D-optimal design of split-split-plot experiments

The minimum aberration 32-run two-level split-plot design from Bingham et al. (2004) can be used as a 32-run split-split-plot design by assigning one of its four subplot columns to the hard-to-change factor s.

“The minimum aberration design can be used as a 32-run split-split-plot design by using one of its four subplot columns for the hard-to-change factor s.”

• cites Bingham, Schoen and Sitter (2004) sourcePartly supported

  Table 2 (reproduced under the Conclusion) lists minimum-aberration 32-run split-plot designs with four subplot columns. The paper does not describe assigning one of those columns to a hard-to-change factor or using the result as a split-split-plot design; that is the citing paper’s proposed adaptation. (LLM, conf=0.84)

  Experimental — accuracy not yet measured

The modified coordinate-exchange algorithm can construct orthogonal split-split-plot designs with diagonal information matrices for many nesting structures in which the numbers of whole plots and subplots within whole plots are powers of two or multiples of four.

“Using the modified coordinate-exchange algorithm sketched in Section 3, we have been able to construct orthogonal split-split-plot designs with diagonal information matrices for many nesting structures for which the numbers of whole plots and subplots within the whole plots are powers of two or multiples of four.”

Those six nonzero off-diagonal elements correspond to covariances between the estimated main effect of one easy-to-change factor and its interactions with the two very-hard-to-change factors w1 and w2.

“They correspond to the covariances between the estimates of the main effect of one of the easy-to-change factors and its interactions with the two very-hard-to-change factors w 1 and w 2 .”

In the 32-run design, all main effects are estimated independently.

“First, all main effects are estimated independently.”

Those covariances are small relative to the variances of the fixed parameter estimates shown in Table 4.

“These covariances are small compared to the variances of the fixed parameter estimates, which are displayed in the column labelled “D-optimal” in Table 4.”

The 32-run design has a D-criterion value of 4.80132 × 10^26 when the three variance components are all one.

“It has a D-criterion value of 4.80132 × 10 26 when σ 2 γ 1 , σ 2 γ 2 and σ 2 ε are all one.”

In the 16-run design, the error variance σ2ε cannot be estimated because the twelve sub-subplot degrees of freedom are used to estimate the main effects of the twelve easy-to-change factors.

“Also, it is impossible to estimate the error variance, σ 2 ε , because the twelve degrees of freedom at the sub-subplot level are used up for estimation the main effects of the twelve easy-to-change factors.”

The 16-run design cannot estimate the whole-plot error variance σ2γ1 because it has only two whole plots.

“Note, however, that this design does not allow for the estimation of the whole plot error variance, σ 2 γ 1 , because it only has two whole plots.”

The six nonzero off-diagonal elements of the variance-covariance matrix of the parameter estimates are all equal to ±1/96 = ±0.01042 when the three variance components are all one.

“The six non-zero off-diagonal elements of the variance-covariance matrix of the parameter estimates are all equal to ±1/96 = ±0.01042when σ 2 γ 1 , σ 2 γ 2 and σ 2 ε are all one.”

For the 24-run design in Table 2, with six whole plots each having two subplots and two observations, the information matrix equals diag(3.4286I2, 8, 24I12) when the three variance components are all one.

“The information matrix of that design equals diag(3.4286I 2 , 8, 24 I 12 ) when σ 2 γ 1 , σ 2 γ 2 and σ 2 ε are all one”

In the 32-run design, exactly one of the six main effects-one belonging to an easy-to-change factor-is not estimated independently of the two-factor interaction effects.

“Only one of the six main effects, namely one corresponding to an easy-to-change factor, is not estimated independently of the two-factor interaction effects.”

For the 16-run design in Table 1, with two whole plots and two subplots per whole plot, the information matrix equals diag(1.2308I2, 3.2, 16I12) when the three variance components are all one.

“The information matrix of that design, which is a projection of a Hall type IV orthogonal array (see Sun and Wu 1993), equals diag(1.2308I 2 , 3.2, 16 I 12 ) when σ 2 γ 1 , σ 2 γ 2 and σ 2 ε are all one.”

• cites Sun and Wu 1993Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The 32-run split-split-plot design in Table 3 is the best design found by the algorithm for estimating all parameters of an interaction model with two very-hard-to-change factors w1 and w2, one hard-to-change factor s, and three easy-to-change factors t1, t2, and t3 when η1 = η2 = 1.

“This design is the best one obtained using our algorithm for estimating all the parameters of an interaction model in two very-hard-to-change factors w 1 and w 2 , one hard-to-change factor s and three easy-to-change factors t 1 , t 2 and t 3 when η 1 = η 2 = 1.”

In the 32-run design, only six of the 462 off-diagonal elements of the information matrix and of the variance-covariance matrix of the parameter estimates are nonzero.

“Second, only six of the 462 off-diagonal elements of the information matrix and of the variancecovariance matrix of the parameter estimates are not equal to zero.”

For η2 = 0.1, there are three runs at two levels and six runs at the other level of the easy-to-change factor.

“For η 2 = 0.1 there are three runs at two levels and six runs at the other.”

For the alternative design, the three two-factor interaction effects between the easy-to-change factors are raised to higher strata.

“However, this is at the expense of the three two-factor interaction effects between the easy-to-change factors, which are raised to one of the higher strata.”

The determinant of the information matrix of the D-optimal design for η2 = 1 is 3978.7.

“The determinant of the information matrix of the D-optimal design for η 2 = 1 is 3978.7”

The differences among the three designs are in the sub-subplot factor levels.

“The differences in the designs come in the sub-subplot factor levels.”

The relative efficiency of the η2 = 10 design is 99.88%.

“The relative efficiency of this design is 99.88%.”

In the second sensitivity design problem, the designs vary depending on both η1 and η2.

“Here the story is more involved as the designs vary depending on both η 1 and η 2 .”

The D-efficiency of the combinatorially constructed design relative to the D-optimal design is 98.14%.

“The D-efficiency of the combinatorially constructed design, relative to the Doptimal design, is 98.14%.”

The determinant of the design with 3, 3 and 6 runs per level (obtained for η2 = 0.1) is 3672.6.

“The determinant of the design with 3, 3 and 6 runs per level (obtained for η 2 = 0.1) is 3672.6”

Two of those easy-to-change two-factor interactions are raised to the subplot stratum and estimated with variance 3/32 = 0.09375.

“Two of these interaction effects are raised to the subplot stratum and thus estimated with variance 3/32 = 0.09375”

When σ2γ1 = σ2γ2 = 0.1 and σ2ε = 1, the D-efficiencies of the four distinct designs relative to the optimal design for η1 = η2 = 0.1 range from 96.36% to 100%.

“When σ 2 γ 1 = σ 2 γ 2 = 0.1 and σ 2 ε = 1, the D-efficiencies of the four distinct designs found, relative to the optimal design for η 1 = η 2 = 0.1, range from 96.36% to 100%.”

The six designs that were the same all had full efficiency for these values of the variance components.

“The six designs that were the same all had full efficiency for these values of the variance components.”

In all other subplots, the levels of the easy-to-change factors are balanced.

“Notice that, in all other subplots, the levels of the easy-to-change factors are balanced.”

The relative efficiency of the η2 = 0.1 design is 98.86%.

“with a relative efficiency of 98.86%.”

The design with four runs at each level (obtained for η2 = 10) has a determinant of 3944.7.

“while the design with four runs at each level (obtained for η 2 = 10) has a determinant of 3944.7.”

For η2 = 10, the arrangement of the three levels of the easy-to-change factor in six subplots of two runs is a duplicated balanced incomplete block design for three treatments with three blocks of size two.

“In this case, the arrangement of the three levels of the easy-to-change factor in six subplots of two runs takes the form of a duplicated balanced incomplete block design for three treatments with three blocks of size two.”

The D-criterion value of the alternative design, when σ2γ1, σ2γ2 and σ2ε are all one, is 3.17836 × 10^26.

“the D-criterion value of the alternative design, when σ 2 γ 1 , σ 2 γ 2 and σ 2 ε are all one, is 3.17836 × 10 26 .”

Four distinct designs were found over the nine possibilities, distinguishing designs by the determinant of their information matrix while holding the covariance matrix V fixed at one value of η1 and η2.

“We found four distinct designs over the nine possibilities where we distinguish between designs based on the determinant of their information matrix while holding the covariance matrix, V , fixed at one value of η 1 and η 2 .”

When σ2γ1 = σ2γ2 = σ2ε = 1, the D-efficiencies of the designs relative to the optimal design for η1 = η2 = 1 range from 95.69% to 100%.

“When σ 2 γ 1 = σ 2 γ 2 = σ 2 ε = 1, the D-efficiencies of the designs found, relative to the optimal design for η 1 = η 2 = 1, range from 95.69% to 100%.”

The combinatorially constructed alternative design has a diagonal information matrix.

“Desirable features of the design, which has a diagonal information matrix, are that all whole-plot and subplot effects are estimated with maximum precision”

For the alternative design, all but three sub-subplot effects are estimated with the best possible variance, 1/32 = 0.03125.

“all but three sub-subplot effects are estimated with the best possible variance, 1/32 = 0.03125.”

One of the designs was optimal for all six scenarios where η2 < 10.

“One of the designs was optimal for all six scenarios where η 2 < 10.”

In the design in Table 3, the low and high levels of easy-to-change factor t3 are unbalanced in each of subplots 7-10.

“This is illustrated by the design in Table 3, where the low and the high level of the easy-to-change factor t 3 are unbalanced in each of the subplots 7-10.”

For η2 = 10, there are four runs at each level.

“For η 2 = 10 there are four runs at each level.”

There is not much literature on the design of three-stratum experiments.

“There is not much literature on the design of such studies.”

• cites Edmondson (1991) sourceNot checked — no full text of the source
• cites Trinca and Gilmour (2001) sourceNot checked — no full text of the source
• cites Schoen (1999) sourceNot checked — no full text of the source

D-optimal split-split-plot designs with the same nesting structures but fewer easy-to-change factors can be obtained from Tables 1 and 2 by deleting columns for easy-to-change factors.

“Of course, D-optimal split-split-plot designs with identical nesting structures and fewer easyto-change factors can easily be obtained from Tables 1 and 2 by removing columns corresponding to easy-to-change factors.”

The information matrix of the 32-run design is very nearly diagonal.

“The information matrix is thus very nearly diagonal.”

In the 32-run design, only interactions involving pairs of easy-to-change factors are estimated substantially less precisely.

“Only the interactions involving pairs of easy-to-change factors are estimated substantially less precisely.”

In the 24-run design, no information for estimating easy-to-change factor effects is lost because of the nested unit structure.

“so that, here too, no information is lost for estimating the easy-to-change factor effects due to the nested unit structure of the design.”

For two-level split-split-plot designs, a diagonal information matrix may not be optimal.

“In this section, we provide an example with a counter-intuitive result indicating that, for two-level split-split-plot designs, a diagonal information matrix may not be optimal.”

• AssumptionRests on an assumption: In completely randomized two-level designs, a diagonal information matrix guarantees an optimal design.

In the 16-run design, no information about easy-to-change factor effects is lost because of the nested unit structure.

“This demonstrates that no information on the easy-to-change factors is lost because of the nested unit structure of the design.”

Jones and Goos (2007a) showed how to avoid constructing a candidate set when searching for optimal split-plot designs.

“Jones and Goos (2007a), who showed how to avoid the construction of a candidate set in the search for optimal split-plot designs.”

• cites Jones and Goos (2007a)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Split-plot response surface designs showed some sensitivity to changes in that ratio.

“Split-plot response surface designs did, however, show some sensitivity to changes in this ratio.”

• cites Goos (2002)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Although the easy-to-change factor levels look very different for the three designs, there is no substantial difference among them for estimating the model parameters.

“While the easy-to-change factor levels look very different for the three designs, there is thus no substantial difference among them for the purpose of estimating the parameters of the model.”

Optimal design construction algorithms like the authors' attempt to limit that increase by introducing imbalance in the levels of easy-to-change factors within subplots.

“Optimal design construction algorithms like ours, however, attempt to limit that increase by introducing some imbalance in the levels of the easy-to-change factors within the subplots.”

In both sensitivity studies, it is more important to correctly specify the ratio of subplot variance to error variance, η2, than η1.

“In both of the sensitivity studies reported here, it is more important to correctly specify the ratio of the subplot variance to the error variance, η 2 .”

The higher determinant of the D-optimal design is achieved by sacrificing some precision of three subplot effect estimates and the independence between these estimates, in order to obtain reasonably small variances for the easy-to-change two-factor interaction estimates.

“This is achieved by sacrificing some of the precision of the estimates of three subplot effects and the independence between these estimates in order to obtain reasonably small variances for the estimates of the two-factor interaction effects between the easy-to-change factors.”

Larger variances for estimates of main and interaction effects of some easy-to-change factors seem very difficult to avoid in many split-plot type design problems, especially when the number of runs per subplot is as small as two.

“The larger variances for the estimates of the main and interaction effects of some of the easyto-change factors seem very difficult to avoid in many split-plot type of design problems, especially when the number of runs in every subplot is as small as two.”

In cases where the numbers of whole plots, subplots, and runs are not constrained to integer multiples, the time-saving update formulas for the information matrix described in Section 3.2 cannot be used because Theorem 1 is valid only for the scenario focused on in the article.

“It should be noted, however, that in such cases the time-saving update formulas for the information matrix described in Section 3.2 can no longer be used because Theorem 1 is valid only for the scenario we focused on in this article.”

• AssumptionThe authors state this as an assumption: The number of subplots is an integer multiple of the number of whole plots, and the total number of runs is an integer multiple of the number of subplots, n = b1 b2 k.

D-optimal split-split-plot designs depend on the two variance ratios η1 and η2.

“As mentioned in Section 2, the D-optimal split-split-plot designs depend on the two variance ratios, η 1 and η 2 .”

This explains the small variance inflation for the main-effect estimate of t3 compared to t1 and t2.

“This explains the small variance inflation for the main-effect estimate of that factor when compared to the main-effect estimates for t 1 and t 2 .”