# Effects of Quantitative Ordinal Scale Design on the Accuracy of Estimates of Mean Disease Severity

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## Abstract

**:**

## 1. Introduction

^{+}–3, 3 = 3

^{+}–6, 4 = 6

^{+}–12, 5 = 12

^{+}–25, 6 = 25

^{+}–50, 7 = 50

^{+}–75, 8 = 75

^{+}–88, 9 = 88

^{+}–94, 10 = 94

^{+}–97, 11 = 97

^{+}–100, and 12 = 100% disease [5]. Several other similar scales have been developed that subdivide the percent scale into different numbers of intervals and interval sizes [6,7,8,9,10,11,12,13]. For subsequent data analysis, it is recommended that the mid-point of the scale interval be used if applying parametric methods [2,14], although the ordinal ratings may be analyzed directly if non-parametric methods are used [14].

## 2. Materials and Methods

#### 2.1. Assessment Methods

^{+}–3, 3

^{+}–6, 6

^{+}–12, 12

^{+}–25, 25

^{+}–50, 50

^{+}–75, 75

^{+}–88, 88

^{+}–94, 94

^{+}–97, 97

^{+}–99% and 100% [5].

^{+}–1, 1

^{+}–4, 4

^{+}–10, 10

^{+}–20, 20

^{+}–30, 30

^{+}–40, 40

^{+}–50, 50

^{+}–70, 70

^{+}–100% disease) [17]. Here, only severities <50% were emphasized because leaves often abscise if the disease becomes too severe, making it difficult to obtain samples with severity >50% [22]. Thus, only two classes (50

^{+}–70, 70

^{+}–100% disease) account for severities ≥50% for assessment in methods 4 through 7.

^{+}–1, 1

^{+}–4, 4

^{+}–10, 10

^{+}–30, 30

^{+}–50, 50

^{+}–70, 70

^{+}–100% disease).

^{+}–1, 1

^{+}–4, 4

^{+}–10, 10

^{+}–15, 15

^{+}–20, 20

^{+}–25, 25

^{+}–30, 30

^{+}–35, 35

^{+}–40, 40

^{+}–45, 45

^{+}–50, 50

^{+}–70, 70

^{+}–100% disease).

^{+}–1, 1

^{+}–3, 3

^{+}–6, 6

^{+}–10, 10

^{+}–20, 20

^{+}–30, 30

^{+}–40, 40

^{+}–50, 50

^{+}–70, 70

^{+}–100% disease).

#### 2.2. Simulation Method

#### 2.3. Criterion for Comparison: Mean Squared Error (MSE)

#### 2.4. Simulation Framework

- We simulated n sample size values (from 10 to 100 in increments of 5) from a beta-distribution with the preselected specific mean severity and variance for that sample (which might represent a plot in a field). These n simulated values on the continuous percentage scale, defined by the beta distribution of a random variable on the closed unit interval 0–1, represent the NPEs.
- The resulting NPEs were converted to the appropriate classes for assessment methods 2–7. These scale data were subsequently converted to the appropriate midpoint value of each class for subsequent analysis [2].
- The MSEs of mean disease severity estimates for each of the different scales were calculated (Equation (1)).
- The corresponding variances and biases were calculated (equations 2 and 3, respectively).
- The Monte Carlo simulation process was repeated 10,000 times. To present the results comparing assessment methods, we plot MSEs (or variance or bias) on the y-axis against sample size values (from n = 10 to 100 in n = 5 increments) on the x-axis at each of a range of “actual” mean disease severities (1%, 5%, 10%, 20%, 30%, and 40% [6 mean disease severities]) and disease severity variances (representing large, medium, and small variation [3 variances]). Thus, the results are presented in a montage (a 6 × 3 array of figures). For example, R1C2 (chart in row 1, column 2) indicates the chart in the second column in the first row, and presents the relationships between the MSE, variance, or bias of the mean disease severity estimates and sample size (n) for each of the different scales used, at an “actual” mean disease severity of 1%, and the medium variation.

#### 2.5. Pear scab Assessment

## 3. Results

#### 3.1. Comparison to Determine the Effect of Scale Structure

^{+}to 25% and 25

^{+}to 50% are so wide (13% and 25%, respectively) that when mid-points are taken for analysis they more often result in less accurate specimen estimates compared with what is achieved based on NPEs, the EI10 or the AM10 scales.

#### 3.2. Comparison to Determine the Effect of Number of Classes

#### 3.3. Analysis of Severity of Pear Scab Data

## 4. Discussion

#### 4.1. “Optimal” Design for an Ordinal Scale

#### 4.2. Effects of Statistical Distributions of Diseased Leaves

#### 4.3. Properties of the AM10 Quantitative Ordinal Scale

^{+}–70, 70

^{+}–100% disease) account for severities ≥50%. Thus, as some diseases can exceed 50% severity, maintaining a linear 10% interval from 10% to 100% will help avoid undesirable bias in the upper range (but increasing the number of classes).

#### 4.4. Mean Squared Error (MSE), Variance, and Bias

^{+}–70, 70

^{+}–100%) which include severities ≥50% for assessment methods 4 through 7; this resulted from the use of a coarse measurement scale at the right as opposed to a finer scale at the left side of a normal distribution.

#### 4.5. Rationale for Simulation Studies

#### 4.6. Direction for Future Studies

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**A beta-distribution was used to simulate the random variables representing disease severity. The means of the random variables of the beta-distributions were set at 1%, 5%, 10%, 20%, 30%, and 40% severities to demonstrate a range of possible “actual” mean disease severities (%). Furthermore, we selected three different variance parameter settings (representing large, medium, and small variation) for the fixed means to illustrate the effect of the variance of estimates with the different assessment methods.

**Figure 2.**The relationships between the mean squared error (MSE) of mean disease severity estimates and sample size (n) (from n = 10 to 100 in n = 5 increments) for each of the different scales used, over a range of “actual” mean disease severities (1%, 5%, 10%, 20%, 30%, and 40%) and variance (represented by large, medium, and small variation). The simulation process was repeated 10,000 times. Assessment methods: (i) NPE (nearest percentage estimate); (ii) HB (Horsfall-Barratt scale); (iii) AM10 (10% linear scale emphasizing severities ≤50% disease, and with additional intervals at severities <10%); (iv) EI10 (10% linear scale).

**Figure 3.**The relationships between the variance of mean disease severity estimates and sample size (n) (from n = 10 to 100 in n = 5 increments) for each of the different scales used, over a range of “actual” mean disease severities (1%, 5%, 10%, 20%, 30%, and 40%) and variance (represented by large, medium, and small variation). The simulation process was repeated 10,000 times. Assessment methods: (i) NPE (nearest percentage estimate); (ii) HB (Horsfall-Barratt scale); (iii) AM10 (10% linear scale emphasizing severities ≤50% disease, and with additional intervals at severities <10%); (iv) EI10 (10% linear scale).

**Figure 4.**The relationships between the bias of mean disease severity estimates and sample size (n) (from n = 10 to 100 in n = 5 increments) for each of the different scales used, over a range of “actual” mean disease severities (1%, 5%, 10%, 20%, 30%, and 40%) and variance (represented by large, medium, and small variation). The simulation process was repeated 10,000 times. Assessment methods: (i) NPE (nearest percentage estimate); (ii) HB (Horsfall-Barratt scale); (iii) AM10 (10% linear scale emphasizing severities ≤50% disease, and with additional intervals at severities <10%); (iv) EI10 (10% linear scale).

**Figure 5.**The relationships between the mean squared error (MSE) of mean disease severity estimates and sample size (n) (from n = 10 to 100 in n = 5 increments) for each of the different scales used, over a range of “actual” mean disease severities (1%, 5%, 10%, 20%, 30%, and 40%) and variance (represented by large, medium, and small variation). The simulation process was repeated 10,000 times. Assessment methods: (i) NPE (nearest percentage estimate); (ii) AM10 (10% linear scale emphasizing severities ≤50% disease, and with additional intervals at severities <10%); (iii) AM20: similar to the structure of AM10 (both scales have the same structure at ≤10%), but with 20% intervals for disease severities from 10% to 50%; (iv) AM5: similar to the structure of AM10 (both scales have the same structure at ≤10%), but with 5% intervals for disease severities 10 to 50%; and (v) AM10f: similar to the structure of the AM10 scale, but with finer intervals at ≤10% disease.

**Figure 6.**The relationships between the variance of mean disease severity estimates and sample size (n) (from n = 10 to 100 in n = 5 increments) for each of the different scales used, over a range of “actual” mean disease severities (1%, 5%, 10%, 20%, 30%, and 40%) and variance (represented by large, medium, and small variation). The simulation process was repeated 10,000 times. Assessment methods were: (i) NPE (nearest percentage estimate); (ii) AM10 (10% linear scale emphasizing severities ≤50% disease, and with additional intervals at severities <10%); (iii) AM20: Similar to the structure of AM10 (both scales have the same structure at ≤10%), this scale has wider ordinal scale intervals for disease severities 10 through 50% based on a 20% linear scale; (iv) AM5: Similar to the structure of AM10 (both scales have the same structure at ≤10%), this scale has narrower ordinal scale intervals for disease severities 10 through 50% based on a 5% linear scale; (v) AM10f: Similar to the structure of AM10, this scale has 10% linear intervals at >10%, but has finer intervals at ≤10% disease.

**Figure 7.**The relationships between the bias of mean disease severity estimates and sample size (n) (from n = 10 to 100 in n = 5 increments) for each of the different scales used, with conditioning on a range of “actual” mean disease severities (1%, 5%, 10%, 20%, 30%, and 40%) and variance (represented by large, medium, and small variation). The simulation process was repeated 10,000 times. Assessment methods were: (i) NPE (nearest percentage estimate); (ii) AM10 (10% linear scale emphasizing severities ≤50% disease, and with additional intervals at severities <10%); (iii) AM20: Similar to the structure of AM10 (both scales have the same structure at ≤10%), this scale has wider ordinal scale intervals for disease severities 10 through 50% based on a 20% linear scale; (iv) AM5: Similar to the structure of AM10 (both scales have the same structure at ≤10%), this scale has narrower ordinal scale intervals for disease severities 10 through 50% based on a 5% linear scale; (v) AM10f: Similar to the structure of AM10, this scale has 10% linear intervals at >10%, but has finer intervals at ≤10% disease.

**Figure 8.**Results in relation to the actual mean severity of peach scab (caused by Venturia nashicola) as measured using image analysis on leaves from six plots in Dongshi District, Taichung County, Taiwan. Plots were sampled from September to November 2018. The results show the relationships between the mean squared error (MSE), or variance, or bias of the mean disease severity estimates and sample size (n) (from 10 to 100 in 5 increments) for each of the different scales used, at the six mean disease severities of 8%, 13%, 20%, 32%, 41%, and 49% with corresponding variances of 0.0091%, 0.0146%, 0.0146%, 0.0192%, 0.0287%, and 0.0231%, respectively. Assessment methods: (i) NPE (nearest percentage estimate); (ii) HB (Horsfall-Barratt scale); (iii) EI10 (equal interval 10% linear scale); (iv) AM10 (10% linear scale emphasizing severities ≤50% disease, and with additional intervals at severities <10%); (v) AM20: Similar to the structure of AM10 (both scales have the same structure at ≤10%), this scale has wider ordinal scale intervals for disease severities 10 through 50% based on a 20% linear scale; (vi) AM5: Similar to the structure of AM10 (both scales have the same structure at ≤10%), this scale has narrower ordinal scale intervals for disease severities 10 through 50% based on a 5% linear scale; (vii) AM10f: Similar to the structure of AM10, this scale has 10% linear intervals at >10%, but has finer intervals at ≤10% disease.

**Table 1.**The mean square error (MSE) for the seven assessment methods summed from n = 10 to 100 in n = 5 increments for each specific mean severity and variance (large, medium, and small) for the simulated data sets and the estimated pear scab severity data. The sums of the MSEs for all the mean disease severities were calculated (“Overall”), and a ranking (1 to 7) assigned for each assessment method, where 1 = lowest MSE.

Data Set | Assessment Method ^{1} | Mean Disease Severities | Overall | Rank | |||||

0.01 | 0.05 | 0.10 | 0.20 | 0.30 | 0.40 | ||||

Simulation study (large variance) | NPE | 8.54 | 22.55 | 78.35 | 239.49 | 437.87 | 451.96 | 1238.75 | 1 |

HB | 42.39 | 35.98 | 100.32 | 250.65 | 429.65 | 438.37 | 1297.36 | 6 | |

EI10 | 388.52 | 105.69 | 111.06 | 241.90 | 427.15 | 442.51 | 1716.84 | 7 | |

AM10 | 12.78 | 26.60 | 83.61 | 255.08 | 434.88 | 445.73 | 1258.68 | 4 | |

AM20 | 15.15 | 43.92 | 101.50 | 261.36 | 432.26 | 439.79 | 1293.98 | 5 | |

AM5 | 12.19 | 23.62 | 81.04 | 254.71 | 435.83 | 447.02 | 1254.41 | 2 | |

AM10f | 12.38 | 25.47 | 83.28 | 255.18 | 435.43 | 446.17 | 1257.90 | 3 | |

Simulation study (medium variance) | NPE | 4.71 | 7.98 | 29.17 | 98.64 | 181.68 | 262.46 | 584.63 | 1 |

HB | 32.69 | 13.50 | 51.56 | 139.14 | 201.28 | 259.75 | 697.92 | 5 | |

EI10 | 354.06 | 30.54 | 31.62 | 100.54 | 184.71 | 265.91 | 967.38 | 7 | |

AM10 | 8.61 | 14.12 | 34.26 | 105.27 | 202.16 | 296.57 | 661.00 | 4 | |

AM20 | 11.40 | 34.03 | 86.93 | 123.69 | 200.86 | 290.04 | 746.96 | 6 | |

AM5 | 7.91 | 10.08 | 29.40 | 104.11 | 202.57 | 297.94 | 652.01 | 2 | |

AM10f | 7.91 | 11.77 | 34.21 | 105.50 | 202.25 | 296.62 | 658.27 | 3 | |

Simulation study (small variance) | NPE | 1.12 | 4.08 | 5.18 | 10.96 | 21.54 | 147.22 | 190.10 | 1 |

HB | 13.31 | 7.73 | 27.22 | 40.17 | 236.39 | 157.86 | 482.68 | 6 | |

EI10 | 306.01 | 9.16 | 15.50 | 15.94 | 25.77 | 151.53 | 523.92 | 7 | |

AM10 | 3.65 | 10.66 | 18.19 | 15.74 | 26.04 | 177.88 | 252.16 | 3 | |

AM20 | 3.84 | 21.35 | 189.88 | 7.44 | 54.30 | 170.53 | 447.35 | 5 | |

AM5 | 3.57 | 7.48 | 6.59 | 12.05 | 22.91 | 180.01 | 232.61 | 2 | |

AM10f | 2.19 | 7.09 | 26.22 | 15.65 | 26.04 | 177.82 | 255.02 | 4 | |

Mean disease severities | |||||||||

0.08 | 0.13 | 0.20 | 0.32 | 0.41 | 0.49 | ||||

Based on estimates of pear scab | NPE | 48.74 | 77.60 | 77.90 | 103.34 | 151.03 | 122.66 | 581.27 | 1 |

HB | 91.44 | 146.95 | 199.68 | 115.86 | 132.38 | 122.26 | 808.58 | 6 | |

EI10 | 111.89 | 89.24 | 70.20 | 116.81 | 160.47 | 129.65 | 678.25 | 2 | |

AM10 | 59.36 | 77.02 | 74.96 | 127.37 | 206.95 | 211.12 | 756.80 | 5 | |

AM20 | 106.09 | 117.79 | 110.30 | 133.63 | 199.47 | 189.06 | 856.32 | 7 | |

AM5 | 49.55 | 76.98 | 77.99 | 123.25 | 208.75 | 201.55 | 738.07 | 3 | |

AM10f | 57.02 | 77.07 | 74.62 | 127.37 | 206.95 | 211.12 | 754.16 | 4 |

^{1}Assessment methods: NPE (nearest percentage estimate); HB (Horsfall-Barratt scale); EI10 (equal interval 10% linear scale); AM10 (10% linear scale emphasizing severities ≤50% disease, and with additional intervals at severities <10%); AM20: Similar to the structure of AM10 (both scales have the same structure at ≤10%), this scale has wider ordinal scale intervals for disease severities 10 through 50% based on a 20% linear scale; AM5: Similar to the structure of AM10 (both scales have the same structure at ≤10%), this scale has narrower ordinal scale intervals for disease severities 10 through 50% based on a 5% linear scale; AM10f: Similar to the structure of AM10, this scale has 10% linear intervals at >10%, but has finer intervals at ≤10% disease.

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**MDPI and ACS Style**

Liu, H.I.; Tsai, J.R.; Chung, W.H.; Bock, C.H.; Chiang, K.S. Effects of Quantitative Ordinal Scale Design on the Accuracy of Estimates of Mean Disease Severity. *Agronomy* **2019**, *9*, 565.
https://doi.org/10.3390/agronomy9090565

**AMA Style**

Liu HI, Tsai JR, Chung WH, Bock CH, Chiang KS. Effects of Quantitative Ordinal Scale Design on the Accuracy of Estimates of Mean Disease Severity. *Agronomy*. 2019; 9(9):565.
https://doi.org/10.3390/agronomy9090565

**Chicago/Turabian Style**

Liu, Hung I., Jia Ren Tsai, Wen Hsin Chung, Clive H. Bock, and Kuo Szu Chiang. 2019. "Effects of Quantitative Ordinal Scale Design on the Accuracy of Estimates of Mean Disease Severity" *Agronomy* 9, no. 9: 565.
https://doi.org/10.3390/agronomy9090565