# Development of Enumerative and Binomial Sequential Sampling Plans for Monitoring Lymantria dispar (L.) (Lepidoptera Erebidae) in Mediterranean Oak Forests

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Data Collection

#### 2.2. Enumerative Sampling Plan

^{−1}) of 0.10 and 0.25 using the following equation proposed by Karandinos [36]:

^{2}) by a power function as the following equation:

^{2}by the linear regression, which is as follows:

_{n}is the cumulative number of egg masses sampled and n is the total number of sampled trees. Stop lines were then generated by plotting the values of T

_{n}against the correspondent values of n.

#### 2.3. Relationship between Egg Masses Density and Occupied or Infested Trees

#### 2.4. Development of Binomial Sampling Plans

_{1}) and lower (θ

_{2}) boundaries for the decision AT and α (type I) and β (type II) errors, which indicate either the probability of treating when pest density is below the defined AT or the probability of not treating when pest density exceeds the AT, respectively. The θ

_{1}and θ

_{2}were set at 10% above and below the AT, respectively, whereas a value of 0.10 was used for both α and β errors. The validation process was carried out using 500 resampling iterations with replacement, and the minimum sample size was arranged considering the x-axis intercept of the lower stop line.

#### 2.5. Validation of Binomial Sampling Plans

_{2}gives the actual α, whereas the OC value at θ

_{1}gives the actual β [56].

_{i}is the proportion of n datasets represented by dataset i, A

_{i}is the probability of making the correct decision to treat, and D

_{i}is the probability of making the correct decision to not treat.

## 3. Results

#### 3.1. Spatial Distribution

#### 3.2. Enumerative Sampling Plan

#### 3.3. Relationship between Egg Masses Density and Percentage of Infested Trees

#### 3.4. Binomial Sampling Plan

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Sample sizes (

**A**,

**B**) and sequential stop lines (

**C**,

**D**) for the assessment of Lymantria dispar egg mass density on cork oak trees using Green’s method at D = 0.25 (

**A**,

**C**) and D = 0.10 (

**B**,

**D**).

**Figure 2.**Validation of enumerative sampling plans to assess the infestation of Lymantria dispar on cork oak trees, based on Green’s plan showing actual precision levels (

**A**,

**B**) and sample sizes (

**C**,

**D**) calculated at a fixed level of 0.25 (

**A**,

**C**) and 0.10 (

**B**,

**D**). Dotted lines indicate the desired precision levels of 0.10 (

**A**) and 0.20 (

**B**).

**Figure 3.**Relationship between the cumulative number of Lymantria dispar egg masses and the percentage of occupied (with one or more egg masses) and infested (with three or more egg masses) trees. Linear regressions were fitted separately for each phase of gypsy moth population development: progradation (

**A**,

**D**), culmination (

**B**,

**E**), retrogradation (

**C**,

**F**).

**Figure 4.**Decision stop lines for binomial sequential sampling plans of trees occupied (

**A**) or infested (

**B**) by Lymantria dispar in Mediterranean cork oak forests. Binomial plans were obtained from resampling validation analysis based on action thresholds of 58.5% (

**A**) and 40.1% (

**B**) of infested sample units, α and β = 0.1 and a tally threshold of 1 egg mass per tree (

**A**) or 3 egg masses per tree (

**B**).

**Table 1.**Dispersion indices for Lymantria dispar egg masses on cork oak trees in Sardinia (Italy) in 1999–2010.

Stage | Dataset (n) | Pest Density Range (Egg Masses/Tree) | Taylor’s Power Law | |||
---|---|---|---|---|---|---|

ln(a) ± SEM | a | b ± SEM | R^{2} | |||

Overall | 547 | 0.05–59.4 | 1.16 ± 0.04 | 3.18 | 1.45 ± 0.02 | 0.90 |

Progradation | 181 | 0.07–38.2 | 1.15 ± 0.06 | 3.15 | 1.45 ± 0.04 | 0.88 |

Culmination | 174 | 0.20–59.4 | 0.95 ± 0.11 | 2.61 | 1.51 ± 0.04 | 0.88 |

Retrogradation | 192 | 0.01–45.1 | 1.22 ± 0.05 | 3.42 | 1.42 ± 0.03 | 0.91 |

**Table 2.**Validation of Green’s sequential sampling plan based on resampling approach at fixed-precision levels of 0.25 and 0.10.

Pest Density Range (Egg Masses/Tree) | Dataset (n) | Fixed-Precision Level = 0.25 | Fixed-Precision Level = 0.10 | ||
---|---|---|---|---|---|

Mean Precision (Range) | Mean Sample Size (Range) | Mean Precision (Range) | Mean Sample Size (Range) | ||

1.02–40.55 | 55 | 0.183 (0.040–0.395) | 27.3 (10–52) | 0.190 (0.040–0.433) | 164.7 (41–315) |

**Table 3.**Results of linear regression analyses aimed at exploring the relationship between the number of Lymantria dispar egg masses (y) and the percentage of occupied (i.e., trees with 1 or more egg masses) or infested (i.e., trees with 3 or more egg masses) trees (x). Linear regressions were fitted separately for each phase of gypsy moth population development.

Model | Stage | n | Equation | R^{2} | F | p |
---|---|---|---|---|---|---|

Egg mass density~% of occupied trees | Progradation | 181 | ln(y) = 1.9284x − 0.8952 | 0.77 | 602.5 | <0.001 |

Culmination | 174 | ln(y) = 2.1276x − 0.8466 | 0.74 | 490.8 | <0.001 | |

Retrogradation | 192 | ln(y) = 1.9350x − 0.8733 | 0.81 | 789.9 | <0.001 | |

Egg mass density~% of infested trees | Progradation | 181 | ln(y) = 4.1957x − 2.9113 | 0.82 | 842.7 | <0.001 |

Culmination | 174 | ln(y) = 4.0853x − 3.0704 | 0.88 | 1310.6 | <0.001 | |

Retrogradation | 192 | ln(y) = 4.3744x − 2.7541 | 0.86 | 1134.3 | <0.001 |

**Table 4.**Comparison of operation characteristics (OC value) and probabilities of correct (A and D) and incorrect (B and C) control decisions for sequential binomial sampling plans for cork oak trees occupied (AT = 58.5%) or infested (AT = 40.1%) by Lymantria dispar in progradation, culmination, and retrogradation phases.

AT ^{1} | TT ^{2} | Stage | Dataset (n) | OC ^{3} | Actual α ^{4} | Actual β ^{5} | ASN (range) ^{6} | A ^{7} | D ^{8} | A + D ^{9} | B ^{10} | C ^{11} | B + C ^{12} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

58.9 | 1 | Progradation | 181 | 0.496 | 0.093 | 0.091 | 31.4 (8–93) | 0.425 | 0.354 | 0.779 | 0.188 | 0.033 | 0.221 |

Culmination | 174 | 0.503 | 0.089 | 0.089 | 26.9 (11–98) | 0.787 | 0.098 | 0.885 | 0.006 | 0.109 | 0.115 | ||

Retrogradation | 192 | 0.505 | 0.092 | 0.096 | 27.4 (8–95) | 0.415 | 0.409 | 0.824 | 0.016 | 0.160 | 0.176 | ||

40.1 | 3 | Progradation | 181 | 0.488 | 0.098 | 0.090 | 42.1 (17–199) | 0.387 | 0.519 | 0.906 | 0.072 | 0.022 | 0.094 |

Culmination | 174 | 0.509 | 0.091 | 0.099 | 35.3 (17–212) | 0.770 | 0.196 | 0.966 | 0.023 | 0.011 | 0.034 | ||

Retrogradation | 192 | 0.486 | 0.101 | 0.092 | 42.4 (17–193) | 0.409 | 0.508 | 0.917 | 0.021 | 0.062 | 0.083 |

^{1}Action thresholds (%).

^{2}Tally threshold: minimum number of egg masses necessary to classify a tree either as occupied (1) or infested (3) by L. dispar.

^{3}Probability of not treating when the pest population density reaches the AT.

^{4}Probability of treating when the pest density is below the AT (type I error).

^{5}Probability of not treating when the pest density is above the AT (type II error)

^{. 6}Number of samples required to make a pest control decision (i.e., to treat or not to treat).

^{7}Correct decision to treat.

^{8}Correct decision not to treat.

^{9}Overall probability of making a correct pest control decision (i.e., to treat or not to treat).

^{10}Incorrect decision to treat.

^{11}Incorrect decision not to treat.

^{12}Overall probability of making an incorrect pest control decision (i.e., to treat or not to treat).

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Mannu, R.; Olivieri, M.; Cocco, A.; Lentini, A. Development of Enumerative and Binomial Sequential Sampling Plans for Monitoring *Lymantria dispar* (L.) (Lepidoptera Erebidae) in Mediterranean Oak Forests. *Agronomy* **2022**, *12*, 1501.
https://doi.org/10.3390/agronomy12071501

**AMA Style**

Mannu R, Olivieri M, Cocco A, Lentini A. Development of Enumerative and Binomial Sequential Sampling Plans for Monitoring *Lymantria dispar* (L.) (Lepidoptera Erebidae) in Mediterranean Oak Forests. *Agronomy*. 2022; 12(7):1501.
https://doi.org/10.3390/agronomy12071501

**Chicago/Turabian Style**

Mannu, Roberto, Maurizio Olivieri, Arturo Cocco, and Andrea Lentini. 2022. "Development of Enumerative and Binomial Sequential Sampling Plans for Monitoring *Lymantria dispar* (L.) (Lepidoptera Erebidae) in Mediterranean Oak Forests" *Agronomy* 12, no. 7: 1501.
https://doi.org/10.3390/agronomy12071501