# Poisson Parameters of Antimicrobial Activity: A Quantitative Structure-Activity Approach

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Probability Distribution Analysis

#### 2.2. QSAR Models

_{K–S}= 0.7948; A–D = 0.3874; Crit

_{A–D5%}= 2.5018 (critical values associated for Anderson-Darling test); C–S

_{df = 2}= 0.9403; p

_{C–S}= 0.6249).

_{critical–5%}= 2.7338). After natural logarithm transformation of the Poisson parameters, seen as an overall antimicrobial activity of investigated compounds, no other outlier was identified (the highest Z value was of 2.528; Z

_{critical–5%}= 2.758) and the normality hypothesis of the ln(λ) values could not be rejected (p > 0.05). Further testing on ln(λ) under the normal distribution assumption gave no reason to reject the normality of the data in the training test (K–S = 0.14351, p

_{K–S}= 0.917; A–D = 0.37751, p

_{A–D}= 0.686; C–S = 0.62246, p

_{C–S}= 0.430; F–C–S = 1.307; p

_{F–C–S}= 0.727) nor in test set (K–S = 0.2301, p

_{K–S}= 0.779; A–D = 0.3860, p

_{A–D}= 0.679; F–C–S = 0.637; p

_{F–C–S}= 0.727).

#### 2.2.1. Based on DRAGON Descriptors

_{i–piID}= 0.5643, h

_{i–R3m+}= 0.7602, where piID and R3m+ are Dragon descriptors) for Sulfametrole compound.

^{2}= determination coefficient; TR = training set; loo = leave-one-out analysis; TS = test set; Ext = external set; R

^{2}

_{Adj}= adjusted determination coefficient; F = F-value (from ANOVA table); p = p-value associated to F-value; se = standard error of estimate; Dragon descriptors: piID = conventional bond order ID number-walk and path counts; R3m+ = R maximal autocorrelation of lag 3/weighted by mass GETAWAY descriptors; T = Tolerance; VIF = Variance Inflation Factor; R = correlation coefficient.

#### 2.2.2. Based on SAPF Descriptors

^{2}= determination coefficient; R = correlation coefficient; TR = training set; loo = leave-one-out analysis; TS = test set; R

^{2}

_{Adj}= adjusted determination coefficient; F = F-value (from ANOVA table); p = p-value associated to F-value; se = standard error of estimate; QSMHIMGP and LSSIIETD = SAPF descriptors; T = tolerance; VIP = Variance Inflation Factor. The abilities in estimation (training set) and prediction (test set) of the model from Equation (2) are presented in Figure 5.

#### 2.2.3. Models Comparison

## 3. Discussion

_{F-C-S}value provided a global agreement of 12% for “Is Poisson the distribution of any compound on bacteria and fungi species?”, enough to assure us that the Poisson distribution is the true distribution of compounds’ antimicrobial activities on the studied bacteria and fungi species. The situation is somehow reversed for oils and mixtures; if the Poisson distribution is the only one not rejected for compounds, then the Negative Binomial distribution also cannot be rejected for oils and mixtures. A deeper investigation on factors influencing antimicrobial activities may reveal that the negative binomial distribution should be rejected for the whole data presented in Table 4. The reason for this fact should be foundd in the distribution of the compounds series activities on a given bacteria (columns data in Table 4).

- Compounds series:
- ○ Without any exception, the antimicrobial effects of all investigated compounds proved to follow Poisson distribution. Moreover, the hypothesis that any compound has a Poisson distribution of antimicrobial activity on bacteria population could not be rejected by F-C-S statistics (F-C-S statistics = 28.79, p = 0.12, Figure 1). Starting with this result, the Poisson λ parameter has been obtained to reflect what happen in the population, this parameter being an estimate for both central tendency and variability of antibacterial effects. The analysis of the obtained Poisson parameters showed to follow more likely a log-normal distribution and a logarithm transformation was applied on these values before quantitative structure-activity relationship search. This transformation was applied to avoid the presence of outliers and to assure the normality assumption needed for linear regression analysis [35,36].
- ○ Negative binomial distribution was rejected by 55% of compounds while Binomial distribution was rejected in 70% of cases. Negative binomial distribution, also known as the Pascal distribution or Pólya distribution, is a twin of Poisson distribution [37,38] widely used in analysis of count data [39,40]. The negative binomial distribution could be obtained by superposition of a continuous distribution over Poisson distribution (Fisher showed the convolution between Chi-Square and Poisson distribution [41]). Other authors showed that the negative binomial distribution might derive from a convolution between the Gamma distribution (Chi-Square distribution is a particular case of Gamma distribution) and Poisson distribution [42,43]. Whenever the separation of factors is possible, it is also possible to separate the convolutions of distributions [44], and this separation give the possibility to analyze separately the factors. The results presented by Jäntschi et al. [44] sustained and/or are sustained by convolution of Poisson distribution with a continuous distribution in regards of both factors (bacteria and chemical compounds) in the expression of antimicrobial activity. The results showed that antimicrobial activity follow a negative binomial distribution under the influence of both factors (bacteria and chemical compound) and Poison distribution under the influence of the bacteria factor [44]. Furthermore, the negative binomial distribution might be obtained by convolution of log-normal with Gamma distribution; although a high number of observations are needed (n > 250) in order to statistically assure the difference between Log-normal and Gamma distributions [45].

- Oils and mixture series:
- ○ Negative Binomial distribution cannot be rejected for oils. Moreover, Negative Binomial distribution for oils had a higher likelihood than Poisson distribution (p
_{F-C-S}for Negative Binomial: 0.56; p_{F-C-S}for Poisson: 0.23) while the Binomial distribution was rejected. - ○ Negative Binomial distribution cannot be rejected for mixtures either. Moreover, Negative Binomial distribution for mixtures had also higher likelihood than Poisson distribution (p
_{F-C-S}for Negative Binomial = 0.66; p_{F-C-S}for Poisson = 0.44) while the Binomial distribution was rejected. - ○ The above-presented facts suggest that in the case of oils and mixtures, the factors of the antibacterial activity are not completely separated when oil/mixture name are taken as factor; this appears to be because the Negative Binomial distribution often occurs when a convolution/superposition of Poisson distributions characterize the observed data [46].

- One compound proved to be influential in the model (CID = 64939, Figure 2). This compound obtained the value of leverage for both Dragon descriptors higher than the accepted threshold (0.41). This compound, which belongs to the training set, was withdrawn, and a model based on 12 compounds in training set was obtained, Equation(1).
- Two descriptors were able to describe the linear relation between overall antimicrobial activities of investigated compounds. One descriptor belongs to the walk and path counts and relates the conventional bond order ID number while the second descriptor relates the maximal autocorrelation of lag 3 divided by mass (R3m+). According with associated coefficients, the R3m+ had a higher contribution in the model compared with piID descriptor, but its contribution is to the significance level threshold (5.8% compared to imposed 5% significance level).
- QSAR-Dragon model proved to be statistically significant (F = 39, p = 3.62 × 10
^{−5}). A low value of root mean square error was obtained in leave-one-out analysis (0.1276). The contribution of R3m+ descriptor to the model is questionable since the significance associated to its coefficient is very close to 0.05 but since it has a real contribution in the r^{2}value its significance of 5.8% was accepted. Moreover, the R3m+ proved not significantly correlate with Poisson parameter (r = −0.2410). - Multicollianearity is not present in the model since the tolerance value 0.1 < T < 1 and the variance inflation factors (VIF) < 10 even if a significant correlation coefficient was obtained between Dragon descriptors.
- The model proved its abilities in estimation (R
^{2}_{TR}= 0.897) as well as in prediction (internal validity of the model in leave-one-out analysis, R^{2}_{loo}= 0.845 and external validation in test set R^{2}_{TS}= 0.652) with a difference in the goodness-of-fit from 0.052 (training vs. interval validation - leave one out analysis) to 0.245 (training vs. external validation-test set). However, the difference of 0.245 proved not statistically significant (p > 0.05). - Unfortunately, external abilities in prediction were away from the expected abilities. The trend is significant far from the expected line-Figure 3.
- The abilities in estimation (training set) proved not statistically significant from the abilities in prediction (test set) since a probability of 0.3598 was obtained in comparison.

- The values of SAPF descriptors associated to compounds proved that no compound had significant influence on the model (all leverage values where lower than threshold −0.41, Figure 4).
- SAPF model proved statistically significant (F = 24, p = 1.48 × 10
^{−4}). The contribution of both descriptors to the model proved statistically significant (p-values associated to coefficients <0.05). - According to descriptors from Equation(2), the global model of antibacterial activity is related to both molecular geometry and topology: one descriptor identified a relation between the geometry of compounds and the overall antimicrobial activity while the second descriptor identified a relation with compounds’ topology. Moreover, the atomic mass and electronegativity proved to be related to the overall antimicrobial activity by the same split ratio in the expression of the model descriptors.
- Multicollianearity was not identified in the QSAR-SAPF model, even if a statistically significant correlation coefficient between descriptors exists (the tolerance values were higher than 0.1 and smaller than 1 and the variance inflation factors (VIF) had values smaller than 10).
- The model proved its abilities in estimation (R
^{2}_{TR}= 0.829) as well as in prediction (internal validity of the model in leave-one-out analysis, R^{2}_{loo}= 0.700 and external validation in test set R^{2}_{TS}= 0.862) with a difference in the goodness-of-fit from −0.034 (training vs. external validation - test set) to 0.129 (training vs. interval validation-leave one out analysis). Moreover, none of these differences were statistically significant (p > 0.05). - External abilities in prediction proved to be close to expected abilities for QSAR-SAPF model (Figure 5).

- Dragon model has slightly better abilities in estimation compared to SAPF model, but these abilities proved not statistically significant. The determination coefficient obtained both in training set and in leave-one-out analysis was higher compared to SAPF model with 0.068 and respectively 0.145. Moreover, the abilities of prediction seem to be better for SAPF model compared to Dragon model (a difference of 0.211, not statistically significant p < 0.05). This observation is also sustained by the lowest value of residuals in training set for Dragon model and in two compounds from training set and all compounds from test set for SAPF model (Table 2).
- The SAPF model systematically obtained smallest values of parameters presented in Table 3: best explaining the variability in the observation; smallest typical errors; smallest standard error of prediction as well as smallest relative error of prediction. The highest difference is observed with regards to standard error of prediction that is almost 4 times higher for Dragon model compared to SAPF model.
- The analysis of predictive power of the models demonstrated that SAPF model had significantly higher power of prediction (Table 3). According to the obtained results, the Q
^{2}values for Dragon model are smaller than 0.6, being considered unacceptable while all Q^{2}values for SAPF model are higher than 0.77. These results show that the Dragon model can be rejected from a statistical point of view, taking also into consideration that the relative error of prediction is almost 2 times higher compared to SAPF model. - Furthermore, the mean of residuals for training, external and external + test set proved not statistically different by zero when the SAPF model was analyzed. The Fisher’s predictive power identified statistically difference by zero of the residuals obtained by Dragon model in both training and test sets (9 compounds) (p < 0.05, Table 3).
- The model with a higher concordance between observed and estimated/predicted could be considered the best model. The analysis of concordance correlation coefficient revealed a substantial strength of agreement for training set but a very poor agreement in test set for Dragon model. A moderate strength of agreement was obtained by SAPF model in both training and test sets (Table 3).
- Steiger’s test was not able to identify any statistically significant differences between Dragon and SAPF model regarding goodness-of-fit neither in training set nor in external set.

## 4. Experimental Section

#### 4.1. Compounds, Oils and Mixtures

#### 4.2. Distribution Analysis

#### 4.3. Molecular Descriptors Calculation

#### 4.4. Identification and Characterization of Linear Regression Models

^{th}observation and the means of the descriptor-values for all observations was computed to identify the leverage in descriptors (leverage value, h

_{i}). Whenever h

_{i}> 3·(k + 1)/n (where k = number of independent variables in the model, n = sample size) compound was considered influential in the model [60] and was excluded from further analysis of the model. The response outliers were defined as compounds with absolute standardized residuals higher than 2.5. Leverage values (h

_{i}) vs. standardized residuals for compounds in training set was plotted to identify response outliers as well as independent variables with leverage values higher than threshold value (see Figures 2 and 3).

## 5. Conclusions

## Supplementary Information

ijms-13-05207-s001.pdf## Acknowledgments

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**Figure 1.**Results of probability distribution functions analysis. X: Compounds (

**1**–

**21; 1**= Citral,

**2**= Geraniol,

**3**= Geranyl formate,

**4**= Geranyl acetate,

**5**= Geranyl butyrate,

**6**= Geranyl tiglate,

**7**= Neral,

**8**= Nerol,

**9**= Nerol acetate,

**10**= Neryl butyrate,

**11**= Neryl propanoate,

**12**= Citronellal,

**13**= Citronellyl formate,

**14**= Citronellyl acetate,

**15**= Citronellyl butyrate,

**16**= Citronellyl isobutyrate,

**17**= Citronellyl propionate,

**18**= Hydroxycitronellal,

**19**= Rose oxide,

**20**= Eugenol,

**21**= Sulfametrole,

**32**= Citronellol), Oils (

**22**–

**29; 22**= Citronella,

**23**= Geranium Africa,

**24**= Geranium Bourbon,

**25**= Geranium China,

**26**= Helichrysum,

**27**= Palmarosa,

**28**= Rose,

**29**= Verbena), Mixtures (

**30**–

**31; 30**= Tetracycline hydrochloride,

**31**= Ciproxin); Y: Binomial (◆), NegBino (■), Poisson (▴); “Is Y the distribution of any X on bacteria and fungi species?”.

**Figure 3.**Observed vs. calculated parameter: QSAR-Dragon (Equation (1)R

^{2}

_{TS}= determination coefficient in test set).

**Figure 5.**Observed vs. calculated parameter: QSAR-SAPF (Equation (2)R

^{2}

_{TS}= determination coefficient in test set).

**Figure 6.**SAPF descriptors (v = value, ln = natural logarithm, V = vector, T = topology, G = geometry, x, y, z = geometric atomic coordinates, i = atom, refD = modality to calculate coordinates—from average, refP = modality to calculate coordinates—from property center formula, t = topological atomic coordinate.

λ | Mode | Mean | Var | StDev | Skew | EKurt | Median | |
---|---|---|---|---|---|---|---|---|

Compound (CID) | ||||||||

Citral (638011) | 14.125 | 14 | 14.125 | 14.125 | 3.758 | 0.266 | 0.071 | 13.457 |

Geraniol (637566) | 13.750 | 13 | 13.750 | 13.750 | 3.708 | 0.270 | 0.073 | 13.082 |

Geranyl formate (5282109) | 8.875 | 8 | 8.875 | 8.875 | 2.979 | 0.336 | 0.113 | 8.207 |

Geranyl acetate (1549026) | 8.200 | 8 | 8.200 | 8.200 | 2.864 | 0.349 | 0.122 | 7.531 |

Geranyl butyrate (5355856) | 8.714 | 8 | 8.714 | 8.714 | 2.952 | 0.339 | 0.115 | 8.046 |

Geranyl tiglate (5367785) | 11.625 | 11 | 11.625 | 11.625 | 3.410 | 0.293 | 0.086 | 10.957 |

Neral (643779) | 13.500 | 13 | 13.500 | 13.500 | 3.674 | 0.272 | 0.074 | 12.932 |

Nerol (643820) | 11.250 | 11 | 11.250 | 11.250 | 3.354 | 0.298 | 0.089 | 10.582 |

Nerol acetate (1549025) | 7.333 | 7 | 7.333 | 7.333 | 2.708 | 0.369 | 0.136 | 6.664 |

Neryl butyrate (5352162) | 10.714 | 10 | 10.714 | 10.714 | 3.273 | 0.306 | 0.093 | 10.046 |

Neryl propanoate (5365982) | 10.714 | 10 | 10.714 | 10.714 | 3.273 | 0.306 | 0.093 | 10.046 |

Citronellal (7794) | 14.600 | 14 | 14.600 | 14.600 | 3.821 | 0.262 | 0.068 | 13.932 |

Citronellyl formate (7778) | 12.143 | 12 | 12.143 | 12.143 | 3.485 | 0.287 | 0.082 | 11.475 |

Citronellyl acetate (9017) | 7.286 | 7 | 7.286 | 7.286 | 2.699 | 0.370 | 0.137 | 6.617 |

Citronellyl butyrate (8835) | 8.167 | 8 | 8.167 | 8.167 | 2.858 | 0.350 | 0.122 | 7.498 |

Citronellyl isobutyrate (60985) | 8.200 | 8 | 8.200 | 8.200 | 2.864 | 0.349 | 0.122 | 7.531 |

Citronellyl propionate (8834) | 14.333 | 14 | 14.333 | 14.333 | 3.786 | 0.264 | 0.070 | 13.665 |

Hydroxycitronellal (7888) | 18.750 | 18 | 18.750 | 18.750 | 4.330 | 0.231 | 0.053 | 18.083 |

Rose oxide (27866) | 12.800 | 12 | 12.800 | 12.800 | 3.578 | 0.280 | 0.078 | 12.132 |

Eugenol (3314) | 28.250 | 28 | 28.250 | 28.250 | 5.315 | 0.188 | 0.035 | 27.583 |

Sulfametrole (64939) | 19.200 | 19 | 19.200 | 19.200 | 4.382 | 0.228 | 0.052 | 18.533 |

Oil | ||||||||

Citronella | 9.750 | 9 | 9.750 | 9.750 | 3.122 | 0.320 | 0.103 | 9.082 |

Geranium Africa | 13.250 | 13 | 13.250 | 13.250 | 3.640 | 0.275 | 0.075 | 12.582 |

Geranium Bourbon | 12.500 | 12 | 12.500 | 12.500 | 3.536 | 0.283 | 0.080 | 11.832 |

Geranium China | 13.625 | 13 | 13.625 | 13.625 | 3.691 | 0.271 | 0.073 | 12.957 |

Helichrysum | 10.667 | 10 | 10.667 | 10.667 | 3.266 | 0.306 | 0.094 | 9.999 |

Palmarosa | 11.625 | 11 | 11.625 | 11.625 | 3.410 | 0.293 | 0.086 | 10.957 |

Rose | 12.750 | 12 | 12.750 | 12.750 | 3.571 | 0.280 | 0.078 | 12.082 |

Verbena | 16.500 | 16 | 16.500 | 16.500 | 4.062 | 0.246 | 0.061 | 15.833 |

Mixture | ||||||||

Tetracycline hydrochloride | 15.143 | 15 | 15.143 | 15.143 | 3.891 | 0.257 | 0.066 | 14.476 |

Ciproxin | 26.000 | 26 | 26.000 | 26.000 | 5.099 | 0.196 | 0.038 | 25.333 |

Set | CID | Y | Ŷ_{Dragon} | Res_{Dragon} | Ŷ_{SAPF} | Res_{SAPF} |
---|---|---|---|---|---|---|

Training | 1549025 | 1.9924 | 2.0070 | −0.0146 | 2.0761 | −0.0836 |

Training | 8835 | 2.1001 | 2.0564 | 0.0437 | 2.1461 | −0.0460 |

Training | 60985 | 2.1041 | 2.0768 | 0.0273 | 2.0553 | 0.0488 |

Training | 5282109 | 2.1832 | 2.2596 | −0.0764 | 2.3267 | −0.1435 |

Training | 643820 | 2.4204 | 2.6106 | −0.1902 | 2.7127 | −0.2923 |

Training | 7778 | 2.4968 | 2.4132 | 0.0835 | 2.2816 | 0.2151 |

Training | 27866 | 2.5494 | 2.5905 | −0.0411 | 2.4957 | 0.0538 |

Training | 637566 | 2.6210 | 2.6106 | 0.0104 | 2.7127 | −0.0917 |

Training | 638011 | 2.6479 | 2.7061 | −0.0582 | 2.6042 | 0.0437 |

Training | 8842 | 2.6741 | 2.6435 | 0.0307 | 2.5713 | 0.1029 |

Training | 7794 | 2.6810 | 2.6929 | −0.0118 | 2.6430 | 0.0380 |

Training | 7888 | 2.9312 | 2.7346 | 0.1966 | 2.8638 | 0.0674 |

Training | 64939 | 2.9549 | 2.8674 | 0.0875 | ||

Test | 1549026 | 2.1041 | 2.0070 | 0.0971 | 2.2012 | −0.0971 |

Test | 5355856 | 2.1650 | 1.9271 | 0.2379 | 2.2830 | −0.1180 |

Test | 5352162 | 2.3716 | 1.9271 | 0.4445 | 2.7847 | −0.4132 |

Test | 5367785 | 2.4532 | 1.8661 | 0.5870 | 2.4642 | −0.0111 |

Test | 643779 | 2.6027 | 2.7061 | −0.1034 | 2.6006 | 0.0021 |

Test | 8834 | 2.6626 | 2.4108 | 0.2518 | 2.6207 | 0.0418 |

Test | 3314 | 3.3411 | 2.7843 | 0.5568 | 3.3685 | −0.0274 |

External | 9017 | 1.9859 | 2.1432 | −0.1572 | 2.0053 | −0.0194 |

External | 5365982 | 2.3716 | 2.2688 | 0.1028 | 2.2889 | 0.0827 |

Parameter (Abbreviation) | Dragon–Equation(1)–n = 21 | SAPF–Equation(2)–n = 22 | ||||
---|---|---|---|---|---|---|

Root-mean-square error (RMSE) | 0.2314 | 0.1357 | ||||

Mean absolute error (MAE) | 0.1582 | 0.0967 | ||||

Mean Absolute Percentage Error (MAPE) | 0.0628 | 0.0403 | ||||

Standard error of prediction (SEP) | 0.2371 | 0.0628 | ||||

Relative error of prediction (REP%) | 9.2964 | 5.4523 | ||||

Predictive Power of the Model | ||||||

Q^{2}_{F1} | 0.2121 * | 0.8436 * | ||||

Q^{2}_{F2} | 0.2041 * | 0.8421 * | ||||

Q^{2}_{F3} | n.a. | 0.7742 * | ||||

ρ_{c-TR} | 0.9457 a | 0.9063 c | ||||

ρ_{c-TS} | 0.4885 b | 0.9219 d | ||||

Fisher’s Predictive Power | TS | EX e | TS + EX f | TS | EX | TS + EX |

n | 7 | 2 | 9 | 7 | 2 | 9 |

t-value | 3.1148 | −0.2095 | 2.5071 | −1.5344 | 0.6198 | −1.2830 |

p-value | 0.0104 | 0.4343 | 0.0230 | 0.0879 | 0.3234 | 0.1234 |

^{*}= test set include also external compounds; ρ

_{c}= concordance correlation coefficient; TR = training set; TS = test set;

^{a}accuracy = 0.9985, precision = 0.9471;

^{b}accuracy = 0.7357, precision = 0.6639;s

^{c}accuracy = 0.9956, precision = 0.9103;

^{d}accuracy = 0.9867, precision = 0.9344;

^{e}= external set (two compounds);

^{f}= training and external sets.

SA | EF | EC | PV | PA | Ss | KP | CA | n | ||
---|---|---|---|---|---|---|---|---|---|---|

Compound (CID) | ||||||||||

1 | Citral (638011) | 15 | 23 | 11 | 9 | 10 | 8 | 9 | 28 | 8 |

2 | Geraniol (637566) | 15 | 12 | 15 | 12 | 11 | 10 | 10 | 25 | 8 |

3 | Geranyl formate (5282109) | 10 | 9 | 7 | 8 | 8 | 7 | 7 | 15 | 8 |

4 | Geranyl acetate (1549026) | 10 | 8 | 7 | NIO | NIO | 7 | NIO | 9 | 5 |

5 | Geranyl butyrate (5355856) | 10 | 11 | 7 | NIO | 9 | 7 | 7 | 10 | 7 |

6 | Geranyl tiglate (5367785) | 17 | 10 | 11 | 9 | 8 | 8 | 15 | 15 | 8 |

7 | Neral (643779) | 15 | 20 | 10 | 6 | 12 | 10 | 10 | 25 | 8 |

8 | Nerol (643820) | 11 | 8 | 10 | 10 | 10 | 7 | 7 | 27 | 8 |

9 | Nerol acetate (1549025) | 8 | NIO | 7 | 7 | 7 | 8 | 7 | NIO | 6 |

10 | Neryl butyrate (5352162) | 25 | 8 | 8 | 8 | NIO | 8 | 8 | 10 | 7 |

11 | Neryl propanoate (5365982) | 17 | 10 | NIO | 7 | 8 | 9 | 10 | 14 | 7 |

12 | Citronellal (7794) | 25 | 18 | NIO | 9 | NIO | 7 | 14 | NIO | 5 |

13 | Citronellyl formate (7778) | 18 | 20 | 10 | 8 | 9 | 7 | NIO | 13 | 7 |

14 | Citronellyl acetate (9017) | 10 | 6 | NIO | 6 | 7 | 6 | 7 | 9 | 7 |

15 | Citronellyl butyrate (8835) | 8 | 8 | NIO | NIO | 8 | 7 | 8 | 10 | 6 |

16 | Citronellyl isobutyrate (60985) | 8 | 10 | 9 | 7 | NIO | NIO | 7 | NIO | 5 |

17 | Citronellyl propionate (8834) | 15 | 20 | NIO | NIO | 10 | 15 | 11 | 15 | 6 |

18 | Hydroxycitronellal (7888) | 20 | 20 | 23 | 16 | 17 | 15 | 14 | 25 | 8 |

19 | Rose oxide (27866) | 8 | 10 | NIO | 11 | 7 | NIO | NIO | 28 | 5 |

20 | Eugenol (3314) | 30 | 30 | 28 | 28 | 25 | 25 | 28 | 32 | 8 |

21 | Sulfametrole (64939) | 27 | 27 | 11 | 23 | NIO | 8 | NIO | NIO | 5 |

32 | Citronellol (8842) | 25 | 18 | NIO | 8 | NIO | 7 | NIO | NIO | 4 |

Oil | ||||||||||

22 | Citronella | 10 | 10 | 7 | 10 | 7 | 7 | 7 | 20 | 8 |

23 | Geranium Africa | 16 | 12 | 10 | 10 | 10 | 9 | 11 | 28 | 8 |

24 | Geranium Bourbon | 13 | 12 | 8 | 12 | 10 | 10 | 10 | 25 | 8 |

25 | Geranium China | 20 | 13 | 14 | 9 | 9 | 9 | 10 | 25 | 8 |

26 | Helichrysum | 20 | 13 | 8 | NIO | 9 | NIO | 7 | 7 | 6 |

27 | Palmarosa | 8 | 13 | 12 | 9 | 11 | 10 | 10 | 20 | 8 |

28 | Rose | 20 | 15 | 10 | 10 | 8 | 9 | 10 | 20 | 8 |

29 | Verbena | 27 | 25 | 10 | 13 | 10 | 12 | 10 | 25 | 8 |

Mixture | ||||||||||

30 | Tetracycline hydrochloride | 15 | 22 | 11 | 13 | 15 | 10 | 20 | NIO | 7 |

31 | Ciproxin | 35 | 33 | 22 | 25 | 32 | 10 | 25 | NIO | 7 |

Parameter (Abbreviation) | Formula [ref] | Remarks |
---|---|---|

Root-mean-square error (RMSE) | $\text{RMSE}=\sqrt{\frac{{\sum}_{\text{i}=1}^{\text{n}}{({\text{y}}_{\text{i}}-{\widehat{\text{y}}}_{\text{i}})}^{2}}{\text{n}}}$ | RMSE > MAE → variation in the errors exist |

Mean absolute error (MAE) | $\text{MAE}=\frac{{\sum}_{\text{i}=1}^{\text{n}}\mid {\text{y}}_{\text{i}}-{\widehat{\text{y}}}_{\text{i}}\mid}{\text{n}}$ | |

Mean Absolute Percentage Error (MAPE) n | $\text{MAPE}=\frac{{\sum}_{\text{i}=1}^{\text{n}}\mid ({\text{y}}_{\text{i}}-{\widehat{\text{y}}}_{\text{i}})/{\text{y}}_{\text{i}}\mid}{\text{n}}$ | MAPE ~ 0 → perfect fit |

Standard error of prediction (SEP) | $\text{SEP}=\sqrt{\frac{{\sum}_{\text{i}=1}^{\text{n}}{({\widehat{\text{y}}}_{\text{i}}-{\text{y}}_{\text{i}})}^{2}}{\text{n}-1}}$ | Lower value indicate a good model |

Relative error of prediction (REP%) | $\text{REP(}\%\text{)}=\frac{100}{\overline{\text{y}}}\sqrt{\frac{{\sum}_{\text{i}=1}^{\text{n}}{({\widehat{\text{y}}}_{\text{i}}-{\text{y}}_{\text{i}})}^{2}}{\text{n}}}$ | Lower value indicate a good model |

Concordance analysis (ρ_{c}) | ${\rho}_{\text{c}}=\frac{2{\sum}_{\text{i}=1}^{\text{n}}\left({\text{y}}_{\text{i}}-\overline{\text{y}}\right)\left({\widehat{\text{y}}}_{\text{i}}-\overline{\widehat{\text{y}}}\right)}{{\sum}_{\text{i}=1}^{\text{n}}{\left({\text{y}}_{\text{i}}-\overline{\text{y}}\right)}^{2}+{\sum}_{\text{i}=1}^{\text{n}}{\left({\widehat{\text{y}}}_{\text{i}}-\overline{\widehat{\text{y}}}\right)}^{2}+\text{n}{\left(\overline{\text{y}}-\overline{\widehat{\text{y}}}\right)}^{2}}$ [61] | Strength of agreement [62]: >0.99 almost perfect; (0.95; 0.99) substantial; (0.90; 0.95) moderate; <0.90 poor |

Predictive Power of the Model Prediction is considered accurate if the predictive power of the model is > 0.6 [66] | ${\text{Q}}_{{\text{F}}_{1}}^{2}=1-\frac{{\sum}_{\text{i}=1}^{{\text{n}}_{\text{TS}}}{({\widehat{\text{y}}}_{\text{i}}-{\text{y}}_{\text{i}})}^{2}}{{\sum}_{\text{i}=1}^{{\text{n}}_{\text{TS}}}{({\text{y}}_{\text{i}}-{\overline{\text{y}}}_{\text{TR}})}^{2}}$ [63] | Prediction power relative to mean value of observable in training set |

${\text{Q}}_{{\text{F}}_{2}}^{2}=1-\frac{{\sum}_{\text{i}=1}^{{\text{n}}_{\text{TS}}}{({\widehat{\text{y}}}_{\text{i}}-{\text{y}}_{\text{i}})}^{2}}{{\sum}_{\text{i}=1}^{{\text{n}}_{\text{TS}}}{({\text{y}}_{\text{i}}-{\overline{\text{y}}}_{\text{TS}})}^{2}}$ [64] | Prediction power relative to mean value of observable in test set | |

${\text{Q}}_{\text{F}3}^{2}=1-\frac{\left[{\sum}_{\text{i}=1}^{{\text{n}}_{\text{TS}}}{({\widehat{\text{y}}}_{\text{i}}-{\text{y}}_{\text{i}})}^{2}\right]/{\text{n}}_{\text{TS}}}{\left[{\sum}_{\text{i}=1}^{{\text{n}}_{\text{TS}}}{({\text{y}}_{\text{i}}-{\overline{\text{y}}}_{\text{TR}})}^{2}\right]/{\text{n}}_{\text{TR}}}$ [65] | Overall prediction weighted by test set sample size relative to observable weighted by mean of observed value in training set weighted by sample size in training set | |

Predictive Power: Fisher’s approach | $\begin{array}{l}\text{t}=\frac{{\overline{\text{res}}}_{\text{TS}}-0}{\text{StDev}({\text{res}}_{\text{TS}})/\sqrt{{\text{n}}_{\text{TS}}}}\\ \text{p}=\text{TDIST}(\text{abs}(\text{t}),\hspace{0.17em}{\text{n}}_{\text{TS}}-1,1)\end{array}$ [67] | Evaluate if the mean of residual is statistically different by the expected value (0) |

_{i}= observed ln(λ) for i

^{th}compound; ŷ

_{i}= estimated/predicted ln(λ) by model from Equation(1), respectively Equation(2); n = sample size; ȳ = arithmetic mean of the observed ln(λ); $\overline{\widehat{\text{y}}}$ = arithmetic mean of estimated/predicted ln(λ); ρ

_{c}= concordance correlation coefficient; TR = training set; TS = test set; $\overline{\text{res}}$ = arithmetic mean of residuals; res = residuals; StDev = standard deviation; abs = absolute value.

© 2012 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Sestraş, R.E.; Jäntschi, L.; Bolboacă, S.D.
Poisson Parameters of Antimicrobial Activity: A Quantitative Structure-Activity Approach. *Int. J. Mol. Sci.* **2012**, *13*, 5207-5229.
https://doi.org/10.3390/ijms13045207

**AMA Style**

Sestraş RE, Jäntschi L, Bolboacă SD.
Poisson Parameters of Antimicrobial Activity: A Quantitative Structure-Activity Approach. *International Journal of Molecular Sciences*. 2012; 13(4):5207-5229.
https://doi.org/10.3390/ijms13045207

**Chicago/Turabian Style**

Sestraş, Radu E., Lorentz Jäntschi, and Sorana D. Bolboacă.
2012. "Poisson Parameters of Antimicrobial Activity: A Quantitative Structure-Activity Approach" *International Journal of Molecular Sciences* 13, no. 4: 5207-5229.
https://doi.org/10.3390/ijms13045207