Copula-based robust optimal block designs
CCopula-based robust optimal block designs
W.G. M¨uller a , A. Rappold b and D.C. Woods ca Department of Applied Statistics, Johannes Kepler University Linz, Austria b Plasser & Theurer Connected G.m.b.H, Hagenberg im M¨uhlkreis, Austria c Southampton Statistical Sciences Research Institute, University of Southampton, UKBlocking is often used to reduce known variability in designed experiments by collecting togetherhomogeneous experimental units. A common modelling assumption for such experiments is thatresponses from units within a block are dependent. Accounting for such dependencies in boththe design of the experiment and the modelling of the resulting data when the response is notnormally distributed can be challenging, particularly in terms of the computation required tofind an optimal design. The application of copulas and marginal modelling provides a computa-tionally efficient approach for estimating population-average treatment effects. Motivated by anexperiment from materials testing, we develop and demonstrate designs with blocks of size twousing copula models. Such designs are also important in applications ranging from microarrayexperiments to experiments on human eyes or limbs with naturally occurring blocks of size two.We present methodology for design selection, make comparisons to existing approaches in theliterature and assess the robustness of the designs to modelling assumptions.Key words: Binary response; pseudo-Bayesian D -optimality; equivalence theorem; generalizedlinear model; marginal model. Statistical design of experiments underpins much quantitative work in the biological, physicaland engineering sciences, providing a principled approach to the efficient allocation of (typicallysparse) experimental resources to address the aims of the study. Often, experiments aim tounderstand a process by modeling discrete data, for example arising from the observation ofa binary or count response. For completely randomized experiments, assuming homogeneousexperimental units, a generalized linear model (GLM) may provide an appropriate descriptionand there has been much research into the construction of optimal and efficient designs formulti-factor GLMs, including Woods et al. (2006). Dror and Steinberg (2006, 2008) and Russellet al. (2009). See Atkinson and Woods (2015) for a comprehensive review.When heterogeneous experimental units can be grouped into more homogenous groups,or blocks, accounting for this grouping can improve the precision of inferences made fromthe experimental data. Methods to find block designs for discrete data have recently beenproposed by, amongst others, Woods and van de Ven (2011), Niaparast and Schwabe (2013)and Waite and Woods (2015). Two modeling paradigms have been adopted in the designliterature: conditional models where the joint distribution of the data is derived by explicitlyincluding block-specific random effects (e.g. generalized linear mixed models, Breslow andClayton, 1993); and marginal models, where the dependence structure of the data is specifiedseparately from the marginal distribution of each response (e.g. with parameters estimated viageneralized estimating equations (GEEs), Liang and Zeger, 1986). For the linear model, thesetwo modeling approaches coincide. In this paper, we find optimal designs under a marginal Corresponding author: Department of Applied Statistics, Johannes Kepler University Linz, 4040 Linz, Aus-tria; [email protected] a r X i v : . [ s t a t . M E ] N ov odeling approach when the intra-block dependence structure is defined via a copula. Suchmodels are particularly appropriate when block effects are not of interest in themselves and theaim of the experiment is to understand the effects of treatment factors averaged across blocks.Optimal designs for marginal models using alternative definitions of the dependence structurehave been found by Hughes-Oliver (1998), Atkinson and Ucinski (2004) and van de Ven andWoods (2014).Although our methods can be generalized to arbitrary block sizes, we focus on the importantspecial case of experiments with blocks of size two (see Godolphin, 2018). Such blocks occurroutinely in microarray experiments (Bailey, 2007; Kerr, 2012) and in experiments on people,for example with eyes or arms as experimental units (David and Kempton, 1996). Practicalmotivation for our work comes from a materials science experiment. In Section 3 we finddesigns appropriate for aerospace materials testing experiments similar to those performed byour collaborators at the UK Defence Science and Technology Laboratory. The aim of theseexperiments is to compare the thermal properties of a set of novel materials against a referencematerial. In particular, one aim is to assess the probability of failure due to the exposure toextreme (high) temperatures. The experiment is performed using a arc jet to heat materialsamples which are held in one of six “wedges”, each of which holds a pair of samples on a strutattached to a circular carousel, see Figure 1. Hence, the experiment can be considered as a blockdesign with six blocks, each containing two units. In the particular experiment considered here,six materials were tested, a reference and five novel samples. A variety of measures are madeon each tested sample, including a visual inspection of quality to assess material failure whichleads to a binary (pass/fail) response. It is this response for which we find optimal designs.
4 Experimental set-up
The material samples were exposed to a high enthalpy flow generated by an arc jet heater. They weare inserted in pairs in a wedge configuration in series by means of a revolving carousel. The dwell time each sample pair can be held in the flow can be specified. Figure 4 shows the experimental set-up.
Figure 4 Arc jet experimental set-up
Note there are 6 struts on the carousel, each holding a wedge of two material samples. There are an additional 2 strut stations. One is empty to permit the arc jet to start, and the second holds a dwell calorimeter on a strut and a swept calorimeter on a sub-strut. Both these monitor arc jet flow conditions. Figure 5 gives a labelled view of the arc jet test station components.
Figure 5 Arc jet components
Figure 6 shows a view of the relative strut positions. The view has the direction of flow into the page. Note the model positioning system (MPS) rotates the carousel of samples in an anticlockwise direction with strut location numbering down from 8 to 1.
Figure 6 Relative strut positioning
The output from the thermocouples are sampled at a rate of 1000 samples per second. A typical insertion time is 7 seconds. Thus 7000 temperature measurements make up a time series. The start and end of each insertion time is given by a timing pulse captured with the data. Figure 7 and Figure 7 show the form of the timing pulse, R-POS.
Figure 7 shows two pulses; the time between the pulses represents the insertion time for a sample. Figure 7 is an expanded view of a single pulse and is used to identify start and end points of a sample.
Figure 7 Timing pulse R-POS
Figure 8 Expanded view of a single pulse
A typical three time series set is shown in Figure 9. These measurements were obtained using the same material type and arc jet run conditions. A variation between each series can be seen. It is this variation that is of interest, since it represents the
Figure 1: Arc jet carousel, struts and “wedges” (left) and schematic (right). In addition tothe six wedges for holding material samples, the carousel had two further wedges used fortemperature measurement.In common with most nonlinear models, the performance of a given design for a copula-based GLM model may depend on the values of the model parameters that define both themarginal model and the dependence structure. If strong prior information is available, thenlocally optimal designs can be sought for given values of the model parameters. Otherwise,Bayesian (e.g. Overstall and Woods, 2017) or maximin (e.g. King and Wong, 2000) approachescan be adopted. In common with much of the recent literature on designs for GLMs, we findoptimal designs robust to the values of the model parameters via a pseudo-Bayesian approach(e.g. Atkinson et al., 2007, ch. 18), with a classical quantity for design performance averaged2ith respect to a prior distribution on the parameters. Here, we adopt variants of D -optimalityfor design selection.The remainder of the paper is organized as follows. In Section 2 we introduce the statisticalmodels we employ, including copulas, and develop design methods for blocked experiments. Anillustrative comparison is made to previous design approaches based on GEEs using an examplefrom Woods and van de Ven (2011). In Section 3 we demonstrate and assess our methods viaapplication to the materials testing example. In particular, we show how prior information onthe parameters influences the choice of optimal design. We provide a brief discussion and someareas for future work in Section 4. Suppose the experiment varies m treatment factors, x T = ( x , . . . , x m ), and the experiment has b blocks of size k ; throughout, our examples will assume k = 2. The j th unit in the i th blockreceives treatment x Tij = ( x ij , . . . , x mij ) ( i = 1 , . . . , b ; j = 1 , . . . , k ) and realizes observation Y ij . The x ij are chosen from a standardized design space X = [ − , m and are not necessarilydistinct. Independence of observations Y ij , Y i (cid:48) j (cid:48) , for i, i (cid:48) = 1 , . . . , b ; j, j (cid:48) = 1 , . . . , k , is assumedacross blocks ( i (cid:54) = i (cid:48) ) but we allow dependence within a block ( i = i (cid:48) ), which we describe via acopula model. The problem of specifying a probability model for dependent random variables Y i , . . . , Y jk canbe simplified by expressing the corresponding k -dimensional joint distribution F Y i ,...,Y ik in termsof marginal distributions F Y i , . . . , F Y ik , and an associated k -copula (or dependence function) C defined as follows (cf. Nelsen, 2007). Definition 2.1. A k -copula is a function C : [0 , k → [0 , , k ≥ , with the following proper-ties:1. ( uniform margins ) for every u ∈ [0 , k , if at least one coordinate of u is , then C ( u ) = 0 , and if all coordinates of u are except u i , then C ( u ) = u i .
2. ( k-increasing ) for all a , b ∈ [0 , k such that a ≤ b , V C ([ a , b ]) ≥ , where V C is the measure induced by C on [0 , k . The connection between a copula and a joint probability distribution is given by Sklar’sTheorem (Sklar, 1959), which affirms that for every k -dimensional joint distribution F Y i ,...,Y ik with marginal distributions F Y i , . . . , F Y ik , there exists a k -copula C , defined as in Definition 2.1,such that F Y i ,...,Y ik ( y , . . . , y k ) = C ( F Y i ( y ) , . . . , F Y ik ( y k )) , (1)for all y , . . . , y k ∈ R . Conversely, if C is a k -copula and F Y , . . . , F Y k are distribution functions,then the function F Y ,...,Y k given by (1) is a joint distribution with marginals F Y , . . . , F Y k . The3opula C may not be unique for discrete margins, however the practical limitations for statisticalpurposes are little, cf. Genest and Neˇslehov´a (2007).Owing to Sklar’s theorem, parametric families of copulas represent a powerful tool to de-scribe the joint relationship between dependent random variables. Selecting the appropriatedependence within an assumed parametric copula family reduces to the selection of copulaparameters, which correspond, for example, to a specific measure of association for the mod-eled random variables. Assuming Y i , . . . , Y ik are continuous random variables with associatedcopula C ( · ; α ), one measure of association proposed by Joe (1990) is given by τ k = 12 k − − k (cid:90) [0 , k C ( · ; α ) dC ( · ; α ) − . (2)Equation (2) is a generalized version of Kendall’s τ , and hence establishes a correspondencebetween a scalar copula parameter α and the degree of dependence. More details and propertiesof this quantity, and another more traditional measure of concordance, can be found in Genestet al. (2011). In common with most work on optimal design of experiments, we base our criterion on the Fisherinformation matrix (FIM), the inverse of which provides an asymptotic approximation to thevariance-covariance matrix of the maximum likelihood estimators of the model parameters.Let ζ i = ( x i , . . . , x ik ) ∈ X k denote the k treatment vectors assigned to the units in block i ( i = 1 , . . . , b ; j = 1 , . . . , k ). We will work within a class of normalized block designs defined as ξ = (cid:26) ζ , . . . , ζ n w , . . . , w n (cid:27) , < w i ≤ , n (cid:88) i =1 w i = 1 , with n ≤ b distinct (support) blocks. As defined, bw i must be integer and represents thereplication of the i th support block ( i = 1 , . . . , n ). Without loss of generality, we assume the first n blocks in the design correspond to ζ , . . . , ζ b , with the remaining b − n blocks being replicates.We relax the assumption that bw i is integer to find so-called approximate or continuous designs;see also Cheng (1995) and Waite and Woods (2015). Let Ξ denote the space of all possibledesigns of this form.Denote the vector of responses from the i th block as Y i = ( Y i , . . . , Y ik ) T , i = 1 , . . . , b , with corresponding expectation vector η i = [ η ( x i ; β ) , . . . , η ( x ik ; β )] T , where η ( · ; · ) is a known function and β = ( β , . . . , β r ) T is a vector of unknown parametersrequiring estimation. Denote the marginal distribution function for the j th entry in the blockas F Y ij ( y ij ; x ij , β ), j = 1 , . . . , k , and denote the joint distribution, derived via a copula trans-formation, for the k responses in the i th block as C ( F Y i , . . . , F Y ik ; α ) where α = ( α , . . . , α l ) T are unknown (copula) parameters.The FIM M ( ζ i ; γ ) for the i th block is an ( r + l ) × ( r + l ) matrix with vw th element M ( ζ i ; γ ) vw = E (cid:18) − ∂ ∂γ v ∂γ w log c Y i ( η i , α ) (cid:19) , (3)4here γ = ( γ , . . . , γ r + l ) T = ( β , . . . , β r , α , , . . . , α l ) T and c Y i ( η i , α ) = ∂ k ∂y i . . . ∂y ik C ( F Y i , . . . , F Y ik ; α )is the joint density function represented through a copula C in accordance with Equation (1).The FIM for an approximate block design ξ is then given by M ( ξ ; γ ) = n (cid:88) i =1 w i M ( ζ i ; γ ) . An optimal design ξ (cid:63) maximizes a scalar function ψ { M ( ξ ; γ ) } of the information matrix.Previous work on optimal designs for copulas has focussed on finding completely randomizedlocally-optimal designs for multivariate responses, which can be considered as a block designwhere every unit within a block must receive the same treatment. Denman et al. (2011) found D -optimal designs for a bivariate response ( k = 2) that maximized ψ D { M ( ξ ; γ ) } = det M ( ξ ; γ ),and Perrone and M¨uller (2016) developed a corresponding equivalence theorem. These methodswere extended to the local D A -criterion, and, as a special case, for the D s -criterion in Perroneet al. (2016). Other relevant uses of design of experiments in copula models are Deldossi et al.(2018) and Durante and Perrone (2016), but until now all relied on the availability of a single“best guess” vector of parameter values.To overcome this dependence on assumed parameter values, here we adopt a pseudo-Bayesianapproach to constructing block designs. Furthermore, our primary interest is typically in s meaningful linear combination of the parameters. Such combinations can be defined as elementsof the vector A T γ , where A T is an s × ( r + l ) matrix of rank s < ( r + l ). If M ( ξ ; γ ) is non-singular,the variance-covariance matrix of the maximum likelihood estimator of A T γ is proportional to A T { M ( ξ ; γ ) } − A . Hence, we define a robust D A -optimal block design ξ (cid:63) as the design thatmaximizes Ψ D ( ξ ; G, A ) = (cid:90) Γ log det[ A T { M ( ξ ; γ ) } − A ] − d G ( γ ) , (4)where G ( γ ) is a proper prior distribution function for γ and Γ ⊂ R r + l is the support of G . Seealso Woods and van de Ven (2011).Most often the main interest is in an s < ( r + l )-dimensional subset of the parameters. Insuch a case, a robust D s -optimal block design can be found by maximizingΨ D ( ξ ; G ) = (cid:90) Γ log det (cid:8) M − M M − M T (cid:9) d G ( γ ) , (5)following the partition of the information matrix as M ( ξ ; γ ) = (cid:18) M M M T M (cid:19) . Here, M is the ( s × s ) partition related to the parameters of interest. This criterion follows asa special case of the D A -criterion with A T = ( I s s × ( r + l − s ) ), with I s the s × s identity matrixand 0 s × ( r + l − s ) the s × ( r + l − s ) zero matrix.We evaluate a design ξ via its Bayesian efficiencies under a given criterion, relative to anappropriate reference design ξ ∗ (see, for example, Waite, 2018). Under robust D s -optimality,this efficiency is given by:eff( ξ, ξ ∗ ) = (cid:32) exp (cid:82) B log det[ M ( ξ, γ ) − M ( ξ, γ ) M − ( ξ, γ ) M T ( ξ, ˜ γ )] d F ( γ )exp (cid:82) B log det[ M ( ξ ∗ , γ ) − M ( ξ ∗ , γ ) M − ( ξ ∗ , γ ) M T ( ξ ∗ , γ )] d F ( γ ) (cid:33) /s .
5e find designs that maximize (4) and (5) numerically using a version of the Fedorov-Wynnalgorithm (Wynn, 1970; Fedorov, 1971), as implemented in R package docopulae (Rappold,2018).The optimality of a block design ξ (cid:63) under the robust D A -criterion, regardless of how itwas found, can be assessed via application of the following Kiefer-Wolfowitz-type equivalencetheorem. The proof is similar to that for completely randomized experiments with multivariateresponse, see Perrone et al. (2016) for the locally-optimal design case. Theorem 2.2.
The following properties are equivalent:1. ξ (cid:63) is D A -optimal;2. for every ζ ∈ X k , (cid:90) B tr [ M ( ξ (cid:63) ; γ ) − A ( A T M ( ξ (cid:63) ; γ ) − A ) − A T M ( ξ (cid:63) ; γ ) − M ( ζ ; γ )] d G ( γ ) ≤ s ;
3. over all ξ ∈ Ξ , the design ξ (cid:63) minimizes the function max ζ ∈X k (cid:90) B tr [ M ( ξ (cid:63) , γ ) − A ( A T M ( ξ (cid:63) , γ ) − A ) − A T M ( ξ (cid:63) , γ ) − M ( ζ ; γ )] d G ( γ ) , where Ξ is the set of all possible block designs. We demonstrate robust optimal block designs for copula models using a simple example fromWoods and van de Ven (2011), which allows comparison to the designs found by those authorsfor a GEE model. We find robust designs for a single-factor log-linear regression model assumingPoisson marginal distirbutions and quadratic linear predictor, implying log { η ( x ; β ) } = β + β x + β x . The prior distribution G is uniform on the parameter space [ − , × [4 , × [0 . , . k = 2 and intra-block dependencedefined according to one of the following bivariate copula functions.1. Product Copula , which represents the independence case, C ( u , u ) = u u , with generalized Kendall’s τ of τ = 0.2. Clayton Copula , C α ( u , u ; α ) = (cid:2) max (cid:0) u − α + u − α − , (cid:1)(cid:3) − α , with α ∈ (0 , + ∞ ) and generalized τ = αα +2 .3. Gumbel Copula , C α ( u , u ; α ) = exp (cid:0) − (cid:2) ( − ln u ) α + ( − ln u ) α (cid:3) α (cid:1) , with α ∈ [1 , + ∞ ) and generalized τ = α − α .6he first copula is chosen for reference purposes; the latter two represent opposing depen-dencies in the tails (lower tail dependence for the Clayton versus upper tail dependence forthe Gumbel). To isolate the effect of the copula structure from the strength of the depen-dence, we set α for each copula such that the values for Kendall’s τ coincide at three level,s τ = (cid:15) > , / , / (cid:15) = 10 − is a small number to approximate the zerocase, but avoid singularity issues.To find robust D -optimal designs, objective function (4) was evaluated using quadrature(Gotwalt et al., 2009). Optimal designs under the Clayton and Gumbel copulas are shown inFigure 2, and demonstrate that increasing the generalized dependence (i.e. increasing τ ) leadsto designs placing more weight on support blocks with points on the edge of the design space.All the designs display a “mirror-image” structure, with all design points having x >
0. Thesefeatures are common in designs for Poisson regression (see Russell et al., 2009). The designsfound under the Gumbel copula tend to include more support blocks but the pattern in thechanges to these blocks as τ is increased is similar for both copulas.Figure 2: Optimal designs for the comparative example; rows: Clayton and Gumbel copula;columns levels τ = (cid:15) > , / , / τ = 0 for a particular copula. In particular the D-efficiencies for the Clayton andGumbel model were 96.3% and 99.7% respectively. This efficiency expectedly decreases as theassociation within the block increases, for τ = 1 / D -optimal designs were found under the samePoisson marginal models and prior distribution but with the dependence described using aGEE approach with an exchangeable correlation matrix and pairwise working correlation of7 .
5. The optimal design found was given by: ξ (cid:63) = (cid:26) ( . ,
1) (1 , .
60) ( − . , . . . . (cid:27) . (6)That is, for example, the first support block is ζ = (0 . , τ = (cid:15) > τ = 1 / τ = (cid:15) > τ = 1 / In this section we return to the materials testing example to find and assess designs for comparingsix materials in block of size two under a variety of modelling assumptions. The measuredresponse is binary, with each material sample either passing or failing a visual check. We labelthe five novel materials as “treatments”, with the reference material considered as a control.Marginally, we assume a logistic regression to model the differences between materials set up as Y ij ∼ Bernoulli { η ( x ij ; β ) } ; η ( x ij ; β ) = expit (cid:32) β + (cid:88) l =1 β i x ijl (cid:33) , where expit( u ) = 1 / { − u ) } , Y ij is the binary response from the i th unit in the j th block( i = 1 , j = 1 , . . . , b ), η ( x ij ; β is the associated probability of success, x ijl is an indicatorvariable taking the value 1 if the i th unit in the j th block was assigned treatment l ( l = 1 , . . . , β , . . . , β are unknown parameters to be estimated. Here, β is the logitfor the reference material, with β l being the difference in expected response, on the logit scale,between the reference material and the l th novel material or treatment.The choice of copula and the strength of intra-block association makes little difference to thedesign selected. However, assuming different marginal models and adopting a local or pseudo-Bayesian approach has a strong impact on the designs. Example designs for the Gumbel copulaare shown in Figure 3.With a null marginal model, i.e. β T = (0 , , , , , β T = (0 , − , , − , , − − ,
1] for each β l ( l =0 , . . . ,
5) yields designs with unequal weights spread across all material combinations. Changingto a continuous uniform prior on the space [ − , × [ − , × [1 , × [ − , − × [3 , × [ − , − β T = (0 , − , , − , , − τ = 0 .
33; rows - local and pseudo-Bayesian; columns - assumed parameters or prior mean of β T = (0 , , , , ,
0) and β T = (0 , − , , − , , − The modeling of block effects by copulas seems a natural choice and allows for elegant separationof the block and the marginal effects. Experimental designs for such models are now readily cal-culable. The pseudo-Bayesian D A -optimality criterion was added to the R package docopulae version 0.4 (see Rappold, 2018) with the functions wDsensitivity and wDefficiency , both re-lying on a prespecified quadrature scheme for evaluation of the integrals. In this paper we haveconcentrated on finding designs to estimate the complete parameter vector but the implemen-tation provides flexibility for checking for symmetry, model discrimination, etc., as investigatedin Perrone et al. (2016).Our examples are confined to the case k = 2. Whilst there is no theoretical necessity forthat it is difficult to specify high-dimensional parametric copulas with a sufficient range ofdependence, for details see the excellent survey of Nikoloulopoulos (2013). However, work onthis issue would go well beyond the scope of this paper. It might also be interesting to contrastour findings with some known analytic results for blocks of size two as, for example, given inCheng (1995) where a Gaussian copula is implicitly assumed.9 cknowledgements We are grateful to Keith Warburton and Rob Ashmore from the UK Defence Science and Tech-nology Laboratory for providing details of the materials testing example. W.G. M¨uller wouldlike to acknowledge the hospitality of the Southampton Statistical Sciences Research Instituteduring his sabbatical, when this research was initiated. He was partially supported by projectgrants LIT-2017-4-SEE-001 funded by the Upper Austrian Government, and Austrian ScienceFund (FWF): I 3903-N32 and D.C. Woods was partially supported by Fellowship EP/J018317/1from the UK Engineering and Physical Sciences Research Council.
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