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Research Article | | Peer-Reviewed

Performance Evaluation of Custom A-, D-, and I-Optimal Designs for Non-Standard Second-Order Models

Iwundu Mary Paschal,  Israel Chinomso Fortune*
Published in American Journal of Theoretical and Applied Statistics (Volume 13, Issue 5)
Received: 14 July 2024     Accepted: 12 August 2024     Published: 26 September 2024
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Abstract

The performances of Custom A-, D-, and I-optimal designs on non-standard second-order models are examined using the alphabetic A-, D-, and G-optimality efficiencies, as well as the Average Variance of Prediction. Designs of varying sizes are constructed with the help of JMP Pro 14 software and are customized for specified non-standard models, optimality criteria, prespecified experimental runs, and a specified range of input variables. The results reveal that Custom-A optimal designs perform generally better in terms of G-efficiency. They show high superiority to A-efficiency as the worst G-efficiency value of the created Custom-A optimal designs exceeds the best A-efficiency value of the designs, and also does well in terms of D-efficiency. Custom-D optimal designs perform generally best in terms of G-efficiency, as the worst G-efficiency value exceeds all A- and D-efficiency values. Custom-I optimal designs perform generally best in terms of G-efficiency as the worst G-efficiency value is better than the best A-efficiency value and performs generally better than the corresponding D-efficiency values. For the Average Variance of Prediction, Custom A- and I-optimal designs perform competitively well, with relatively low Average Variances of Prediction. On the contrary, the Average Variance of Prediction is generally larger for Custom-D optimal designs. Hence when seeking designs that minimize the variance of the predicted response, it suffices to construct Custom A-, D-, or I-optimal designs, with a preference for Custom-D optimal designs.

Published in American Journal of Theoretical and Applied Statistics (Volume 13, Issue 5)
DOI 10.11648/j.ajtas.20241305.11
Page(s) 92-114
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group
Previous article
Keywords

Custom A-, D-, and I-optimal Designs, Non-Standard Second-Order Model, Average Variance of Prediction, D-, G-, A-efficiency

1. Introduction
In most response surface experiments, the full second-order model is assumed to be a true approximation of the underlying mechanism. Montgomery noted that experimenters may have a foreknowledge of the process being studied, which could suggest the need for a non-standard model in Montgomery, D.C.
[13].
In like manner, Gaifman, H.
[2]
observed that one may have doubts about standard models and opt for an alternative model. According to Myers et al.
 [14]
, non-standard second-order models are reduced models which is a result of the removal of some insignificant terms in the full second-order models. Iwundu, M. P.
 [7]
observed that model fitting may reveal that not all model parameters are significant after data have been collected. Hence, there is a need to remove insignificant parameters thus resulting in a reduced model. Some literature on the Design of Experiments (DOE) sees non-standard models as models that contain different types of variables (both categorical and continuous) or models that have underlying forms that may not comply with the usual standard linear models Johnson et al.
 [9]
.
As far back as the 20th century, Kiefer and Wolfowitz
[11]
, Kiefer and Wolfowitz
[12]
introduced the use of computers to create optimal designs in most non-standard situations. Hence, with the availability of computers, statistical software, and programming languages, experimenters can go beyond the limitations of classical or standard designs to create optimal designs that align with specific experimental objectives. Warren F. K.
[19]
supported that when faced with non-standard situations, the computer can be used to generate efficient designs whereby the precision of the parameter estimate is maximized. Furthermore, when a suitable orthogonal design does not exist, computer-generated non-orthogonal designs can be used as a good alternative.
Although some standard designs are robust to model misspecification it is necessary to use a more appropriate design, computer algorithms can be used to generate optimal designs for any specified model but with strict adherence to design optimality criterion depending on the goal of the experiment Montgomery, D.C.
[13]
. Computer-generated designs (CGDs) have also shown usefulness in mixture experiments, constrained design spaces, and situations where a botched experiment needs to be salvaged Kiefer and Wolfowitz
[11]
.
As its name, computer-generated designs are designs that are generated automatically using a computer algorithm or software programs that are based on predefined statistical principles and do not require manual input from the researcher. The software has built-in mathematical optimization techniques that enable it to generate efficient experimental designs under given design conditions. Closely related to the computer-generated design is custom design which allows experimenters to carefully choose the factors, the levels, and how the treatments should be arranged to solve the research questions effectively. Custom designs are situated in most high-level statistical software such as JMP and Design Expert. They provide a platform for flexibility in the design of experiments. Custom designs involve designing an experiment either manually or with the help of computer programs, based on specific research objectives, constraints, and available resources in a way that allows experimenters to gather relevant data to draw meaningful insights. Rather than force a standard design into the space of the research problem, custom designs can be readily tailored to the problem and resource limitations Johnson et al.
 [9]
. Researchers can bring in their knowledge of the process under study and make appropriate adjustments based on practical considerations to effectively address their research questions.
According to Zhou and Xu
[22]
, an essential part of process optimization is the need to obtain an approximation that expresses the relationship between the response variable and the set of factors (independent variables). The second-order model provides a robust approximation and it is widely used for approximating a continuous underlying relationship between the response (y) and a set of experimental factors (k) Walsh et al.
[18]
. For second-order experiments, the assumed model is of the form
y= β0+ ∑i=1kβixi+ ∑i=1kβiixii2+ ∑i<j∑βijxixj+ ε(1)
where y represents the predicted response, β0 represents the model intercept, βi,  βii,  βij are the regression coefficients for the linear, quadratic, and interactive effects of the model respectively.
The error term (ε) is usually assumed to be independently and identically distributed with mean 0 and variance δ2. The variables Xi and Xj are the factors, and could be k dimensional, where k is the number of factors considered by the experimenter. The model in equation (1.1) is usually known as a p parameter model and an interest could be to obtain the best model parameter estimate containing some desirable statistical properties including unbiasedness of the estimators, minimum variance of the estimators, etc. Interestingly, these properties may be easily obtained if a good design of the experiment is achieved.
Standard models are intended models for a process under study using standard designs. However, there are certain situations when standard designs are unsuitable for expressing the relationship between the dependent variable and the independent variables in these models. Johnson et al.
[9]
observed that when the experimental problem involves unusual resource restrictions, when there is a presence of constraints on the design region, or when the model is non-standard, a standard experimental design becomes unfit to solve such research problem and hence an alternative design is required to explain the response appropriately. Smucker et al.
[17]
stated that the presence of constraints in some experiments may prevent the use of standard designs. Akinlana D. M.
[1]
described some experimental conditions where standard designs are unsuitable such as, when the model of interest is in a particular order, when a smaller sample size is required, or when there are constraints on the design region. For such situations listed by Akinlana D. M.
[1]
, Johnson et al.
[9]
, and Smucker et al.
[17]
, it is in doubt how well a standard response surface design is appropriate. Although some standard response surface designs are robust to model misspecification, there is a need for alternative designs that are more economical and efficient. For example, Akinlana D. M.
[1]
considered the estimation of multiple responses using a computer algorithm based on D-optimality and compared the performance of the design to that of a Unique Factor-Central Composite Design (UF-CCD).
To avoid misapplication of experimental designs when standard designs do not seem appropriate, it is good to opt for custom designs. Researchers can customize their designs in two ways; (i) by manually specifying the factors, levels, and other properties of the design or (ii) by using the custom design platform in some statistical software that involves specifying the factors, levels, and other experimental parameters to create a tailored experimental design. The JMP software includes a platform for custom design to provide users with the flexibility and capability to create experimental designs that are tailored to their specific research objectives and constraints Akinlana D. M
[1]
. The custom design platform constructs an optimal design to fit the research problem, taking into account one’s ability to manipulate factors, constraints on factor settings, information from covariates, and other experimental conditions and resource restrictions. To use the custom design platform in JMP, a first-step requirement is to define the goal of the experiment. Also, the optimality criterion must be specified as it helps to reflect the objective of the experiment.
The purpose of this research is to evaluate the performance of custom A-, D-, and I-optimal designs for non-standard second-order models and this will be achieved by;
i. Creating custom designs of varying sizes for non-standard second-order models based on the A-optimality criterion;
ii. Creating custom designs of varying sizes for non-standard second-order models based on the D-optimality criterion;
iii. Creating custom designs of varying sizes for non-standard second-order models based on the I-optimality criterion; and
iv. Evaluating the quality or performance of the custom designs using design efficiency metrics.
2. Methodology
2.1. The Experimental Models
This research considers three non-standard models in three input variables and one non-standard model in four input variables. The non-standard models considered are;
ŷ x1, x2, x3= β̂0+ β̂1x1+ β̂2x2+β̂3x3+ β̂11x12(2)
Source: Myers et al.
[14]
.
ŷ x1, x2, x3= β̂0+ β̂1x1+ β̂2x2+β̂3x3+ β̂23x2x3+ β̂33x32 (3)
Source: Ossia and Big-Alabo
[15]
ŷ x1,x2,x3= β̂0+ β̂1x1+ β̂2x2+ β̂3x3+ β̂12x1x2+ β̂13x1x3+ β̂11x12(4)
Source: Iwundu M. P.
 [7]
.
ŷ x1,x2,x3, x4=β̂0+β̂1x1+β̂2x2+β̂3x3+ β̂4x4+ β̂12x1x2+ β̂23x2x3+ β̂11x12+ β̂44x42(5)
Source: Iwundu and Otaru
[5]
.
When constructing custom designs using JMP Pro 14 statistical software, some general rules are followed. The rules include stating the model, defining the region of interest, selecting the number of experimental runs, choosing an optimality criterion, and from a set of candidate points, choosing the design points one would like to consider, and which should satisfy the goal of the experimentation.
2.2. Design Optimality and Design Efficiencies
Design optimality are mathematical functions that reflect the objective of the experimental designs and are closely related to Design efficiency, being a function of design optimality. As in the literature of optimal design of experiments, design efficiencies are measures used to assess, evaluate, and compare the qualities of different experimental designs. They offer experimenters insight into the performance of designs such that the most appropriate designs can be selected for a given experimental objective Iwundu and Cosmos
[8]
. In this study, the A-, D-, G-, design efficiencies, and the Average Variance of Prediction (AVP) are employed to assess the qualities of the custom A-, D-, and I-optimal designs for non-standard second-order models.
2.2.1. A-efficiency
The A-efficiency is related to minimizing the individual variances of the model parameters. It allows comparison of designs across different sample sizes and is given b
A-efficiency= 100 *ptrace N(X'X)-1  (6)
where (X'X)-1 is the variance-covariance matrix, N is the sample size, and p denotes the number of parameters in the model.
2.2.2. D-efficiency
The D-efficiency as in Goos and Jones
[3]
compares the determinant of the information matrix of a design to an “ideal” determinant related to an orthogonal design. The D−efficiency serves as a useful tool for evaluating the quality of the estimated model parameters and it is usually expressed as a percent. The D-efficiency is symbolically written as
D-efficiency=100 * X'X1pN(7)
where p denotes the number of parameters in the model, N is the sample size, and X'X is the determinant of the information matrix.
2.2.3. G-efficiency
The G-efficiency which is expressed in percentage compares the maximum value of the scaled prediction variance within the design region to its theoretical minimum variance, p Iwundu M.
[6]
. Myers et al.
 [14]
stated that “the G-efficiency emphasizes the use of designs for which the maximum scaled prediction variance, v(x) in the design region is not too large”. That is, it handles worst-case prediction variance. Iwundu M.P.
[7]
defined the G-efficiency as
G-efficiency=100  pN* SPVmax(8)
where p denote the parameters in the model, N represent the design size, and SPVmax is the maximum scaled prediction variance at any point, x̲ in the design region and is given as;
SPV=Nx̲'(X'X)-1x̲
2.2.4. I-optimal Designs and the Average Variance of Prediction (AVP)
Johnson et al.
[9] 
defined the Average Variance of Prediction as “a single measure of prediction performance that is created by the integration of the prediction variance”. The I-optimal designs are designs that minimize the Average Variance of Prediction. Hence, the smaller the value of the AVP, the better the design. According to Goos et al.
[4]
, it can be computed as
Average Variance= ∫χf'xM-1fxdx∫χdx= 1∫χdx ∙tr M-1B (9)
over the design region χ.
where B= ∫χfxf'xdx is the moment matrix over the design region.
3. Numerical Illustrations
3.1. Custom A Designs and Efficiency Values
Case 1: An Illustration using Equation (2)
Considering the three-variable non-standard second-order model having p = 5 model parameters given in Equation (2), an N-point custom design is obtained from a continuum of points Ν̃ on the design space, Ω. The design size N satisfies p ≤N ≤ Ν̃. The N points of the custom designs need not be discrete as in the case of standard designs. This allows flexibility in the choice of the points. However, if an experimenter desires to have a discrete point, a mathematical approximation to an integer-valued number is recommended. The construction of custom designs requires specifying the model, the number of factors to be included, the input variable constraints, and the number of center points. The center point is a very important factor as it helps in the estimation of pure error, thereby providing information at a minimum cost and also in detecting model adequacy or inadequacy through a test for lack of fit.
For this illustration, N=9, 10, . . . 15 well as N=27 design points are considered. The choice of each N is to allow a good understanding of the effect of small-sized designs. In this section, the custom designs are created to satisfy the A-optimality criterion. The custom A-optimal designs for N=9, 10, 11, 12, 13, 14, 15, and 27, with nc=1 are given as;
ξ9=0111-1-101-1000-1-110-11-11-10-1-1111
ξ10=1-1-1111-0.08-1-1-1110-11-11-101100001-1-1-11
ξ11=-1-1-11110110.03-111-1-1-0.031-1-11-10-1-111-1000-1-11
ξ12=1-1-1-1-11-11-11110-1-1-1-110111-1101-10000-110-11
ξ13=000111-1-1-1-11-111-10.03-111-1-1-1110-1-1-1-1101-1-0.041-1011
ξ14=1-110001-1-10.071-1111-1-1101111-1-11-10110-1-1-11-10-1-1-0.05-11
ξ15=-1-111-1100001-10-11-1-1-111-1-1111-1101-10-11-11-10-1-11-1-1011
ξ27=0-1-10-1101-10110-110-11-11-100001-1-1111-1-11-1-10-1-111-1-11-11111-11111-1-110-1-11-11011011-1-1-1-1-1-10.031-1-1-11
Case 2: An Illustration using Equation (3)
The illustration considers a three-variable non-standard model with 6 parameters given in Equation (3). For N=9, 10, . . . 15 and 27, the custom A-optimal designs for each of the N design points with nc=1 is given as;
ξ9=000111-11011-1-1-11-110-1-1-11-101-10
ξ10=110-111-11-1000-1-1-0.051-1-11-11-1-101101-1-1
ξ11=11-0.041-1-1-11-1-111110-1-101-11-11-10001-1-1-1-10
ξ12=0-11000-1-10111-110-1111101-1-111-11-10.01-11-1-1-1-1
ξ13=11-1000-1101-101-11110-11-11-1-1-1-10-1-11-111111-1-1-1
ξ14=-1-101101-1-11-10-1-10-1-1-10001-11111-110-11-111-1-111-1-11
ξ15=111110-1-10110-110-1-1-11-1-100011-11-11-1-11-1-10-11-1-1111-10
ξ27=111000-1-1-1-1-111111-1-1-11-11-11110110-11011011-11-11-1101-10-1-1-1-11-1-1-101-10-111-1-10-1-1011-1-1111-1-1-1-11
Case 3: An Illustration using Equation (4)
The illustration considers a three-variable non-standard model with 7 parameters given in Equation (4). For N=9, 10, . . ., 15, and 27, the custom A-optimal designs for the N design points with nc=1 are given as;
ξ9=0-11-110.15-1-1-101-1000111-1-111-0.35-11-11
ξ10=00001-10-111-1-0.37-111-1-1-1111-1-11-11-111-1
ξ11=-1-1111-1011-11-11110-1-1-1110001-1-1-1-1-11-11
ξ12=11-1000-1-1-1-1-11-1111-11-11-10-1-0.2911100101-11-1-1
ξ13=1-1-1-11-111-1-111-1-1-101100001-10-11111-1-111-110-1-1
ξ14=1110-110-1-100001-10111-1111-1-1-11-1-1-1-1110-1-11-1-1-11-1
ξ15=11-11-11-1-1-101-11-1-10-1-1-111-11-10-11000-1-110110-1-1111011
ξ27=11-10-1-101-10-111110-1-11-11-1-1101-1-1-1-1011-11-1-1-1-10-111-11-11-10-1-1-1110111-1-1011000-1-1111-11-1-1-111111
Case 4: An Illustration using Equation (5)
The illustration considers a four-variable non-standard model with 9 parameters, given in Equation (5).
For N=13, 14, . . ., 18, 21 and 25 the custom A-optimal designs for the N design points with nc=1 are given as;
ξ13=-1-11101-10-11-10.11110-111-111-1-11-11001110-11-100000-1-1-0.281-1-11-1-1-10
ξ14=11100-1-1-11-110-111-10110-11-10-1-1-1-0.0601010-11-111-1-1-1-111000001-101-1-11
ξ15=0-11-10-1-1-11-1-11-1-1-10-1-11101-1100000-10011-1-1-11-1001000111-111-111101-110
ξ16=0-1-10-1-1101110-1-1-11-11-10-111-11-1-101-1110-110011111-1-1000001-1101000-11-10-1-1-1
ξ17=-11-1001-1-10-1100-1-101-1-1101-11000011-100-1-10-1110011-10-1100111-1-1-1-11-11-11110-1-111
ξ18=-1-1-1-1-0.1110-11-10-111-1-11-111-11-10-1110-1-10111100000-11011-10-1-11001100-1-111-1-1001-1-101-1-1
ξ21=0000-1-1-100-110011101-10-111-101-101-1-111110-11101-1-1-1-1-111-11-110-1-1111111-11011-1-1011-10-11-10-1-10-1-1-1-1
ξ25=000011-1-11-1-1100-110-11-10-1-1-11-1-1-1111101-10-1-11101100-110-1-1-10-1-1-10-111011-10-1-11-10-111-11-11-11-1-10-1-1001-101-11-0.061-110011-1
Table 1 below shows the results of the efficiency measures and AVP values of the custom A-optimal designs for non-standard models obtained from John’s Macintosh Project (JMP) statistical software. The result shows that the A-optimal design which is a function of the A-optimality criterion had the highest A-efficiency values compared to the custom D- and I-optimal designs. The result also showed that the A-optimal design had high D- and G-efficiency values (above 50%). This means that the A-optimal designs are also D- and G- efficient. Lastly, the result of the AVP showed that the design produced smaller prediction variances compared to the D-optimal design. Hence, they are more appropriate for prediction than the custom D-optimal designs.
Table 1. Efficiency Measures of A-optimal designs for Non-standard models.

Number of factors (k)

Number of parameters (p)

Number of design points (N)

D-efficiency (%)

G-efficiency (%)

A-efficiency (%)

Average Variance of Prediction (AVP)

3

5

9

61.32132

74.07407

48.30918

0.323333

10

61.77937

71.05871

47.20342

0.298081

11

62.80609

79.03737

46.8311

0.27298

12

61.17072

69.28839

47.25415

0.25488

13

63.43927

92.31354

47.5631

0.231168

14

63.14763

60.47968

48.04881

0.208608

15

64.68813

80

48.9083

0.191468

27

65.15811

83.22195

49.24383

0.105461

3

6

9

58.11824

66.66667

47.61905

0.351111

10

58.70716

60.00001

46.51177

0.326014

11

59.90774

58.2449

46.2004

0.300606

12

61.81098

89.0614

47.27213

0.271109

13

65.1364

85.2071

49.01293

0.242778

14

63.71614

82.20551

49.09363

0.22378

15

62.64027

73.84615

49.5941

0.205357

27

65.38677

85.82375

49.74763

0.114556

3

7

9

57.44635

57.60852

43.30225

0.432184

10

61.98699

58.6284

44.39605

0.385574

11

66.98854

84.84848

46.28099

0.341667

12

64.36823

81.63543

48.55338

0.29485

13

64.6098

80.76923

50.48077

0.256667

14

62.73272

77.35849

50.10183

0.237669

15

61.21732

74.66667

50.09585

0.219246

27

64.47407

80.65844

50.9151

0.121501

4

9

13

57.28011

65.42829

42.28961

0.352008

14

56.75164

80.16152

42.78596

0.323169

15

56.19102

74.49687

43.4051

0.295676

16

56.00956

73.65331

43.93409

0.273887

17

55.95376

70.8992

44.33641

0.25519

18

56.24087

70.28544

43.77273

0.24469

21

58.53976

72.40759

43.80603

0.208551

25

57.67718

79.99164

44.21452

0.176553
3.2. Custom D Designs and Efficiency Values
Case 1
Considering the 5-parameter non-standard model in three variables stated in Equation (2). For N=9, 10, . . . 15 and 27, the custom D-optimal designs for each of the N design points are given as;
ξ9=1-11-111000-1-1-11-1111-10110-1-111-1
ξ10=11-1-111000-11-11-11-1-1-11-1-10.071-1111-0.07-11
ξ11=11-1111-11-1000-1-110-11-1-1-101-11-11-1111-1-1
ξ12=1-110-1-10-1111111-1-1111-1-1-11-1-1-1-101-1000-1-11
ξ13=1-1-101-1-11111-10111-11-1-1-10-11-1-110-1-1-11-1111000;
ξ14=-1-111-11-11-1000-11111-10110-1-10-1-1-1-1-111-1-0.0511-1-1-11-11
ξ15=1-11-11-1011-1-111111-1-10-1-1111000-1-1-1-0.04-111-1-1-1-11-11-10.051-1
ξ27=011-11-111-10-1-11-1-10-1-10001-11-111011-11-10-111-11-1-11111-1-111-11-0.0211-1-1-10-1-1-11-111111-1-1-11-1-1-11-1-101-1
Case 2
The illustration considers a three-variable non-standard model with 6 parameters given in Equation (3). For N=9, 10, . . . 15 and 27, the custom D-optimal designs for each of the N design points with nc=1 is given as;
ξ9=-1-11-1111-1-1-1-1-1111-11-11-1100011-1
ξ10=-11-1-1111-11-1-111-1-1000-1-1-111-1111110
ξ11=000-1-1-1-1101-1-1-11-11-11-1111-1011-1-1-11111
ξ12=-1-11-1-1-1-11-11-1111111-1-1101-1-1110-111000-1-11
ξ13=-1101-10-1-11-111-1-11-1-1-1-11-10001-1-11-11-1-1-111111-1
ξ14=11-1-11-1-1-1-11-11-1111-11-1-10-111-1-111-1-11101-1-1000111
ξ15=-11-1-1-10000-11-11-1-11-1-1-1111101-1111-1-111-1-111111-11-1-1-1
ξ27=011-11-111-10-1-11-1-10-1-10001-11-111011-11-10-111-11-1-11111-1-111-11-0.0211-1-1-10-1-1-11-111111-1-1-11-1-1-11-1-101-1
Case 3
The illustration considers a three-variable non-standard model with 7 parameters given in Equation (4). For N=9, 10, . . ., 15, and 27, the custom D-optimal designs for the N design points with nc=1 are given as;
ξ9=-1-1111-11110001-1-1-111-1-1-1-11-11-11
ξ10=-11-1000-1-1-1111-11111-11-1-10-11-1-111-11
ξ11=-1-1-1-11111-1011111-11-10001-110-1-1-1-111-1-1
ξ12=-1-1-111-1-1-110-11-11-101-1-1111-110001111-1-1-1-1-1
ξ13=-1-1-1-111111000-1-1-11-1-101-1-1-110-1111-11-11-11-11-1-1
ξ14=01-11-11-1110001-1-11-1-1-1-11-1-1-1-11-1-11111-1111-1-1-10-11
ξ15=0-1101-1-1-1-11111-1-11-1-1-111-1-1-1000-1-11-11111-11111-11-11-1
ξ27=-1-1-1-1-11-1111-1-1-11-1-11-10-111-111-1-1-1-1-111-1111-111-11101-101-11-110-111-1-1000-1-1-111111111-1-1-11-1-1111-1
Case 4
The illustration considers a four-variable non-standard model with 9 parameters given in Equation (5). For N=13, 14, . . ., 18, 21, and 25 the custom D-optimal designs with nc= 1 are given as;
ξ13=-1-1100-1-110-111-1-1-1-1-11-101110000011-1101-1-11-11-11-1-10011-1-1111
ξ14=-1110111-1-1-11001-11-1-1-1100001-1-1-0.040.13-11-111-10-11-1-101111-1110-1-1-1-1-11-1
ξ15=1-11-1-11-110-1111-1-10111-1-1-1-11-1-110000011-1001-1-11-1-111111-1110.08-1-1-1-1-111-1
ξ16=01-1-11-111-1111-1-1-111-1100-111-111-1-1-11-11-1-1-10-1-10111-100001110.03-1-11-0.08-11-1011-11
ξ17=11-111-1-1-1-11-10111-10-1-11-1-1-11011101-1-1-1-1-1-1-1-110-111-100001-1110-11-11-1-101110-1111
ξ18=-1-1-1101-11-1110-11-11-1-11-111-1-11-110-0.1111-1111-111111-1-1-11-1-110-1-1-1-1-1-10000011-100-111-11-1-1
ξ21=-11-1-11-11-111-10-1-1-1-1-1111-1-111111-1-1-1101-111111-1-11-1-10-1-1-10-11-111-11011101-111-1-11-1-1-110000-11101-1-10
ξ25=1-1-101-1-11011111-11111-11110-0.06-110-1-1-101-11001-11-1-1-11-11-10-111-1-1111-1-111-1-11-11-1111-11-1-11-1-111-1-10-1-1-10-1-1-111100000-11-10
Results of the efficiency measures and AVP values of custom A-optimal designs for non-standard models are tabulated in Table 2 below. From the result, it is observed that for each of the non-standard models, the custom D-optimal design which is constructed to satisfy the D-optimality criterion produces high (above 50%) D-efficiency values which increase slightly as the design size increases. This indicates that the custom D-optimal designs are D-efficient that is, they produce designs that minimize the variance and covariance of the parameter estimates. In terms of G-efficiency, the result showed that they performed very well with G-efficiency values higher than their related D-efficiency values. But, for the A-efficiencies, the result revealed low efficiency values (<50%). This means that the D-optimal design does not fare well in terms of A-efficiency. Again, the AVP values of the D-optimal design are high compared to other custom designs. This implies that the D-optimal design is not appropriate for prediction purposes.
Table 2. Efficiency Measures of Custom D-optimal designs for Non-standard models.

Number of factors (k)

Number of parameters (p)

Number of design points (N)

D-efficiency (%)

G-efficiency (%)

A-efficiency (%)

Average Variance of Prediction (AVP)

3

5

9

63.59863

74.07407

43.01075

0.603333

10

64.0722

82.85965

41.49901

0.336215

11

64.82191

77.92208

40.40404

0.313889

12

65.13999

71.9697

43.18182

0.269883

13

65.88084

92.30769

47.09576

0.22889

14

65.59811

90.40147

45.50675

0.21977

15

65.64985

93.83373

44.2516

0.210578

27

67.00405

97.65296

44.96045

0.115005

3

6

9

62.85394

53.33333

25.39683

0.697222

10

64.75482

60

34.83871

0.455556

11

66.07711

87.27273

43.63636

0.327778

12

65.6757

81.25

41.6

0.315171

13

65.66822

76.0181

39.96828

0.303042

14

65.9953

71.42857

38.6681

0.29142

15

66.61465

80

37.64706

0.280093

27

67.79991

95.2381

43.31013

0.134206

3

7

9

66.04419

62.22222

28.28283

0.711111

10

66.74042

70

37.89474

0.469444

11

66.98854

84.84848

46.28099

0.341667

12

66.51685

77.77778

44.14414

0.328968

13

66.51036

85.34107

42.47021

0.41627

14

66.70295

79.24528

41.07579

0.303968

15

67.23916

80

40

0.291667

27

68.71212

86.9281

39.11621

0.167515

4

9

13

59.49831

82.48773

39.16084

0.376323

14

59.18462

77.6559

38.06724

0.363664

15

59.39471

67.37465

32.53158

0.408748

16

59.76306

81.60601

38.02047

0.326061

17

60.1428

77.51599

36.68532

0.320008

18

60.21471

75.41108

35.99499

0.311856

21

61.47599

82.6686

34.77996

0.283406

25

61.47872

89.83834

37.93374

0.213251
3.3. Custom I Design and Efficiency Values
Case 1
Consider the three-variable non-standard second-order model having p = 5 model parameters given in Equation (2), for N=9, 10, . . . 15 and 27, the custom I-optimal designs for each of the N design points are given as;
ξ9=00011-1-11101-10110-1-11-110-11-1-1-1,ξ10=1-1-1-11-1-1-11-0.08-110-1-10111110.151-1-11-1000
ξ11=00001-10-11-1-1-1-0.15-111-1-1-1111110.081-11-1-1-11-1,ξ12=-11-10110001-11-1-1-10-11-1-110-1-1-0.18111-1-10.261-1111
ξ13=0.48-1-10-1-1-0.311111-11-111-1-1-1-10001-11011-1-1101-10-11ξ14=-11-1-1-110-11-1-110.14110.2111-1-11110-1-1-11-10-1-100001-11-1-1
ξ15=01-101-100011-10-111-110111-11-1-1-10-1-1-1-1-111-10-11-111-111ξ27=0-111-1-10-1-10001-11-11101-101-11-1-10110-1111111-10-1111-10-1-1-1-1-101-1111-1-1-1-1-1-1-111-1-11-0.0511-11-101-10-11
Case 2
The illustration considers a three-variable non-standard model with 6 parameters given in Equation (3). For N=9, 10, . . . 15 and 27, the custom I-optimal designs for each of the N design points with nc=1 is given as;
ξ9=110-11-1-111-1-10-1-101100001-111-1-1ξ10=-11-1-111110.1911-1-1-1-0.011101-1-1000-1-101-11
ξ11=000-1-10.26-111-11-111111-0.17-1-1-11-1-1-1-101101-11ξ12=1101-1111-1-1-101-100111-1-1-11-1-110.14000-1-1-1-1-11
ξ13=-1-111-10-111-11-11-111-1-111-1-110111110-1-10-1-1-1000ξ14=-111-11-11-1-1-1-1100011-1-1-10110-1-1-11-10-1-101-11-110111
ξ15=-1-11110-1-10-1-10-11-111-1-111-1-1-11-1-11-10-1101100001111-11ξ27=1-1-1-111-1-1-1-1-10-1-11-1-10-1111-10-1-10.121-11-11-11-11-1-1011-11101101100001-1-11-10-1-1-1-110-111110-11-1-110111
Case 3
Considering the three-variable non-standard model with 7 parameters, given in Equation (4), for N=9, 10, . . ., 15, and 27, the custom I-optimal designs for the N design points with nc=1 are given as;
ξ9=-1-110-1-1-1-1-11-1110.35-1000111-11-0.37011
ξ10=1-11-0.8211-11-110-101-11110-110-1-1-1-10000ξ11=-1-1-10.34110-1-1-11-1-111-1-1111-10.53-1-10110001-11
ξ12=-1-111-1-10000-11-11-11-110.18110-1-111-0.2401-1-1-1-1-111;ξ13=-1-1-111-10-1-11-1-1-11-1-1-11011-11101-10001110-111-11,
ξ14=-11-1-11100011-10-1-11-11-1-1-101-10110111110-111-1-1-1-11ξ15=0-11-1-11-11-1-1-1-10-110-1-11-1-101-111111-11-1101-1000-111011,
ξ27=-1-11-11-10-1-111-11-1-11-11111-1-1-1-1-1-111-10-110-1-101-10000110-11-1-11-11-101-11-110111-1-1111-111-1110-1-1011
Case 4
This illustration considers the 9-parameter non-standard model in four variables given in Equation (5). For N=13, 14, . . ., 18, 21 and 25 the custom I-optimal designs for the N design points with nc= 1 are given as;
ξ13=-11-10.03-1-11-0.570-1-1-1000001111-1-100-11001-10-111-1-1-1-111-111111011-1-1
ξ14=00001-11-1011-111-11-1-110-111101-1-11-1-100-1-10.420100-1-1-1-1-11-1-0.0411100-111
ξ15=-1-1-111-11101110-100-111-10000-11-1001000-1-1-11-1-1011100-11-111-1-101-11-1-110ξ16=01-10-11110-11-1000001100-1-11-1-11001000-1111-110-11-1-111-11-1-1-10111-10-1-1-11-1-10
ξ17=01-1-1-1111000001-100-1-10011-10-1-1111-11-11-101110-1-1-101-1-1-10-11001100-1111-110-1-11-1,
ξ18=1-1-1-101-1111-1-1-1-1100-11-1-11-100-110-0.25-1-1-1-1-1-110-1-1001110000-111-111101-1111-1-10011001-10ξ21=11-100-11-111-1-101-1-11-11-101-111-111-1-1110-1-11-11-10011011110-1-100110-111-1-1-1100000-1-1-1-11-1-100-110-11-11
ξ25=0000-11-10-1-1-1-1111001-1101-1-1-1-1111-11011-1-111-100-1-10-11-1001100111-1-1110-110-0.17-1-100-1-101-11-1-111-11-1-110-11-1011001-111-1-11
Table 3 below shows the result of the efficiency measures and AVP values of custom I-optimal designs for non-standard models. From the result, it is observed that the I-optimal design performed well (above 50%) in terms of D- and G-efficiency. It also performed well in terms of A-efficiency and the values in most cases are similar to that of A-optimal design. Thus, it can be said that the I-optimal designs are D, A, and G efficient. Also, from the result, it was observed that the I-optimal designs had the smallest average variance of prediction compared to other custom designs thus, making it a good choice for prediction.
Table 3. Efficiency Measures of I-optimal designs for Non-standard models. Efficiency Measures of I-optimal designs for Non-standard models. Efficiency Measures of I-optimal designs for Non-standard models.

Number of factors (k)

Number of parameters (p)

Number of design points (N)

D-efficiency (%)

G-efficiency (%)

A-efficiency (%)

Average Variance of Prediction (AVP)

3

5

9

61.32132

74.07407

48.30918

0.323333

10

61.81643

66.77774

46.9793

0.297374

11

62.87172

66.42324

46.73369

0.272618

12

61.87742

72.80956

45.73915

0.249532

13

61.96404

69.21453

46.55971

0.226197

14

63.18278

85.76883

47.71272

0.208198

15

64.68813

80

48.9083

0.191468

27

65.15931

83.36667

49.24094

0.105458

3

6

9

58.11824

66.66667

47.61903

0.351111

10

58.86504

59.82963

46.20002

0.324658

11

60.30197

58.83521

45.5094

0.297414

12

61.88824

89.08194

47.11375

0.270582

13

65.1364

85.2071

49.01293

0.242778

14

63.71614

79.12088

49.09363

0.22378

15

62.64027

73.84615

49.5941

0.205357

27

62.43969

68.53836

49.21173

0.113965

3

7

9

57.69692

51.59114

43.3002

0.429518

10

53.80632

53.15039

41.48247

0.378536

11

58.18175

54.23887

43.80323

0.328886

12

59.70504

84.75852

46.83452

0.292052

13

64.6098

80.76923

50.48077

0.256667

14

62.73272

77.35849

50.10183

0.237669

15

61.21732

70

50.09585

0.219246

27

64.47407

77.77778

50.9151

0.121501

4

9

13

57.21456

77.00078

41.93254

0.350826

14

56.7514

77.71704

42.55255

0.322023

15

56.19102

74.49687

43.4051

0.295676

16

56.00956

73.65331

43.93409

0.273887

17

55.95376

70.8992

44.33641

0.25519

18

56.33369

68.83533

43.70558

0.244087

21

58.53976

72.40759

43.80603

0.208551

25

57.08659

66.10358

43.44466

0.175454
4. Discussion of Findings
Based on the computation of the efficiency properties of the custom D-, A-, and I-optimal design for the non-standard models, it is seen that the D, G, and A-efficiency values of the custom designs are less than 100%. This is in line with Kiefer and Wolfowitz
[11]
who proposed that the optimality criteria consider experimental design as an N-point design. Therefore, optimal designs will generally be less than 1.0 (or 100%).
Again, from the results, it is observed that without much loss in efficiency, small-size designs are as efficient as large-size designs. For example, the custom A 13-point design for the 5-parameter non-standard model has the highest G-efficiency of 92.31354%, D-efficiency of 63.43927%, A-efficiency of 47.5631%, and AVP of 0.231168. In comparison, the custom A 27-point design has a G-efficiency of 65.15811%, D-efficiency of 83.22195%, A-efficiency of 49.24383%, and AVP of 0.105461. Hence, small-size economical designs can be selected to meet specific research objectives.
For custom A-optimal designs, the result in Table 1 showed that the custom A-optimal designs had the highest A-efficiency value compared to other custom designs. This owes to the fact that the design is created based on the A-optimality criterion thus, they are A-efficient. In terms of D-efficiency, it was observed that the values were moderately high but less than those of a D-optimal design. This implies that custom A-optimal designs are also D-efficient. Hence, they can be used to identify both the main factors and interactions in second-order experiments. Again, custom A-optimal design is efficient in terms of G-efficiency. This is in contrast with Wong W.K.
 [20]
. This implies that custom A-optimal designs minimize the worst-case prediction variance. Lastly, the result of the Average Variance of Prediction (AVP) showed a smaller variance compared to that of the D-optimal design for each of the design points. Hence, the custom A-optimal design is more suitable for prediction than the custom D-optimal design. Therefore, for the non-standard models, we can say that the custom A-optimal design is D-efficient, G-efficient, and A-efficient, and is more suitable for prediction than the custom D-optimal design.
The result in Table 2 showed that the custom D-optimal design which was created based on the D-optimality criterion had the highest D-efficiency values compared to other custom designs. This implies that custom D-optimal designs are good at estimating all the parameters of interest in the model. Also, it was observed that the custom D-optimal design produced the best G-efficiency (60% and above) compared to other custom designs. This means that the custom D-optimal designs are G-optimal hence, minimize the worst-case prediction variance. This corresponds with the work of Kiefer and Wolfowitz
[11]
who proposed that a D-optimal design is also G-optimal.
In terms of A-efficiency and Average Variance of Prediction (AVP), the result revealed that the D-optimal design does not fare well. This is contrary to the work of Wong W. K.
[20]
that D-optimal designs are A-efficient. The A-efficiency values were generally less than 50% this implies that the D-optimal design is not an appropriate design for identifying only the significant factors in a model. Also, the AVP values were high compared to other custom designs. Indicating high prediction variance thereby, making the D-optimal design unsuitable for prediction purposes.
Table 3 shows the efficiency and AVP values of the I-optimal design. From the result, it is observed that custom I-optimal design performed moderately well (above 50%) in terms of D- and G-efficiency. In terms of A-efficiency, they performed as well as the A-optimal design. This corresponds with the work of Jones and Goos
[10]
, and Rady et al.
 [16].
For the AVP values, it is observed that the custom I-optimal design had the lowest values compared to custom D- and A-optimal designs. This corresponds with the works of Johnson et al.
 [9]
, Jones and Goos
[10]
, and Yeh et al.
 [21]
, that the I-optimal design has a lower average prediction variance than the D-optimal design. This justifies the fact that the I-optimality criterion seeks designs that minimize the Average Variance of Prediction. Generally, for non-standard models, the I-optimal designs performed well in terms of D-efficiency, G-efficiency, and A-efficiency and have the smallest Average Variance of Prediction.
From the results of the analysis, it is appropriate to describe custom designs as efficient designs for fitting second-order non-standard models. As earlier mentioned, custom designs are suitable in situations that involve non-standard models, fewer experimental runs, and irregularly shaped design regions where standard designs are not suitable to use or not economical to utilize. The efficiency value is a function of the optimality criterion. The custom D-optimal design performed best in terms of D- and G-efficiency. Custom A-optimal design had the best A-efficiency value. The custom I-optimal design had the smallest average variance of prediction.
5. Conclusion
The efficiency of custom D-, A-, and I-optimal designs have been thoroughly examined and demonstrated in this research work in second-order non-standard models under varying design points. This study is applied mostly in situations where the experimenter knows the model. Also, in situations where some constraints (such as time, materials, and resources) may limit the use of commonly known classical designs. Thus, custom designs can be used in non-standard models to effectively construct designs that align with the goal of the experiment and available resources.
Design efficiency metrics such as D-, G-, A-efficiency, and AVP are functions of design optimality criterion and they help experimenters evaluate the quality of designs based on the specific goal and objectives of the experiment even before the experiment is conducted.
Abbreviations

AVP

Average Variance of Prediction

DOE

Design of Experiment

CGD

Computer-generated Designs

JMP

John’s Macintosh Project

SPV

Scaled Prediction Variance
Author Contributions
Iwundu Mary Paschal: Datacuration, Formal Analysis
Israel Chinomso Fortune: Methodology, Resources
Conflicts of Interest
The authors declare no conflicts of interest.
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    Paschal, I. M., Fortune, I. C. (2024). Performance Evaluation of Custom A-, D-, and I-Optimal Designs for Non-Standard Second-Order Models. American Journal of Theoretical and Applied Statistics, 13(5), 92-114. https://doi.org/10.11648/j.ajtas.20241305.11

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    Paschal, I. M.; Fortune, I. C. Performance Evaluation of Custom A-, D-, and I-Optimal Designs for Non-Standard Second-Order Models. Am. J. Theor. Appl. Stat. 2024, 13(5), 92-114. doi: 10.11648/j.ajtas.20241305.11

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    AMA Style

    Paschal IM, Fortune IC. Performance Evaluation of Custom A-, D-, and I-Optimal Designs for Non-Standard Second-Order Models. Am J Theor Appl Stat. 2024;13(5):92-114. doi: 10.11648/j.ajtas.20241305.11

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  • @article{10.11648/j.ajtas.20241305.11,
  author = {Iwundu Mary Paschal and Israel Chinomso Fortune},
  title = {Performance Evaluation of Custom A-, D-, and I-Optimal Designs for Non-Standard Second-Order Models
},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {13},
  number = {5},
  pages = {92-114},
  doi = {10.11648/j.ajtas.20241305.11},
  url = {https://doi.org/10.11648/j.ajtas.20241305.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20241305.11},
  abstract = {The performances of Custom A-, D-, and I-optimal designs on non-standard second-order models are examined using the alphabetic A-, D-, and G-optimality efficiencies, as well as the Average Variance of Prediction. Designs of varying sizes are constructed with the help of JMP Pro 14 software and are customized for specified non-standard models, optimality criteria, prespecified experimental runs, and a specified range of input variables. The results reveal that Custom-A optimal designs perform generally better in terms of G-efficiency. They show high superiority to A-efficiency as the worst G-efficiency value of the created Custom-A optimal designs exceeds the best A-efficiency value of the designs, and also does well in terms of D-efficiency. Custom-D optimal designs perform generally best in terms of G-efficiency, as the worst G-efficiency value exceeds all A- and D-efficiency values. Custom-I optimal designs perform generally best in terms of G-efficiency as the worst G-efficiency value is better than the best A-efficiency value and performs generally better than the corresponding D-efficiency values. For the Average Variance of Prediction, Custom A- and I-optimal designs perform competitively well, with relatively low Average Variances of Prediction. On the contrary, the Average Variance of Prediction is generally larger for Custom-D optimal designs. Hence when seeking designs that minimize the variance of the predicted response, it suffices to construct Custom A-, D-, or I-optimal designs, with a preference for Custom-D optimal designs.
},
 year = {2024}
}

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  • TY  - JOUR
T1  - Performance Evaluation of Custom A-, D-, and I-Optimal Designs for Non-Standard Second-Order Models

AU  - Iwundu Mary Paschal
AU  - Israel Chinomso Fortune
Y1  - 2024/09/26
PY  - 2024
N1  - https://doi.org/10.11648/j.ajtas.20241305.11
DO  - 10.11648/j.ajtas.20241305.11
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 92
EP  - 114
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20241305.11
AB  - The performances of Custom A-, D-, and I-optimal designs on non-standard second-order models are examined using the alphabetic A-, D-, and G-optimality efficiencies, as well as the Average Variance of Prediction. Designs of varying sizes are constructed with the help of JMP Pro 14 software and are customized for specified non-standard models, optimality criteria, prespecified experimental runs, and a specified range of input variables. The results reveal that Custom-A optimal designs perform generally better in terms of G-efficiency. They show high superiority to A-efficiency as the worst G-efficiency value of the created Custom-A optimal designs exceeds the best A-efficiency value of the designs, and also does well in terms of D-efficiency. Custom-D optimal designs perform generally best in terms of G-efficiency, as the worst G-efficiency value exceeds all A- and D-efficiency values. Custom-I optimal designs perform generally best in terms of G-efficiency as the worst G-efficiency value is better than the best A-efficiency value and performs generally better than the corresponding D-efficiency values. For the Average Variance of Prediction, Custom A- and I-optimal designs perform competitively well, with relatively low Average Variances of Prediction. On the contrary, the Average Variance of Prediction is generally larger for Custom-D optimal designs. Hence when seeking designs that minimize the variance of the predicted response, it suffices to construct Custom A-, D-, or I-optimal designs, with a preference for Custom-D optimal designs.

VL  - 13
IS  - 5
ER  -

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Author Information

  • Iwundu Mary Paschal

    Department of Mathematics and Statistics, University of Port Harcourt, Port Harcourt, Nigeria

    Research Fields: Iwundu Mary Paschal: Modelling and Response Surface opti-mization, Sequential design of experiments, Missing observations and loss functions, Optimal design and efficiency, Exploration of nonstandard model, Development of search algorithms for global optimization.

    Contact Email

    http://orcid.org/0000-0001-6770-131X

  • Israel Chinomso Fortune

    Department of Mathematics and Statistics, University of Port Harcourt, Port Harcourt, Nigeria

    Research Fields: Israel Chinomso Fortune: Optimal experimental design, Exploration of Non-standard Models, Design Optimality and efficiency, Response Surface Methodology, Design of Ex-periments, Statistics.

    Contact Email

    http://orcid.org/0009-0004-8923-1362

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