# Statistical Assessment of Solvent Mixture Models Used for Separation of Biological Active Compounds

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## Introduction

## Material and Methods

#### Experimental Measurements

^{th}experiment was a result of an optimization method applied on an objective function that suggests the composition of the optimum mobile phase as 55:19:26 (trichloromethane:propanone:petroleum ether).

_{1},x

_{2},x

_{3}) = a

_{1}x

_{1}+ a

_{2}x

_{2}+ a

_{3}x

_{3}+ a

_{4}x

_{1}x

_{2}+ a

_{5}x

_{1}x

_{3}+ a

_{6}x

_{2}x

_{3}+ a

_{7}x

_{1}x

_{2}x

_{3}

_{1},x

_{2},x

_{3}) = a

_{1}x

_{1}+ a

_{2}x

_{2}+ a

_{3}x

_{3}+ a

_{4}x

_{1}x

_{2}+ a

_{5}x

_{1}x

_{3}+ a

_{6}x

_{2}x

_{3}

_{1}, x

_{2}, and x

_{3}= molar fractions of the three solvents (where x

_{1}+ x

_{2}+ x

_{3}= 1); a

_{1}, a

_{2}, a

_{3}, a

_{4}, a

_{5}, a

_{6}, and a

_{7}= model coefficients; first determined based on the best estimation of the selected chromatographic parameter (using 7 experiments for M7; 6 experiments for M6) and then used to predict used chromatographic parameter for any composition of the mobile phase (used for not included in model experiments).

_{254}(Merck) (Table 1).

No. | TCM:Prop:PE^{*} | L | l_{1} | w_{1} | l_{2} | w_{2} | l_{3} | w_{3} | l_{4} | w_{4} | l_{5} | w_{5} |

1 | 33:33:33 | 8.70 | 6.65 | 0.48 | 7.36 | 0.35 | 7.26 | 0.23 | 4.00 | 0.38 | 4.76 | 0.98 |

2 | 0:0:100 | 8.83 | 0.00 | 0.42 | 0.00 | 0.44 | 0.00 | 0.22 | 0.00 | 0.25 | 0.00 | 0.21 |

3 | 0:100:0 | 8.75 | 8.29 | 0.37 | 8.49 | 0.26 | 8.49 | 0.11 | 7.93 | 0.28 | 7.79 | 0.59 |

4 | 100:0:0 | 9.00 | 1.21 | 0.62 | 2.05 | 0.45 | 1.43 | 0.41 | 0.05 | 0.23 | 0.19 | 0.30 |

5 | 50:0:50 | 8.93 | 0.54 | 0.56 | 0.98 | 0.38 | 0.68 | 0.27 | 0.00 | 0.26 | 0.00 | 0.25 |

6 | 50:50:0 | 8.84 | 6.71 | 0.55 | 7.12 | 0.31 | 7.05 | 0.20 | 5.31 | 0.36 | 5.56 | 0.69 |

7 | 0:50:50 | 8.76 | 8.44 | 0.36 | 8.56 | 0.11 | 8.56 | 0.05 | 7.35 | 0.31 | 7.20 | 1.38 |

8 | 10:10:80 | 8.86 | 3.49 | 0.60 | 4.71 | 0.42 | 4.51 | 0.28 | 0.53 | 0.27 | 0.64 | 1.41 |

9 | 80:10:10 | 8.87 | 5.08 | 0.69 | 6.71 | 0.51 | 6.06 | 0.34 | 1.01 | 0.32 | 2.32 | 0.63 |

10 | 10:80:10 | 8.82 | 8.24 | 0.52 | 8.41 | 0.24 | 8.46 | 0.14 | 7.38 | 0.32 | 7.27 | 0.96 |

11 | 55:19:26 | 18.95 | 3.43 | 0.82 | 5.86 | 1.16 | 11.52 | 1.43 | 13.44 | 1.25 | 14.38 | 1.32 |

#### Statistical Validation

_{0}).

_{0}) is rejected at a significance level of 5% in the investigation of the response factors (see Table 2). Pearson (r), Spearman (ρ), Kendall (τ-a,b,c) and Gamma (Γ) correlation coefficients [23,24,25] were used in order to identify and to quantify the nature of the link (quantitative, categorical, semi-quantitative, quantitative and categorical) between experimental and estimated values. The correlation approach was choose for analysis of the quality of the models (Eq (1.1), and Eq (1.2), respectively) due to its ability of identification of linear relationship between two variable (in our case the experimental and estimated values by the proposed mathematical models from Eq (1.1) and Eq (1.2), respectively).

Parameter | Formula | Eq. | Notes | |
---|---|---|---|---|

Retardation factors (RF) matrix | RF(i,e) = l(i,e)/l(e) | (2) | i | a separated compound |

e | the mobile phase | |||

l(i,e) | migration distance of i in e | |||

l(e) | migration distance of e | |||

Ordered RF | RFO(i,e) = 2·(l _{o}(i+1,e)-l_{o}(i,e))/l(e) | (3) | l_{o}(i,e) | i^{th} migration coordinate in the list of migration, ordered by length |

Resolution matrix | RSM(i,j,e) = 2·(l(i,e)-l(j,e))/(w(i,e)+w(j,e)) | (4) | j | a separated compound |

w(i,e) | spot width of i | |||

w(j,e) | spot width of j | |||

Resolution of adjacent spots matrix | RSO(i,e) = 2·(l _{o}(i+1,e)-l_{o}(i,e))/(w(i+1,e)+w(i,e)) | (5) | l_{o}(i,e) | i^{th} migration coordinate in the list of migration, ordered by length |

Number of components | nc(e) = Σ _{i} 1 | l_{o}(i+1,e)-l_{o}(i,e)>(w(i+1,e)+w(i,e))/8 | (6) | nc(e) | number of components observed in e at least 1σ (σ = standard deviation) |

Maximum number of components | mnc = max _{e} nc(e) | (7) | mnc | from all experimented mobile phases (or previous knowledge) |

Retardation factors deviation | RFD(e) = √(∑ _{i} (ΔRF(i,e)-1/mnc)^{2}/√nc(e)(nc(e)+1) | (8) | 1/mnc | theoretical difference between two retardation factors |

ΔRF(i,e) | RFO(i+1,e)-RFO(i,e) | |||

Informational energy | IEne(e) = mnc ^{2} - Σ_{i }(n_{i})^{2} | (9) | n_{i} | number of compounds that migrate into i^{th} equidistant interval from mnc intervals |

Informational entropy | IEnt(e) = Σ _{i }(n_{i})log_{2}(n_{i}) | (10) | ||

Resolution sum | RSS(e) = ∑ _{i} RSO(i,e) | (11) | RSS(e) | average indicator for separation |

Effective plates number squared root | QN_{eff}(e) =4·l(e)/(Σ _{i} w(e,i)) | (12) | QN_{eff}(e) | average indicator for a hypothetic quantitative analysis |

Resolution divided by the number of effective plates | RSP(e) = 25·RSS(e)/QN _{eff}(e) | (13) | RSP(e) | composite indicator for separation expressed as proportion; note that 4·RSS(e) → QN_{eff}(e) for an ideal separation |

Average resolution for separation | RSA(e) = RSS(e)/nc(e) | (14) | RSA(e) | average indicator for separation |

Relative resolution product | RRP(e) = Π _{i} RSO(i,e)/ Σ_{i} RSO(i,e) | (15) | RRP(e) | average indicator for separation |

Minkowski type mean of resolutions | RSR(e) = (∑ _{i} (RSO(i,e))^{1/p}/nc(e))^{p}; p = 2 | (16) | RSR(e) | is better descriptor for separation than RSA |

Quality factor | QF(e) = min _{i,j} RSM(i,j,e) = min_{i} RSO(i,e) | (17) | QF(e) | worst one define the resolution of separation |

## Results and Discussion

No. | Experimental | Estimated by Eq(1.1) | Estimated by Eq(1.2) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

AI1 | AI2 | AI3 | AI4 | AI5 | AI1 | AI2 | AI3 | AI4 | AI5 | AI1 | AI2 | AI3 | AI4 | AI5 | |

1 | 0.764 | 0.845 | 0.834 | 0.460 | 0.547 | 0.006 | 0.021 | 0.134 | 0.159 | 0.228 | |||||

2 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | ||||||||||

3 | 0.947 | 0.970 | 0.970 | 0.906 | 0.890 | ||||||||||

4 | 0.134 | 0.228 | 0.159 | 0.006 | 0.021 | ||||||||||

5 | 0.060 | 0.110 | 0.076 | 0.000 | 0.000 | ||||||||||

6 | 0.759 | 0.805 | 0.798 | 0.601 | 0.629 | ||||||||||

7 | 0.963 | 0.977 | 0.977 | 0.839 | 0.822 | ||||||||||

8 | 0.394 | 0.532 | 0.509 | 0.060 | 0.072 | 0.291 | 0.314 | 0.308 | 0.203 | 0.216 | 0.215 | 0.219 | 0.271 | 0.279 | 0.284 |

9 | 0.573 | 0.756 | 0.683 | 0.114 | 0.262 | 0.309 | 0.393 | 0.347 | 0.139 | 0.174 | 0.157 | 0.172 | 0.289 | 0.317 | 0.364 |

10 | 0.934 | 0.954 | 0.959 | 0.837 | 0.824 | 1.017 | 1.052 | 1.053 | 0.878 | 0.891 | 0.882 | 0.903 | 0.997 | 1.024 | 1.022 |

11 | 0.181 | 0.309 | 0.608 | 0.709 | 0.759 | 0.505 | 0.591 | 0.565 | 0.253 | 0.324 | 0.309 | 0.323 | 0.438 | 0.465 | 0.491 |

^{-2}) revealed that the models from Eq (1.1) and Eq (1.2) have good abilities in estimation of chromatographic retardation factor.

Name | Correlation coefficient | p-value | Statistical parameter |
---|---|---|---|

Eq(1.1), n = 20 | |||

Pearson | r = 0.7214 | 3.31·10^{-4} | t_{Prs,1} = 4.42 |

Spearman | ρ = 0.7789 | 5.19·10^{-5} | t_{Spm,1} = 5.27 |

Semi-Q | r_{sQ} = 0.7496 | 1.42·10^{-4} | t_{sQ} = 4.80 |

Kendall τa | τ_{Ken,a} = 0.6316 | 9.89·10^{-5} | Z_{Ken,τa} = 3.89 |

Kendall τb | τ_{Ken,b} = 0.6316 | 9.89·10^{-5} | Z_{Ken,τb} = 3.89 |

Kendall τc | τ_{Ken,c} = 0.6000 | 2.17·10^{-4} | Z_{Ken,τc} = 3.70 |

Gamma | Γ = 0.6316 | 1.39·10^{-2} | Z_{Γ} = 2.46 |

Eq (1.2), n = 25 | |||

Pearson | r = 0.8292 | 3.02·10^{-7} | t_{Prs,1} = 7.11 |

Spearman | ρ = 0.9008 | 8.45·10^{-10} | t_{Spm,1} = 9.95 |

Semi-Q | r_{sQ} = 0.8642 | 2.58·10^{-8} | t_{sQ} = 8.24 |

Kendall τa | τ_{Ken,a} = 0.7667 | 7.80·10^{-8} | Z_{Ken,τa} = 5.37 |

Kendall τb | τ_{Ken,b} = 0.7667 | 7.80·10^{-8} | Z_{Ken,τb} = 5.37 |

Kendall τc | τ_{Ken,c} = 0.7360 | 2.51·10^{-7} | Z_{Ken,τc} = 5.16 |

Gamma | Γ = 0.7667 | 3.82·10^{-5} | Z_{Γ} = 4.12 |

^{st}peak, experiment no. 8, Eq (1.2)) to 0.363 (5

^{th}peak, experiment no. 9, Eq (1.1)) and 0.675 (4

^{th}peak, experiment no. 1, Eq (1.2)). Systematically, the estimated values were higher than experimental values for experiment no. 10, for both Eq (1.1) and Eq (1.2). The correlation analysis between experimental and estimated values (by Eq (1.1) and Eq (1.2), respectively) leads to the results presented in Table 6.

**Table 5.**Matrix of retardation factors ordered by the chromatographic peak: experimental vs estimated.

No. | Experimental peak | Estimated peak by Eq(1.1) | Estimated peak by Eq(1.2) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1^{st} | 2^{nd} | 3^{rd} | 4^{th} | 5^{th} | 1^{st} | 2^{nd} | 3^{rd} | 4^{th} | 5^{th} | 1^{st} | 2^{nd} | 3^{rd} | 4^{th} | 5^{th} | |

1 | 0.460 | 0.547 | 0.764 | 0.834 | 0.845 | 0.006 | 0.021 | 0.134 | 0.159 | 0.228 | |||||

2 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | ||||||||||

3 | 0.890 | 0.906 | 0.947 | 0.970 | 0.970 | ||||||||||

4 | 0.006 | 0.021 | 0.134 | 0.159 | 0.228 | ||||||||||

5 | 0.000 | 0.000 | 0.060 | 0.076 | 0.110 | ||||||||||

6 | 0.601 | 0.629 | 0.759 | 0.798 | 0.805 | ||||||||||

7 | 0.822 | 0.839 | 0.963 | 0.977 | 0.977 | ||||||||||

8 | 0.060 | 0.072 | 0.394 | 0.509 | 0.532 | 0.200 | 0.219 | 0.291 | 0.308 | 0.314 | 0.215 | 0.219 | 0.271 | 0.279 | 0.284 |

9 | 0.114 | 0.262 | 0.573 | 0.683 | 0.756 | 0.141 | 0.172 | 0.309 | 0.347 | 0.393 | 0.157 | 0.172 | 0.289 | 0.317 | 0.364 |

10 | 0.824 | 0.837 | 0.934 | 0.954 | 0.959 | 0.866 | 0.902 | 1.017 | 1.053 | 1.052 | 0.882 | 0.903 | 0.997 | 1.024 | 1.022 |

11 | 0.181 | 0.309 | 0.608 | 0.709 | 0.759 | 0.256 | 0.321 | 0.505 | 0.565 | 0.591 | 0.309 | 0.323 | 0.438 | 0.465 | 0.491 |

^{-5}). Thus, it can be concluded that the link between experimental and estimated by Eq (1.1) and Eq (1.2) data are linear related and sustain the validity of the models from Eq (1.1) and Eq (1.2) for this chromatographic response function.

**Table 6.**Correlation analysis on retardation factor ordered ascending by the chromatographic peak (experimental vs estimated values).

Name | Correlation coefficient | p-value | Statistical parameter |
---|---|---|---|

Eq(1.1), n = 20 | |||

Pearson | r = 0.8654 | 8.38·10^{-7} | t_{Prs,1} = 7.33 |

Spearman | ρ = 0.9579 | 3.39·10^{-11} | t_{Spm,1} = 14.15 |

Semi-Q | r_{sQ} = 0.9105 | 2.53·10^{-8} | t_{sQ} = 9.34 |

Kendall τa | τ_{Ken,a} = 0.8526 | 1.47·10^{-7} | Z_{Ken,τa} = 5.26 |

Kendall τb | τ_{Ken,b} = 0.8526 | 1.47·10^{-7} | Z_{Ken,τb} = 5.26 |

Kendall τc | τ_{Ken,c} = 0.8100 | 5.94·10^{-7} | Z_{Ken,τc} = 4.99 |

Gamma | Γ = 0.8526 | 7.42·10^{-6} | Z_{Γ} = 4.48 |

Eq(1.2), n = 25 | |||

Pearson | r = 0.8292 | 3.02·10^{-7} | t_{Prs,1} = 7.11 |

Spearman | ρ = 0.9008 | 8.45·10^{-10} | t_{Spm,1} = 9.95 |

Semi-Q | r_{sQ} = 0.8642 | 2.58·10^{-8} | t_{sQ} = 8.24 |

Kendall τa | τ_{Ken,a} = 0.7667 | 7.80·10^{-8} | Z_{Ken,τa} = 5.37 |

Kendall τb | τ_{Ken,b} = 0.7667 | 7.80·10^{-8} | Z_{Ken,τb} = 5.37 |

Kendall τc | τ_{Ken,c} = 0.7360 | 2.51·10^{-7} | Z_{Ken,τc} = 5.16 |

Gamma | Γ = 0.7667 | 3.82·10^{-5} | Z_{Γ} = 4.12 |

Eq(1.1), n = 20 | |||||||

Pearson | 1.0000 | 0.3510 | 0.7287 | 0.5805 | 0.5805 | 0.4346 | 0.5805 |

Spearman | 0.0824 | 1.0000 | 0.5559 | 0.1408 | 0.1408 | 0.0903 | 0.1408 |

Semi-Q | 0.2305 | 1.0000 | 0.3699 | 0.3699 | 0.2614 | 0.3699 | |

Kendall τa | 0.2468 | 1.0000 | 0.3699 | 0.2614 | 0.3699 | ||

Kendall τb | 1.0000 | 1.0000 | 1.0000 | 0.8178 | |||

Kendall τc | 0.5803 | 1.0000 | 0.8178 | ||||

Gamma | 0.5803 | 1.0000 |

No. | AI1-AI2 | AI1-AI3 | AI1-AI4 | AI1-AI5 | AI2-AI3 | AI2-AI4 | AI2-AI5 | AI3-AI4 | AI3-AI5 | AI4-AI5 |
---|---|---|---|---|---|---|---|---|---|---|

Experimental | ||||||||||

1 | 1.687 | 1.718 | 6.163 | 2.589 | 0.310 | 9.178 | 3.895 | 10.689 | 4.132 | 1.118 |

2 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

3 | 0.635 | 0.833 | 1.108 | 1.042 | 0.000 | 2.074 | 1.647 | 2.872 | 2.000 | 0.322 |

4 | 1.570 | 0.427 | 2.729 | 2.217 | 1.442 | 5.882 | 4.960 | 4.313 | 3.493 | 0.528 |

5 | 0.936 | 0.337 | 1.317 | 1.333 | 0.923 | 3.063 | 3.111 | 2.566 | 2.615 | 0.000 |

6 | 0.953 | 0.907 | 3.077 | 1.855 | 0.275 | 5.403 | 3.120 | 6.214 | 3.348 | 0.476 |

7 | 0.511 | 0.585 | 3.254 | 1.425 | 0.000 | 5.762 | 1.826 | 6.722 | 1.902 | 0.178 |

8 | 2.392 | 2.318 | 6.805 | 2.836 | 0.571 | 12.116 | 4.448 | 14.473 | 4.580 | 0.131 |

9 | 2.717 | 1.903 | 8.059 | 4.182 | 1.529 | 13.735 | 7.702 | 15.303 | 7.711 | 2.758 |

10 | 0.447 | 0.667 | 2.048 | 1.311 | 0.263 | 3.679 | 1.900 | 4.696 | 2.164 | 0.172 |

11 | 2.455 | 7.191 | 9.671 | 10.234 | 4.371 | 6.290 | 6.871 | 1.433 | 2.080 | 0.732 |

Estimated by Eq(1.1) | ||||||||||

8 | 0.512 | 0.456 | 1.968 | 0.891 | 0.178 | 3.211 | 1.402 | 3.639 | 1.409 | 0.207 |

9 | 1.515 | 0.786 | 3.447 | 2.253 | 1.062 | 6.405 | 4.542 | 5.912 | 3.770 | 0.586 |

10 | 0.872 | 1.082 | 3.081 | 1.620 | -0.004 | 5.028 | 2.327 | 6.269 | 2.734 | 0.521 |

11 | 1.681 | 1.380 | 5.042 | 2.480 | 0.677 | 7.971 | 4.292 | 8.665 | 4.142 | 0.901 |

Estimated by Eq(1.2) | ||||||||||

1 | 0.822 | 0.673 | 2.973 | 1.688 | 0.372 | 5.440 | 2.847 | 6.091 | 2.885 | 0.196 |

8 | 0.325 | 0.231 | 1.279 | 0.696 | 0.191 | 2.404 | 1.176 | 2.646 | 1.14 | 0.008 |

9 | 1.328 | 0.56 | 2.758 | 2.058 | 1.076 | 5.597 | 4.316 | 4.919 | 3.501 | 0.387 |

10 | 0.685 | 0.857 | 2.392 | 1.426 | 0.01 | 4.22 | 2.101 | 5.276 | 2.465 | 0.322 |

11 | 1.046 | 0.613 | 2.702 | 1.819 | 0.722 | 5.228 | 3.523 | 5.292 | 3.228 | 0.225 |

^{st}peak - experiment no. 9, Eq (1.1)) and 2.172 (2

^{nd}peak - experiment no. 10, Eq (1.1)). In most cases (four out of five), the estimated by Eq (1.1) values were greater than the experimental values for experiment no. 10.

Name | Correlation coefficient | p-value | Statistical parameter |
---|---|---|---|

Estimated by Eq(1.1), n = 40 | |||

Pearson | r = 0.5173 | 6.30·10^{-4} | t_{Prs,1} = 3.72 |

Spearman | ρ = 0.6214 | 1.88·10^{-5} | t_{Spm,1} = 4.89 |

Semi-Q | r_{sQ} = 0.5670 | 1.36·10^{-4} | t_{sQ} = 4.24 |

Kendall τa | τ_{Ken,a} = 0.4462 | 5.02·10^{-5} | Z_{Ken,τa} = 4.05 |

Kendall τb | τ_{Ken,b} = 0.4462 | 5.02·10^{-5} | Z_{Ken,τb} = 4.05 |

Kendall τc | τ_{Ken,c} = 0.4350 | 7.71·10^{-5} | Z_{Ken,τc} = 3.95 |

Gamma | Γ = 0.4462 | 7.05·10^{-2} | Z_{Γ} = 1.81 |

Estimated by Eq(1.2), n = 50 | |||

Pearson | r = 0.6185 | 1.70·10^{-6} | t_{Prs,1} = 5.45 |

Spearman | ρ = 0.6786 | 6.12·10^{-8} | t_{Spm,1} = 6.40 |

Semi-Q | r_{sQ} = 0.6478 | 3.67·10^{-7} | t_{sQ} = 5.89 |

Kendall τa | τ_{Ken,a} = 0.4939 | 4.18·10^{-7} | Z_{Ken,τa} = 5.06 |

Kendall τb | τ_{Ken,b} = 0.4939 | 4.18·10^{-7} | Z_{Ken,τb} = 5.06 |

Kendall τc | τ_{Ken,c} = 0.4840 | 7.07·10^{-7} | Z_{Ken,τc} = 4.96 |

Gamma | Γ = 0.4939 | 1.24·10^{-2} | Z_{Γ} = 2.50 |

No. | Experimental peak | Estimated peak by Eq(1.1) | Estimated peak by Eq(1.2) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1^{st} | 2^{nd} | 3^{rd} | 4^{th} | 5^{th} | 1^{st} | 2^{nd} | 3^{rd} | 4^{th} | 5^{th} | 1^{st} | 2^{nd} | 3^{rd} | 4^{th} | 5^{th} | |

1 | 1.118 | 2.589 | 1.718 | 0.310 | 0.845 | 0.196 | 2.487 | 0.673 | 0.372 | 0.196 | |||||

2 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | ||||||||||

3 | 0.322 | 1.108 | 0.833 | 0.000 | 0.970 | ||||||||||

4 | 0.528 | 2.217 | 0.427 | 1.442 | 0.228 | ||||||||||

5 | 0.000 | 1.317 | 0.337 | 0.923 | 0.110 | ||||||||||

6 | 0.476 | 1.855 | 0.907 | 0.275 | 0.805 | ||||||||||

7 | 0.178 | 3.254 | 0.585 | 0.000 | 0.977 | ||||||||||

8 | 0.131 | 2.836 | 2.318 | 0.571 | 0.532 | 0.207 | 1.293 | 0.456 | 0.178 | 0.314 | 0.008 | 1.271 | 0.231 | 0.191 | 0.008 |

9 | 2.758 | 4.182 | 1.903 | 1.529 | 0.756 | 0.586 | 2.143 | 0.786 | 1.062 | 0.393 | 0.387 | 2.121 | 0.560 | 1.076 | 0.387 |

10 | 0.172 | 2.048 | 0.447 | 0.263 | 0.959 | 0.521 | 2.064 | 1.082 | -0.004 | 1.052 | 0.322 | 2.042 | 0.857 | 0.01 | 0.322 |

11 | 2.455 | 4.371 | 1.433 | 0.732 | 0.759 | 0.901 | 2.238 | 1.380 | 0.677 | 0.591 | 0.225 | 2.163 | 0.613 | 0.722 | 0.225 |

^{-3}). With one exception (Kendall τ

_{c}, τ

_{Ken,c}= 0.4900), the values of correlation coefficient were higher than 0.5, indicating moderate to good correlations between experimental and estimated values. Four out of seven correlation coefficients (Kendall τa, Kendall τb, Kendall τc, and Gamma) had values less than or equal with 0.4737, indicating a weak correlation between experimental and estimated by Eq (1.2) values. There could not be identified any statistically significant differences between correlation coefficients presented in Table 11. The lower p-values (0.2348 - Eq (1.1), and 0.2592 - Eq (1.2)) were obtained when Pearson and Kendall τc correlation coefficients were compared.

**Table 11.**Results of correlation analysis: resolution matrix of successive chromatographic peaks (experimental vs estimated).

Name | Correlation coefficient | p-value | Statistical parameter |
---|---|---|---|

Estimated by Eq(1.1), n = 20 | |||

Pearson | r = 0.7446 | 1.66·10^{-4} | t_{Prs,1} = 4.73 |

Spearman | ρ = 0.6692 | 1.25·10^{-3} | t_{Spm,1} = 3.82 |

Semi-Q | r_{sQ} = 0.7059 | 5.06·10^{-4} | t_{sQ} = 4.23 |

Kendall τa | τ_{Ken,a} = 0.5158 | 1.47·10^{-3} | Z_{Ken,τa} = 3.18 |

Kendall τb | τ_{Ken,b} = 0.5158 | 1.47·10^{-3} | Z_{Ken,τb} = 3.18 |

Kendall τc | τ_{Ken,c} = 0.4900 | 2.52·10^{-3} | Z_{Ken,τc} = 3.02 |

Gamma | Γ = 0.5158 | 1.01·10^{-1} | Z_{Γ} = 1.64 |

Estimated by Eq(1.2), n = 25 | |||

Pearson | r = 0.6821 | 9.24·10^{-4} | t_{Prs,1} = 3.96 |

Spearman | ρ = 0.6361 | 2.57·10^{-3} | t_{Spm,1} = 3.50 |

Semi-Q | r_{sQ} = 0.6587 | 1.59·10^{-3} | t_{sQ} = 3.71 |

Kendall τa | τ_{Ken,a} = 0.4737 | 3.50·10^{-3} | Z_{Ken,τa} = 2.92 |

Kendall τb | τ_{Ken,b} = 0.4737 | 3.50·10^{-3} | Z_{Ken,τb} = 2.92 |

Kendall τc | τ_{Ken,c} = 0.4500 | 5.54·10^{-3} | Z_{Ken,τc} = 2.77 |

Gamma | Γ = 0.4737 | 1.67·10^{-1} | Z_{Γ} = 1.38 |

- ÷
- The number distinct compounds on chromatogram DCN - Eq (7);
- ÷
- The string of standard deviation of retardation factors ordered ascending and estimated by Eq (1) compared with ideal positions of the peaks obtained through experiment RFD - Eq (9);
- ÷
- The string of sum of the peak resolutions obtained through experiment RSS - Eq (12);
- ÷
- The squared of effective plate number QN- Eq (13);
- ÷
- Average peaks separation (into experiment) RSA - Eq (15);
- ÷
- The string of mean resolution calculated with Minkowski experimental peaks RSR - Eq (17);
- ÷
- The string of experimental peaks with minimal resolution QF- Eq (18).

No. | Experimental | Estimated by Eq(1.1), n = 4 | Estimated by Eq(1.2), n = 5 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

DCN | RFD | RSS | QN | DCN | RFD | RSS | QN | DCN | RFD | RSS | QN | |

1 | 5 | 0.047 | 5.730 | 71.900 | 4 (4.222) | 0.057 | 3.73 | 83.857 | ||||

2 | 1 | 0.283 | 0.000 | 114.680 | ||||||||

3 | 4 | 0.081 | 2.260 | 108.700 | ||||||||

4 | 5 | 0.055 | 4.610 | 89.550 | ||||||||

5 | 4 | 0.078 | 2.580 | 103.840 | ||||||||

6 | 5 | 0.057 | 3.510 | 83.790 | ||||||||

7 | 3 | 0.097 | 4.020 | 79.280 | ||||||||

8 | 4 | 0.067 | 5.860 | 59.460 | 2 (2.368) | 0.181 | 2.135 | 98.554 | 2 (2.200) | 0.183 | 1.703 | 101.14 |

9 | 5 | 0.036 | 10.370 | 71.240 | 5 (5.168) | 0.042 | 4.574 | 85.744 | 5 (5.000) | 0.044 | 4.142 | 88.326 |

10 | 4 | 0.076 | 2.930 | 80.920 | 4 (4.328) | 0.062 | 3.661 | 89.591 | 4 (4.160) | 0.064 | 3.229 | 92.174 |

11 | 5 | 0.040 | 8.990 | 63.380 | 5 (5.220) | 0.039 | 5.192 | 79.123 | 5 (4.650) | 0.046 | 3.725 | 87.895 |

No | Experimental | Estimated by Eq(1.1), n = 4 | Estimated by Eq(1.2), n = 5 | ||||||
---|---|---|---|---|---|---|---|---|---|

RSA | RSR | QF | RSA | RSR | QF | RSA | RSR | QF | |

1 | 1.434 | 1.285 | 0.310 | 0.932 | 0.636 | 0.075 | |||

2 | 0.000 | 0.000 | 0.000 | ||||||

3 | 0.566 | 0.401 | 0.000 | ||||||

4 | 1.153 | 1.035 | 0.427 | ||||||

5 | 0.644 | 0.452 | 0.000 | ||||||

6 | 0.878 | 0.778 | 0.275 | ||||||

7 | 1.004 | 0.559 | 0.000 | ||||||

8 | 1.464 | 1.169 | 0.131 | 0.533 | 0.380 | 0.028 | 0.425 | 0.240 | 0.000 |

9 | 2.593 | 2.498 | 1.529 | 1.144 | 1.021 | 0.344 | 1.035 | 0.881 | 0.293 |

10 | 0.733 | 0.573 | 0.172 | 0.916 | 0.696 | 0.105 | 0.807 | 0.556 | 0.054 |

11 | 2.248 | 2.038 | 0.732 | 1.299 | 1.180 | 0.311 | 0.931 | 0.704 | 0.138 |

Name | Correlation coefficient | p-value | Statistical parameter | Name | Correlation coefficient | p-value | Statistical parameter |
---|---|---|---|---|---|---|---|

DCN | RSA | ||||||

Pearson | r = 0.8165 | 1.80·10^{-1} | t_{Prs,1} = 2.0 | Pearson | r = 0.5905 | 4.09·10^{-1} | t_{Prs,1} = 1.03 |

Spearman | ρ = 0.9428 | 5.72·10^{-2} | t_{Spm,1} = 4.0 | Spearman | ρ = 0.6000 | 4.00·10^{-1} | t_{Spm,1} = 1.06 |

Semi-Q | r_{sQ} = 0.8457 | 1.23·10^{-1} | t_{sQ} = 2.59 | Semi-Q | r_{sQ} = 0.5952 | 4.05·10^{-1} | t_{sQ} = 1.05 |

Kendall tau-a | τ_{Ken,a} = 0.6667 | 1.75·10^{-1} | Z_{Ken,τa} = 1.36 | Kendall tau-a | τ_{Ken,a} = 0.3333 | 4.97·10^{-1} | Z_{Ken,τa} = 0.68 |

Kendall tau-b | τ_{Ken,b} = 0.7303 | 1.49·10^{-1} | Z_{Ken,τb} = 1.44 | Kendall tau-b | τ_{Ken,b} = 0.3333 | 4.97·10^{-1} | Z_{Ken,τb} = 0.68 |

Kendall tau-c | τ_{Ken,c} = 0.5000 | 2.79·10^{-1} | Z_{Ken,τc} = 1.08 | Kendall tau-c | τ_{Ken,c} = 0.2500 | 6.10·10^{-1} | Z_{Ken,τc} = 0.51 |

Gamma | Γ = 1.0000 | 4.15·10^{-2} | Z_{Γ} = 2.04 | Gamma | Γ = 0.3333 | 8.21·10^{-1} | Z_{Γ} = 0.23 |

RFD | RSR | ||||||

Pearson | r = 0.5434 | 4.56·10^{-1} | t_{Prs,1} = 0.92 | Pearson | r = 0.7118 | 2.88·10^{-1} | t_{Prs,1} = 2.05 |

Spearman | ρ = 0.6000 | 4.00·10^{-1} | t_{Spm,1} = 1.06 | Spearman | ρ = 0.6000 | 3.46·10^{-1} | t_{Spm,1} = 1.06 |

Semi-Q | r_{sQ} = 0.5710 | 4.29·10^{-1} | t_{sQ} = 0.98 | Semi-Q | r_{sQ} = 0.6535 | 5.73·10^{-1} | t_{sQ} = 1.22 |

Kendall tau-a | τ_{Ken,a} = 0.3333 | 4.97·10^{-1} | Z_{Ken,τa} = 0.68 | Kendall tau-a | τ_{Ken,a} = 0.3333 | 4.47·10^{-1} | Z_{Ken,τa} = 0.68 |

Kendall tau-b | τ_{Ken,b} = 0.3333 | 4.97·10^{-1} | Z_{Ken,τb} = 0.68 | Kendall tau-b | τ_{Ken,b} = 0.3333 | 4.97·10^{-1} | Z_{Ken,τb} = 0.68 |

Kendall tau-c | τ_{Ken,c} = 0.250 | 6.10·10^{-1} | Z_{Ken,τc} = 0.51 | Kendall tau-c | τ_{Ken,c} = 0.2500 | 6.10·10^{-1} | Z_{Ken,τc} = 0.51 |

Gamma | Γ = 0.3333 | 8.21·10^{-1} | Z_{Γ} = 0.23 | Gamma | Γ = 0.3333 | 8.21·10^{-1} | Z_{Γ} = 0.23 |

RSS | QF | ||||||

Pearson | r = 0.5906 | 4.09·10^{-1} | t_{Prs,1} = 1.04 | Pearson | r = 0.8936 | 1.06·10^{-1} | t_{Prs,1} = 2.82 |

Spearman | ρ = 0.6000 | 4.00·10^{-1} | t_{Spm,1} = 1.13 | Spearman | ρ = 1.0000 | 5.47·10^{-2} | t_{Spm,1} = 4.10 |

Semi-Q | r_{sQ} = 0.5953 | 4.05·10^{-1} | t_{sQ} = 1.05 | Semi-Q | r_{sQ} = 0.9453 | 6.68·10^{-2} | t_{sQ} = 2.82 |

Kendall | τ_{Ken,a} = 0.3333 | 1.97·10^{-1} | Z_{Ken,τa} = 0.68 | Kendall | τ_{Ken,a} = 1.0000 | 4.15·10^{-2} | Z_{Ken,τa} = 2.04 |

Kendall | τ_{Ken,b} = 0.3333 | 4.97·10^{-1} | Z_{Ken,τb} = 0.68 | Kendall | τ_{Ken,b} = 1.0000 | 4.15·10^{-2} | Z_{Ken,τb} = 2.04 |

Kendall | τ_{Ken,c} = 0.2500 | 6.10·10^{-1} | Z_{Ken,τc} = 0.51 | Kendall | τ_{Ken,c} = 0.7500 | 1.26·10^{-1} | Z_{Ken,τc} = 1.53 |

Gamma | Γ = 0.3333 | 8.21·10^{-1} | Z_{Γ} = 0.23 | Gamma | Γ = 1.0000 | 4.15·10^{-2} | Z_{Γ} = 2.04 |

QN | |||||||

Pearson | r = -0.1588 | 9.85·10^{-1} | t_{Prs,1} = 0.22 | n = sample size; DCN = number of distinct compounds on chromatogram; RFD = string of standard deviation of retardation factors estimated by Eq(1) ordered ascending compared with ideal positions of the peaks obtained through experiment; RSS = string of sum of the peak resolutions obtained through experiment; QN = squared of effective plate number;RSA = average peaks separation (into experiment); RSR = string of Minkowski mean resolution of experimental peaks; QF = string of experimental peaks with minimal resolution. | |||

Spearman | ρ = -0.2000 | 8.22·10^{-1} | t_{Spm,1} = 0.29 | ||||

Semi-Q | r_{sQ} = 0.1782 | 8.00·10^{-1} | t_{sQ} = 0.25 | ||||

Kendall tau-a | τ_{Ken,a} = 0.0000 | 1.00 | Z_{Ken,τa} = 0.00 | ||||

Kendall tau-b | τ_{Ken,b} = 0.0000 | 1.00 | Z_{Ken,τb} = 0.00 | ||||

Kendall tau-c | τ_{Ken,c} = 0.0000 | 1.00 | Z_{Ken,τc} = 0.00 | ||||

Gamma | Γ = 0.0000 | 1.00 | Z_{Γ} = 0.00 |

Name | Correlation coefficient | p-value | Statistical parameter | Name | Correlation coefficient | p-value | Statistical parameter |
---|---|---|---|---|---|---|---|

DCN | RSA | ||||||

Pearson | r = 0.7454 | 1.48·10^{-1} | t_{Prs,1} = 3.75 | Pearson | r = 0.4698 | 4.25·10^{-1} | t_{Prs,1} = 0.92 |

Spearman | ρ = 0.4722 | 4.22·10^{-1} | t_{Spm,1} = 0.93 | Spearman | ρ = 0.5000 | 3.91·10^{-1} | t_{Spm,1} = 1.00 |

Semi-Q | r_{sQ} = 0.5933 | 2.92·10^{-1} | t_{sQ} = 1.28 | Semi-Q | r_{sQ} = 0.4847 | 4.08·10^{-1} | t_{sQ} = 0.96 |

Kendall tau-a | τ_{Ken,a} = 0.3000 | 4.62·10^{-1} | Z_{Ken,τa} = 0.73 | Kendall tau-a | τ_{Ken,a} = 0.4000 | 3.27·10^{-1} | Z_{Ken,τa} = 0.98 |

Kendall tau-b | τ_{Ken,b} = 0.3162 | 4.49·10^{-1} | Z_{Ken,τb} = 0.76 | Kendall tau-b | τ_{Ken,b} = 0.4000 | 3.27·10^{-1} | Z_{Ken,τb} = 0.98 |

Kendall tau-c | τ_{Ken,c} = 0.2400 | 5.44·10^{-1} | Z_{Ken,τc} = 0.61 | Kendall tau-c | τ_{Ken,c} = 0.3200 | 4.33·10^{-1} | Z_{Ken,τc} = 0.78 |

Gamma | Γ = 0.4286 | 6.53·10^{-1} | Z_{Γ} = 0.45 | Gamma | Γ = 0.4000 | 6.95·10^{-1} | Z_{Γ} = 0.39 |

RFD | RSR | ||||||

Pearson | r = 0.5520 | 3.35·10^{-1} | t_{Prs,1} = 1.15 | Pearson | r = 0.6827 | 2.04·10^{-1} | t_{Prs,1} = 2.62 |

Spearman | ρ = 0.9000 | 3.74·10^{-2} | t_{Spm,1} = 3.58 | Spearman | ρ = 0.9000 | 3.74·10^{-2} | t_{Spm,1} = 3.58 |

Semi-Q | r_{sQ} = 0.7049 | 1.84·10^{-1} | t_{sQ} = 1.72 | Semi-Q | r_{sQ} = 0.7838 | 1.17·10^{-1} | t_{sQ} = 2.19 |

Kendall tau-a | τ_{Ken,a} = 0.8000 | 5.00·10^{-2} | Z_{Ken,τa} = 1.96 | Kendall tau-a | τ_{Ken,a} = 0.8000 | 5.00·10^{-2} | Z_{Ken,τa} = 1.96 |

Kendall tau-b | τ_{Ken,b} = 0.8000 | 5.00·10^{-2} | Z_{Ken,τb} = 1.96 | Kendall tau-b | τ_{Ken,b} = 0.8000 | 5.00·10^{-2} | Z_{Ken,τb} = 1.96 |

Kendall tau-c | τ_{Ken,c} = 0.6400 | 1.17·10^{-1} | Z_{Ken,τc} = 1.57 | Kendall tau-c | τ_{Ken,c} = 0.6400 | 1.17·10^{-1} | Z_{Ken,τc} = 1.57 |

Gamma | Γ = 0.8000 | 1.17·10^{-1} | Z_{Γ} = 1.57 | Gamma | Γ = 0.8000 | 1.17·10^{-1} | Z_{Γ} = 1.57 |

RSS | QF | ||||||

Pearson | r = 0.4691 | 4.25·10^{-1} | t_{Prs,1} = 0.92 | Pearson | r = 0.9871 | 1.76·10^{-3} | t_{Prs,1} = 10.67 |

Spearman | ρ = 0.5000 | 3.91·10^{-1} | t_{Spm,1} = 1.00 | Spearman | ρ = 1.0000 | 1.24·10^{-2} | t_{Spm,1} = 5.41 |

Semi-Q | r_{sQ} = 0.4843 | 4.08·10^{-1} | t_{sQ} = 0.96 | Semi-Q | r_{sQ} = 0.9935 | 6.26·10^{-4} | t_{sQ} = 15.14 |

Kendall | τ_{Ken,a} = 0.4000 | 3.27·10^{-1} | Z_{Ken,τa} = 0.98 | Kendall | τ_{Ken,a} = 1.0000 | 1.43·10^{-2} | Z_{Ken,τa} = 2.45 |

Kendall | τ_{Ken,b} = 0.4000 | 3.27·10^{-1} | Z_{Ken,τb} = 0.98 | Kendall | τ_{Ken,b} = 1.0000 | 1.43·10^{-2} | Z_{Ken,τb} = 2.45 |

Kendall | τ_{Ken,c} = 0.3200 | 4.33·10^{-1} | Z_{Ken,τc} = 0.78 | Kendall | τ_{Ken,c} = 0.8000 | 5.00·10^{-2} | Z_{Ken,τc} = 1.96 |

Gamma | Γ = 0.4000 | 6.95·10^{-1} | Z_{Γ} = 0.39 | Gamma | Γ = 1.0000 | 1.43·10^{-2} | Z_{Γ} = 2.45 |

QN | |||||||

Pearson | r = -0.4189 | 4.82·10^{-1} | t_{Prs,1} = 0.80 | n = sample size; DCN = number of distinct compounds on chromatogram; RFD = string of standard deviation of retardation factors ordered ascending and estimated by Eq(1) compared with ideal positions of the peaks obtained through experiment; RSS = string of sum of the peak resolutions obtained through experiment; QN = squared of effective plate number; RSA = average peaks separation (into experiment); RSR = string of mean resolution calculated with Minkowski experimental peaks; QF = string of experimental peaks with minimal resolution. | |||

Spearman | ρ = -0.3000 | 6.24·10^{-1} | t_{Spm,1} = 0.54 | ||||

Semi-Q | r_{sQ} = 0.3545 | 5.58·10^{-1} | t_{sQ} = 0.66 | ||||

Kendall tau-a | τ_{Ken,a} = 0.2000 | 6.24·10^{-1} | Z_{Ken,τa} = 0.49 | ||||

Kendall tau-b | τ_{Ken,b} = 0.2000 | 6.24·10^{-1} | Z_{Ken,τb} = 0.49 | ||||

Kendall tau-c | τ_{Ken,c} = 0.1600 | 7.05·10^{-1} | Z_{Ken,τc} = 0.39 | ||||

Gamma | Γ = 0.2000 | 9.22·10^{-1} | Z_{Γ} = 0.10 |

No. | Experimental | Estimated by Eq(1.1), n = 4 | Estimated by Eq(1.2), n = 5 | ||||||
---|---|---|---|---|---|---|---|---|---|

RSP | IEnt | IEne | RSP | IEnt | IEne | RSP | IEnt | IEne | |

1 | 31.900 | 4.000 | 16.000 | 17.678 | 10.407 | 2.67 | |||

2 | 0.000 | 11.610 | 0.000 | ||||||

3 | 8.300 | 11.610 | 0.000 | ||||||

4 | 20.600 | 8.000 | 8.000 | ||||||

5 | 9.900 | 11.610 | 0.000 | ||||||

6 | 16.800 | 8.000 | 8.000 | ||||||

7 | 20.300 | 11.610 | 0.000 | ||||||

8 | 39.400 | 4.000 | 16.000 | 11.096 | 10.371 | 2.560 | 8.024 | 11.754 | 0.00 |

9 | 58.200 | 2.000 | 18.000 | 21.652 | 7.338 | 9.280 | 18.58 | 8.722 | 6.40 |

10 | 14.500 | 11.610 | 0.000 | 17.676 | 9.360 | 4.800 | 14.604 | 10.744 | 1.92 |

11 | 56.700 | 4.750 | 14.000 | 27.285 | 5.203 | 13.565 | 16.852 | 9.902 | 3.78 |

**Table 17.**Results of correlation analysis for response functions presented in Table 15: Eq (1.1), n = 4.

Name | Correlation coefficient | p-value | Statistical parameter | Name | Correlation coefficient | p-value | Statistical parameter | |
---|---|---|---|---|---|---|---|---|

RSP | IEnt | |||||||

Pearson | r = 0.5326 | 4.67·10^{-1} | t_{Prs,1} = 0.90 | Pearson | r = 0.3188 | 6.81·10^{-1} | t_{Prs,1} = 0.48 | |

Spearman | ρ = 0.6000 | 4.00·10^{-1} | t_{Spm,1} = 1.06 | Spearman | ρ = 0.0000 | 1.00 | t_{Spm,1} = 0.00 | |

Semi-Q | r_{sQ} = 0.5653 | 4.35·10^{-1} | t_{sQ} = 0.97 | Semi-Q | r_{sQ} = 0.0000 | 1.00 | t_{sQ} = 0.00 | |

Kendall tau-a | τ_{Ken,a} = 0.3333 | 4.97·10^{-1} | Z_{Ken,τa} = 0.68 | Kendall tau-a | τ_{Ken,a} = 0.0000 | 1.00 | Z_{Ken,τa} = 0.00 | |

Kendall tau-b | τ_{Ken,b} = 0.3333 | 4.97·10^{-1} | Z_{Ken,τb} = 0.68 | Kendall tau-b | τ_{Ken,b} = 0.0000 | 1.00 | Z_{Ken,τb} = 0.00 | |

Kendall tau-c | τ_{Ken,c} = 0.2500 | 6.10·10^{-1} | Z_{Ken,τc} = 0.51 | Kendall tau-c | τ_{Ken,c} = 0.0000 | 1.00 | Z_{Ken,τc} = 0.00 | |

Gamma | Γ = 0.3333 | 8.21·10^{-1} | Z_{Γ} = 0.23 | Gamma | Γ = 0.0000 | 1.00 | Z_{Γ} = 0.00 | |

IEne | ||||||||

Pearson | r = 0.2962 | 7.04·10^{-1} | t_{Prs,1} = 0.44 | RSP = resolution divided by the number of effective plates; IEnt = informational energy; IEne = informational entropy; n = sample size. | ||||

Spearman | ρ = 0.0000 | 1.00 | t_{Spm,1} = 0.00 | |||||

Semi-Q | r_{sQ} = 0.0000 | 1.00 | t_{sQ} = 0.00 | |||||

Kendall | τ_{Ken,a} = 0.0000 | 1.00 | Z_{Ken,τa} = 0.00 | |||||

Kendall | τ_{Ken,b} = 0.0000 | 1.00 | Z_{Ken,τb} = 0.00 | |||||

Kendall | τ_{Ken,c} = 0.0000 | 1.00 | Z_{Ken,τc} = 0.00 | |||||

Gamma | Γ = 0.0000 | 1.00 | Z_{Γ} = 0.00 |

**Table 18.**Results of correlation analysis for response functions presented in Table 15: Eq (1.2), n = 5.

Name | Correlation coefficient | p-value | Statistical parameter | Name | Correlation coefficient | p-value | Statistical parameter |
---|---|---|---|---|---|---|---|

RSP | IEnt | ||||||

Pearson | r = 0.2864 | 6.40·10^{-1} | t_{Prs,1} = 0.52 | Pearson | r = 0.3770 | 5.32·10^{-1} | t_{Prs,1} = 0.71 |

Spearman | ρ = 0.5000 | 3.91·10^{-1} | t_{Spm,1} = 1.00 | Spearman | ρ = 0.4104 | 4.92·10^{-1} | t_{Spm,1} = 0.78 |

Semi-Q | r_{sQ} = 0.3784 | 5.30·10^{-1} | t_{sQ} = 0.71 | Semi-Q | r_{sQ} = 0.3934 | 5.12·10^{-1} | t_{sQ} = 0.74 |

Kendall tau-a | τ_{Ken,a} = 0.4000 | 3.27·10^{-1} | Z_{Ken,τa} = 0.98 | Kendall tau-a | τ_{Ken,a} = 0.3000 | 4.62·10^{-1} | Z_{Ken,τa} = 0.73 |

Kendall tau-b | τ_{Ken,b} = 0.4000 | 3.27·10^{-1} | Z_{Ken,τb} = 0.98 | Kendall tau-b | τ_{Ken,b} = 0.3162 | 4.48·10^{-1} | Z_{Ken,τb} = 0.76 |

Kendall tau-c | τ_{Ken,c} = 0.3200 | 4.33·10^{-1} | Z_{Ken,τc} = 0.78 | Kendall tau-c | τ_{Ken,c} = 0.2400 | 5.44·10^{-1} | Z_{Ken,τc} = 0.61 |

Gamma | Γ = 0.4000 | 6.95·10^{-1} | Z_{Γ} = 0.39 | Gamma | Γ = 0.3333 | 7.85·10^{-1} | Z_{Γ} = 0.27 |

IEne | |||||||

Pearson | r = 0.3152 | 6.05·10^{-1} | t_{Prs,1} = 0.58 | RSP = resolution divided by the number of effective plates; IEnt = informational energy; IEne = informational entropy; n = sample size. | |||

Spearman | ρ = 0.4104 | 4.92·10^{-1} | t_{Spm,1} = 0.78 | ||||

Semi-Q | r_{sQ} = 0.3596 | 5.52·10^{-1} | t_{sQ} = 0.67 | ||||

Kendall | τ_{Ken,a} = 0.3000 | 4.62·10^{-1} | Z_{Ken,τa} = 0.73 | ||||

Kendall | τ_{Ken,b} = 0.3162 | 4.48·10^{-1} | Z_{Ken,τb} = 0.76 | ||||

Kendall | τ_{Ken,c} = 0.2400 | 5.44·10^{-1} | Z_{Ken,τc} = 0.61 | ||||

Gamma | Γ = 0.3333 | 7.85·10^{-1} | Z_{Γ} = 0.27 |

#### Experiments Quality Assessment

**Table 19.**Summary of validation the response functions estimated by the model from Eq (1.1) and Eq (1.2), respectively.

Parameter | Pearson | Spearman | Semi-Q | Kendall τa | Kendall τb | Kendall τc | Gamma |
---|---|---|---|---|---|---|---|

Eq (1.1) | |||||||

RF | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |

RFO | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |

RSM | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✕ |

RSO | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✕ |

QF | ✕ | ✕ | ✕ | ✓ | ✓ | ✕ | ✓ |

DCN | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ | ✓ |

RFD | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ |

RSS | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ |

QN | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ |

RSA | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ |

RSR | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ |

RSP | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ |

IEne | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ |

IEnt | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ |

Eq (1.2) | |||||||

RF | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |

RFO | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |

RSM | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |

RSO | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✕ |

QF | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |

DCN | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ |

RFD | ✕ | ✓ | ✕ | ✓ | ✓ | ✕ | ✕ |

RSS | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ |

QN | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ |

RSA | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ |

RSR | ✕ | ✓ | ✕ | ✓ | ✓ | ✕ | ✕ |

RSP | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ |

IEne | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ |

IEnt | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ |

**Table 20.**Summary of validation the response functions estimated by the model from Eq (1.1) and Eq (1.2), respectively.

No | Model | CRF | Estimated | Experimental | Difference (%) | Group Rank | Exp No |
---|---|---|---|---|---|---|---|

1 | Eq (1.1) | RF | 0.505 | 0.181 | 23.62 | 1 | 11 |

2 | Eq (1.1) | RF | 0.253 | 0.709 | 23.70 | 2 | 11 |

3 | Eq (1.1) | RF | 0.216 | 0.072 | 25.00 | 3 | 8 |

4 | Eq (1.1) | RF | 0.203 | 0.060 | 27.19 | 4 | 8 |

5 | Eq (1.1) | RSM | 0.456 | 2.318 | 33.56 | 1 | 8 |

6 | Eq (1.1) | RSM | 1.38 | 7.191 | 33.90 | 2 | 11 |

7 | Eq (1.1) | RSM | 8.665 | 1.433 | 35.81 | 3 | 11 |

8 | Eq (1.1) | RSM | 0.677 | 4.371 | 36.59 | 4 | 11 |

9 | Eq (1.1) | RSM | -0.004 | 0.263 | 51.54 | 5 | 10 |

10 | Eq (1.1) | RFO | 0.393 | 0.756 | 15.80 | 1 | 9 |

11 | Eq (1.1) | RFO | 0.347 | 0.683 | 16.31 | 2 | 9 |

12 | Eq (1.1) | RFO | 0.219 | 0.072 | 25.26 | 3 | 8 |

13 | Eq (1.1) | RFO | 0.200 | 0.06 | 26.92 | 4 | 8 |

14 | Eq (1.1) | RSO | 0.586 | 2.758 | 32.48 | 1 | 9 |

15 | Eq (1.1) | RSO | 0.456 | 2.318 | 33.56 | 2 | 8 |

16 | Eq (1.1) | RSO | -0.004 | 0.263 | 51.54 | 3 | 10 |

18 | Eq (1.1) | QF | 0.028 | 0.131 | 32.39 | 1 | 8 |

19 | Eq (1.2) | RF | 0.311 | 0.709 | 19.51 | 1 | 11 |

20 | Eq (1.2) | RF | 0.322 | 0.759 | 20.21 | 2 | 11 |

21 | Eq (1.2) | RF | 0.438 | 0.181 | 20.76 | 3 | 11 |

22 | Eq (1.2) | RF | 0.215 | 0.072 | 24.91 | 4 | 8 |

23 | Eq (1.2) | RF | 0.220 | 0.060 | 28.57 | 5 | 8 |

24 | Eq (1.2) | RSM | 2.646 | 14.473 | 34.54 | 1 | 8 |

25 | Eq (1.2) | RSM | 1.819 | 10.234 | 34.91 | 2 | 11 |

26 | Eq (1.2) | RSM | 0.196 | 1.118 | 35.08 | 3 | 1 |

27 | Eq (1.2) | RSM | 0.722 | 4.371 | 35.82 | 4 | 11 |

28 | Eq (1.2) | RSM | 0.387 | 2.758 | 37.69 | 5 | 9 |

29 | Eq (1.2) | RSM | 0.325 | 2.392 | 38.04 | 6 | 8 |

30 | Eq (1.2) | RSM | 0.231 | 2.318 | 40.94 | 7 | 8 |

31 | Eq(1.2) | RSM | 0.613 | 7.191 | 42.15 | 8 | 11 |

32 | Eq (1.2) | RSM | 0.008 | 0.131 | 44.24 | 9 | 8 |

33 | Eq (1.2) | RSM | 0.010 | 0.263 | 46.34 | 10 | 10 |

34 | Eq (1.2) | RFO | 0.289 | 0.573 | 16.47 | 1 | 9 |

35 | Eq (1.2) | RFO | 0.364 | 0.756 | 17.50 | 2 | 9 |

36 | Eq (1.2) | RFO | 0.317 | 0.683 | 18.30 | 3 | 9 |

37 | Eq (1.2) | RFO | 0.219 | 0.072 | 25.26 | 4 | 8 |

38 | Eq (1.2) | RFO | 0.215 | 0.060 | 28.18 | 5 | 8 |

39 | Eq (1.2) | RSO | 0.231 | 2.318 | 40.94 | 1 | 8 |

40 | Eq (1.2) | RSO | 0.225 | 2.455 | 41.60 | 2 | 11 |

41 | Eq (1.2) | RSO | 0.008 | 0.131 | 44.24 | 3 | 8 |

42 | Eq (1.2) | RSO | 0.010 | 0.263 | 46.34 | 4 | 10 |

43 | Eq (1.2) | QF | -0.023 | 0.131 | 71.30 | 1 | 8 |

## Conclusions

- The model presented in Eq (1.2) seems to be more reliable for the estimation of chromatographic response functions on investigated androstane isomers. Four response functions (RF - retardation factor; RFO - retardation factor ordered ascending by the chromatographic peak; RSM - resolution of pairs of compounds; QF - string of experimental peaks with minimal resolution) revealed statistically significant linear relationships between experimental and estimated values.
- The models presented in Eq (1.1) is valid and reliable in investigation of retardation factor, retardation factor ordered ascending by the chromatographic response, resolution of pairs of compounds and resolution matrix of successive chromatographic peaks;
- Good performances are obtained in estimation of resolution of pairs of compounds but the relationship between experimental and estimated values by Eq (1.1) and Eq (1.2) could be questionable due to the absence of significantly statistic Gamma correlation coefficient;
- Some estimation abilities were observed in investigation of the string of standard deviation of retardation factors ordered ascending estimated by Eq (1.2) compared with ideal positions of the peaks obtained through experiment; and of the string of Minkowski type mean resolution calculated by Eq (1.2) with experimental peaks. These two chromatographic response functions seem to be qualitative and rank variables.
- Two global response functions for the separation, abbreviated as QF and DCN recorded a weak acceptance in investigation of Eq (1.1). Thus, QF are rejected at 95% confidence by Spearman (with 5.47% error), Semi-Q (6.68% error), Pearson (with 10.6% error) and Kendall τ
_{c}(with 12.6% error) even if the correlations are good (over 0.75). The small dimension of the sample size, not grater enough to provide statistical significance of the obtained correlations, explained with a good confidence the rejection of these correlations. Note that QF chromatographic response function is in fact a minimum function of resolutions of the separation, resolutions that are accepted by the model from Eq (1.1) - see 2^{nd}conclusion. DCN is statistically significant by the Goodman-Kruskal method (concordant vs. discordant) and is near to be statistically significant by the Spearman method (5.72% error). Thus, the rejection is recorded for a quantitative correlation, but a possible acceptance is seen by the qualitative correlation. Again, small sample size is against of a solid statistical conclusion for DCN. - The results presented in Experiments Quality Assessment subsection sustain the hypothesis that the proposed equations (Eq (1.1) and Eq (1.2), respectively) could be used in order to verify the quality of experimental data. The results obtained for deviations of rank sums suggest that the experimental data of the experiment no. 8 are questionable.

## Acknowledgements

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**MDPI and ACS Style**

Bolboacă, S.D.; Pică, E.M.; Cimpoiu, C.V.; Jäntschi, L.
Statistical Assessment of Solvent Mixture Models Used for Separation of Biological Active Compounds. *Molecules* **2008**, *13*, 1617-1639.
https://doi.org/10.3390/molecules13081617

**AMA Style**

Bolboacă SD, Pică EM, Cimpoiu CV, Jäntschi L.
Statistical Assessment of Solvent Mixture Models Used for Separation of Biological Active Compounds. *Molecules*. 2008; 13(8):1617-1639.
https://doi.org/10.3390/molecules13081617

**Chicago/Turabian Style**

Bolboacă, Sorana D., Elena M. Pică, Claudia V. Cimpoiu, and Lorentz Jäntschi.
2008. "Statistical Assessment of Solvent Mixture Models Used for Separation of Biological Active Compounds" *Molecules* 13, no. 8: 1617-1639.
https://doi.org/10.3390/molecules13081617