units is random and the sampling of units is random.
For large lots sufficiently mixed, the likelihood of finding an infested unit is approximated by binomial distribution. The sample size is less than 5% of the lot size. Binomial sampling is based on sampling with replacement. The probability of observing i
infested units in a sample of n
units is given by:
is the proportion of infested items.
The hypergeometric distribution is appropriate for describing the probability of finding a pest in a relatively small lot. A lot is considered to be small when the sample size is more than 5% of the lot size. In this case, sampling of one unit from the lot affects the probability of finding an infested unit in the next unit selected. Hypergeometric-based sampling is based on sampling without replacement. This probability is given by
is the number of infested units in the lot, j
is the number of infested units in the sample, L
is the total number of units in the lot, and n
is the number of units in the sample (Venette et al., 2002
The methodology for extracting the nematodes often cannot achieve 100% extraction efficacy; thus the results will involve one or more assay errors. Generally, there are two types of assay errors (Cowling et al., 1999
; Williams & Moffitt, 2010
). First, there is the probability of falsely detecting defective items when defective items do not exist in reality (denoted by
). Secondly, there is the probability of failing to detect defective items when in reality defective items exist (denoted by
represents the extraction efficacy in the Nematology case study. The first type of error is ignored because the confirmations of the extracting results are visual observations. The specific plant parasitic nematodes to be isolated for research or quarantine purposes usually have distinct morphological characteristics. A well-trained nematologist or inspector will not misidentify the target nematode; therefore the impact of specificity (
) will not be addressed here. So, in this study we determined the probability of detecting at least one infested unit with imperfect extraction efficacy for quarantine sampling.
Based on the above assumption, the true infestation rate of units should be adjusted to
, where p
is the true infestation rate and
is the adjusted infestation rate determined by the assay method. It is easy to show that the adjusted infestation rate (
) decreases as the extraction efficacy
decreases. Therefore, the binomial distribution probability of detecting zero infested units in sample size (n
) when taking the extraction efficacy
into account is:
Also, the probability of detecting at least one infested unit can be written as
For the hypergeometric distribution, the probability of detecting zero infested units in sample size (n
) from a lot of L
units adjusted for imperfect extraction efficacy, which contains K1
infested units, is as follows:
, which is the smallest integer greater than
. Hence, the probability of detecting at least one infested unit using the hypergeometric distribution is as follows:
Expressions (3) and (5) are the probabilities of erroneously accepting (PEA) to assess the performance of the different sampling schemes on binomial and hypergeometric probability distributions, respectively.
Monte Carlo simulation method:
When values for the extraction efficacy and the true infestation rate are both fixed, expression (4) or (6) is employed to calculate the probability of detecting at least one infested unit. However, when these values are not fixed, it is necessary to determine ranges of variation based on real world conditions. Then, using these ranges, a Monte Carlo simulation is performed to account for the uncertainty of the parameters of interest. A Monte Carlo simulation is a technique that involves using random sampling and probability to solve problems (Metropolis and Ulam, 1949
). Since this method is based on repeated computation of random numbers, calculation by computers is required and this method tends to be used when it is unfeasible or impossible to compute an exact result with a deterministic algorithm. In this study, the adjusted infestation rate of the assay method is expressed as a product of the extraction efficacy and the true infestation rate. To take into account the uncertainty both in the extraction efficacy and in the true infestation rate, both of which affect the adjusted infestation rate, the extraction efficacy and the true infestation rate need to be specified separately for each of the particular distributions (Rai and Krewski, 1998
A beta distribution (Beta
)), defining the distribution of a random variable on the closed unit interval
, can be made very flexible by choosing different shape parameters a
based on expert knowledge or previous data (Cowling et al., 1999
; Williams and Moffitt, 2010
). Thus, it would be a logical choice for defining a distribution of the values of the extraction efficacy and the true infestation rate. Traditionally, if r
individuals are positive among n
examined, then the parameters of the distribution can be calculated as
. These parameters can be specified by provided estimates of the mode and 5% or 95% confidence limits, both available from expert opinion or previous data. Previous papers (Branscum et al., 2005
; Messam et al., 2008
) have presented some useful software to deal with the situation. Their free software called “BetaBuster” is available at <http://www.epi.ucdavis.edu
/diagnostictests/betabuster.html> and can be used to determine the parameters of specific beta prior distributions based on scientific input. For example, if a laboratory assay assumes 95% confidence that the extraction efficacy is less than 50% and that the accuracy concentrates around 30%, “Betabuster” will take the information and obtain the unique Beta distribution with mode at 30%, and with 95% of the area of the distribution to the left side of 50%. For the above case, Beta
(6.2809, 13.3221) will be specified. In this study, we obtained information about the extraction efficacy by the experimental results of the modified Baermann's Funnel method for Aphelenchoides besseyi
To account for uncertainty using Monte Carlo simulation, we followed the steps listed below:
- Step 1: Based on the specified beta distributions, generate 50,000 sets of random samples each for the extraction efficacy and for the infestation rate.
- Step 2: Using the results of Step 1, calculate the binomial or hypergeometric distribution probability of detecting at least one infested unit in a sample lot by expressions (4) or (6).
- Step 3: Calculate the medians of the outputs of Step 2.
- Step 4: Repeat Steps 1, 2, and 3 for 100 times and calculate the mean of the resulting medians.
We provide a computer program (see Appendix) based on Monte Carlo simulation, performed in a statistical software R to obtain the probabilities of detecting at least one infestation unit as a function of four factors: lot size, number of samples, level of infestation, and extraction efficacy. R software available at <http://www.r-project.org/
> is a free software environment that includes a set of base packages for graphics, math, and statistics. There are some useful books which introduce R programming environment (Dalgaard, 2002
; Venables and Smith, 2002