Application of Matrices, Regression, and ANOVA in the Design and Analysis of Experimental Data Using Statistical Software for Pharmaceutical Formulation Optimization

The primary goal is to demonstrate how Mathematical and Statistical methods are more appropriate to analyze the response variable using DOE (Design of Experiments) [1]. It can be applied for Matrices & Determinants, Standard Deviation, Coefficient of Variation, Regression and ANOVA. To perform this type of study, DOE is the only statistical software which can be used to Design, Perform the experiment and analyzing the result to verify the level of significance [2]. Before the existence or development of DOE, investigators were using the traditional method of changing one factor at time among the input variable within an appropriate levels by retaining the range of remaining factors as constant. But the traditional method OFAT (Integrated Factory Acceptance Test) approach fails to explain the measure of interaction between the   factors, which may lead to an inadequate conduction of the development and optimization. To overcome these types of problems, DoE has been considered as a better approach to get accurate results with few numbers of experiments [3]. DoE has become one of the main components of pharmaceutical and analytical QbD. In this paper we are providing the theoretical and practical aspect of DoE in pharmaceuticals [4]. There are different software’s such as GraphPad Prism, SPSS, R, and MATLAB are utilized for conducting statistical analyses These software packages are often found to be costly, and their utility extends beyond statistical analyses to encompass curve fitting, modeling, and graphing functionalities, which are more expensive if the user's only objective is to perform the statistical analysis [5].

Table 1: Coded Levels of Ingredients A, B, and C

If the investigator considers the factorial design in DOE to design and analyze the data of a formulation, then 2³ – Factorial design has to be chosen to perform experiments [7].

Table 2: Design of experiment without interaction and response variable.

Mathematical and statistical methods to obtain a Model to analyze the data in DOE

To analyze the experimental results, investigators should follow the matrix method to obtain the mathematical or regression model. To obtain regression model, one has to proceed as per the steps listed [8].

Step 1: Construct three matrices as per ingredients or input variables selected, response variables and regression coefficients need to be computed.

X =     =       Y =      =

Table 3: Design of experiment table with response variable.

Step 2: Obtaining the transpose Matrix of X

X^T =

Step 3: Obtaining the product of  X^TX

X^TX = *      

X^TX =  

X^TX =  

Step 4: Obtaining the co-factor matrix of X^TX

A =

Let                                                                                       

Co-factors are

    _,

_{, ,}

_{, ,}

      The  C.F matrix A =     =

Step 5: Obtaining the adjoint of X^TX matrix as explained in the

      The  C.F matrix A =     =

adj

_{Step 6: Obtaining the value of determinant A}

               According to the property of determinant

a₁₁A₁₁ + a₁₂A₁₂ + a₁₃A₁₃ = |A|

 =

Step 7: Obtaining the inverse of X^TX =(X^TX)^-1

              A = 

               =

The  C.F matrix A =     =

a₁₁A₁₁ + a₁₂A₁₂ + a₁₃A₁₃ = |A|

      1*3 + 4*6 + (-2)*9  = 9

In the given example A^-1 = adjA  =  

Step 8: Obtaining the inverse of X^TY

Step 9: Obtaining the product of (X^TX)^-1X^TY =  =

Then the regression equation can be written as Y¹ =

Step 10: Obtaining the sum of the squares due to regression, due to error and Total sum square

   Y1 + Y2 + Y3 + Y4 + Y5 + Y6 + Y7 + Y8

 =

N= Number of runs

Step 11: Developing of ANOVA table

Table 4: Construction of ANOVA table to verify the level of significance of Regression

Step 12: Analyzing the main and interaction effect of factors:

Investigator has to verify main effect and interaction effect of the ingredients which are chosen for preparing the formulation [9]. The regression equation which is obtained by applying the matrix method is taken as Y¹ = , here  ,  and   are regression coefficients of input variables. Here we have taken the regression equation without an interaction [10].

If the interaction term is statistically significant and p<0.05 the interaction term has to be considered to  analyze the response variables and also if the coefficient interaction term is very high, then the investigator has to consider the interaction effect and the model will be  Y¹ = If P > 0.05, then the interaction effect has to be removed from the model equation [11,12].

Table 5: Combinations of ingredients with interaction

The main (Average) effect of each of the factors can be obtained using the mathematical equations

A =  = 

B =  

C=  =

 here n= number taken as replication.

The interaction effects can be obtained as

AB =  =

BC =  =

AC =  =

ABC =  =

Step 13: The contrast is an important part of the experimental design. It describes the key differences that are being compared within an experiment.

The contrast of the first factor or variable A is obtained using the equation

Usually, this contrast in DOE is called as total effect. Similarly, investigators can obtain the contrast for factor B, C, AB, AC, BC and ABC.

The sum of the squares for each effect are computed, because each effect will have a single degree of freedom contrast [13 -16].

The sum of squares of the factor A =  

                                                           =

The sum of squares of the factor B=  

                                                           =

The sum of squares of the factor C=  

                                                           =

The sum of squares of the factor AB=  

                                                           =

The sum of squares of the factor BC=  

                                                           =

The sum of squares of the factor AC=  

                                                           =

The sum of squares of the factor ABC=  

                                                           =

Table 6: 2³ - Factorial Design

Table 7: The ANOVA table for the Statistical model of thickness of the response variable.

The model is significant, as indicated by the F-value of 8.18. The probability of an F-value of this magnitude by random chance is only 3.50%. Model terms are considered significant when their p-values are less than 0.0500. In this specific instance, "C" emerges as a significant model term, while "B" does not achieve significance, as its p-value exceeds 0.1000.

Table 8: Fit Statistics

The coefficient of variation is obtained as C.V.% =  

                                                                           = (51.54/424.75) = 12.13%

The coefficient of variation is 12.13%, indicating the degree of variation of the of response variables of  a particular series. If the investigators observe, the coefficient of variation is very high, indicates that the design of experiment need to be modified or not appropriate [17,18].

After doing the analysis, model was found significant. Final Equation in Terms of Actual Factors. Y = 123. 50 + 92.00X1 – 0.908X2 +7.2750X3.

The equation, expressed in terms of actual factors, enables predictions about the response for given levels of each factor. In this case, each factor levels ought to be stated in their original units. However, this equation should not be utilized to assess the relative impact of each factor due to the coefficients being scaled to accommodate the units of each factor, and the intercept not being positioned at the center of the design space [19 - 23].

Report of Actual and Predicted value are obtained using the model equation Y = 123. 50 + 92.00X₁ – 0.908X₂ +7.2750X₃.

Table 9: Regression Model Residuals

Figure 1:  Graph to how observed values scattered around the predicted values (around the straight line) in DOE.

Figure 2:  Graph to how observed values scattered around the predicted values (around the straight line) in JMP software.

 The graph which is constructed taking the observed values on X-axis and Predicted Values on Y-Axis indicated the observed /experimental values are very close to the straight line of predicted values. This is an indication to show that model is significant [24]. Distributions of response variables is normal.

Figure 3:  Graph to show the normality of the distribution.

The following conclusion was based on the above study was to explain the important applications of Matrices in developing Regression model or forming the regression equation in establishing the functional the relationship between the input variables (Ingredients which are used with different levels to prepare a formulation) & response variable (Output Variable). After developing the regression model, it is explained how we can verify the effect of each variable, the interaction effects of between the variables or ingredients and explained the importance of ANOVA to verify the level of significance of the mathematical model.

Competing Interests

The authors declare no competing interests

Authors’ Contributions

All authors made contributions to the writing of this article and have approved the final manuscript

1.     Jankovic A, Chaudhary G, Goia F. Designing the design of experiments (DOE) – An investigation on the influence of different factorial designs on the characterization of complex systems. Energy Build. 2021;250(111298):1-7.

2.     Ranga S, Jaimini M, Sanjay K, Sharma B, Singh A. A review on design OF experiments (DOE) 2014;3(1):216-224

3.     Choguill CL. The research design matrix: A tool for development planning research   studies. Habitat Int. 2005;29(4):615–26.

4.     Koukouvinos C, Seberry J. Weighing matrices and their applications. J Stat Plan Inference.  1997;62(1):91–101.

5.     Yu P, Low MY, Zhou W. Design of experiments and regression modelling in food flavour and sensory analysis: A review. Trends Food Sci Technol. 2018; 71:202–15.

6.     Khomchenko AN. Applied topics of control theory, mathematical statistics, and mathematical cybernetics: A version of the projective grid method. J Math Sci. 1993;66(5):2525–7.

7.     Tovmachenko NN, Fedorov VV. Regression analysis in experimental design problems. J  Math Sci. 1992;60(4):1607–10.

8.     Rodriguez-Granrose D, Jones A, Loftus H, Tandeski T, Heaton W, Foley KT, et al. Design of experiment (DOE) applied to artificial neural network architecture enables rapid bioprocess improvement. Bioprocess Biosyst Eng. 2021;44(6):1301–8

9.     Schubert J, Simutis R, Dors M, Havlik I, Lübbert A. Bioprocess optimization and control: Application of hybrid modelling. J Biotechnol. 1994;35(1):51–68.

A.     Sethuramiah, Rajesh Kumar. Chapter 6 - Statistics and Experimental Design in Perspective.Modeling of Chemical Wear.Elsevier.2016 Pages 1-31,

10.   Jankovic A, Chaudhary G, Goia F. Designing the design of experiments (DOE) – An investigation on the influence of different factorial designs on the characterization of complex systems. Energy Build. 2021;250(111298):111298

11.   Fan Y, Chen J, Shirkey G, John R, Wu SR, Park H, et al. Applications of structural equation modeling (SEM) in ecological studies: an updated review. Ecol Process. 2016;5(1)

12.   Fukuda IM, Pinto CFF, Moreira C dos S, Saviano AM, Lourenço FR. Design of experiments (DoE) applied to pharmaceutical and analytical quality by design (QbD). Braz J Pharm Sci. 2018;54:01006

13.   Brown AM. A new software for carrying out one-way ANOVA post hoc tests. Comput Methods Programs Biomed. 2005 Jul;79(1):1-7.

14.   J. D. Faucher and P. Maussion, "Response Surface Methodology for the Tuning of Fuzzy Controller Dedicated to Boost Rectifier with Power Factor Correction," 2006 IEEE International Symposium on Industrial Electronics, Montreal, QC, Canada, 2006, pages-1-6.

15.   Conte N, Díez E, Almendras B, Gómez JM, Rodríguez A. Sustainable recovery of cobalt from aqueous solutions using an optimized mesoporous carbon. J Sustain Met. 2023;9(1):266–79.

16.   Porto de Lima B, da Silva AF, Marins FAS. New hybrid AHP-QFD-PROMETHEE decision-making support method in the hesitant fuzzy environment: an application in packaging design selection. J Intell Fuzzy Syst. 2022;42(4):2881–97.

17.   Sartori MMP, Florentino H de O. Energy balance optimization of sugarcane crop residual biomass. Energy. 2007;32(9):1745–8.

18.   Freij A, Yuan S, Zhou H, Solihin Y. Persist level parallelism: Streamlining integrity tree updates for secure persistent memory. In: 2020 53rd Annual IEEE/ACM International Symposium on Microarchitecture. IEEE; 2020. p. 14–27.

19.   Jauregui M, Vílchez J, Chacón L. A PROCEDURE FOR MAP UPDATING USING DIGITAL MONO-PLOTTING AND DTMs. Isprs.org.

20.   Wazwaz AM. A new method for solving singular initial value problems in the second-order ordinary differential equations. Applied Mathematics and computation. 2002;10;128(1):1-13.

21.   Ghellam M, Koca İ. Autumn olive berries: characterization, antioxidant capacity, optimization of solvents and extraction conditions of lycopene. Journal of Food Measurement and Characterization. 2021:1-11.

22.   Rouberty F, Fournier J. Modelling of GC and HPLC separations of simazin and atrazin by experimental design methodology. Chromatographia. 1995;41(9-10):1-8.

23.   Beane G, Geuther BQ, Sproule TJ, Trapszo J, Hession L, Kohar V, Kumar V. Video based phenotyping platform for the laboratory mouse. bioRxiv. 2022; 18:20 22-01.

24.   Yu P, Low MY, Zhou W. Development of a partial least squares-artificial neural network (PLS-ANN) hybrid model for the prediction of consumer liking scores of ready-to-drink green tea beverages. Food Research International. 2018;103:68 - 75.

25.   Vats A, Sharma VS, Ansari I. Effects of parameters on burr heights and diametral error in dry drilling. International Journal of Additive and Subtractive Materials Manufacturing. 2017;1(3-4):223-39.

26.   Roy C, Chakrabarty J. Quality by design-based development of a stability-indicating RP-HPLC method for the simultaneous determination of methylparaben, propylparaben, diethylamino hydroxybenzoyl hexyl benzoate, and octinoxate in topical pharmaceutical formulation. Scientia Pharmaceutica. 2014;82(3):519-40.

27.   Rao Lakkimsetty N, Karunya S, Kavitha G, Ali Moosa Al Balushi N, Sulaiman Ali Al Rashdi S, Shaik F. RETRACTED: Application of waste tire derived activated carbon for the removal of heavy metals in wastewater using Response surface methodology. Mater Today. 2023.

28.   Bakri MK, Nyuk Khui PL, Rahman MR, Hamdan S, Jayamani E, Kakar A. Dielectric Properties of Acacia Wood Bio-composites. Acacia Wood Bio-composites: Towards Bio-Sustainability of the Environment. 2019:171-86.

29.   Vats A, Sharma VS, Ansari I. Effects of parameters on burr heights and diametral error in dry drilling. International Journal of Additive and Subtractive Materials Manufacturing. 2017;1(3-4):223-39.

30.   Kacan E. Exergetic optimization of basic system components for maximizing exergetic efficiency of solar combisystems by using response surface methodology. Energy and Buildings. 2015;15;91:65-82.

31.   Zhao L, Chen Y, Schaffner DW. Comparison of logistic regression and linear regression in modeling percentage data. Applied and environmental microbiology. 2001;1;67(5):2129-35.