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5 Conclusion and Future Research
The search for optimal designs for generalized linear models is more challenging than for linear models in large measure in that GLM designs depend on the very values of parameters we intend to estimate. This leads to our strategy to employ the so-called locally optimal designs, which are optimal for a priori selected values of the parameters.
The existing literature has provided characterization of certain optimal designs for generalized linear models, in which most linear predictors are simple, multiple, or polynomial regressions though. Comparatively less attention has been received to finding the optimal designs involving qualitative treatment and plot structures.
In this paper, the problem of finding D- and A-optimal designs in the settings of zero- and one-way elimination of heterogeneity for generalized linear models is investigated. An efficient algorithm for finding the D-optimal exact designs for a completely randomized design with v treatments is proposed, which is based on some proven results describing the general relations between unknown variances and their corresponding optimal ni’s.
It is found that, in terms of D-optimality, uniform designs are better when the treatment variances are equal, and a design having larger ni’s with corresponding smaller treatment variances is better. An upper bound of ni, in terms of ratios, is given to indicate the relative size of this inverse proportionality between ni’s and related variances.
The A-optimal approximate designs for v unequal variances can be obtained through the method of Lagrangian multiplier. The finding of the D-optimal
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terpart, in contrast, are limited to the case when values of unknown parameters are split into two or three categories. The derivation of D-optimal exact designs, which is our main result, is even more difficult, so it is confined to the realm of either the largest or the smallest variance being different from the others.
The dependence of unknown parameters of the locally-optimal designs brings us into the discussion of robustness. It is empirically found that for v = 3, uniform designs are highly D-efficient to wide ranges of model parameter values. When the variances are divided into two groups, the upper and lower bounds are established and they are independent of unknown parameters. The bounds also indicate that designs having as equal number of replications for each treatment as possible are efficient in D-optimality.
We are at a very preliminary stage to find the randomized block designs for v treatments. Assuming the size of each block is given, the D-optimal approximate and exact designs for v = 2 and b blocks are constructed. For v = 3, it is proved that for equal treatment variances within a block (variances across blocks may not necessarily be equal), the uniform design is D-optimal.
Promising avenues for future research on optimal designs for GLMs are along the lines of the following directions. The A-optimal exact design of a completely randomized design for v treatments is expected to be decided in a relatively short period of time, for the A-optimal approximate design has been found.
As for the randomized block designs, in addition to keeping on proving the con-jecture in Table 1, we will try simplifying our expression of the reduced information matrix M. For A-optimal block designs, we will continue working analogously on finding the optimal exact and approximate designs as what we did for the
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optimal designs. To begin with, for w1j = w2j = w3j with each block size kj given, 1 ≤ j ≤ b, it is conjectured that the design having nij = [kj/3] or nij = [kj/3] + 1, where [·] is the greatest integer function, is an A-optimal design. Whether a design minimizing λ−11 + λ−12 = 2(m12+ m13+ m23)/3(m12m13+ m12m23+ m13m23) with w1j ≥ w2j ≥ w3j satisfies n1j ≤ n2j ≤ n3j, 1 ≤ j ≤ b, where m12, m13 and m23 have the same definition as in (2) will also be investigated. If for v = 3, both D-and A-optimal block designs can be characterized with the two above-mentioned properties, we would like to take further steps to see if they still hold for arbitrary v treatments.
Another issue of interest is to study the performance of a design having nij as equal as possible in each b block with fixed block size. Once such a design is high efficient within certain ranges of variances or their relative magnitudes, it can be considered a good alternative.
Since it is assumed that P τi = 0 and P βj = 0 in a randomized block design, it is natural to compare a randomized block design with a factorial design without interaction. We thus close this section by an interesting comparison, as well as a plan for future research, between a block design for v = b = 2, with a 22 factorial design without interaction.
Consider the simplest case of a 22 factorial and ηij = µ + τi+ βj, i.e., the two-way main-effects model, where µ is the overall mean, τi is the effect on the response due to the fact that the ith level (low or high) of factor F1 is observed, and βj is the effect on the response due to the fact that the jth level (low or high) of factor F2 is observed. Let Yijh, the response of the hth replicate of observing levels i and j of factors F1 and F2 together, i, j = 1, 2 and h = 1, · · · , nij, n = P2
i,j=1nij
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where θij is the natural parameter, and b(·) and c(·) are known functions.The expectation of Yij, E(Yij), is related to ηij via a canonical link function g, say. That is, ηij = g(E(Yij)). Through standard methods, the Fisher information matrix I for estimating (µ, τ1, τ2, β1, β2) is given as
is a 4 × 4 matrix. Since our focus is on investigating the contributions that each of the factors make individually to the response, the information matrix M for the estimation of (τ1, τ2, β1, β2), can be derived as
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Note that the sum of rows 1 and 2 of M, the sum of column 1 and 2 of M, the sum of rows 3 and 4 of M, and the sum of columns 3 and 4 of M, are all zeros, so there are two non-zero eigenvalues of M. One can show that the product of the two non-zero eigenvalues of M is 4(P
iniwi)−1P
i<j<hninjnhwiwjwh, where i, j, k ∈ {11, 12, 21, 22}.
It follows from the preceding expression that if w11 > w21, then any solution to maximizing λ1λ2, the product of non-zero eigenvalues of M, satisfies n11≤ n21, and this result is contrast with the one obtained by Yang et al. (2012).
Yang et al. (2012) considered, for the binary response, the problem of obtaining locally D-optimal approximate designs for the 22 factorial experiment with main-effects model. Hence, their design problem is similar to ours while they determined the proportion of observations allocated to each of the 4 design points with linear predictor η = β0+ β1x1 + β2x2, where each xi ∈ {−1, 1}, and the parameters of interest are β = (β0, β1, β2)T. They showed that if the (assumed) variance at one design point is substantially larger than the others, then the D-optimal design is to assign the same proportion of observations to the other three of the four points. If w1 > w2, then any solution to maximizing the information matrix satisfies p1 ≥ p2 and if w1 = w2, then any solution satisfies p1 = p2, where pi = ni/n.
It is of interest to observe the results of designs in a 22 factorial design without interaction obtained by expressing the design matrix in terms of a regression model entirely differs from using an ANOVA model and we are more interested in whether there is a difference in the 2k factorial designs with interactions between these two forms.
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