a-comvar: a flexible extension of common variance designs · 21 1 introduction 22 fractional...

A-ComVar: A Flexible Extension of Common Variance Designs1

Shrabanti Chowdhury

Department of Genetics and Genomic Sciences

Icahn School of Medicine at Mount Sinai

New York, NY 10029

2

Joshua Lukemire

Department of Biostatistics and Bioinformatics

Emory University

Atlanta, GA 30322

Abhyuday Mandal

Department of Statistics

University of Georgia

Athens, GA 30606

3

March 22, 20194

Abstract5

We consider nonregular fractions of factorial experiments for a class of linear models. These6

models have a common general mean and main effects, however they may have different 2-7

factor interactions. Here we assume for simplicity that 3-factor and higher order interactions8

are negligible. In the absence of a priori knowledge about which interactions are important, it9

is reasonable to prefer a design that results in equal variance for the estimates of all interaction10

effects to aid in model discrimination. Such designs are called common variance designs and can11

be quite challenging to identify without performing an exhaustive search of possible designs. In12

this work, we introduce an extension of common variance designs called approximate common13

variance, or A-ComVar designs. We develop a numerical approach to finding A-ComVar designs14

that is much more efficient than an exhaustive search. We present the types of A-ComVar15

designs that can be found for different number of factors, runs, and interactions. We further16

demonstrate the competitive performance of both common variance and A-ComVar designs with17

Plackett-Burman designs for model selection using simulation.18

Keywords: Class of Models, Model Identification, Common Variance, Placket-Burman, Adap-19

tive Lasso, Approximate Common Variance, Genetic Algorithm20

1

1 Introduction21

Fractional factorial designs are widely used in many scientific investigations because they provide22

a systematic and statistically valid strategy for studying how multiple factors impact a response23

variable through main effects and interactions. When several factors are to be tested, often the24

experimenter does not know which factors have important interactions. Instead, the experimenter25

will need to perform model selection after conducting the experiment to identify important interac-26

tions. Generally this process will involve fitting different models under consideration and examining27

statistical significance of the interaction terms. Some techniques have been developed concerning28

finding efficient fractional factorial plans for this purpose. There is a rich literature on identifica-29

tion and discrimination to find the model best describing the data (Srivastava, 1976; Srivastava30

and Ghosh, 1976; Srivastava and Gupta, 1979).31

Borrowing notation from Ghosh and Chowdhury (2017), consider the following class of s candi-

date models for describing the relationship between p factors and the n× 1 vector of observations

y,

E (y) = β0jn +X1β1 +X(i)2 β2i, i = 1, . . . , s (1)

V ar(y) = σ2I,

where n is the number of runs, β0 is the general mean, jn is a vector of ones, β1 is the vector of32

p main effects that are common in all s models. The other parameters, β2i, are specific for the33

ith model and hence β2i 6= β2i′ for i 6= i′, i = 1, . . . , s. We call these parameters “uncommon34

parameters.” The design matrices X1 and X(i)2 correspond to the main effects and ith set of two35

factor interactions, respectively.36

Under the above setup, model selection consists of identifying the correct i from the s candidate37

models. This process is complicated by the fact that the variance estimates for the uncommon38

parameters are generally not the same, which can pre-bias the experiment towards identifying39

certain interactions as significant over others, i.e. making some i more likely to be selected than40

others regardless of the true underlying model. To address this issue, Ghosh and Flores (2013)41

introduced the notion of common variance designs for a single uncommon parameter. These designs42

2

estimate the uncommon parameter in all models with equal variance, which is desirable in the43

absence of any a priori information about the true model. Ghosh and Chowdhury (2017) generalized44

this concept of common variance to k (k ≥ 1) uncommon parameters in each model in the class.45

Under the situation of k > 1, Ghosh and Chowdhury (2017) defined a common variance design to46

be the one satisfying |X(i)′X(i)| to be a constant, for all i, X(i) =(jn,X1,X

(i)2

).47

The concept of variance-balancedness is not totally new. Different types of “variance-balanced48

designs” estimating all or some of the treatment-contrasts with identical variance were developed49

by Calvin (1986), Cheng (1986), Gupta and Jones (1983), Hedayat and Stufken (1989), Khatri50

(1982), Mukerjee and Kageyama (1985), among others.51

While common variance designs have been identified for 2 and 3 level factorial experiments with52

a single 2-factor interaction (Ghosh and Flores, 2013; Ghosh and Chowdhury, 2017), it remains to53

develop a method which can find them for general number of factors and interactions. To date,54

these designs have been found using exhaustive searches, which becomes prohibitively expensive55

as the number of factors and runs increases. This leads us to introduce approximate common56

variance (A-ComVar) designs, which relax the requirement that the variance of the uncommon57

parameters be exactly equal. We introduce an objective function that allows us to rank designs58

under consideration, and we develop a genetic algorithm for searching for these designs. Moreover,59

we investigate the performance of both common variance and A-ComVar designs for model selection60

using the adaptive lasso regression technique in simulation (Kane and Mandal, 2019). We find61

comparable performance of common variance and A-ComVar designs to Placket-Burman designs,62

which further demonstrates the usefulness of designs that prioritize having a similar variance for63

the uncommon parameters in the model.64

The rest of the article is organized as follows. In Section 2 we present the current state of65

knowledge for both two level and three level common variance designs. For three level designs66

we also present the exhaustive search result for m = 3. In Section 3 we illustrate the advantage67

of common variance designs for model selection via a comparison of the performance of common68

variance designs with some popular non-regular designs such as Plackett-Burman designs using69

simulations and the adaptive lasso regression technique. In Section 4 we introduce our numerical70

approach for finding A-ComVar designs. In Section 5 we conduct numerical simulations examining71

the algorithm’s ability to find common variance designs as we increase the number of factors and72

3

number of interactions in the model. We also conduct simulations comparing the model selection73

capabilities of A-ComVar designs to Placket-Burman designs. We conclude the article with some74

discussion in Section 6.75

2 Common Variance Designs76

2.1 2 Level Designs77

The term “common variance” for the class of variance balanced designs was first introduced in78

Ghosh and Flores (2013). As a more stringent criteria, the authors also introduced the concept79

of optimum common variance (OPTCV), which is satisfied by designs having the smallest value80

of common variance in a class of common variance designs with m factors and n runs. Several81

characterizations of common variance and optimal common variance designs were presented that82

provide efficient ways for checking the common variance or OPTCV property of a given design.83

These characterizations were obtained in terms of the projection matrix, eigenvalues of the model84

matrix, balancedness, and orthogonal properties of the designs. In Corollary 1 of Ghosh and Flores85

(2013), they stated one sufficient condition of common variance designs in terms of equality of the86

vectors of eigen values of X(i)′X(i), X(i) =(jn,X1,X

(i)2

), for all i. We present one design in87

Table 2 from Ghosh and Flores (2013) for m = 5 and n = 12, that has identical vectors of eigen88

values for all i. In Section 3 we compare the performance of this particular design with that of89

Plackett-Burman designs for model selection to demonstrate further usefulness of such designs.90

In their work, Ghosh and Flores (2013) presented several general series of designs with the91

common variance property. For example, they identified two fold-over designs with the common92

variance property with m factors and n = 2m and n = 2m+ 2 runs respectively:93

d(2m)m =

2Im − Jm

−2Im + Jm

.

4

d(2m+2)m =

j′m

−j′m

2Im − Jm

−2Im + Jm

.

As reported in Ghosh and Flores (2013), both of these designs are balanced arrays of full strength94

and orthogonal arrays of strength 1, ∀ m. Moreover, the design d(2m)m is OPTCV for m = 4 and95

d(2m+2)m is OPTCV for m = 3.96

2.2 3 Level Designs97

Ghosh and Chowdhury (2017) presented common variance designs for 3m fractional factorial exper-98

iments. Consider the following model for a 3m factorial experiment, with one two-factor interaction99

effect in the model, i.e. k = 1:100

E (y) = β0jn +X1β1 +X(i)2 β2i, V ar(y) = σ2.

A design for such an experiment would have the common variance property iff V ar(β2i)σ2 is constant101

for all i = 1, . . . ,(4m2

).102

Ghosh and Chowdhury (2017) presented two general series of 3m fractional factorial common103

variance designs d1 and d2 with n runs. The design d1 has a common variance value given by104

V ar(β(i)2

)σ2 = 2−m+m2

9 , for m ≥ 2 and n = 2m + 2 runs, while design d2 has a common variance105

value given byV ar

(β(i)2

)σ2 = m

9(m−2) , for m ≥ 3 and n = 3m. Also, the design d1 is efficient common106

variance (ECV, as termed in Ghosh and Chowdhury (2017)) design for m = 2, and design d2 is107

ECV for m = 3.108

Ghosh and Chowdhury (2017) also presented several sufficient conditions for general fractional109

factorial designs to have the common variance property, including the special case for 3m designs110

in terms of the projection matrix of the design and the columns of two-factor interaction. For111

example, a design is common variance if (i) PX(i1)2 = PX

(i2)2 , for i1, i2 ∈ {1, . . . , s}, where P is the112

projection matrix defined as In −X1 (X ′1X1)−1X ′1, and X1 contains the columns corresponding113

to the general mean and main effects from the model matrix X(i) =(jn,X1,X

(i)2

), and X

(i)2114

5

corresponds to the ith two-factor interaction, i = 1, . . . ,(4m2

). Another set of sufficient conditions115

for having common variance is, for, i1, i2 ∈ {1, . . . , s}, (i)(X

(i1)2 ±X(i2)

2

)belongs to the column116

space of X1 and (ii) X(i2)2 = FX

(i1)2 holds, where the permutation matrix F obtained from the117

identity matrix satisfies F ′PF = P .118

For 33 fractional factorial experiment, Chowdhury (2016) conducted a complete search of com-119

mon variance designs for n = 8 to n = 27, since n = 8 is the minimum number of runs needed to120

estimate all the parameters (one general mean + 6 main effects + one 2-factor interaction effect).121

The results of this search are presented in Table 1. The complete search revealed that common122

variance designs only exist for n = 8, 9, 10, 11 for 33 factorial experiments. For each of the runs123

multiple groups of common variance designs were obtained, having different common variance val-124

ues, among which 32 designs for n = 11; 48 designs for n = 10; 8256 designs for n = 9; and 9600125

designs for n = 8, are the efficient common variance designs giving the minimum value of common126

variance in the respective classes.127

Table 1: Complete search results for finding common variance designs for 33 factorial experiments.

n Possible Designs =(27n

) Satisfying Rank No. of No. of GroupsCondition Non-CV designs CV designs No. of CV designs in each group CV value

11 13,037,895 6,926,898 6,924,772 2,096 32 0.21512,064 0.2222

10 8,436,285 2,792,387 2,775,747 16,640 48 0.256448 0.266716 0.2837

16,512 0.296316 0.4

9 4,686,825 636,348 588,348 48,000 8,256 0.333332 0.381

13,056 0.416726,640 0.4444

16 0.58 2,220,075 49,628 23,340 26,288 9,600 0.6667

16,688 0.8889

3 Common Variance Designs for Model Selection - An Example128

In this section we use simulation studies to demonstrate the advantages of having common variance129

designs by comparing the performance of the design d(12)5 in Table 2 with two sets of five factors130

from a Plackett-Burman design (Plackett and Burman, 1946) given by factors A−E and F − J in131

Table 3.132

6

Table 2: Design d(12)5 with common variance for 5 factors and 12 runs used for comparison with

the Plackett-Burman design.

A B C D E

-1 -1 -1 -1 11 1 -1 -1 -11 -1 1 -1 -11 -1 -1 1 -1

-1 1 1 -1 -1-1 1 -1 1 -1-1 -1 1 1 -11 1 1 1 -11 1 1 -1 11 1 -1 1 11 -1 1 1 1

-1 1 1 1 1

Table 3: Plackett-Burman Design with 11 factors and 12 runs

A B C D E F G H I J K

1 1 -1 1 1 1 -1 -1 -1 1 -11 -1 1 1 1 -1 -1 -1 1 -1 1

-1 1 1 1 -1 -1 -1 1 -1 1 11 1 1 -1 -1 -1 1 -1 1 1 -11 1 -1 -1 -1 1 -1 1 1 -1 11 -1 -1 -1 1 -1 1 1 -1 1 1

-1 -1 -1 1 -1 1 1 -1 1 1 1-1 -1 1 -1 1 1 -1 1 1 1 -1-1 1 -1 1 1 -1 1 1 1 -1 -11 -1 1 1 -1 1 1 1 -1 -1 -1

-1 1 1 -1 1 1 1 -1 -1 -1 1-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

Our simulation sought to measure how well each design could identify the true underlying133

model, which corresponds to identifying the significant active effects in the model without also134

including any inert effects. To do this, we considered both “big” (B) and “small” (S) effects for135

a series of different models of varying complexity. For the “big” effect we randomly generated136

the coefficient from U(0.5, 1.5), and for the “small” effect the coefficients are randomly generated137

from U(0.1, 0.3). For each design under consideration, we generated the response values from the138

specified model. For example, to fix ideas consider the model specified by the 9th row of Table139

4. For this simulation, data were generated from y = β0 + β1F1 + β2F2 + β12F1F2 + ε, where F1140

and F2 are the factors A and B from Table 2 for the common variance design and Table 3 for the141

7

Plackett-Burman design. We used the adaptive lasso (Kane and Mandal, 2019) to fit the model.142

The model was then said to be correctly identified if the only significant effects were for F1, F2, and143

their interaction. We chose the adaptive lasso method of (Kane and Mandal, 2019) because they144

showed that this technique is suitable for identifying the correct model for designs with complex145

aliasing and that it outperforms other popular variable selection methods including the Dantzig146

Selector (Candes et al., 2007), LARS (Yuan et al., 2007), and the Nonnegative Garotte estimator147

(Breiman, 1995; Yuan et al., 2009). We repeated this process 100 times for each model setup and148

reported the percentage of times the true underlying model was correctly identified.149

Tables 4 and 5 display the percentage of times the correct model is identified by the common150

variance design d(12)5 and the Placket-Burman design presented in Table 3, under different factorial151

effect size combinations and varied choices of the variance (σ2) of the noise. In Table 4 we compare152

the performance of model selection between d(12)5 and the design with factors A − E from the153

Plackett-Burman design, while in Table 5 we compare between d(12)5 and the design with factors154

F − J from the same design given in Table 3.155

As we observe from Tables 4−5, the model selection performance of the two designs are fairly156

comparable across different models and sizes. However, we do observe that as the model variance157

σ2 increases the percentage of correct model selection by d(12)5 is much higher than that of the158

Plackett-Burman designs, and this difference is quite striking for larger variances.159

4 Identifying Common Variance Designs160

4.1 Challenges in Numerically identifying Common Variance Designs161

Ghosh and Flores (2013) and Ghosh and Chowdhury (2017) presented some general series of designs162

satisfying the common variance property for two and three level factorial experiments obtained163

via exhaustive searches of the design space. Such searches become extremely computationally164

challenging as the number of factors increases. For example, for a 33 factorial experiment with one165

2-factor interaction (k = 1) the possible set of candidate designs with 8 runs is(278

)= 2220075, with166

9 runs is(279

)= 4686825, with 10 runs is

(2710

)= 8436285, and so on. For a 34 factorial experiment,167

the cardinality of this set increases to(8110

)= 1.878392×1012, even for the designs with the smallest168

possible number of runs. This rapid growth in the size of the search space makes exhaustive searches169

8

Table 4: Percentage of Correctly Identified Models for design d(12)5 and Plackett-Burman design

(PB) with factors A− E.

Model Size0.1 0.25 0.5 0.75 1

CV PB CV PB CV PB CV PB CV PB

1 F1 B 100 100 0 1 22 34 14 5 6 72 F1 S 92 86 0 1 16 6 99 100 5 03 F1 + F2 B +B 60 68 100 100 3 5 26 27 1 04 F1 + F2 B + S 56 60 92 81 2 2 9 3 100 625 F1 + F2 S + S 56 49 53 55 65 76 5 8 55 36 F1 + F1F2 B +B 82 88 30 26 30 10 6 0 9 17 F1 + F1F2 B + S 9 24 49 27 11 6 25 45 4 08 F1 + F1F2 S + S 6 15 100 97 17 16 12 15 1 19 F1 + F2 + F1F2 B +B +B 3 10 26 15 7 4 3 1 94 410 F1 + F2 + F1F2 B +B + S 3 3 6 13 89 96 0 0 9 211 F1 + F2 + F1F2 B + S +B 100 100 6 2 17 9 0 0 5 012 F1 + F2 + F1F2 B + S + S 70 68 4 1 2 6 91 55 3 013 F1 + F2 + F1F2 S + S + S 49 44 92 88 2 0 7 9 0 014 F1 + F2 + F3 B +B +B 20 22 48 25 0 0 0 1 85 1215 F1 + F2 + F3 B +B + S 19 16 11 8 89 84 0 0 10 416 F1 + F2 + F3 B + S + S 100 39 5 2 15 7 0 0 5 017 F1 + F2 + F3 S + S + S 14 10 1 0 2 2 100 94 0 018 F1 + F2 + F1F3 B +B +B 11 11 100 100 0 1 79 68 0 119 F1 + F2 + F1F3 B +B + S 8 1 85 87 0 1 38 5 87 3320 F1 + F2 + F1F3 B + S +B 2 2 57 36 100 100 13 0 7 121 F1 + F2 + F1F3 B + S + S 86 79 24 25 74 68 5 1 0 022 F1 + F2 + F1F3 S +B + S 14 20 28 1 31 18 46 78 0 023 F1 + F2 + F1F3 S + S + S 0 1 100 65 23 16 41 2 0 0

for common variance designs impossible for anything but small design problems.170

In light of the difficulty in finding common variance designs, we introduce a class of approximate171

common variance (A-ComVar) designs. Instead of having exactly equal variance for the uncommon172

parameters for the s models under consideration, A-ComVar designs try to minimize the ratio of the173

minimum variance to the maximum variance. In doing so, they contain common variance designs174

as a sub-case where the minimum variance is exactly equal to the maximum variance. In relaxing175

the requirement that the variances be exactly equal, we are able to develop an objective function176

and algorithm for identifying these A-ComVar designs without performing an exhaustive search.177

9

Table 5: Percentage of Correctly Identified Models for design d(12)5 and Plackett-Burman design

(PB) with factors F − J .

Model Size0.1 0.25 0.5 0.75 1



4.2 Proposed Algorithm: Genetic Algorithm for Finding A-ComVar Designs178

In this section we propose to use a genetic algorithm to identify A-ComVar Designs. We start by179

defining an objective function that seeks to quantify our goal.180

f(d;φ) =1/min

i{var(β2i)}

1 + φ×[

maxi{var(β2i)} −min

i{var(β2i)}

] , (2)

where β2i corresponds to the interaction effect for the ith model. The value of the objective function181

increases as the variance of the estimates decreases through the numerator, encouraging designs with182

small variances for the interaction terms. However, this value is also strongly penalized towards zero183

as the minimum and maximum variances move apart. The strength of this penalty is controlled by184

the tuning parameter φ, which we recommend setting to a large value. In our experiments we found185

10

φ = 1000 to be adequate. Thus taken together the numerator allows us to differentiate between186

designs with common variance to select the better one, and the denominator encourages common187

variance designs by penalizing differing variance under alternative models under consideration.188

This maximization approach will prefer A-ComVar designs with exactly common variance. Of189

course, in many experimental situations a common variance design may not exist. For example in190

the exhaustive search, Chowdhury (2016) found that common variance designs did not exist for 33191

experiments for 13 runs. This leads us to the principal advantage of our approach: when a common192

variance design does not exist we can still find designs with variance that is as close as possible to193

being equal. To assess the quality of an A-ComVar design, we define the A-ComVar ratio194

rACV =mini{var(β2i)}

maxi{var(β2i)}

, (3)

where the variance is replaced by the determinant of the lower-right k × k sub-matrix for k > 1,195

which bears some similarity to the idea behind D-optimal design of experiments. Clearly when a196

design has common variance, rACV = 1. When a design does not have common variance, rACV197

gives us an idea of how far we are from common variance. For example, if rACV = 0.5 then we198

know that between any two models under consideration, the largest variance of interaction terms199

is twice that of the smallest. This knowledge can hopefully help inform model selection.200

Any off-the-shelf optimization algorithm could be used to try to maximize this objective func-201

tion. We have chosen to use a genetic algorithm, as is common in the design literature (Mandal202

et al., 2015; Lin et al., 2015). Genetic algorithms are optimization techniques mimicking Darwin’s203

idea of natural selection and survival of the fittest. This search expects that a good candidate solu-204

tion will provide good offspring and imitates the way that chromosomes crossover and mutate when205

reproducing. Here, each chromosome is a design, and the fitness of a chromosome is determined by206

the corresponding objective function value. At each iteration the worst chromosomes are replaced207

with offspring generated by combining the settings from two better chromosomes, along with some208

small probability of a mutation. In the context of our problem, a mutation corresponds to randomly209

changing the settings for one of the factors in one of the runs. The algorithm terminates when210

either the maximum number of iterations has been reached, or a design with common variance has211

been found. The steps in our genetic algorithm are outlined in Algorithm 1.212

11

The genetic algorithm requires the user to specify the mutation probability and the maximum213

number of iterations. Our experience with the algorithm suggests using a small mutation probability214

to encourage only one or two mutations each time a new chromosome is created. For our purposes,215

we generally use a maximum of 1000 iterations, although the algorithm is quite fast and this216

number can easily be increased if needed. Our algorithm is implemented in Julia version 1.0.2 and217

is available for download from the author’s website.218

Algorithm 1 Pseudo-code for the genetic algorithm to find A-ComVar designs.

1: function A-ComVarDESIGN(design problem, mutation prob., max iter., φ)2: for Each chromosome do3: initialize chromosome to random design4: Calculate fitness5: end for6: while termination criteria not met do7: Identify worst two chromosomes8: Use a crossover to generate two new chromosomes9: Mutate the two new chromosomes

10: Replace the worst chromosomes with the two new chromosomes11: Calculate fitness for new chromosomes12: end while13: end function

5 Numerical Experiments219

5.1 Simulation 1 - Designs with One 2-Factor Interaction220

We conducted a series of simulations to investigate the ability of our approach to find A-ComVar221

designs and to gain a better understanding of when common variance designs can be found. We222

started by examining designs with a single 2-factor interaction. We consider 2m1 and 3m2 ex-223

periments, with m1 = 4, . . . , 9 and m2 = 3, . . . , 6. For the 2m1 experiments, we considered run224

sizes of nm1 = m1 + 2, . . . ,m1 + 11, and for the 3m2 experiments, we considered run sizes of225

nm2 = 2m1 + 2, . . . , 2m1 + 11. For each combination of settings, we ran our genetic algorithm 1000226

times and stored the rACV results. The tuning parameters used were a mutation probability of227

0.05 and a maximum of 1000 iterations.228

Figure 1 displays the results for the 2m1 simulations and Figure 2 displays the results for the 3m2229

simulations. We first note that our results are consistent with the findings of Ghosh and Chowdhury230

12

(2017), who used exhaustive searches to identify common variance designs. For example, Ghosh231

and Chowdhury (2017) found that common variance designs exist for 33 designs with 8 runs, which232

agrees with the boxplots in the first panel of Figure 2. This supports our use of the genetic233

algorithm approach with the objective function described above. Furthermore, in cases where the234

common variance designs either do not exist or could not be found, our approach was able to find235

designs that attempt to get as close as possible to common variance. For example, it is known from236

exhaustive searches that no common variance design exists for a 33 experiment with 12 factors.237

However, the proposed approach was able to find designs where the smallest variance was greater238

than 0.8 times the largest variance, indicating that the design is quite close to having the common239

variance property.240

5.2 Simulation 2 - Designs with Two 2-Factor Interactions241

For designs with multiple 2-factor interactions (i.e. k > 1), we generalize the objective function242

in (2) by replacing var(β2i) with the determinant of the block of the variance covariance matrix243

corresponding to the interactions terms. That is, we take the determinant of the bottom-right k×k244

sub-matrix of var(β).245

To demonstrate the approach, we conducted another simulation with two 2-factor interactions246

(i.e. k = 2). We consider 2m3 experiments, with m3 = 4, . . . , 7. We considered run sizes of247

nm1 = m3 + 6, . . . ,m3 + 12. For each combination of settings, we ran our genetic algorithm 1000248

times and stored the rACV results. The tuning parameters used were a mutation probability of249

0.05 and a maximum of 1000 iterations.250

Figure 3 shows the results for these simulations. As before, we can see that in many cases251

the genetic algorithm is able to find common variance designs. In cases where common variance252

designs cannot be found, the approach is often able to identify a design resulting in relatively close253

to common variance.254

5.3 Simulation 3 - Comparison to Plackett-Burman Designs255

Next, we examine the performance of two A-ComVar designs that do not achieve the common256

variance property, as well as one common variance design for k = 1 obtained from simulation.257

We perform model selection using the adaptive lasso regression technique as described earlier. We258

13

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0.00.10.20.30.40.50.60.70.80.91.0

8 9 10 11 12 13 14 15 16 17

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0.00.10.20.30.40.50.60.70.80.91.0

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0.00.10.20.30.40.50.60.70.80.91.0

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0.00.10.20.30.40.50.60.70.80.91.0

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Figure 1: Ratios rACV for the 2m1 simulations across 1000 replicates for each experimental setting.

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Number of runs

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V

6 Factors

Figure 2: Ratios rACV for the 3m2 simulations across 1000 replicates for each experimental setting.

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Figure 3: Ratios rACV for the 2m1 simulations across 1000 replicates for each experimental settingwith k = 2.

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A B C D E

-1 -1 -1 -1 -1-1 -1 -1 -1 1-1 -1 -1 1 -1-1 -1 1 1 1-1 1 -1 -1 -1-1 1 1 -1 -11 -1 -1 1 11 -1 1 1 11 1 -1 -1 -11 1 1 -1 11 1 1 1 -11 1 1 1 1

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A B C D E

-1 -1 -1 -1 -1-1 -1 1 1 1-1 1 -1 1 -1-1 1 -1 1 1-1 1 1 -1 1-1 1 1 1 -11 -1 -1 1 11 -1 1 -1 11 -1 1 1 -11 1 -1 -1 11 1 -1 1 -11 1 1 -1 -1

Table 6: Three designs (1) D1(left): Common variance design for k = 1 with m = 5 and n = 12,(2) D2 (middle): Approximate common variance design for k = 2 with m = 4 and n = 12 and (3)D3 (right): Approximate common variance design for k = 2, m = 5, n = 12, used in Simulation 3.For D2 and D3 the ratio of the minimum to the maximum variance is 0.87 and 0.81 respectively.

compare performance of each of the three designs D1, D2 and D3 with the design having five factors259

A− E from Plackett-Burman design presented in Table 2, using simulations. These three designs260

are presented in Table 5. D1 is a common variance design for k = 1 with m = 5 and n = 12 found261

using our approach. D2 is an A-ComVar design for k = 2 with m = 4 and n = 12. D3 (right) is262

an A-ComVar design for k = 2, m = 5, n = 12. For D2 and D3 the ratio of the minimum to the263

maximum variance is 0.87 and 0.81 respectively.264

We consider similar simulation setting as described in Section 3. We generate the data in265

simulation using the three designs presented in Table 5 and using the coefficients (both “big” and266

“small”) as described in Section 3. Tables 7-9 display the percentage of times the correct model is267

identified by the designs D1-D3, respectively, and the Plackett-Burman designs with factors A−E268

under different factorial effect size combinations and varied choices of variance (σ2) of the noise.269

On the whole, in each case designs obtained using our proposed approach perform comparably with270

the Plackett-Burman design, with the better design often depending on the amount of variability271

in the model in combination with the true underlying model.272

17

Table 7: Percentage of Correctly Identified Models for common variance design D1 and Plackett-Burman design with factors A− E.

Model Size0.1 0.25 0.5 0.75 1



18

Table 8: Percentage of Correctly Identified Models for A-ComVar design D2 and Plackett-Burmandesign with factors A− E.

Model Size0.1 0.25 0.5 0.75 1

A-ComVar PB A-ComVar PB A-ComVar PB A-ComVar PB A-ComVar PB


Table 9: Percentage of Correctly Identified Models for A-ComVar design D3 and Plackett-Burmandesign with factors A− E.

Model Size0.1 0.25 0.5 0.75 1

A-ComVar PB A-ComVar PB A-ComVar PB A-ComVar PB A-ComVar PB


19

6 Discussion273

In this work we introduced A-ComVar Designs, an extension of common variance designs. These274

designs address the difficulties associated for finding common variance designs via exhaustive search275

and allow a relaxation of the common variance property for cases where a common variance design276

does not exist. Through simulation, we demonstrated that the proposed algorithmic approach277

allows us to quickly find common variance designs that overlap with those known in the literature.278

Furthermore, in cases where common variance designs do not exist or cannot be found, our approach279

allows identification of designs with close to common variance. Comparisons to Plackett-Burman280

designs demonstrated that such designs perform quite well in practice, and that in many cases these281

A-ComVar designs were as good as common variance designs.282

There are several avenues here for future work. First, we considered only cases with 2-level283

and 3-level factors. Future work could consider finding A-ComVar designs with a mixture of284

both. Second, we utilized a genetic algorithm to find these designs. There are numerous other285

optimization approaches that could be used to maximize the objective function in (2). In some286

cases, these other approaches may succeed in finding designs with a much better ratio of minimum287

to maximum variance of the uncommon parameters. Third, there is another approach to finding288

common variance designs through hierarchical designs (Chowdhury, 2016). These designs are found289

by identifying a common variance design for a smaller number of runs and then adding runs while290

trying to preserve the common variance property. It is possible that a similar idea could be291

developed for A-ComVar designs. Finally, future work could study the types of A-ComVar designs292

that can be found when the number of interactions in the model increases beyond two.293

20

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On behalf of all authors, the corresponding author states that there is no conflict of interest.

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