Abstract
Wang and Stufken (J Stat Theory Practice 14(2):1–15, 2020) identified locally D-optimal designs for generalized linear models with factorial effects and one continuous covariate. Using an approximate design approach, the design problem consists of selecting values for the covariate and design weights for each group formed by the various factors. For the logistic and probit link, the optimal designs in Wang and Stufken (2020) use two covariate values for each of the groups and equal weights. We establish that smaller D-optimal designs can often be obtained by using orthogonal arrays so that an optimal design uses only some of the groups with at most two covariate values in those groups. The general theory is illustrated through an application.
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Research was partially supported through NSF Grant DMS-1935729.
This article is part of the topical collection “Special Issue: State of the art in research on design and analysis of experiments” guest edited by John Stufken, Abhyuday Mandal, and Rakhi Singh.
Appendix
Appendix
Proof of Theorem 1
From Eq. (8), we have that
where for any \((i_1, \ldots , i_L) \in H\), the matrix \(\varvec{D^{i_1 \cdots i_L j}} (\varvec{D^{i_1 \cdots i_L j}})^T\) is
Since H has N rows, the top-left element of \(M_{\xi _g}(\varvec{\theta _1^g})\) is
while the bottom-right element is
All other elements in the last column of \(M_{\xi _g}(\varvec{\theta _1^g})\) are 0 because each cell has c-values \(c^*\) and \(-c^*\) with equal weights.
Other off-diagonal blocks in \(\varvec{D^{i_1 \cdots i_L j}} (\varvec{D^{i_1 \cdots i_L j}})^T\) are of the form \((\varvec{Z^{i_{l}}_{l}})^T\), \((\varvec{Z^{i_{l_1}i_{l_2}}_{l_1l_2}})^T\), \(\varvec{Z^{i_{l_1}}_{l_1}} (\varvec{Z^{i_{l_2}}_{l_2}})^T\) for \(l_1 \not = l_2\), \(\varvec{Z^{i_{l}}_{l}} (\varvec{Z^{i_{l_1}i_{l_2}}_{l_1l_2}})^T\), \(\varvec{(Z^{i_{l_1}i_{l_2}}_{l_1 l_2})}(\varvec{Z^{i_{l_3}i_{l_4}}_{l_3 l_4}})^T\) for \((l_1,l_2) \not = (l_3,l_4) \), or their transposes. Considering what this means for \(M_{\xi _g}(\varvec{\theta _1^g})\), first, from the definition of \(\varvec{Z^{i_{l}}_{l}}\) and since each level of factor l appears equally often, we see that
Now, corresponding to \((\varvec{Z^{i_{l_1}i_{l_2}}_{l_1l_2}})^T\), since all level combinations for any two factors \((l_1,l_2)\) appear equally often, we have
Next, using that the elements of H form an OA of strength 2 and that \(l_1 \not = l_2\),
Further, if \(l=l_1\), again using that the elements of H form an OA of strength 2,
A similar argument applies for \(l=l_2\). If \(l \ne l_1\) and \(l\ne l_2\), then
where we have used that \((l, l_1, l_2) \in C_3\) and H is an OA of strength 3 for such a set of three columns.
Finally, for the off-diagonal blocks corresponding to \(\varvec{(Z^{i_{l_1}i_{l_2}}_{l_1 l_2})}(\varvec{Z^{i_{l_3}i_{l_4}}_{l_3 l_4}})^T\) for \((l_1,l_2) \not = (l_3,l_4) \), we could have two situations: the two interactions have one factor in common, say \(l_1=l_3\) (and \(l_2 \ne l_4\)), or they represent four different factors. Looking at the first case, we have
where we have used that \((l_1, l_2, l_4) \in C_3\) and H is an OA of strength 3 for such a set of three columns.
For the second case, with no common factors, \((l_1,l_2, l_3,l_4) \in C_4\) and H is an OA of strength 4 for such a set of factors. Hence,
For a diagonal block of \(M_{\xi _g}(\varvec{\theta _1^g})\) that corresponds to a main effect, say for factor l, we obtain
where \(B_l=\frac{1}{(s_l -1)^2} (s_l I - J)\). The first equality follows since every level of factor l comes \(N/s_l\) times. The second equality follows as in the proof of Lemma 1 in [11].
For a diagonal block of \(M_{\xi _g}(\varvec{\theta _1^g})\) that corresponds to a two-factor interaction, say for factors \((l_1,l_2)\), we get
where the penultimate equality follows as in [11].
Combined, the previous steps show that \(M_{\xi _g}(\varvec{\theta _1^g})\) is identical to \(M_{\xi ^*}(\varvec{\theta _1^g})\) in Lemma 2, so that \(\xi _g\) is also D-optimal for \(\varvec{\theta _1^g}\). \(\square \)
Proof of Theorem 2
As in the proof of Theorem 1, we now need to show that \(M_{\xi _{1a}}(\varvec{\theta _1^1})\) is the same as \(M_{\xi ^*}(\varvec{\theta _1^1})\) in Lemma 2. Since H is an OA as in Corollary 1, our work is simplified. Only the entries in \(M_{\xi _{1a}}(\varvec{\theta _1^1})\) that depend on the c-values need verification. In other words, we only need to consider the last column in \(M_{\xi _{1a}}(\varvec{\theta _1^1})\).
First, the bottom-right diagonal element in \(M_{\xi _{1a}}(\varvec{\theta _1^1})\) is
Further, for the entry in the final column of \(M_{\xi _{1a}}(\varvec{\theta _1^1})\) that corresponds to the main effect of factor l, using Eq. (8), we get
The first equality follows because every level of factor l appears equally often in both \(H_1\) and \(H_2\).
The only other entry in the final column of \(M_{\xi _{1a}}(\varvec{\theta _1^1})\) corresponds to the interaction of factors 1 and 2. Because H is an orthogonal array of strength 3 for factors 1, 2 and the additional 2-level column (the \((L+1)\)st column), it follows that every level combination \((i_1,i_2)\) for the first two factors appears equally often in \(H_1\) and \(H_2\). Hence,
Therefore, all entries in the last column of \(M_{\xi _{1a}}(\varvec{\theta _1^1})\) are also equal to those of \(M_{\xi ^*}(\varvec{\theta _1^1})\). This concludes the proof. \(\square \)
Proof of Theorem 3
As in the proof of Theorem 2, it suffices to verify that entries in the final column of \(M_{\xi _{1b}}(\varvec{\theta _1^1})\)) are equal to those in \(M_{\xi ^*}(\varvec{\theta _1^1})\) in Lemma 2. For the bottom-right diagonal element in \(M_{\xi _{1b}}\), using Eq. (8), we get
by counting the sizes of the \(H_i\)’s.
Further, for the entry in the final column of \(M_{\xi _{1b}}(\varvec{\theta _1^1})\) that corresponds to the main effect of factor l, we get
where we have used that each level of factor l appears \(\frac{uN}{(2u+1)s_l}\) times in each of \(H_1\) and \(H_2\) and \(\frac{N}{(2u+1)s_l}\) times in \(H_3\).
The only other entry in the final column of \(M_{\xi _{1b}}(\varvec{\theta _1^1})\) corresponds to the interaction of factors 1 and 2. Because H is an orthogonal array of strength 3 for factors 1, 2 and the additional \((2u+1)\)-level column (the \((L+1)\)st column), it follows that every level combination \((i_1,i_2)\) for the first two factors appears \(\frac{Nu}{(2u+1)s_1s_2}\) times in each of \(H_1\) and \(H_2\) and \(\frac{N}{(2u+1)s_1s_2}\) times in \(H_3\). Using Eq. (8), we obtain that the final entry in the final column is equal to
Therefore, all entries in the last column of \(M_{\xi _{1b}}(\varvec{\theta _1^1})\) are also equal to those of \(M_{\xi ^*}(\varvec{\theta _1^1})\). This concludes the proof. \(\square \)
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Wang, Z., Stufken, J. Orthogonal Array Based Locally D-Optimal Designs for Binary Responses in the Presence of Factorial Effects. J Stat Theory Pract 15, 87 (2021). https://doi.org/10.1007/s42519-021-00224-w
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DOI: https://doi.org/10.1007/s42519-021-00224-w