The procedures for analyzing the data were detailed comprehensively. Reference [2] aimed to predict outcomes using Model 1 with 12 molecules for MOLT3 cell lines and seven for HepG2 ones.

Case 1

The activity (IC50) values of N-Tosyl-1,2,3,4-Tetrahydroisoquinoline derivatives were measured. They utilized LOO-CV to create training and test sets, identifying outliers with standardized residual cut-offs of ±2. The modelling was performed with Weka [12] . Descriptors were computed using Gaussian 03 DFT/B3LYP/6-31(d) and Dragon, with redundant indicators removed by the Unsupervised Forward Selection algorithm. Key predictors were pinpointed employing stepwise SPSS analysis, and the main findings were summarized. Two models were involved in Case 1. The expression for Model 1 was

MOLT3 = 2.01312*Mor32u + 64.533*Gu − 12.2097 (3)

Model 2’s wording was

HepG 2 = −892.215*PJI3 + 1.1322*Mor32u − 1.048 3*Mor31v + 6.6454 (4)

Gu referred to the symmetric index, Mor32u, Mor31v, and Mor32u were 3D-MoRSE indexes, and PJI3 was the Petitjean 3D index. Case 2 aimed to predict the antiproliferative activity of 13 5-Arylidene Rhodamines and compare descriptors.

Case 2

Nine molecules were modelled for inhibiting human lung tumours (NCI-H727) and ductal carcinoma (MDA-MB-231), with four used for testing. MLR was performed with Excel and Gaussian 09. It generated IVs at the DFT/B3LYP/6-31 G(d) level. Model 3 for the MDA-MB-231 cell line had the expression of:

ActMDA (PCI50) = −459.091 76 + 1.892 18*ELUMO + 0.08176*ν_C=O + 39.433 17*d_C-N (5)

Model 4 for the NCI-H727 cell line was documented as follows:

ActNCI (PCI50) = −612.154 55 + 2.145 18*ELUMO + 0.11987*ν_C=O + 30.62007*d_C-N (6)

ELUMO represented the lowest unoccupied molecular orbital energy, ν_C=O indicated the CO frequency of the five-membered ring, and d_C-N referred to its CN distance. The third study analyzed 14 compounds related to Thiazoline and Thiazine for their antitumour properties.

Case 3

The case 3 predicted their pulmonary effects on A-549 cells. Model 5 used 10 molecules for training and four for testing with Gaussian 09 at DFT/B3LYP/6-31+G(d, p) levels. MLR in Excel calculated IVs from their expressions. Model 5 from this case was described as follows:

PIC50 = 2.26432 − 0.77981*mu + 0.44572*LogP (7)

LogP measured lipophilicity, and mu represented the molecular dipole moment.

The research undertaken by [2] [5] [6] supplied secondary data concerning both independent and dependent variables. These are organized in the Appendix across Table A1 , Table A2 , Table A3 and Table A4 . They were employed to verify that Models 1 to 5 adhered to MLR premises. Their compliance with this latter was conducted using SPSS Statistics version 27.

Cook’s distance identified outliers utilizing Analysis > Regression > Linear > Save, then selecting “Cooks” in the “Distance” box. Calculate CV with Descriptive Analyze > Descriptives to find the mean and standard deviation. The KS test assessed normality of DV via Analyze > Nonparametric > Univariate tests > KS test, with the null hypothesis assuming its normal distribution. Multicollinearity was checked by employing VIF: Analyze > Regression > Linear > Statistics > Collinearity Diagnostics. For linearity, a scatter plot of predicted values (x-axis) and residuals (y-axis) was generated: Analysis > Regression > Linear > Graphs, plotting ZRESID vs. ZPRED. In graph editor, select Elements, Fit Curve to Total, Loess; a curve oscillating around Y = 0 indicates linearity. The sample size was verified by comparing the molecules’ quantity exploited to the estimates in Equations (1) [3] and (2) [4] . Homoscedasticity of error term was examined as per [12] . The process included standard regression analysis, saving non-standardized predicted values and residuals, and performing the test. Unstandardized residuals were squared and utilized in a subsequent regression model. Achieve another one with RES_squared as the DV employing the same IVs. Realize the Breusch-Pagan test via: Analyze > Regression > Linear > Save, ensuring that Unstandardized for Predicted Values and Residuals boxes are checked, which will generate PRE1 and RES1 columns in Data View. The residuals were squared via the Transform menu: Compute Variable, naming RES_squared and entering RES1*RES1 as the formula. The p-value from the Breusch-Pagan test in the ANOVA table indicates heteroscedasticity if below 0.05. The normal distribution of residuals is assessed with a P-P graph using linear regression. The author [9] suggests configuring statistics by selecting: “Estimation,” “Hypothesis test,” and “Residual predictions.” After choosing “Histograms” and “Normal P-P Chart” in the chart menu, the PP plot is generated. Points near the diagonal suggest a normal distribution of residuals. The Durbin-Watson D. identifies error term independence [9] . References [1] and [4] describe the standard regression procedure: select the “Statistics” option and check the “Durbin-Watson” box under “Residuals.” Durbin-Watson D. close to 2 indicates independent residuals, next to 0 shows positive autocorrelation, and around 4 proposed a negative one [9] . MLR premise conformity analyses were performed for all models. The findings are presented and discussed.

In Model 1, molecule 8 (4h) had a Cook’s distance of 0.507 [1] . In Model 2, compound 4 (4h) had a statistic of 4.372, above 0.5, while molecule 6 (4k) had 0.812. These data were outliers [1] . They were removed before linear regression analysis [4] . Those for models 1 and 2 were heterogeneous, with CVs over 15% [7] . For Model 1, DV MOLT3 was 39%, IV Mor32u was 41%. In Model 2, DV HepG2 was 16%, IV Mor32u was 59%, and IV Mor31v was 19%.

The DVs and IVs in models 1 and 2 were continuous; they were quantitative with no constraints [1] [4] . DV data were normally distributed (KS test: p = 0.730). MOLT3 and HepG2 activities originated from different molecules, ensuring their independence from each other. The non-zero variance premises were violated. Furthermore, Model 1 had VIFs of 1.028. Those of Model 2 were worth 1.899 for IV Mor32u, 1.872 for Mor31v and 1.030 for PIJ3. All values were below 10, indicating an absence of multicollinearity between the IVs of the two models [1] [11] . Figure 1 indicates that Model 1 has deviated from the y=0 line, thereby compromising its linearity [9] [13] . Similarly, Figure 2 , associated with Model 2, depicts a comparable scenario.

The models used small samples. Model 1 employed 11 molecules, less than the 66 and 106 obtained with Equations (1) [3] and (2) [4] . Likewise, Model 2 exploited compounds instead of the needed 74 and 107. The Breusch-Pagan test revealed heteroscedasticity in Model 1 (F = 0.117, p = 0.891) and Model 2 (F = 0.545, p = 0.665) [9] [12] .

The points in the normal P-P plot of Model 1 didn’t align along the diagonal, as shown in Figure 3 . Similarly, those in Model 2, illustrated in Figure 4 , didn’t align either. In both cases, the residuals weren’t normally distributed around zero [9] . Model 1 produced a Durbin-Watson statistic of 2.668, while Model 2 generated 2.736. These values suggest that errors in both models are independent [8] . A summary of the results can be found in Table 1 . The models respected the same premises.

The two models followed the assumptions of continuity, normal distribution, and independence concerning the DVs. They adhered to the continuity of the IVs and avoided multicollinearity. They violated basic hypotheses related to non-zero variances of IVs, overall linearity of the model, and adequate sample size. They also transgressed premises regarding homoscedasticity and independence of the error term. The effectiveness of Models 1 and 2 is determined by their capacity to accurately predict the cytotoxic activity of N-Tosyl-Tetrahydroisoquinoline molecules.

Respect for the premises underlines this possibility. Without multicollinearity, the relationship between the DV and its IVs can be pinpointed. This precisely identifies the β_X coefficients and accounts for variations in the IVs. In concert with [1] [4] , the DVs MOLT3 and HepG2 can be exactly predicted due to their continuity, normal distribution, and independence. Independent residuals ensure accurate confidence intervals [9] [14] and parameter estimates for both models [9] . According to [9] [14] , this improves the reliability of their outcomes and forecasts. Evaluating confidence intervals correctly avoids biases in estimations and predictions. Deviation from premises reduces the projection precision of the two models.

The nonlinearity impacts β_X estimates, making it difficult to understand the relationship between HepG2 and its predictors PIJ3, Mor32u, and Mor31v, leading to unreliable projections [9] [15] . A small sample size may result in incorrect R² and β_X coefficients [1] [4] [15] [16] . It raises the variance of estimators, complicating the detection of IV effects [1] [3] . According to [9] , heteroscedasticity affects hypothesis testing and confidence intervals by misestimating variances, resulting in incorrect β_X coefficients. Non-normally distributed residuals increase the risk of type I errors, leading to inaccuracies in model predictions [1] [9] .

Violating global linearity, homoscedasticity, and normal distribution of residuals reduces the precision of projections from Models 1 and 2. Respecting multicollinearity doesn’t sufficiently compensate for these issues. The quality of each model is also evaluated based on their ability to generalize results.

The small sample size increases sensitivity to fluctuations and lower external validity, making it harder to generalize predictions to additional N-Tosyl-Tetrahydroisoquinoline molecules [1] [3] [4] . The residuals ’heteroscedasticity can lead to unstable MOLT3 and HepG2 predictions, causing incorrect conclusions about IVs variations [9] . However, their independence ensures that each model is robust and applicable to new derivatives from N-Tosyl-Tetrahydroisoquinoline compounds [17] . Generalizing the outcomes of both models is difficult despite the contribution of their quality. Models 1 and 2 don’t meet the premises, allowing the activity of the molecules studied to be accurately predicted and generalized to others in the same family. Consequently, their validity remains uncertain. Additionally, Models 3 and 4 related to 5-Arylidene Rhodanine compounds were analyzed thoroughly.

In Model 3, the DV ActMDA score for molecule 4 was an outlier with a Cook’s distance of 0.662, exceeding 0.5. It was removed [4] . Cook distances in Model 4 ranged from 0.007 to 0.236, all below 0.5. No outlier was found in DV ActNCI [1] . Furthermore, IVs data were homogeneous for each model [7] . The CV of IV ELUMO was 8%, and those of IV d_CN and νC=O were identical at 0.2%. The DVs data were heterogeneous [7] , with CVs of 57.6% for DV ActNCI and 58.8% for DV ActMDA.

The DVs data of both models exhibited a normal distribution (p = 0.200) [1] [4] . They were continuous as they were quantitative and didn’t face any limitations, such as the IVs ELUMO, ν_C=O and d_CN [4] . The DVs data originated from different 5-Arylidene Rhodanine compounds that were independent. In Models 3 and 4, ELUMO and ν_C=O displayed non-zero variances, while d_CN presented zero variance. Model 3 indicated VIFs of 1.134, and Model 4 had VIFs of 1.077, 1.659, and 1.63, all below 10, suggesting no multicollinearity [1] [11] . Figure 5 shows Model 3’s Loess line deviating significantly from zero. Figure 6 depicts Model 4’s deviation. The condition for overall linearity wasn’t met [13] . The sample sizes for the two models were insufficient, with 13 compounds instead of the required 74 or 107 as outlined by Equations (1) [3] and (2) [4] .

The Breusch-Pagan distribution tests showed no significant results, with a Fisher coefficient of F = 1.049 (p = 0.418) for Model 3 and F = 0.284 (p = 0.836) for Model 4. Neither model met the assumption of residuals’ homoscedasticity [4] [12] . Figure 7 , related to Model 3, demonstrates the non-alignment of the points on the P-P graph with the diagonal. Similarly, Figure 8 , associated with Model 4, shows the same pattern. Consequently, under these conditions, the errors for DV ActMDA and ActNCI weren’t normally distributed [4] [9] . According to [1] [9] , the Durban-Watson D. suggested positive autocorrelation (ActMDA: 1.184; ActNCI: 0.997). The residues for models 3 and 4 weren’t independent. A summary of the results can be found in Table 2 .

Models 3 and 4 met data homogeneity criteria, ensuring continuity and avoiding multicollinearity of the IVs. However, they fall to respect non-zero variances, linearity, adequate sample size, homoscedasticity, normal distribution, or independence of residual requirements. Their effectiveness is linked to their ability to accurately predict the antiproliferative activity of 5-Arylidene Rhodanine molecules. Compliance with certain assumptions underlying MLR predisposes them.

As noted by references [1] and [4] , the normal distribution and independence of DV data in both models enable precise estimation of molecular activities, enhancing consistency between dependent and independent variables. Continuity in IVs ELUMO, ν_C=O, and d_CN achieves similar outcomes. The absence of multicollinearity simplifies models 3 and 4 [18] . However, predicting the antiproliferative effects of 5-Arylidene Rhodanines remains challenging due to certain premise violations.

The non-zero variance of IVs (ELUMO, freq. CO, and d_CN) limits models 3 and 4 in assessing the accuracy of relationships between variables. Moreover, the absence of linearity can cause unclear predictions for DVs ActMDA and ActNCI [9] . As highlighted in references [9] [15] , it introduces bias into β_X coefficients. This lack of precision doesn’t significantly improve the accuracy of forecasts. Consequently, parameter estimates become incorrect, resulting in unreliable values for IV ELUMO, ν_C=O, and d_CN, which increases the risk of type II errors [1] . Thus, the determination of R² and β_X remains imprecise [15] . Heteroscedasticity reduces the variance of ActMDA and ActNCI, impacting hypothesis tests and confidence intervals similarly to β_X [9] . It may lead to false conclusions regarding the effects of IVs ELUMO, ν_C=O, and d_CN [9] . According to [9] [14] , error dependence results in incorrect parameter estimates and affects significance tests and confidence intervals, indicating unincorporated structures in both models. Furthermore, non-normally distributed residuals elevate the likelihood of type I errors, resulting in model prediction inaccuracies [1] [9] .

Models 3 and 4’s prediction accuracy is reduced due to violations of linearity, homoscedasticity, normal distribution, and independence of residuals. Additionally, their effectiveness also depends on the generalizability of the results.

The small sample size prevents extrapolating the results of the two models to another 5-Arylidene Rhodanine molecule [3] . For [9] , Heteroscedasticity may introduce instability in the effects of the IVs ELUMO, ν_C=O, and d_CN. Models 3 and 4 can’t measure variations in VI. Moreover, the error term dependence can decrease performance and restrict the generalizability of results to new molecules from 5-Arylidene Rhodanine [9] [17] . Additionally, these two models don’t accurately predict their antiproliferative activity. Hence, it’s advised that they be re-evaluated. The study further examined Model 5 in relation to Thiazine and Thiazoline compounds.

The cook’s distance for outliers ranged from 0.003 to 0.346, all below 0.5. Model 5 had none [1] . IV mu data were uniform (CV 3%), while those of DV PIC50 and IV logP weren’t (CVs 23% and 30%) [7] . DV PIC50 and its IVs were continuous [4] ; their data were quantitative and unrestricted. The KS test confirmed the DVPIC50’s normal distribution (p = 0.328) [1] [4] . Data from distinct Thiazoline and Thiazine compounds were independent. Variances were 0.013 for IV mu and 0.450 for IV LogP, meeting non-nullity assumption. VIFs were 1.072, indicating no multicollinearity [1] [11] . According to [9] [13] , the Loess curve approached zero. Figure 9 illustrates that the model exhibited overall linearity [9] [13] . With only 14 compounds studied, the sample size was insufficient as Equations (1) [3] and (2) [4] required 66 or 106 molecules. The Breush-Pagan test wasn’t significant (F = 1.546; p = 0.256), confirming the heteroscedasticity of this model [12] . The normal P-P plot in Figure 10 shows the Model 5 error distribution, but the points didn’t align with the diagonal.

Model 5 residues weren’t normally distributed around zero [4] [9] , although they were independent [1] [9] ; Durbin D. value was 1.916, indicating no autocorrelation [9] . The primary findings of this analysis are presented in Table 3 .

Model 5 met the DV PCI50 criteria for continuity, independence, and normality. It adheres to IVs continuity without multicollinearity issues. Although linearity and normally distributed error terms were followed, data homogeneity, sample size, and residuals’ homoscedasticity were inadequate. Model 5’s prediction accuracy for molecules’ antitumour activity depends on specific premises.

The continuity of the DV PIC50 and its IVs mu and LogP, along with non-zero variances and the absence of multicollinearity, facilitate a clearer understanding of the structural relationships between them [1] [4] [18] . Non-zero variances and no multicollinearity ensure small confidence intervals and reliable significance tests [1] [18] . According to [10] , Model 5’s linearity allows for appropriate correlations between variables and valid confidence intervals, which enhances forecast accuracy. Independent residuals result in precise parameter estimates [9] and confidence intervals [14] . This improves the reliability of its outcomes and forecasts. Correctly evaluated confidence intervals avoid biases in parameter estimation and predictions [1] . Nonetheless, violating several premises can restrict the model’s precision.

The limited sample size compromises the reliability of parameter estimators, rendering them inadequate for accurately representing Thiazoline or Thiazine activity [19] . According to [9] , error term’s heteroscedasticity affects the correct understanding of the effects of IVs LogP and mu. It can misestimate the variance of DV PIC50. This results in less precise βx coefficients.

The non-normality of the residual distribution impacts reliability and invalidates confidence intervals and hypothesis tests [1] [4] [9] . Model 5’s validation evaluates outcome generalizability, but the insignificant sample size may influence reliability. Changes in DV PIC50 and IVs mu and LogP increase sensitivity, affecting adaptability and predictive accuracy [3] . Results vary by Thiazoline and Thiazine studied. On the other hand, the independence of residuals makes Model 5 robust and applicable to different datasets [17] . The small sample size remains a significant concern, which hinders the generalization of the findings of Model 5 to additional molecules from Thiazine or Thiazolines [3] [4] . This limitation invalidates it. Model 5 excels in forecasting the pulmonary effects of Thiazine or Thiazoline on A-549 cells. However, it struggles to apply this prediction to similar molecules. Increasing the sample size could corroborate it. Table 4 outlines how the five models conform to MLR premises. It offers a perspective to conclude the article.