Throughout, “#Balance” denotes the number of balanced columns of a design. Let ${\mathbf{1}_{n}}$ denote the n-dimensional unit vector with $+1$’s. Let I denote the identity matrix. For a matrix A with entries in $\{-1,+1\}$ and an even number of rows, let ${\mathbf{A}^{\ast }}$ denote a matrix obtained by changing the signs of entries in all odd rows and ${\mathbf{A}^{\ast \ast }}$ denote a matrix obtained by changing the signs of entries in all even rows, where ∗ and $\ast \ast $ are two matrix operators. Note that ${(-\mathbf{A})^{\ast \ast }}={\mathbf{A}^{\ast }}$. An $n\times n$ Hadamard matrix H is a matrix with entries in $\{-1,+1\}$ and ${\mathbf{H}^{\top }}\mathbf{H}=n\mathbf{I}$, where n can only be 1, 2 or a multiple of 4.

Our construction method expands a $6m\times p$ supersaturated design ${\mathbf{D}_{0}}$ with $\mu \le 1/3$ to a $12m\times 4p$ design with $\mu \le 1/3$, where $m,p$ are positive integers with $6m\lt p$. Partition ${\mathbf{D}_{0}}$ as where U and L have $4m$ and $2m$ rows, respectively. The proposed method has three steps.

If ${\mathbf{D}_{1}}$ is viewed as a block matrix with four rows, Step 2 applies the operator ∗ to the second row and changes the signs of all entries in the fourth row to obtain ${\mathbf{D}_{2}}$. If ${\mathbf{D}_{2}}$ is viewed as a block matrix with four rows and two columns, Step 3 applies the operator $\ast \ast $ to the first and fourth rows and applies the operator ∗ to the third row to obtain ${\mathbf{D}_{3}}$. It can be proved that the coherence of ${\mathbf{D}_{1}}$, ${\mathbf{D}_{2}}$ and ${\mathbf{D}_{3}}$ is no greater than $1/3$ by Lemmas 1, 2, 3 and Theorem 1 in Appendix A.

Since the design ${\mathbf{D}_{3}}$ from the construction is a $12m\times 4p$ design with $\mu \le 1/3$, repeating the above procedure multiple times with ${\mathbf{D}_{0}}$ being ${\mathbf{D}_{3}}$ produces a class of designs with $({2^{k}}\times 6m)$ rows, $({4^{k}}\times p)$ columns and $\mu \le 1/3$ for every positive integer k.

The only requirement for ${\mathbf{D}_{0}}$ is to be a $6m\times p$ two-level supersaturated design with $\mu \le 1/3$. For a given pair of m and p, different choices of ${\mathbf{D}_{0}}$ yield different forms of ${\mathbf{D}_{3}}$ but all of them have guaranteed small coherence. As suggested by Chen and Lin [6] and Liu, Ruan and Dean [20], a supersaturated design with coherence of $1/3$ can identify most of the important factors.

We now provide several examples for the proposed method with different choices of ${\mathbf{D}_{0}}$. In the following examples, we show the first several designs constructed by the proposed method and choose ${\mathbf{D}_{0}}$ to have as many columns as possible.

Whenever necessary, one can select some columns of the designs constructed for use. Selecting a subset of columns could remain coherence the same or further decrease it, but will never increase coherence. For example, for a given p, first, one can select as many balanced columns as possible. If there are no sufficient balanced columns, then one can select some unbalanced columns to achieve p columns. In addition, one may also select the columns according to an additional criterion such as $E({s^{2}})$.

We now generalize the proposed method to obtain a supersaturated design with μ smaller than $1/3$. Recall that our method partitions ${\mathbf{D}_{0}}$ into two parts. The upper part U has $6mr$ rows and the lower part L has $6m(1-r)$ rows with a partition ratio $r=2/3$. We generalize the original construction by using a different partition ratio and stopping at Step 2. Suppose that ${\mathbf{D}_{0}}$ is an $n\times p$ two-level supersaturated design with $\mu \le t/n$ for $t\in \{2,4,\dots ,n/2\}$ and an even n. Partition ${\mathbf{D}_{0}}$ as in (2.1), where U has $2t$ rows and L has $n-2t$ rows with $r=2t/n$. Define ${\mathbf{D}_{2}}$ as in (2.2). Then ${\mathbf{D}_{2}}$ is the supersaturated design constructed by our generalized method. It can be proved that the coherence of ${\mathbf{D}_{2}}$ constructed here is no greater than $t/n$ by Theorem 2 in Appendix A.

The generalization indicates the possibility of expanding any two-level supersaturated design with $\mu \le 1/2$ and an even number of runs while retaining its coherence. For the original construction, coherence must be no greater than $1/3$ and the initial design must have $6m$ runs but the initial design with p columns can be expanded to a design with $4p$ columns and $\mu \le 1/3$. The generalization only requires that the coherence is no greater than $1/2$ and the initial design has an even number of runs but the initial design with p columns can only be expanded to a design with $2p$ columns and the same coherence. In addition, the number of balanced columns can be fewer than the number of runs for the designs constructed by the generalization. Here is an example.

We obtain the $12\times 27$ proposed design by selecting the intercept, the 24 balanced columns and the first two unbalanced columns of the $12\times 64$ design in Example 1. We use Lin’s $14\times 27$ design directly. The result is shown in Table 6, where and in other tables below, the smaller values of the four criteria are in boldface. Table 6 indicates that the proposed design outperforms Lin’s design in terms of variable selection and parameter estimation with the Lasso. In addition, the proposed design has fewer runs than Lin’s design.

To verify that this result is not due to some better property of the selected active factors in one design versus another, we also conducted a follow-up comparison with randomly selected active factors. More specifically, for a given setting in Table 5, we randomly select an #Active number of active factors with the same values given in the setting, and repeat 10 times to get 10 sets of active factors. Then, for each set of randomly selected active factors, we repeat the above data generation and variable selection procedure $N=30$ times for the proposed design and Lin’s design. The final criteria are averaged over the criteria from the 10 sets of active factors, and they can be found in Table 7. According to Table 7, the proposed design still generally outperforms Lin’s design, even with randomly selected active factors. To save computational cost and keep our paper concise, we will omit such a verification for the following comparisons.

We use the $12\times 64$, $24\times 256$ proposed designs in Example 1 directly. We obtain the $48\times 300$ proposed design by selecting the intercept and the first 299 balanced columns of the $48\times 8192$ design in Example 2. We obtain Wu’s $12\times 64$, $24\times 256$ and $48\times 300$ designs by selecting the first 64 columns of Wu’s $12\times 67$ design, the first 256 columns of Wu’s $24\times 277$ design and the first 300 columns of Wu’s $48\times 1129$ design, respectively. The result shown in Table 8 indicates that the proposed designs are generally better than Wu’s designs in terms of variable selection and parameter estimation with the Lasso. In almost all cases, the proposed designs have better performance while in the first two cases, Wu’s designs perform relatively better.

We use the $12\times 64$ proposed design in Example 1 directly. We obtain the $24\times 253$ proposed design by selecting the intercept, the 168 balanced columns and the first 84 unbalanced columns of the $24\times 256$ design in Example 1. We first search a $12\times 64$ and a $24\times 256$ Bayesian D-optimal supersaturated design by the JMP^ő software [13] with 500 random starting designs. We use the $12\times 64$ design directly. We remove the 22th, 134th and 209th columns of the $24\times 256$ design to obtain the $24\times 253$ design used in the simulations since the original searched $24\times 256$ Bayesian D-optimal supersaturated design has $\mu =0.83$. The result shown in Table 9 indicates that the proposed designs generally outperform the Bayesian D-optimal supersaturated designs in terms of variable selection and parameter estimation with the Lasso.

According to Jones and Majumdar [15], for given n and p, there are a large number of $UE({s^{2}})$-optimal designs. In practical scenarios, people may randomly generate only one $UE({s^{2}})$-optimal design and use it to screen important factors. In our comparison, we want to mimic such a scenario, i.e., we want to know whether people will have a higher chance ($\gt 0.5$) of getting a worse design (compared with our design) if they generate a $UE({s^{2}})$-optimal design in this way. Therefore, we randomly generate 500 $UE({s^{2}})$-optimal designs and for each of them, repeat the simulation procedure $N=30$ times (rather than $N=300$ in comparisons with other competing designs). We take the median values for the four criteria introduced in the beginning of the section over the 500 randomly generated designs and compare these median values with the corresponding criteria of the proposed design. If the AFDR of a proposed design is less than the median AFDR (MAFDR) of all the 500 $UE({s^{2}})$-optimal designs, it means the AFDR of the proposed design is less than that of the $UE({s^{2}})$-optimal design with more than half probability. The coherence, $E({s^{2}})$ and number of balanced columns in the table are also the corresponding medians. In addition, it is possible that a generated $UE({s^{2}})$-optimal design has $\mu =1$ and we will avoid using these designs in the simulations, because in practice if people get a bad design with $\mu =1$, they will probably regenerate another one.

We use the $12\times 64$ and $24\times 256$ proposed designs in Example 1 directly. We obtain the $UE({s^{2}})$-optimal designs by randomly sampling 12 rows of a $64\times 64$ Hadamard matrix 500 times and 24 rows of a $256\times 256$ Hadamard matrix 500 times, respectively. The result shown in Table 10 indicates that with more than half probability, the proposed designs outperform the $UE({s^{2}})$-optimal designs in terms of variable selection and parameter estimation with the Lasso. Furthermore, one may want to see the distributions of the criteria of 500 $UE({s^{2}})$-optimal designs. We take Case 8 for an example and show the distributions of the criteria in Figure 1.

To help people better understand how the proposed design performs compared to competing designs with the same size and under the same active coefficient setting, we collect the comparisons between designs with exactly the same size and active coefficient setting all together in Table 11. Lin’s design is not of size $12\times 64$, so it is not included in the table (it is actually impossible to construct a design of this size with Lin’s method).

According to Table 11, the proposed design significantly outperforms the Bayesian D-optimal design and the $UE({s^{2}})$-optimal design. In this scenario, Wu’s design performs slightly better than the proposed design, but there are also scenarios where the proposed design performs better than Wu’s design (see Table 8).

In summary, for Wu’s design, the proposed design is generally comparable to it, because of the same coherence they have. For Lin’s design, the Bayesian D-optimal design and the $UE({s^{2}})$-optimal design, the proposed design generally outperforms them, except that it sometimes tends to select more factors and gives a larger AFDR. However, the proposed design always gives a smaller AMR when compared to these three designs. In practice, a small AMR is more important than a small AFDR, because missing active factors is a more serious problem than falsely discovering inactive factors. This is because we can conduct follow-up experiments to screen out spurious factors, but we can never detect active factors that were already removed in the screening phase [21].