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Robust second-order rotatable designs invariably applicable for some lifetime distributions
Communications for Statistical Applications and Methods 2021;28:595-610
Published online November 30, 2021
© 2021 Korean Statistical Society.

Jinseog Kima, Rabindra Nath Das1,b, Poonam Singhc, Youngjo Leed

aDepartment of Big data and Applied Statistics, Dongguk University, Korea;
bDepartment of Statistics, The University of Burdwan, India;
cDepartment of Statistics, University of Delhi, India;
dSeoul National University and Dankook University, Korea
Correspondence to: 1 Department of Statistics, The University of Burdwan, Rajbati, Bardhaman, West Bengal 713104, India. E-mail: rabin.bwn@gmail.com
Received March 21, 2021; Revised August 5, 2021; Accepted September 10, 2021.
 Abstract
Recently a few articles have derived robust first-order rotatable and D-optimal designs for the lifetime response having distributions gamma, lognormal, Weibull, exponential assuming errors that are correlated with different correlation structures such as autocorrelated, intra-class, inter-class, tri-diagonal, compound symmetry. Practically, a first-order model is an adequate approximation to the true surface in a small region of the explanatory variables. A second-order model is always appropriate for an unknown region, or if there is any curvature in the system. The current article aims to extend the ideas of these articles for second-order models. Invariant (free of the above four distributions) robust (free of correlation parameter values) second-order rotatable designs have been derived for the intra-class and inter-class correlated error structures. Second-order rotatability conditions have been derived herein assuming the response follows non-normal distribution (any one of the above four distributions) and errors have a general correlated error structure. These conditions are further simplified under intra-class and inter-class correlated error structures, and second-order rotatable designs are developed under these two structures for the response having anyone of the above four distributions. It is derived herein that robust second-order rotatable designs depend on the respective error variance covariance structure but they are independent of the correlation parameter values, as well as the considered four response lifetime distributions.
Keywords : correlated errors, invariant designs, lifetime distributions, mean lifetime, response surface, robust second-order rotatability
1. Introduction

Generally, response surface methodology (RSM) assumes that the response follows normal distribution with independent and homoscedastic errors (Box and Hunter 1957; Box and Draper 2007). RSM is frequently used in lifetime improvement (or quality engineering) experiments for locating the optimal level combinations to attain the specific target (Nair et al., 1992; Myers et al., 2002), while the response lifetime generally follows exponential, lognormal Weibull, gamma distributions (Lawless, 1982; Das and Lee 2009; Das 2013). In addition, lifetime observations can be correlated (Myers et al., 2002). Thus, usual RSM is not appropriate for lifetime improvement experiments. Second-order rotatable designs with correlated errors, assuming a response that follows normal distribution was introduced by Das (1997). First-order D-optimal and rotatable designs with correlated errors for some lifetime distributions have been studied by Das and Lin (2011), Das and Huda (2011), Das, Kim and Park (2015), and Das, Kim and Lee (2015). Note that second-order D-optimal slope-rotatable designs under normal distributions have been studied by Das et al., (2010). First-order models are not appropriate for unknown experimental regions, whereas second-order models are always suitable. The present article considers a correlated second-order lifetime model having the above mentioned four lifetime distributions, and errors having intra and inter-class structures. Invariant robust second-order rotatable designs (RSORDs) have been derived for these situations.

The article considers four lifetime distributions such as lognormal, gamma, exponential, Weibull and to be added. Suppose the lifetime (T1) follows a Weibull distribution with the probability density function (p.d.f.):

δ1α1(t1α1)δ1-1exp [-(t1α1)δ1],         t10.

Suppose k concomitant control experimental factors (or variables) noted as z = (z1, z2, . . . , zk)′ are adopted to explain the heterogeneity in lifetime T1. Commonly used Weibull regression models are those for which just α1, but not δ1, depends on z, and α1 can be replaced by α1(z). Therefore, the p.d.f. of X0 = ln T1, given z is as follows,

f(x0z)=1σexp [xo-μ(z)σ-exp (x0-μ(z)σ)],         -<x0<,

where σ = 1/δ1 and μ(z) = ln α1(z). From the above distribution of X0|z, it can be written as

X0=μ(z)+σh1,

where h1 follows the standard extreme value distribution with p.d.f.: exp(h1eh1 ), −∞ < h1< ∞. Note that E(h1) = −ν, where ν = 0.5772, . . . , known as Euler’s constant and Var(h1) = π2/6. The model (1.1) is known as a location-scale regression model with random component h1. A class of models can be derived for different choices of μ(z) = ln α1(z).

If δ1=1 (σ=1), the above Weibull distribution reduces to exponential lifetime distribution, and (1.1) reduces to,

X0=μ(z)+h1.

For gamma lifetime (T1) distribution with p.d.f.: 1/Γ(p)t1p-1/{α1(z)}pexp[-t1/α1(z)], t1 ≥ 0, the p.d.f. of X0 = ln T1, given z, is given by,

f(x0z)=1Γ(p)exp[(x0-μ(z))p]exp[-exp(x0-μ(z))],         -<x0<,

where Γ(p) is a gamma function of p. Similarly, from the above distribution, it can be expressed as

X0=μ(z)+h2,

where h2 follows log-gamma distribution with p.d.f.: 1/Γ(p) exp(h2peh2), −∞ < h2< ∞.

For the lifetime (T1) follows lognormal distribution (ln T1 (= X0) follows normal distribution with mean ln α1(z) and variance δ22) with p.d.f.: 1/t1δ2(2π)1/2exp[-(ln t1-ln α1(z))2/2δ22], t1 ≥ 0, δ2 ≥ 0, then

X0=μ(z)+δ2τ,

where τ follows the standard normal distribution.

In the following section, it is shown that for a real lifetime data set, the lifetime distribution is important for locating the optimal level combinations from its fitted second-order model with constant variance. The article is organized as follows. The following section presents a motivating example, while Sections 3 and 4 present correlated second-order model and rotatability conditions, respectively. Section 5 presents robust second-order rotatable designs, which is then followed by the conclusion.

2. Motivating second-order real lifetime model example

Firth (1988) has shown that the regression parameter estimates from the log-normal, or the gamma model have almost identical results with constant variance. In this section, a real lifetime experimental data set is presented which has different regression parameter estimates for these two models with constant variance. For non-constant variance, regression parameter estimates for these two models may be different which has been shown by Das and Lee (2009).

Example

Watkins et al. (1994) studied the lifetime of a cutting tool which was used to minimize the tooling costs of deducting threads on pipes to make electrical conduit. The cutting tool is known as a chaser and the angle at which it accesses the pipe is known as the rake angle. The experimental operating factors are chaser type (A), coolant type (B), speed (C), conduit manufacturer (D), and rake angle (E). For each factor, two levels were used using a 251 design with E = ABCD. The threading machine had two spindles (right and left denoted by R and L); each spindle can handle a pipe. The study variable is the time until poor thread quality is observed or the product has to be scrapped. The lifetime data and design matrix are displayed in Watkins et al. (1994). It is noted that no observation was censored, and the setting number 4 had three runs, while the setting number 15 had two runs.

The lognormal and gamma model analyses of the above data set have been derived by Das and Park (2012), where the estimates of the two models are not similar, and also the model selection criteria (based on Akaike information criterion (AIC)) are not same. For reference, both analyses results are reproduced in Table 1. From Table 1, it is clear that the regression parameter estimates are not the same in both the models. In addition, the lognormal model gives a better fit than the gamma based on AIC. It is clear that DE (5%) and AB (9%) are almost significant in the lognormal model, while they are not significant in the gamma model. It has been shown that many second-order significant terms are present in the lognormal mean model, while the variance is constant. Thus, for a lifetime analysis, an appropriate second-order model is important for locating the optimal level combinations along with its distribution. Again some runs are replicated that may introduce correlation in the response lifetime data, which was pointed out by Myers et al. (2002). For these situations, appropriate second-order designs are very important in order to obtain better information regarding the regression parameters. To the best of our knowledge these problems (second-order correlated lifetime models and the respective designs) are little studied in the quality engineering literature. In the following sections, these problems are addressed.

3. Correlated lifetime second-order models

First-order models are often used if the experimenters are remote from the optimum operating conditions of the process. The direction of steepest ascent indicates the direction in which the estimated response () increases most rapidly, under the assumption that response x is a larger-the-better characteristic. If there is a curvature in the process, then a higher degree polynomial such as the second-order model must be used. Each of the above location-scale model (equations (1.1) to (1.4)) has two components: μ(z) (systematic part) and a random component (connected to the original lifetime (T1) distribution). Note that the random component (in equations (1.1) to (1.4)) distribution is completely different from the original lifetime distribution of T1, and it is not a usual ‘random effect’ (generally used in random effects model) such as unobserved effect, missing information, censored data, unknown future prediction etc. (Lee et al., 2017). Correlated second-order linear natural logarithm lifetime models under Weibull, exponential, lognormal and gamma distributions are derived as follows.

For a lifetime T1 which follows Weibull distribution as in (1.1) with mean α1(z) (= eg(z, β), say as t1> 0) indicating μ(z) = ln α1(z) = g(z, β). This is the same as X0 = ln T1|z = g(z, β) + σh1 (from (1.1)), where h1 follows the standard extreme value distribution. Assuming that the response surface g(z, β) is second-order, we adopt the model,

ln T1z=β0+i=1kβizi+ijkβijzizj+σh1,

where h1 follows the standard extreme value distribution whose mean and variance are given in (1.1). For estimating the parameters β0, β1, . . . , βk, β11, . . . , βkk, β12, . . . , β(k1)k one needs to conduct an experiment to collect the data on the basis of which the estimation is to be done. As soon as an experiment is conducted, the experiment provides some noise factors which may be numerous, some of which may be even unidentifiable or undefinable. The whole effect of all these noise factors on the response ‘ln T1’ expressed by the experimental condition is denoted by ‘e’ and is known as an experimental error. It is written as,

y=lnT1z+e=β0+i=1kβizi+ijkβijzizj+σh1+e,or,yu=β0+i=1kβizui+ijkβijzuizuj+σh1u+eu,         1uN,

where βi are unknown regression parameters and zui (for the uth run, 1 ≤ uN, of ith factor, 1 ≤ ik) are experimental levels which are non-stochastic, h1 follows standard extreme value distribution, and eu’s represent experimental errors that arise from the noise factors. Note that there are two random variables ‘h1’ and ‘e’ in the model (3.1), where ‘h1’ is associated with the original lifetime T1 distribution, and ‘e’ is the random error which accounts for all sources of variation due to noise factors included in the experiment. The model (3.1) is a linear mixed model. Note that not one of these two random variables is the same as random effects generally used in random effect models such as generalized linear mixed models, hierarchical generalized linear models (HGLMs) or double HGLMs (Lee et al., 2017). Usual second-order response surface models do not consider the random component related to the lifetime distribution. It is assumed that h1u’s are all uncorrelated and h1u and eu’s are uncorrelated. Following usual response surface models (Box and Hunter 1957), it is assumed that eu’s follows normal distribution. In addition, this article assumes that eu’s are not independent but they are correlated, which is frequently observed in practice (Myers et al., 2002). Therefore, e = (e1, . . . , eN)T follows the multivariate normal distribution with E(e) = 0, dispersion (variance-covariance) matrix, Die(e)=σ12W1, and rank (W1) = N, where W1 is any unknown general error.

The model (3.1) can be expressed as

xu=yu+σν=β0+i=1kβizui+ijkβijzuizuj+ξu,         1uN,         or,X=Zβ+ξ,

considering σ to be known for the time being, ν is known as Euler’s constant, where ξu = σh1u + σν + eu is the composite error (Das, Kim and Park, 2015). Note that xu = yu + σν, and X = (x1, x2, . . . , xN)′, is the vector of recorded origin changed natural logarithm of lifetime observations, β = (β0, β11, . . . , βkk, β1, . . . , βk, β12, . . . , β(k1)k)′ is the vector of regression coefficients of order (k+22)×1, Z = (1 : Z1) is the design matrix, Z1 = (z11z1, . . . , zk1zk, z1, . . . , zk, z11z2, . . . , zk11zk), zi = (z1i, . . . , zNi)′, 1 ≤ ik, zi1zj = (z1iz1 j, . . . , zNizN j)′ 1 ≤ i, jk, and ξ = (ξ1, ξ2, . . . , ξN)′. Then E(ξu) = 0, Var(ξu)=(σ2π2/6+σ12)=σ22, Cov(ξu, ξu′) = Cov(eu, eu′), E(ξ) = 0, Dis(ξ)=(σ2π2/6)IN+σ12W1=σ22W say, and ‘W’ is the composite correlated error structure (Das, Kim and Park, 2015), where,

W=(1Cov(e1,e2)σ22Cov(e1,e3)σ22Cov(e1,eN)σ22Cov(e2,e1)σ221Cov(e1,e3)σ22Cov(e2,eN)σ22Cov(eN,e1)σ22Cov(eN,e2)σ22Cov(eN,e3)σ221).

For the deriving model (3.2) under theWeibull distribution, we assume σ to be known. Practically, σ is unknown, and it is replaced by its estimate σ̂ based on the original lifetime observations, where lifetime T1 follows the Weibull distribution as in (1.1). For all computations, we consider, xu = yu + σ̂ ν, u = 1, 2, . . . , N.

As in (1.2), for δ1 = 1, implying σ = 1, the second-order correlated lifetime model for exponential distribution is identical to (3.2), with xu = yu + ν, ξu = h1u + ν + eu, E(ξu)= 0, and Var(ξu)=π2/6+σ12=σ22, u = 1, 2 . . . , N, E(ξ) = 0, Dis(ξ)=(π2/6)IN+σ12W1=σ22W, where W is as in (3.2).

For the gamma distribution as in (1.3), we have the same second-order correlated lifetime model as in (3.2) with xu = yuα0, ξu = h2uα0 + eu, α0 = E(h2u), E(ξu) = 0, and Var(ξu)=α22+σ12=σ22, u = 1, 2 . . . , N, E(ξ) = 0, Dis(ξ)=α22IN+σ12W1=σ22W, assuming α0 to be known for the time being, where α22=Var(h2), h2 follows the log-gamma distribution as in (1.3) and W is as in (3.2). The same assumptions as h1u are satisfied by h2u. For all theoretical derivations with the gamma distribution, we assume α0 is known. Practically α0 is unknown, and it is replaced by its estimate α̂0 with the original lifetime observations, where lifetime T1 follows gamma distribution as in (1.3). It is considered that xu = yuα̂0, u = 1, 2, . . . , N.

For the lognormal lifetime distribution as in (1.4), we have the same model as in (3.2) with xu = yu, ξu = δ2τu+eu, E(ξu) = 0 and Var(ξu)=δ22+σ12=σ22, u = 1, 2 . . . , N, E(ξ) = 0, Dis(ξ)=δ22IN+σ12W1=σ22W, where τ follows the standard normal distribution and W is as in (3.2). The same assumptions as h1u are satisfied by τu.

Note that for the considered four lifetime distributions, the second-order response surface model is same as in (3.2), but only the unknown composite error (ξ) and its dispersion structure are different for different lifetime distributions. Also, Dis(ξ)=δ22IN, if Cov(ei, ej) = 0 for all ij, W = IN. Generally, W is unknown but for all the calculations W is assumed to be known. In practice, however, W contains a number of unknown parameters, and in the calculations which follow, the expressions for W and W1 are replaced by those obtained after replacing the unknown parameters with appropriate estimates or some assumed values. Under this situation, with the appropriate selection of the experimental levels zui’s, Z = ((zui), 1 ≤ uN, 1 ≤ ik), the design matrix is the most important part for getting maximum information regarding the unknown regression parameters. This situation is little studied for the second-order response surface models in Das (1997). The present article aims to develop robust second-order rotatable designs for the considered four lifetime distributions under the intra-class and inter-class correlation structure of W1. It is shown herein that the derived second-order rotatable designs are independent of the considered four lifetime distributions, as well as the correlation parameter values in W1, and it only depends on the pattern of the correlation structure W1. Therefore, the derived designs are termed as invariant (free of these considered four lifetime distributions) and robust (free of correlation parameter values) second-order rotatable designs.

4. Second-order rotatability conditions with correlated errors

For the second-order correlated lifetime model (3.2), we assume (Z′W1Z) is positive definite for known W, which is the best linear unbiased estimator of β is β̂ = (Z′W1Z)1(Z′W1X) with,

Dis(ξ^)=(ZW-1Z)-1=(ABCBPQCQR)(k+22)×(k+22),

where A, P and R are all symmetric matrices given by

A(k+1)×(k+1)=(f00f0.11f0.22f0.kkf11.0(fii.jj),1i,jkfkk.0),

f00 = 1′W11, f0. j j = 1′W1(zj1zj) = (zj1zj)′W11 = f j j.0, 1 ≤ jk, fii. j j = (zi1zi)′ W1(zj1zj), 1 ≤ i, jk,

B(k+1)×k=(f0.1f0.2f0.k(fii.j)k×k,         1i,jk),

f0. j = 1′W1zj, fii. j = (zi1zi)′W1zj, 1 ≤ i, jk,

C(k+1)×(k2)=(f0.12f0.13f0.(k-1)k(fii.jl)k×(k2),         1i,j<lk),f0.jl=1W-1(zj1zl),         fii.jl=(zi1zi)W-1(zj1zl),         1ik,1j<lk,Pk×k=(fi.j)),fi.j=ziW-1zj,         1i,   jk,Qk×(k2)=(fi.jl)),         fi.jl=ziW-1(zj1zl),         1i,   j<lk,R(k2)×(k2)=(fij.lt)),fij.lt=(zj1zl)W-1(zl1zt),         1i<jk,         1l<tk.

Note that f0. j = f j.0, f0. j j = f j j.0, f0. jl = f jl.0, fi. j = f j.i, fii. j = f j.ii, fi. jl = f jl.i, fii. j j = f j j.ii , fi j.lt = flt.i j. In the inverse matrix (Z′W1Z)1, the element corresponding to fm in (Z′W1Z) is denoted by fm for all ‘m’ included in the preceding expressions.

Following Das (1997), the necessary and sufficient conditions of second-order rotatability for the model (3.2) (with correlation structure W) are the following,

  • (I)

    • (i) f0. j = f0. jl = 0, 1 ≤ j < lk,

    • (ii) fi. j = 0, 1 ≤ i, jk, i ≠ j,

      • (iii) (1) fii. j = 0, 1 ≤ ik, 1 ≤ jk,

      • (2) fi. jl = 0, 1 ≤ ik, 1 ≤ j < lk,

      • (3) fii. jl = 0, 1 ≤ ik, 1 ≤ j < lk,

      • (4) fi j.lt = 0, 1 ≤ i < jk, 1 ≤ l < tk, (i, j) ≠ (l, t),

  • (II)

    • (i) f0. j j = constant = a0(> 0), 1≤ jk,

    • (ii) fi,i=constant=1e(>0), 1≤ ik,

    • (iii) fii,ii=constant=(2c+d)(>0), 1 ≤ ik,

  • (III)

    • (i) fii. j j = constant = d(> 0), 1 ≤ i, jk, i ≠ j,

    • (ii) fij,ij=constant=1c(>0), 1 ≤ i < jk,

  • (IV)

    fii.ii=2fij.ij+fii.jj,         1i<jk.

The non-singularity condition of correlated second-order rotatable design is

  • (V)

    2c+k(d-a02f00)>0,

    where f00 and f ’s are in (4.1), and a0, 1/c, and d are as in (4.2).

Following Das (1997), the variance of the estimated response at z (variance function) of a robust second-order rotatable design is given by,

Var(x^z)=f00+(2a1+e)r2+(c2+d1)r4,

which is a function of r2=Σi=1kzi2, where

f00=(2c+kd){f00(2c+kd)-ka02},         a1=-a0{f00(2c+kd)-ka02},         d1=c(a02-df00)2{f00(2c+kd)-ka02},

and f00, c, d, e are as in (4.2).

Note that a design ‘d’ of k factors for the correlated lifetime model (3.2) remains second-order rotatable for all the variance-covariance matrices belonging to a well defined class W0 = W positive definite: WN×N defined by a particular correlation structure possessing a definite pattern is called a robust second-order rotatable design (RSORD), with reference to the variance-covariance class W0.

5. RSORDs for correlated lifetime models under intra and inter-class error structures

The present section derives the necessary and sufficient conditions for second-order rotatability following Das (1997) for lognormal, gamma, exponential, andWeibull lifetime distributions under intraclass and inter-class error variance-covariance structure W1. These two correlation structures are very similar, where the inter-class structure is an extension of intra-class structure. RSORDs are derived for both the correlation structures. Practically, inter-class structure is mostly observed in many data sets.

5.1. Intra-class correlation structure

The simplest dispersion matrix, or errors variance-covariance structure, is known as ‘intra-class’ or ‘uniform’ correlation structure. This dispersion structure occurs if any two observational errors have the identical correlation coefficient (ρ*) and each with identical variance (σ12). This arises if all the observations are examined under identical conditions, or from the same run in a batch, or from the same furnace. It is presented by,

Dis(e)=σ12[(1-ρ*)IN+ρ*EN×N]=σ12W1(ρ*),

where σ1> 0, −(N − 1)1< ρ*< 1, IN is an identity matrix of order N, and EN×N is a matrix of order N × N with all elements 1 (unity). The combined error dispersion structure (σ22W) as in (3.2) reduces to the intra-class structure σ22W1(ρ) as in (5.1), where ρ=σ12/σ22ρ* and σ22=(σ2π2/6+σ12) for Weibull distribution, σ22=(π2/6+σ12) for exponential distribution, σ22=(α22+σ12) for gamma distribution and σ22=(δ22+σ12) for lognormal distribution.

For the second-order model (3.2) (for all the four considered lifetime distributions) with errors having intra-class structure σ12W1(ρ*) as in (5.1), the simplified correlated second-order rotatability and non-singularity conditions (4.2) and (4.3) give the following designs along with all necessary and sufficient conditions.

The necessary and sufficient conditions of a robust second-order rotatable design (for all the four considered lifetime distributions) under the intra-class error structure (5.1), for all values of ρσ22W(ρ) as in (5.1), in the model (3.2) are (for all 1 ≤ i, jk)

  • (I) Σu=1Nzui1γ1zui2γ2zui3γ3zui4γ4=0, for any γi is odd and Σi=14γi4,

  • (II) (i) Σu=1Nzui2=constant, (ii) Σu=1Nzui4=constant

  • (III) Σu=1Nzui2zuj2=constant, i ≠ j,

  • (IV) Σu=1Nzui4=3Σu=1Nzui2xuj2, ij,

  • (V) λ4λ22>kk+2         where,         Nλ2=u=1Nzui2         and         3Nλ4=u=1Nzui4.

The above conditions (I)–(V) as in (5.2) are independent of intra-class correlation coefficient ρ and are the same as the necessary and sufficient conditions of usual independent errors second-order rotatable designs (SORDs). Therefore, usual SORDs are invariant (independent of any one of the considered four lifetime distributions) RSORDs for the intra-class error structure (5.1).

5.2. Inter-class correlation structure

Inter-class error dispersion matrix, or errors correlation structure is often observed in practice. For instance, products from different furnaces, or in batch products, where it is noted that observations are grouped into some batches such that within each batch there is the same intra-class correlation and between batches there is no correlation. Let ‘m’ be the number of groups and ‘n’ be the number of observations in each group. Similar situations can indicate inter-class correlation structure in a production system, which is displayed as follows.

Dis(e)=σ12[ImA]=σ12W1(ρ*,0),

where ⊗ denotes Kronecker product, A = (1 − ρ*)In + ρ*En×n (the intra-class structure as in (5.1)), En×n is as in (5.1) and N = mn. For the error structure (5.3), the composite error structure (σ22W) as in (3.2) reduces to the inter-class structure σ22W1(ρ,0) as in (5.3), where ρ = σ12/σ22ρ* and σ22 is as in (5.1), and its values are different for the different distributions.

For all the considered four lifetime distributions with second-order model (3.2), the second-order rotatability (4.2) and non-singularity (4.3) conditions under the inter-class structure (5.3) reduce to

    • (I) (i) f0.j=0u=1Nzuj=0, 1 ≤ jk, f0.jl=0u=1Nzujzul=0, 1 ≤ j < lk,

    • (ii) fi.j=011-ρu=1Nzuizuj-ρ(1-ρ){1+(n-1)ρ}{u=1nzuiu=1nzuj++u=n(m-1)+1mnzuiu=n(m-1)+1mnzuj}=0,, 1 ≤ i, jk, i ≠ j,

      • (iii) (1) fii.j=011-ρu=1Nzui2zuj-ρ(1-ρ){1+(n-1)ρ}{u=1nzui2u=1nzuj++u=n(m-1)+1mnzui2u=n(m-1)+1mnzuj}=0, 1 ≤ ik, 1 ≤ jk,

      • (2) fij.l=011-ρu=1Nzuizujzul-ρ(1-ρ){1+(n-1)ρ}{u=1nzuizuju=1nzul++u=n(m-1)+1mnzuizuju=n(m-1)+1mnzuj}=0, 1 ≤ i < jk, 1 ≤ lk,

      • (3) fii.jl=011-ρu=1Nzui2zujzul-ρ(1-ρ){1+(n-1)ρ}{u=1nzui2u=1nzujzul++u=n(m-1)+1mnzui2u=n(m-1)+1mnzujzul}=0, 1 ≤ ik, 1 ≤ j < lk,

      • (4) fij.lt=011-ρu=1Nzuizujzulzut-ρ(1-ρ){1+(n-1)ρ}{u=1nzuizuju=1nzujzut++u=n(m-1)+1mnzuizuju=n(m-1)+1mnzulzut}=0, 1 ≤ i < jk, 1 ≤ l < tk, (i, j) ≠ (l, t)

    • (II) (i) f0.jj=constant{σ2(1+(n-1)ρ)}-1u=1Nzuj2=a0(>0), 1 ≤ jk,

    • (ii) fi.i=constantσ-2[1(1-ρ)u=1Nzui2-ρ(1-ρ){1+(n-1)ρ}{(u=1nzui)2+(u=n(m-1)+1mnzui)2}]=1e(>0), 1 ≤ ik,

    • (iii) fii.ii=constantσ-2[1(1-ρ)u=1Nzui4-ρ(1-ρ){1+(n-1)ρ}{(u=1nzui2)2++(u=n(m-1)+1mnzui2)2}]=(2c+d)(>0), 1 ≤ ik,

    • (III) (i) fii.jj=constantσ-2[1(1-ρ)u=1Nzui2zuj2-ρ(1-ρ){1+(n-1)ρ}{u=1nzui2u=1nzuj2++u=n(m-1)+1mnzui2u=n(m-1)+1mnzuj2}]=d(>0), 1 ≤ i, jk, i ≠ j,

    • (ii) fij.ij=constantσ-2[1(1-ρ)u=1Nzui2zuj2-ρ(1-ρ){1+(n-1)ρ}{(u=1nzuizuj)2++(u=n(m-1)+1mnzuizuj)2}]=1c(>0), 1 ≤ i < jk,

  • (IV) fii.ii = 2 fi j.i j + fii. j j; 1 ≤ i < jk,

  • (V) 2c+k(d-a02f00)>0,

    where fii.ii, fii. j j and fi j.i j are as in (II) (iii) and (III) (i), (ii), f00 = N/1 + (n − 1)ρ and 1/c, d, a0 are as in (5.4).

5.3. Construction of RSORDs under inter-class structure

Method I

For all the considered four lifetime distributions with model (3.2) and errors having the inter-class structure (5.3), RSORDs can be constructed from ‘m’ usual SORDs (each with k factors and n runs) by arranging each SORD as the design points in each group with ‘n’ runs, such that the design points in each group satisfy the second-order rotatable structure in the usual sense (with uncorrelated errors). Then all these ‘m’ usual SORDs from ‘m’ groups form a RSORD for this situation.

  • It can be readily examined that (5.4) holds for these designs.

  • For a non-singularity condition, the following design parameters can be noted.

    a0=mnλ2σ2{1+(n-1)ρ}=mnλ2σ2{1+(n-1)ρ},         1e=Nλ2σ2(1-ρ),         d=1σ2[Nλ4(1-ρ)-mρ(nλ2)2(1-ρ){1+(n-1)ρ}],   1c=1σ2[Nλ4(1-ρ)],f00=Nσ2{1+(n-1)ρ}.

Using these design parameters, the non-singularity condition 2/c+k(d-a02/f00) in (5.4) simplifies to,

λ4λ22>kk+2,

which is the non-singularity condition for each set of observations of ‘m’ sets. Therefore, the designs described by Method I satisfy the non-singularity condition for all values of ρ.

Method II

For all the considered four lifetime distributions with model (3.2) and errors having the inter-class structure (5.3), RSORDs can be constructed as follow. Let D0:k × 2p be a design matrix which is obtained from the 1/2kp fraction of 2k-experiment with levels −1 and +1, where the defining equation in obtaining the fraction does not include any main effect or interactions containing four or fewer factors (Resolution V). Suppose the array of treatment combinations is L(k×G) = D0 with G = 2p, where L can be partitioned as L(k×G) = [L1(k×G0):L2(k×G0): . . . :Ls(k×G0)], where sG0 = G, and the inner product of any two distinct rows of Li is zero, i = 1, 2, . . . , s.

Let us consider a matrix H1 of order (k + 1) × s written with elements −1 and +1 such that the inner product of every pair of rows of the matrix is 0 (zero). Based on the conditions of H1, it can be arbitrarily assumed that the first row of H1 consists of all +1’s. By deleting the first row from H1, we get,

H10=(h11h12h13h1sh21h22h23h2shk1hk2hk3hks).

Suppose Di = ς Diag (h1i, h2i, . . . , hki), Fi = [Li:Di], i = 1, 2, . . . , s and F = (F1:F2: . . . , :Fs). The k rows of matrix F form a RSORD under the inter-class structure (5.3) of k factors with m = s groups, with each group of size n = (G0 + k) and number of design points, N = s(G0 + k) = sG0 + sk = G + sk = 2p + mk, s = m being the number of groups in which the observations are divided. Note that ‘ς’ is determined from the condition (IV) of (5.4), and the design points which are obtained from the part Li of Fi are from the appropriate fractional factorial 2k experiment, are given by Di, i = 1, 2, . . . , m are the so called star points. Thus, the overall design so constructed, forms a kind of central composite design usually constructed.

Obviously, in this design construction method, a factor point (obtained either from the appropriate fraction of a 2k experiment or the star points) may have to be repeated. In certain situations, the repetition has to be accepted so that different parameters of the design match in order to give an overall rotatable design in the present context. The parts of the design belonging to the different groups do not give a SORD.

The RSORDs obtained by the Method II satisfy condition (I) of (5.4). The expressions (i), (ii) and (iii) in (II) of (5.4) simplify respectively to,

a0=(G0+ς2)mσ2{1+(n-1)ρ},         1e=1σ2[(G0+ς2)m1-ρ-ρς2m(1-ρ){1+(n-1)ρ}],(2c+d)=1σ2[(G0+ς4)m1-ρ-ρ(G0+ς2)2m(1-ρ){1+(n-1)ρ}]

and they are the same for i = 1, 2, . . . , k.

Note that n = (G0 + k), N = mn = (G0 + k)m = G + mk = 2p + mk, s = m. Again, the expressions (i) and (ii) in (III) of (5.4) reduce respectively to,

d=1σ2[G0m1-ρ-ρ(G0+ς2)2m(1-ρ){1+(n-1)ρ}],1c=1σ2(G0m1-ρ),

and they are the same for 1 ≤ i < jk.

Note that ‘ς’ is determined from (IV) of (5.4) (using the above results), in consideration of the following equation,

2G0m1-ρ+[G0m1-ρ-ρ(G0+ς2)2m(1-ρ){1+(n-1)ρ}],=(G0+ς4)m1-ρ-ρ(G0+ς2)2m(1-ρ){1+(n-1)ρ}

or, 3G0 = G0 + ς4 or, ς = (2G0)1/4.

Thus, with ς = (2G0)1/4, rotatability conditions (5.4) are satisfied by the design points described by Method II for all ρ.

Using the above design parameters and noting f00 = N/σ2{1 + (n − 1)ρ}, the expression 2/c+k(d-a02/f00) simplifies to,

mσ2(1-ρ)[(k+2)G0-k(G0+ς2)2n].

Thus, the non-singularity condition (V) in (5.4) reduces to

(k+2)G0-k(G0+ς2)2n>0nG0(G0+ς2)2>kk+2G0+kG0+2{1+(2G0)12}>kk+2.

Therefore, the designs described by Method II satisfy the non-singularity condition (5.5) for all values of ρ as G0 is always greater than k. So, the designs derived by using Method II are RSORDs under the inter-class structure (5.3).

Note that the above Method II for constructing RSORDs is quite powerful as the original design Do can also be selected from different sources other than factorial experiments such as balanced incomplete block (BIB) designs, which have been utilized extensively by many authors in the construction of SORDs with uncorrelated errors. Construction of a RSORD with k = 7 factors using BIB design with parameters v = b = 7, r = k = 3, λ = 1 is described as follows. Let the symmetrical BIB design with parameters v = b = 7, r = k = 3, λ = 1 be given as (0, 1, 3), (1, 2, 4), (2, 3, 5), (3, 4, 6), (4, 5, 0), (5, 6, 1), (6, 0, 2). Let us consider the 7 factors corresponding to the 7 treatments in the BIB design. Construct a 7×8 matrix Li, i = 1,2, . . . , 7 from the ith block as follows. Suppose the treatments in the ith block of the design are (i1, i2, i3). Then the three rows of the 7 × 8 matrix Li, which correspond to the factors represented by the treatments i1, i2 and i3 of the block give a full factorial experiment with 3 factors each having two levels +1 and −1. Let us write 0 as the level of every other factor in each of the columns. The 7 × 8 matrix of −1, +1 and 0’s thus obtained is called Li. The following example illustrates the above method.

Example

Using the BIB design for Method II, let us consider an experimental design with seven factors, eight independent groups, each of fifteen observations i.e., k = 7, m = 8, n = 15 and Dis(e) = I8W15×15(ρ), N = mn = 120. The level combinations of 7 factors in 8 groups are given.

Group totals for xi’s are given in the following table.

Note that the design points within each group do not satisfy the usual SORD conditions, but the overall design forms a RSORD under the inter-class structure (5.3). In addition, the number of design points by Method II is less than the number of points by Method I design.

6. Concluding remarks

The present article considers the univariate response lifetime random variable having distributions such as gamma, lognormal, Weibull and exponential, which are often observed in practice. Regressor variables are considered in lifetime models for handling heterogeneity. First-order lifetime location-scale regression models with constant variance are extended to second-order correlated models. Mixed linear logarithms design of lifetime correlated models are derived from two random components, whereas one component is associated with the response lifetime distribution and the other is the error term. Second-order rotatable designs are derived under these four lifetime distributions with errors having intra and inter-class structures. For the intra-class error structure, usual SORDs are robust SORDs and invariant for these four lifetime distributions. For inter-class structure, two design construction methods have been suggested which give invariant robust SORDs.

Lifetime data sets can be homogeneous, or heterogeneous with lognormal, or gamma distributed (Das and Lee, 2009; Das and Park, 2012; Das, 2013; Das, Kim and Park, 2015). In industrial design of experiments, experimental units can not be treated as independent, as they produce correlation among observations via a repeated measures scenario as in split plot design (Myers et al., 2002). From a practical point of view, the current manuscript has considered correlated non-linear second-order models (as the fitted model is non-linear, t^1u=exp(β^0+Σi=1kβ^izui+Σi1kβijzuizuj)) for some useful lifetime distributions. Two errors correlation structures such as intra-class and inter-class have been considered in the report, while intra-class is a particular form of inter-class structure that is relevant in many practical situations such as blocked experiments and split-plot experiments (Goos and Jones, 2011). Under these situations, improved designs are orthogonally blocked designs, which are D-optimal, and they are commonly used in quality engineering (Cheng, 1988). There are many more important errors correlation structures such as autocorrelated, tri-diagonal, compound symmetry, which are very realistic and useful. We propose to discuss these structures in our subsequent articles.

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x2−1−111−1−1110200000
x3−1−1−1−111110020000
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x51−11−1−11−110000200
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Group2123456789101112131415

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x6000000000000020
x700000000000000−2

The developed designs in the article can be used to predict the optimal operating conditions that achieves the target mean value, while reducing the variance. Dual response surface (DRS) approach was introduced by Myers and Carter (1973) for this purpose, whereas second-order response surface designs are used for separate modeling of mean and variance. For this purpose, the joint generalized linear models (JGLMs) have been proposed by Nelder and Lee (1991). Therefore, for both the DRS and JGLMs approaches, the present developed designs can be used to attain the specific target mean value, and simultaneously reduce the process variability.

Group12345678
x12−22−22−22−2
x222−2−222−2−2
x32−2−222−2−22
x42222−2−2−2−2
x52−22−2−22−22
x622−2−2−2−222
x72−2−22−222−2
TABLES

Table 1

Results for mean models of threading machine lifetime data from log-normal and gamma fits

Log-normal model Gamma model


Covar. est. s.e. t P-value est. s.e. t P-value
Mean const. 4.10 0.11 37.46 <0.01 4.37 0.11 39.73 <0.01
A −0.84 0.11 −7.66 <0.01 −0.84 0.11 −7.55 <0.01
B 0.48 0.11 4.32 <0.01 0.47 0.11 4.24 <0.01
C 0.44 0.11 4.01 <0.01 0.42 0.11 3.83 <0.01
D −0.18 0.11 −1.66 0.10 −0.14 0.11 −1.28 0.21
Model E 0.04 0.11 0.38 0.71 0.07 0.11 0.59 0.55
CE −0.36 0.11 −3.30 <0.01 −0.31 0.11 −2.76 0.01
DE 0.22 0.11 1.96 0.05 0.18 0.11 1.65 0.10
AB −0.19 0.11 −1.71 0.09 −0.17 0.11 −1.58 0.12

Var model const. −0.44 0.20 −2.19 0.03 −0.54 0.20 −2.64 0.01

AIC 599.0 604.4

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