Food Quality and Preference

Transcription

1 Food Qulity nd Preference () Contents lists ville t ScienceDirect Food Qulity nd Preference journl homepge: Interpreting sensory dt y comining principl component nlysis nd nlysis of vrince Giorgio Lucino, Tormod Ns,, * NOFIMA FOOD, Mtforsk, Oslovegen, Ås, Norwy Deprtment of Mthemtics, University of Oslo, Blindern, Oslo, Norwy rticle info strct Article history: Received June Received in revised form August Accepted August Aville online Septemer Keywords: PCA ANOVA Sensory profiling ASCA This pper compres two different methods for comining PCA nd ANOVA for sensory profiling dt. One of the methods is sed on first using PCA on rw dt nd then relting dominting principl components to the design vriles. The other method is sed on first estimting ANOVA effects nd then using PCA to nlyse the different effect mtrices. The properties of the methods re discussed nd they re compred on dt set sed on sensory nlysis of cndy product. Some new plots re lso proposed for improved interprettion of results. Ó Elsevier Ltd. All rights reserved.. Introduction * Corresponding uthor. Tel.: + ; fx: +. E-mil ddress: tormod.nes@mtforsk.no (T. Ns). Sensory pnel dt cn lwys e looked upon s three-wy dt tles with ssessors, ojects/smples nd ttriutes s the three wys. In order to nlyse differences nd similrities etween smples nd ssessors s well s the correltion structure mong ttriutes, the three-wy structure of the dt needs to e tken into ccount. This cn e done in vrious wys using different underlying ides nd philosophies. A technique tht cn e useful in some cses is regulr multivrite nlysis of vrince (MANOVA, Kent & Biy, ) for testing the effect of smples nd/or ssessors for ll ttriutes simultneously. Usully one is, however, interested in more insight thn this method cn give nd therefore other techniques re to e preferred. A much used method within the re of sensory nlysis is the generlised procrustes nlysis (GPA), treting ech ssessor slice s mtrix, followed y principl component nlysis of the verge or consensus mtrix (Dijksterhuis, ). GPA is sed on the ide of mking individul ssessor dt mtrices s similr s possile to ech other y scling nd rottion. Another possile pproch is regulr principl components nlysis (PCA) of ll individul sensory profiles followed y two-wy ANOVA of the most importnt components with ssessor nd products effects s independent vriles. (Ellekjr, Ilseng, & Ns, ). The rows in the dt tle used for this nlysis correspond to ll smples * ssessor comintions nd the columns correspond to sensory ttriutes. This method cn e modified using the MANOVA (Lngsrud, ) method which hndles significnce testing in more elegnt wy. Using prtil lest squres regression (PLS-) of ll sensory profiles versus the two independent design vriles ssessors nd products nd their interction is closely relted pproch (Mrtens & Mrtens, ). PCA sed on the lterntive unfolding with ojects nd ssessor * ttriutes s columns nd rows hs een tested in for instnce Dhl nd Ns (). In the sme pper generlised cnonicl correltion nlysis CCA (Crroll, ) nlysis of individul sensory dt ws tested nd compred to PCA. Clssicl three-wy fctor nlyses such s Tucker- nd PARAFAC hve lso found useful pplictions within the frmework of sensory nlysis (Bro, Qnnri, Kiers, Ns, & Frøst, ; Brockhoff, Hirst, & Ns, ). Recently n lterntive method for three-wy nlysis of vrince (ANOVA) hs een proposed in the chemometric literture (ASCA, Jnsen et l., ), ut the method is not yet tested for sensory dt. ASCA is method which first uses regulr two-wy ANOVA for ech ttriute seprtely, estimtes the effects (under regulr ANOVA restrictions) nd then uses PCA on the min effects mtrices nd interction mtrix seprtely for interprettion of results. The method hs recently een comined with PARAFAC in the so-clled PARAFASCA (Jnsen et l., ). Other importnt pproches nd overviews of lterntive methods cn e found in Qnnri, Wkeling, Courcoux, nd McFie (, ) nd in Hnfi nd Kiers (). The present pper is comprison study of two of the ANOVA sed methods descried ove. In prticulr we will e interested in compring the newly developed ASCA method with trditionl PCA of the unfolded three-wy dt tle followed y ANOVA (here -/$ - see front mtter Ó Elsevier Ltd. All rights reserved. doi:./j.foodqul...

2 G. Lucino, T. Ns / Food Qulity nd Preference () clled PC-ANOVA). As cn e noted, the two methods re closely relted in the sense tht they re oth sed on the sme two sic methodologies, two-wy ANOVA nd PCA, ut with the difference tht the two methodologies re used in opposite order. These pproches hve the dvntge over other methods tht they focus oth on the multivrite spects of the sensory profiles nd the explicit reltion of the sensory dt to the design of the study. The methods will e compred conceptully nd lso with respect to results otined in n empiricl illustrtion.. Theory In the present pper we will consider three-wy dt tle with I * M rows corresponding to M replictes of I smples, K columns corresponding to the ttriutes nd with J slices corresponding to ssessors. We refer to Fig. for n illustrtion of the structure of the dt set. Three wy dt of this type cn lwys, for ech ttriute k, e modelled y the two-wy ANOVA model X k ijm ¼ lk þ k i þ k j þ k ij þ ek ijm ðþ Here the l is the generl men, the s re min effects for products, the s min effects for ssessors, the s the interctions nd e is the rndom error term corresponding to replicte vrition. For ANOVA purposes, the error terms re ssumed to e uncorrelted nd normlly distriuted with the sme vrince. The usul wy of pplying this model is to ssume tht the ssessor nd interction effects re rndom, leding to mixed model (see Ns & Lngsrud, ). Sometimes experimentl designs re used for the smples nd in such studies (Brdseth et l., ), the product effect cn e split in severl components corresponding to the experimentl fctors in the design (Box, Hunder, & Hunter, ). How to hndle this extension within the frmework of the methodologies presented here will e discussed elow. How to hndle structures in the replictes will lso e discussed in the sme sections. In the following we will use the symol X to denote the unfolded three-wy dt tle with I * M * J rows nd K columns. Using this symol it is possile to rewrite the model in Eq. () for ll the ttriutes simultneously s follows X ¼ l t þ D B þ D B þ D B þ E where l is the generl men vector for ll K ttriutes simultneously, is vector of s, the D,D nd D re the dummy design mtrices for the products, ssessors nd interctions etween ssessors nd products respectively nd the B s re the corresponding prmeter mtrices. The B corresponds to the s in Eq. (), the B to the s nd B to the s. The design mtrix D will hve one column for ech ssessor nd consists of s nd s with in column j nd row i if this line corresponds to n oservtion for ssessor ðþ j. The sme structure holds for the other two mtrices. The mtrix E is the mtrix of residuls. Correltions etween the different elements (columns) of this mtrix re possile (Mrdi, Kent, & Biy, ), ut it is lwys ssumed within multivrite ANOVA tht the error terms for different oservtions nd replictes re independent. The model () cn e used directly, either for ech ttriute (column) seprtely or for ll simultneously for testing hypotheses out product nd ssessor effects. An exmple of n importnt hypothesis relted to this model is H :B =, which is the hypothesis of no product effects. For the univrite ANOVA, this hypothesis cn e seprted in K individul hypotheses, one for ech ttriute. Similr hypotheses cn e set up for ssessor nd interction effects. If wnted, one cn lso construct comined hypothesis of for instnce B nd B s is done in the ASCA pper (Jnsen et l., ) nd in Ns & Lngsrud,. The min prolems with regulr ANOVA pproches is tht they only focus on hypothesis tests nd provide little further insight out the reltions etween the ttriutes. Therefore ANOVA will usully e ccompnied with some type of PCA for further interprettion of the reltions etween the vriles. In this pper we will discuss two lterntive pproches proposed in the literture for providing this type of dditionl insight y comining ANOVA with PCA. For the purpose of the methods to e discussed elow, it is of interest to estimte the effects mtrices B in Eq. () ove. This is usully done y lest squres (LS) fitting of the responses to the design mtrices, ut in order to otin unique results, one needs to dd restriction on the prmeter estimtes (see e.g. Le, Ns, & Rødotten, ). This cn e done in vrious wys, ut the most common wy is to use the restriction tht ll min effects of ssessors nd min effects of products sum to nd tht the sme is true for the interctions summed either over ssessors or products. In this pper min ttention will e given to lnced designs, ut how to extend the pproch to more generl dt sets will lso e discussed. For the lnced cse, the min effects nd interctions hve prticulrly simple expression sed on simple verges nd sutrction, i.e. ^ k i ¼ X k i X k ðþ ^ k j ¼ X k j X k ^ k ij ¼ Xk ij X k i X k j þ X k where X k i is the verge for product i nd ttriute k, X k j is the verge for ssessor j nd ttriute k. nd X k is the totl verge for ttriute k. Note tht the interctions cn e considered s otined y doule centring of the originl dt mtrix. When PCA is used in this pper we will lwys use it on centred dt, i.e. dt for which the verge hs een sutrcted for ech column. ðþ ðþ.. PCA-ANOVA Smple Assessor Attriute Fig.. The dt structure for descriptive sensory dt. The simplest wy of comining ANOVA with PCA is to use PCA directly on the unfolded dt mtrix X in Eq. () where the numer of columns corresponds to the numer of ttriutes nd the numer of rows corresponds to ll ssessor, product nd replicte comintions. This implies tht the PCA gives components tht re comintions of ll the effects in the model (). A possiility is to verge over replictes efore computtion of principl components, ut generlly this is not nturl since the replictes re needed for testing purposes in the susequent ANOVA. Exmples of the use of this nd similr methodologies cn e found in Ellekjr et l. () nd in Lngsrud ().

3 G. Lucino, T. Ns / Food Qulity nd Preference () The PCA model, with let use sy A components, for this dt set (men centred) cn e written s X ¼ TP T þ E where T now represents the A first scores for the M * I * J product * ssessor comintions nd P represents the lodings for the K ttriutes for the sme components. The E represents the rest, i.e. the components with smll vrince, sometimes thought of s noise. If we sort the X mtrix ccording to ssessors (in the verticl direction), the first I * M rows of T will contin the scores for ll the products for the first ssessor, the next I * M lines will contin the scores for ll oservtions for ssessor etc. A similr sorting cn e done for the smples. As soon s the scores re computed, they cn e interpreted y the use of sctter plots nd lso relted to the design mtrix using regulr two-wy ANOVA. The ANOVA model for the scores cn e written s in Eq. () with X replced y T. All clcultions of effects nd hypothesis tests re done s for regulr ANOVA. Since the T- scores re uncorrelted, running seprte ANOVAs for ech response cn now e justified. For exct sttements of joint significnces etc. they cn, however, should lso e considered in joint pproch. The scores in Eq. () cn lwys e written s (using Eq. ()) T ¼ XP ¼ðD B þ D B þ D B þ EÞP ðþ ðþ The lst prt of the eqution clerly shows tht the scores re functions of ll the effects in the dt tle including the errors E. Note lso tht the noise prt of T, i.e. EP, cn e written s liner function of the originl noise mtrix E. Since the error terms contriution of the scores re liner functions of the originl errors, there re resons to expect n error distriution closer to the norml for the principl components thn for the originl ttriutes. The product, ssessor nd interction effects cn s ove e computed from verges of the T s. For the purpose of improved interprettion of the multivrite vriility in the dt nd its reltion to the design vriles we will in this pper propose to plot these effects in the sme PCA scores plot s the originl oservtions (see lso Lngsrud, ). This is done y simply plotting the fctor effects for the different components ginst ech other in the PCA scores plot. For instnce for the min effects for products, the estimted vlues (see Eq. ()) otined y ANOVA of score, re plotted ginst the corresponding estimted -vlues for score. Note tht this is identicl to doing the corresponding verging over X-vlues, projecting these verges onto P nd then plotting them in the sme wy s for the originl scores T. This implies tht it is meningful to interpret them vs. the sme lodings s used for the interprettion of the originl scores. Another feture tht will e proposed here for enhnced interprettion of the scores plots, is the superimposition of line segments in oth plotting directions (for PCA) corresponding to the level of noise in the sme two directions. In the plots presented here we use the squre root of the residul error vrince for the ANOVA models for his purpose. For component numer this mens tht we first compute the squre root of the residul error vrince for the ANOVA model of the first component vs. the design vriles nd then present this vlue s line segment long the first component. The line segment is centred t. The sme is then done for component numer. Alterntively, one cn use the lest significnt difference (LSD) vlues from multiple testing using the sme ANOVA model. In oth cses, the line segments provide the user with visul tool for getting quick nd direct impression of the importnce of the effects seen in the plot. Note tht ll the usul tools of ANOVA, such s different types of sums of squres nd corresponding tests (Type, Type II sums of squres (SS) etc.) cn e used lso for the t s. In this pper, however, with lnced dt, ll the SS-types will give the sme results. It should lso e mentioned tht s in regulr ANOVA, multiple comprison tests my e used for ssessing which individul products nd ssessors tht re different from ech other long the different PCA directions. Comined hypotheses relted to D nd D s proposed in Ns nd Lngsrud () nd for ASCA (Jnsen et l., cn lso e used here. The method cn lso esily e extended to situtions where the product effect is composed of different experimentl fctors, for instnce ccording to fctoril design. A simple exmple where the product effect is composed of two experimentl fctors cn e modelled s X k ijlm ¼ lk þ / k i þ d k l þ k j þ /d k il þ /k ij þ dk lj þ ek ijlm Where u nd d re now the two experimentl fctors in the product design nd is s efore the ssessor effect. A three-wy interction is lso possile to incorporte in the model. The PCA of X goes s efore nd the ANOVA of the scores T is performed y simply incorporting n extr fctor in the model. All tests nd computtions of min effects nd interctions etc. go s usul. Another extension which is esily hndled y the PC-ANOVA is the use of more complex error structure s discussed in e.g. Le et l. (). An exmple of such structure is the quite common replicte error structure d k im þ ek ijm where the d s correspond to the systemtic replicte effect within ech product nd the e s correspond to the regulr rndom error noise. This model is quite typicl in situtions where the sme physicl smple, i.e. replicte within product) (for instnce fish) is served to ll pnellists. In such cses, ech individul fish, m, is replicte within product (for instnce specil tretment) nd will thus correspond to one of the d-terms in Eq. (). The superscripts nd suscripts hve the sme menings s efore; k denotes ttriute, i denotes product, j denotes ssessor nd m denotes replicte. Multiplying this effect y P, one otins new error vector with the sme structure d im P þ e ijm P ðþ ðþ ðþ The two error terms in the sum now represent the vectors of the error contriutions in Eq. () for ll ttriutes considered simultneously. In other words, one cn esily see tht the more complex error structure in Eq. () is split in the sme wy for the scores s for the originl vriles. Imlnce with respect to the numer of replictes is lso simple to hndle. The sme effects cn e used in the model (see Eq. ()) nd the sme ANOVA cn e used with the pproprite correction for the degrees of freedom. The clcultion of the effects is, however, slightly more complex, ut this is esily hndled y modern ANOVA progrmmes. In situtions with missing product nd ssessor comintions the sitution is more complex, ut not more complex thn for regulr ANOVA. The prolem is then how to define nd compute interction nd min effects. Different suggestion is proposed. One simple possiility is to eliminte the interctions, ut this is not dvisle for sensory dt t lest unless pre-tretment hs een done to reduce the scling effect (Romno, Brockhoff, Hersleth, Tomic, & Ns, in press). Another possile modifiction of the PC-ANOVA ws proposed in Lngsrud (), where more elegnt wy of testing significnce ws proposed. In tht cse, however, less emphsis ws put on the effects of the fctors for the different principl components. Thus, this technique is merely to e considered s modifiction of clssicl MANOVA nd lies somewhere etween the present pproch nd the clssicl MANOVA. Another interesting possiility is the PLS- method dvocted in Mrtens nd Mrtens () with the sensory profile s the multivrite response nd

4 G. Lucino, T. Ns / Food Qulity nd Preference () the dummy design vriles s the independent vriles. This method is lso grphiclly oriented, ut less developed when concerns significnce testing. There is lso close reltion of PC-ANOVA to n extension of the Tucker- model proposed y vn der Kloot nd Kroonenerg (). The clssicl Tucker- model is one with oth common scores nd common lodings nd cn e written s X j ¼ TW j P T þ E ðþ where now X j indictes ssessor or slice numer in the dt mtrix in Fig.. The model () ssumes tht the different ssessors shre the sme dimensions nd their reltion to the externl dt, ut it llows for different weight to the two dimensions, represented y the individul mtrices W j. The modifiction proposed in vn der Kloot nd Kroonenerg () is to relte the common product scores T to externl dt, for instnce product design mtrix s ws done for the PC-ANOVA. Representing T s liner function of the externl design vriles D for the products, the model in () cn e written s X j ¼ DBW j P T þ E ðþ where B is mtrix of regression coefficients nd the D is the design mtrix for the products. Note tht the wy the ssessors nd products comine in this model is different from the PC-ANOVA. Here the joint effect is comintion of n dditive effect for the products nd multiplictive effect for the ssessors. An ovious dvntge of the PC-ANOVA is its simplicity nd tht it provides oth significnce tests sed on ANOVA nd visul tools for direct interprettion of the tests (min effects, interction plots nd multiple comprisons). The min disdvntge is tht if the fctors spn different multivrite spces, the numer of components to interpret my e high... ASCA Compred to the PC-ANOVA, the ASCA method is sed on reversing the order of the two opertions ANOVA nd PCA. The first step is to use the regulr two-wy ANOVA model for ech ttriute (model () ove) nd estimte the effects using the regulr ANOVA restrictions (sum equl to over the levels). Considering these vlues for ll ttriutes t the sme time gives us mtrix of effects for the smples (dimension I * K), mtrix of effects for the ssessors (J * K) nd three-wy mtrix (I * J * K) for the interctions. The two former re regulr two-dimensionl mtrices nd cn then e nlysed directly y the use of PCA. The ltter is nlysed y the use of PCA (lso so-clled Tucker-. see Tucker ()) on n unfolded mtrix s descried ove. Three different wys of unfolding re possile, ut here we will focus on the sme unfolding s for the PC-ANOVA, i.e. the unfolded mtrix hs dimensions I * M * J nd K. Note tht the mtrices used in ASCA correspond to the estimted versions of the mtrices B, B nd B in the model () ove. As n exmple, the B mtrix cn e written s X X X K X K : : : ^B ¼ : : ðþ B : : A X I X X K I X K As n lterntive to using Tucker- for the interctions one cn use the PARAFAC s ws suggested in Jnsen et l. (). This corresponds to using the PARAFAC on the doule centred mtrix, i.e. fter sutrction of min effects. The method is clled PARAFASCA. Note tht this pproch is direct extension to severl dimensions of the pproch proposed in Mndel (). The difference etween ASCA nd the PC-ANOVA pproch is tht here the verging is tken efore PCA, while ove it ws tken fter PCA. This is generlly n dvntge for ASCA since one otins PCA plots which re focused on the different effects nd not influenced y everything t the sme time. It my thus possily give clerer conclusions for ech of the seprte effects. As will e seen from the exmple elow, however, the multivrite spces spnned y the different effects re rther similr for this dt set. Tle Percentge explined vrince y the PCA model of the unfolded dt orgnised s oject * ssessors * replictes vs. ttriutes Numer of components PC % exp vr. PC. % exp vr PC % exp vr Percentge vrince explined Bites Hrd Elstic Sticky Trnsp Cumultive percentge vrince explined Acid Sweet Rsp. Sugr PC. % exp vr Fig.. PC-ANOVA. Scores nd Lodings plot for component nd of the unfolded sensory dt.

5 G. Lucino, T. Ns / Food Qulity nd Preference () Tle Mixed model ANOVA performed on components,,,, (effect of ssessors nd interctions considered rndom nd effect of smple considered fixed) Source Sum Sq. d.f. Men Squre Squre root of men squre F Pro > F PC Assessor.... Smple... Assessor * smple.... Error... Totl. PC Assessor.... Smple... Assessor * smple.... Error... Totl. PC % exp vr PC Assessor... Smple.... Assessor * smple.... Error... Totl. PC Assessor.... Smple.... Assessor * smple.... Error... Totl. All three replictes included in the nlysis. Since no informtion out rndom vrition is ville in the plot, it is not ovious how to mke direct ssessment of significnce of the differences etween products or ssessors long the different xes s could e done for the PC-ANOVA using multiple comprisons. A possile extension of ASCA would e to dd some type of confidence ellipses sed on for instnce the ootstrp (see e.g Pges & Husson, ). For the sme reson it is nturl to use MANOVA nd individul ANOVA s efore the ASCA to provide dditionl insight out significnt effects tht cn e used to interpret the ASCA plots. Regrding imlnce, the sme s stted for the PC-ANOVA cn e stted lso here. Also with respect to more complex error structure, the ASCA method cn e used. The only modifiction tht hs to e done is tht restricted mximum likelihood (REML) estimte is needed for improved estimtion of effects. REML is method tht tkes the more complex error structure into ccount when estimting the effects. The stndrd LS estimtes cn lso e used since they re unised, ut the REML estimtes re more precise. Also incorportion of fctoril designs in the product structure is possile. As long s the effects cn e estimted, the method cn e used. One simply ends up with more thn three mtrices to sumit to PCA. For instnce in the model () ove, one will end up with different PCA nlyses. As the numer of fctors increses, the numer of plots lso increses. The ASCA method cn lso e used to nlyse the joint effect of for instnce the min effects of products nd the interctions s ws demonstrted in Jnsen et l. (). PC % exp vr PC % exp vr PC % exp vr. Fig.. PC-ANOVA. Plot of scores verged over ssessors fter performing PCA. () PC vs. PC. () PC vs. PC. The horizontl nd verticl rs close to the centre represent the squres root of the MSE. All vriles were tested using two-wy ANOVA (model ()) nd ll ttriutes were found to e significnt for the seprtion of the smples nd therefore kept during the study.. Results nd discussion All clcultions were performed using MinitÓ, UnscrmlerÓ. nd custom mde Mtl/Octve routines which re freely ville for downlod from the first uthor s wesite Dt set The dt set chosen for this pper is from sensory nlysis of cndy product. different cndies (I = ). There re K = sensory ttriutes nd J = ssessors in the pnel nd M = replictes. The ttriutes were trnsprency, cidity, sweet tste, rsperry flvour, sugr coted texture tested with spoon, iting strength in the mouth, hrdness, elsticity in the mouth, stick to teeth in the mouth... PC-ANOVA As cn e seen from Tle, the explined vrinces re quite high for this dt set (% explined fter components). The first fctor is totlly dominting with percentge of explined vrince equl to %. Scores nd lodings plot of ll the oservtions re presented in Fig.. The scores re mrked ccording to the products. As cn e seen from Fig., there is some disgreement mong the ssessors, ut the overll greement mong the ssessors

6 G. Lucino, T. Ns / Food Qulity nd Preference () nd replictes seems to e quite good for ech product s compred to the difference etween the products. The first component distinguishes etween two groups of ojects, ojects, nd on one side nd ojects nd on the other. The ltter group hs more sugr flvour, higher sweetness, rsperry nd cidity tste, while the former group hs more stickiness, hrdness, trnsprency etc. The second xis primrily distinguishes etween smples nd nd nd. Ojects nd re the less cidic nd the sweetest nd most rsperry flvoured mong them. All the results were in ccordnce with wht could e expected from the design of the smples. As cn e seen from the ANOVA tles in Tle, the smple effect is the dominting effect for oth the first two components, while for the third component, the ssessor effect is the strongest. For the fourth component, none of the effects re significnt. ANOVA ws lso conducted for the rest of the components with some significnt effects here nd there, ut these results re not considered further here due to their very low explined vrince (less thn % in totl). The interction effect is significnt for oth the first components. All these results correspond well to wht is seen in the plots (Figs. ). The dvntge of the plots, however, is tht one utomticlly sees which smples those re similr nd lso their reltion to the ttriutes. The product nd ssessor effects (see Eqs. () ()) re plotted long the sme xes nd with the sme units s for the originl scores. These results re presented in Figs.. The Fig. represents the products nd shows very cler tendency. As compred to the squre root of the corresponding MSE, i.e the stndrd devition of the noise (presented s stright line round in oth directions), one cn lso get visul interprettion of the differences s compred to the noise level. Multiple testing (Tukey s method) of the second xis (verticl) in Fig., shows tht smple is significntly different from ll except smple nd tht smple is only slightly different from nd not significntly different from the rest. The only smple which is significntly different from ll is smple. For this dt set, the verge score plot of the products does not provide new insight s compred to the overll plot, ut in more complex sitution with severl more ojects nd smller differences etween them this type of plot my simplify interprettion considerly. In Fig. is presented the differences mong the ssessors nd s cn e seen, the differences re much smller thn for the product effects. Along the third xis there is some more vriility mong the ssessors, however, which is lso reflected in the F-test for ssessors (Tle ). PC % exp vr. - - PC % exp vr PC % exp vr PC % exp vr. - - PC % exp vr PC % exp vr PC % exp vr. Fig.. PC-ANOVA. Plots of scores verged over smples fter performing PCA on the rw dt. The horizontl nd verticl rs close to the centre represent the squres root of the MSE PC % exp vr. Fig.. PC-ANOVA. PCA performed on dt fter doule centring. Point is mrked ccording to ssessors. Figure is mgnifiction of the figure.

7 G. Lucino, T. Ns / Food Qulity nd Preference () The interction plot in Fig. is presented oth for the sme units s ove nd for units showing the results more clerly. The min dvntge of the interction plot is proly tht it cn shed some light onto which ssessors tht re the most similr to the verge nd which re the most different. It is for instnce cler here tht ll points elonging to ssessor lie close to the centre of the plot while ssessor numer hs some of the points fr wy from the centre. This mens tht ssessor is much closer to the pnel verge for ll the smples thn ssessor numer. The ltter is then more responsile for the interction effect thn ssessor numer. More specificlly, one cn lso identify for which ssessor nd product comintions the interctions re lrgest. Compring this with the lodings plot one cn lso otin informtion out which ttriutes tht re involved in the interctions. A study of the normlity of the residuls of the originl dt nd the PCA scores ws done ccording to the ides mentioned in Section.. Generlly, the residuls from the scores hve distriution closer to normlity thn the residuls from the originl vriles. Two exmples re presented in Fig. nd. in Fig. is given typicl exmple relted to one of the originl vriles QQ Plot of Smple Dt versus Stndrd Norml (stickiness) while in Fig. is presented the normlity plot for the residuls from the ANOVA of the first principl component. As cn e seen, the plot for the ltter follows much strighter line thn for the former except from few outlying points on ech side... ASCA The product plots, ssessor plots nd the interction plots re presented in Figs.. For the products in Fig., more or less the sme interprettion s for the PC-ANOVA cn e mde. The plot is lmost identicl except for degrees switch of the first component which hs no effect on the interprettion. Since no rndom vriility is directly relted to the plot, it is, however, hrd to tell wht is significnt, in prticulr long xis. For the ssessors plot in Fig., one cn see tht there re strong similrities etween the lodings plot here nd the lodings plot in Fig., indicting much of the sme structure in the multivrite differences etween the ssessors s etween the products. There re, however, some smll differences relted to for instnce the ttriute sugr coting. This my indicte possile dvntge of the ASCA. It is, however, hrd to tell from the scores plot whether these differences re significnt or not. For the interction plot in Fig., the sme s stted ove is the cse for the lodings. The interction scores plot cn e used for the sme purpose s ove. Agin we see tht ssessor Quntiles of Input Smple PC % exp vr Stndrd Norml Quntiles QQ Plot of Smple Dt versus Stndrd Norml PC % exp vr. Sweet Rsp. Quntiles of Input Smple - - PC. % exp vr Sugr Bites Sticky Hrd Trnsp Elstic Stndrd Norml Quntiles Fig.. Norml proility plot for () stickiness nd the () first principl component. -. Acid PC. % exp vr Fig.. ASCA. PCA scores nd lodings plot for the oject effects mtrix.

8 G. Lucino, T. Ns / Food Qulity nd Preference () PC % exp vr. - - PC % exp vr PC % exp vr PC % exp vr. PC. % exp vr Trnsp Sugr Elstic Sticky Hrd Bites Rsp. Acid PC. % exp vr Sweet Rsp. Trnsp Sugr Sticky Hrd Bites Elstic -. Sweet Acid PC. % exp vr Fig.. ASCA. PCA scores nd lodings plots for the ssessor effects mtrix PC. % exp vr Fig.. ASCA. PCA scores nd lodings plot for the interctions dt mtrix. numer is much closer to the verge thn is ssessor numer. The min conclusion is tht for this dt set, the two methods gve prcticlly the sme results nd interprettions. The fct tht ll the three mtrices give similr lodings is strongly relted to the fct tht so few components re needed for the PCA on the rw dt (PC-ANOVA), In sitution with different multivrite structure for the different model effects, the numer of components for PC-ANOVA would siclly e sum of the dimensions for ech of the effects nd this is clerly not the cse here. An interesting reserch question is whether this finding is generlly true for sensory dt. For the two min effects plots, % nd % of the vrition is explined y the first two xes while for the interction plot, the corresponding vlue is % (See Tles ).. Conclusions In this pper we hve compred two pproches for nlysis of sensory dt which comine PCA nd ANOVA. One of them is sed on PCA of for the originl dt with susequent ANOVA of the scores to test the effects of the design fctors on the scores (PC- ANOVA). The other one is sed on using PCA on the mtrices of estimted min effects nd interctions (ASCA). The difference Tle ASCA. Percentge vrince explined y PCA performed on mtrix of products effects Numer of components Tle ASCA. Percentge vrince explined y PCA performed on mtrix of ssessor effects Numer of components Percentge vrince explined Percentge vrince explined Cumultive percentge vrince explined Cumultive percentge vrince explined

9 G. Lucino, T. Ns / Food Qulity nd Preference () Tle ASCA. Percentge vrince explined y PCA performed on the interction effects Numer of components Percentge vrince explined Cumultive percentge vrince explined etween the methods lies in the fct tht they use ANOVA nd PCA in opposite order. Both methods hve multivrite focus since they consider ll ttriutes simultneously nd they provide informtion out ssessor effects nd products effects s well s their interctions on the multivrite structure. In this sense oth methods re highly useful for nlysing sensory dt. Both methods re esily computed using regulr sttisticl softwre pckges nd none of them need ny itertion or other complex numericl procedures. Complex replicte structure nd severl fctors in the experiment cn lso esily e hndled without modifiction of the methods. Reltions to other methods in the literture were highlighted in the method section. The min dvntge of PC-ANOVA is tht it is esier to use for direct ssessment of significnt differences, lso multiple comprison, directly relted to the interprettion plots, i.e. PCA plots, otined. Another dvntge is tht the PCA plots cn esily e equipped with dditionl informtion tht cn e used to give direct ssessment of the size of the effects s compred to the noise level in the dt, s mesures y either the stndrd devition of the rndom noise or the LSD vlues from multiple comprisons. The min disdvntge of PCA-ANOVA is tht in cses where the different effects hve different multivrite profile, one my end up with mny PCA components to nlyse nd interpret which cn e oth time-consuming nd complex tsk. This is prticulrly true in situtions with severl fctors in the design. This is exctly the point where ASCA hs its min dvntge. Since it provides seprte PCA plot for ech fctor seprtely, ech of the PCA models will generlly hve lower dimension thn for PC-ANOVA. Testing significnce for the effects in the plots, is, however, less ovious for the ASCA, lthough some recent ttempts hve een mde to crete such tests (Vis, Westerhuis, Smilde, nd vn der Greef ()). As ws demonstrted in the exmple, the two methods gve the sme overll interprettion for this prticulr dt set. It ws detected tht the ssessor differences nd product differences spnned the sme low-dimensionl multivrite correltion structure leding to very similr lodings plots for the different effects for the ASCA method. This phenomenon leds to smll numer of importnt components lso for the PC-ANOVA. If this is not the cses, the ltter method will need severl components nd the ASCA will need different interprettion for ech of the effect mtrices. An interesting prolem to e investigted in the future is how often this sitution occurs in prctice in sensory nlysis. In the exmple, only product nd ssessor effects were considered, ut oth methods discussed cn e extended to situtions with severl design fctors nd complex replicte structure. Acknowledgments We would like to thnk Norwegin Reserch Council (NFR) for finncil support for this study. We would lso like to thnk Asgeir Nilsen nd Grete Hyldig for mking the dt ville to us. References Brdseth, P., Nes, T., Mielnik, J., Skrede, G., Høllnd, S., & Eide, O. (). Diry ingredients effects on susge sensory properties studied y principl component nlysis. Journl of Food Science, (),. Box, G. E. P., Hunder, W., & Hunter, S. (). Sttistics for experimenter. NY: Wiley. Bro, R., Qnnri, E. M., Kiers, H. A., Ns, T., & Frøst, M. B. (). Multi-wy models for sensory profiling dt. Journl of Chemometrics,,. Brockhoff, P. M., Hirst, D., & Ns, T. (). Anlysing individul profiles y threewy fctors nlysis. In T. Ns & E. Risvik (Eds.), Multivrite nlysis of dt in sensory science. Elsevier Science Pulishers. Crroll, J. D. (). Generlistion of cnonicl nlysis to three or more sets of vriles. Proceedings of the th convention of the Americn Psychologicl Assocition (vol. ) (pp. ). Dhl, T., & Ns, T. (). A ridge etween Tucker- nd Crroll s generlised cnonicl nlysis. Computtionl sttistics nd dt nlysis, (),. Dijksterhuis, G. (). Procrustes nlysis in sensory reserch. In T. Ns & E. Risvik (Eds.), Multivrite nlysis of dt in sensory science (pp. ). Elsevier Science. Ellekjr, M. R., Ilseng, M. R., & Ns, T. (). A cse study of the use of experimentl design nd multivrite nlysis in product improvement. Food Qulity nd Preference, (),. Hnfi, M., & Kiers, H. (). Anlysis of K sets of dt, with differentil emphsis on greement etween nd within sets. Computtionl sttistics nd dt nlysis,,. Jnsen, J., Hoefsloot, J., vn der Greef, M., Timmermn, E., Westerhuis, J., & Smilde, A. K. (). ASCA: Anlysis of multivrite dt otined from n experimentl design. Journl of Chemometrics, (),. Jnsen, J., Bro, R., Huu, C., Hoefsloot, J., vn den Berg, F. W. J., Westerhuis, J., & Smilde, A. K. (). PARAFASCA: ASCA comined with PARAFAC for the nlysis of metolic fingerprinting dt. Journl of Chemometrics,,. Lngsrud, O. (). multivrite nlysis of vrince for colliner responses. Journl of the Royl Sttisticl Society: Series D,, (The Sttisticin). Le, P., Ns, T., & Rødotten, M. (). Anlyis of vrince of sensory dt. J. Wiley nd sons. Mndel, J. (). A new nlysis of vrince model for non-dditive dt. Technometrics, (),. Mrdi, K., Kent, J., & Biy, J. (). Multivrite nlysis. UK: Acdemic press. Mrtens, H., & Mrtens, M. (). Multivrite nlysis of qulity: An introduction. UK: Wiley Chicester. Ns, T., & Lngsrud, Ø. (). Fixed or rndom ssessors in sensory profiling? Food Qulity nd preference, (),. Pges, J., & Husson, F. (). Multiple fctors nlysis with confidence ellipses: methodology to study the reltionships etween sensory nd instrumentl dt. Journl of Chemometrics,,. Qnnri, E. M., Wkeling, I., Courcoux, P., & McFie, H. J. H. (). Defining the underlying sensory dimensions. Food Qulity nd preference,,. Romno, R., Brockhoff, P. B., Hersleth, M., Tomic, O., & Ns, T. (in press) Correcting for different use of the scle nd the need for further nlysis of individul differences in sensory nlysis, Food Qulity nd Preference, doi:./ j.foodqul... Tucker, L. R. (). Some mthemticl notes on three-mode fctor nlysis. Psykometric,,. Vn der Kloot, W. A., & Kroonenerg, P. M. (). Externl nlysis with threemode principl component nlysis. Psykometrik, (),. Vis, D. J., Westerhuis, J. A., Smilde, A. K., & vn der Greef, J. (). Sttisticl vlidtion of megvrite effects in ASCA. BioMed Centrl (BMC) BioInformtics,,.