Identifying assessor differences in weighting the underlying sensory dimensions EL MOSTAFA QANNARI (1) MICHAEL MEYNERS (2)

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1 Identfyng assessor dfferences n weghtng the underlyng sensory densons EL MOSTAFA QANNARI () MICHAEL MEYNERS (2) () ENITIAA/INRA - Unté de Statstque Applquée à la Caractérsaton des Alents Rue de la Géraudère B. P , NANTES cedex 03 FRANCE (2) Fachberech Statstk, Unversty of Dortund, D-4422 DORTMUND, GERMANY. Abstract In a prevous paper Kunert and Qannar (999) dscussed a sple alternatve to Generalzed Procrustes Analyss to analyze data derved fro a sensory proflng study. After sple pre-treatents of the ndvdual data atrces, they propose to erge the data sets together and undergo Prncpal Coponents Analyss of the atrx thus fored. On the bass of two data sets, t was shown that the results slghtly dffer fro those obtaned by eans of Generalzed Procrustes Analyss. In ths paper we gve a atheatcal justfcaton to ths approach by relatng t to a statstcal regresson odel. Furtherore, we obtan addtonal nforaton fro ths ethod concernng the densons used by the assessors as well as the contrbuton of each assessor to the deternaton of these densons. Ths nforaton ay be useful to characterze the perforance of the assessors and sngle out those assessors who downweght or overweght soe densons. In partcular, those assessors who overweght the last densons should arouse suspcon regardng ther perforance as, n general, the last densons n a prncpal coponents analyss are deeed to reflect rando fluctuatons.

2 Introducton Kunert and Qannar (999) propose a sple ethod for the analyss of sensory proflng data. They suggest sple pre-treatent of the ndvdual data atrces to account for dfferences n scorng, such as shfts and use of dfferent ranges of the scale. In a second step, the data atrces are erged together and a Prncpal Coponents Analyss (PCA) s perfored on the superatrx whch ncludes the attrbutes of all assessors. On the bass of two data sets t was shown that the results of the proposed ethod atch up to a large extent wth those of Generalzed Procrustes Analyss (GPA) (Gower, 975; Arnold and Wllas, 986). Note that the ethod proposed by Kunert and Qannar s applcable to data obtaned by eans of a fxed vocabulary,. e. all assessors use the sae attrbutes, as well as to data derved fro a free choce proflng procedure whch allows each panelst to use hs or her own set of attrbutes. In ths paper, we relate ths approach to a lnear regresson odel whch states that there exst underlyng sensory densons (latent varables) and that the observed attrbutes can be expressed as lnear cobnatons of these densons. Orgnally ths odel was ntroduced by Tucker (966) as a three-way factor analyss technque and was dscussed wthn a sensory analyss fraework by Brockhoff et al. (996). Ths three-way factor analyss can be seen as a generalzaton of PCA and s therefore used to analyze dfferences between products, assessors and attrbutes sultaneously. We show useful propertes of the ethod that enable a panel leader to dentfy dfferences between the assessors regardng ther use of densons. Ths paper can also be seen as an extenson to the case of several data sets of a relatonshp between regresson analyss and PCA exhbted by Jong and Kotz (999). These authors gve a lnk between these two statstcal ethods wthn the usual fraework (two way data set). In sensory proflng analyss, we have several data sets that are slar n the sense that the panelsts assess the sae set of products. The odel has two nterestng propertes. On the one hand, t provdes a sound statstcal background and a strong justfcaton to the ethod proposed by Kunert and Qannar (999). On the other hand, t allows the panel leader to dentfy ndvdual dfferences between the panelsts wth respect to both the nuber of densons used and the portance attached to the varous underlyng densons. Therefore, ths paper can be seen as a sequel to the paper wrtten by Kunert and Qannar (999) as we dscuss useful extensons and ntroduce ndces whch provde the practtoner wth useful nforaton. 2

3 A noteworthy feature of the ethod presented heren s that t does not nvolve heavy coputatons. Both the PCA and the perforance coeffcents, whch consttute the core subject of the present paper, can be sply coputed wth standard statstcal software. Centerng and pre-scalng the data sets The sensory proflng (free choce proflng or fxed vocabulary) of n products by assessors results n atrces X, X 2,... X, where the rows refer to the products and the coluns to the attrbutes. These atrces are centered n order to reove the effects of judges scorng at dfferent levels of the scale. The confguratons X,X 2,... X are also pre-scaled n order to adjust for varatons aong assessors n range of scorng. For ths purpose each confguraton X, =,...,, s ultpled by the sotropc scalng factor α gven by : T α =, ( ) t where s the su of varances of the attrbutes n data atrx X and t T = =. ( 2 ) More detals and a justfcaton of these sotropc scalng factors are gven by Kunert and Qannar (999). In what follows, we consder the pre-scaled data sets denoted by Y = α X for =,...,. Deternaton of the underlyng sensory densons The a of the practtoner n sensory analyss s to depct relatonshps aong products on the bass of a hopefully sall nuber of densons (latent varables) that underle the sensory perceptons expressed by the assessors. Let C (n, q) denote the atrx whch contans the q underlyng latent varables. These varables are assued to be uncorrelated and to have varances equal to one. We assue that each assessor s attrbutes, whch for the atrces Y for =,,, can be expressed as lnear cobnatons of the latent varables gven n C plus soe rando errors. Forally, the odel can be wrtten as : 3

4 Y =C B +E, whereb contans the coeffcents of the lnear cobnatons that lnk the attrbutes n Y to the latent varables n C, and E gves the rando errors. Ths odel relates to the well-known lnear regresson odel. However, unlke ultvarate regresson analyss, we have to estate several addtonal paraeters : besdes the estaton of C, we have to estate also the atrces B, =,...,. In the context of sensory proflng, we are partcularly nterested n C n order to descrbe the relatonshps aong products. However, the dfferent atrces B provde useful nforaton about the ndvdual dfferences between the assessors wth respect to the underlyng densons. Estaton of the paraeters We consder the well-known least squares crteron to estate the paraeters. Ths conssts n nzng the loss-functon : SSQ( Y CB ) = (3 ) wth respect to C and B for =,,. In ths expresson, SSQ(A) denotes the su of squares of all the eleents for a gven atrx A. In the followng, the paraeter estates whch are solutons to the nzaton proble wll also be denoted by C and B, thus avodng the cubersoe notatons Ĉ and Bˆ. It can be shown that the q coluns of the estated atrx C are gven by the frst q prncpal coponents of the atrx Y = (Y Y 2... Y ) whch s fored by ergng the ndvdual data atrces Y, =,,. Moreover, as t s well known fro ultvarate lnear regresson analyss, the atrces B (for =,, ) are gven by : B T T = ( C C) C Y, where C T denotes the transpose of atrx C. Assung that the coluns of C are uncorrelated and ther respectve varances beng equal to one, we have: C T C = n I, 4

5 where I s the dentty atrx. It follows that : T B = C Y. n Hence, the current eleent of atrx B correspondng to the (j, k) th entry s the covarance between the k th attrbute n atrx Y (=, 2,, ) and the j th standardzed prncpal coponent of atrx Y,. e the j th colun of atrx C. The predcted ndvdual assessor atrces are gven by : Y ˆ = CB. We can show that the loss functon (3) can be wrtten as : = SSQ( Y ) SSQ( Yˆ ). ( 4 ) = Fro the pre-scalng procedure, t follows that the quantty SSQ(Y ) s a constant equal to nt where n s the nuber of products and T s defned accordng to (2). Furtherore, SSQ Y ˆ ) s the su of squares explaned by the odel that conssts n regressng Y relatonshp n = SSQ( Yˆ ) = λ + λ2 + L+ λq ( on C. The lnks the latter ter n (4) to the total varance explaned by the q prncpal coponents of Y. In ths expresson, λ j ( j =,, q) are the egenvalues assocated wth the prncpal T T coponents,. e. the egenvalues of YY or equvalent of Y Y. Snce the coluns of C n n are assued to be uncorrelated, we can derve a ore precse and ore useful relatonshp by consderng ˆ ( j ) Y, whch s defned as the atrx obtaned by regressng Y on the j th prncpal coponent of Y (. e. the j th colun of C). We have : ( ) where SSQ( ˆ ) j Y n = SSQ Y ( ˆ ( j) ) = λ j, ( 5 ) can be sply coputed as the su of squared covarances between the attrbutes n Y and the j th prncpal coponent of Y (.e. j th colun of atrx C). We consder the ndex : 5

6 SSQ Y ( ˆ ( j) ) λ j = (6) SSQ( Y ) whch reflects the percentage of total varance n Y ( =,, ) explaned by the j th prncpal coponent of Y. It has been entoned that SSQ( Y ) = nt, thus (6) splfes to : ( Yˆ ( ) j ) SSQ λ j =. ( 7 ) nt Ths ndex reflects the portance that assessor ( =,, ) attaches to the j th denson. If we consder all q prncpal coponents assocated wth Y, t can be shown that q j= λ = (8) j for each assessor. Roughly speakng, (8) expresses that each panelst provdes us wth an nforaton regardng how the products dffer fro each others and the ndces λ j reflect how ths nforaton s parttoned nto the varous underlyng densons. Furtherore fro (5) and (7) t also follows drectly that : λ j λ j =. ( 9 ) = T Snce for =,, the total varaton n Y s equal to T owng two the prescalng procedure and therefore the total varaton of Y =(Y Y 2 Y ) s equal to T, the rght sde of equaton (9) s precsely the percentage of total varance n Y explaned by the j th prncpal coponent of Y. Thus λ j reflects the portance that assessor attaches to the j th denson and the average over assessors reflects the relatve portance of denson j. The varous steps to calculate the underlyng sensory densons and the perforance ndces are suarzed n the appendx. 6

7 Exaples Two exaples wll be gven to llustrate the outcoes of the PCA on erged data sets (PCAMDS). We purposefully use the data that has already been analyzed by Kunert and Qannar (999) and we do not depct relatonshps aong products as ths has already been dscussed n ther paper, but we focus on the assessor perforances. The frst data set s obtaned by eans of a fxed vocabulary proflng procedure appled to 5 Geran beers. A panel of 3 assessors wth no experence n sensory analyss partcpated n the test n whch four attrbutes were consdered (Kunert, 998). The percentages of varaton explaned by the four underlyng densons are 37.0%, 29.7%, 9.9% and 3.5%, respectvely. Table gves the ndces λ j fro equaton (7). They reflect the portance that the assessors attach to the underlyng densons. The rather poor perforance of the untraned panel s reflected by the hgh varaton of the ndces wthn each colun. If we consder the frst denson for nstance, we can see that the portance of ths denson s evaluated to 37.0% on average, whch s exactly the percentage of varaton explaned by the frst prncpal coponent (see the last row of table ). However, ths denson explans only a sall aount of varaton n the confguraton assocated wth assessor 2 (7.0%), and contrarwse t turns out to be very portant for assessors 3 and 7 (75.0%). The sae coents hold for the other densons wth respect to other assessors. If we copare the rows n table, we can see that except for assessor 2 and, to a saller extent, assessor, all assessors have one donant denson. For nstance denson 2 explans 79% of the total varaton n the confguraton assocated wth assessor 3. Beng untraned, these assessors see to anly assess the products accordng to only one latent varable. Consderng assessor 2, the densons, 3 and 4 turn out to be equally portant and the portance he or she gves to denson 2 s also not neglgble. Therefore, the confguraton of assessor 2 s ore or less sphercal (equal portance of all the drectons) whch, consderng the context, ght be a hnt that the assessor under consderaton has just gven rando scores to the products. 7

8 Table. Beer data: Percentage of varaton n each assessor s confguraton explaned by the prncpal coponents. denson Total assessor A 5.0% 5.0% 7.0% 53.0% 00.0% A2 7.0% 66.0% 23.0% 4.0% 00.0% A3 75.0% 4.0% 0.0%.0% 00.0% A4 55.0% 3.0% 4.0% 27.0% 00.0% A5 5.0% 49.0% 9.0% 7.0% 00.0% A6.0% 30.0% 58.0%.0% 00.0% A7 75.0%.0% 9.0% 4.0% 00.0% A8 65.0% 4.0% 4.0% 6.0% 00.0% A9 6.0% 26.0% 0.0% 3.0% 00.0% A0 2.0% 49.0% 22.0% 8.0% 00.0% A 4.0% 24.0% 30.0% 5.0% 00.0% A2 28.0% 4.0% 24.0% 33.0% 00.0% A3.0% 79.0% 9.0% 2.0% 00.0% average 37.0% 29.7% 9.9% 3.5% 00.0% The second data set was obtaned fro an experent nvolvng a free choce proflng procedure. The data are provded and dscussed by Djksterhus and Punter (990) and have also been analyzed by Djksterhus and Gower (99) by eans of GPA. In ths experent, seven assessors rated eght yogurts. Table 2 shows how uch the assessors contrbute to the dfferent densons. There s less dscrepancy between the portance attached by the assessors to each denson than n the prevous case. The ndces assocated wth the assessors wth respect to densons and 2 are plotted n fgure. It can be seen that assessors 2 and 5 bestow uch portance upon denson whereas assessors 6 and 7 gve less portance to ths denson. For assessor 7 we see that densons and 2 together explan less than 50% of the total varaton n the confguraton. In table 2 t can be seen that ths assessor gves even ore portance to denson 3 than to densons and 2, respectvely. 8

9 Table 2. Yogurt data: Percentage of varaton n each assessor s confguraton explaned by the prncpal coponents. Denson Total Assessor 42.0% 34.0% 3.0% 5.0% 8.0% 7.0% 2.0% 00.0% 2 6.0% 5.0% 0.0% 9.0% 7.0% 6.0%.0% 00.0% % 4.0% 8.0% 7.0% 3.0% 6.0% 4.0% 00.0% % 28.0% 8.0% 4.0% 5.0% 5.0% 2.0% 00.0% % 2.0% 4.0% 3.0% 8.0% 6.0%.0% 00.0% % 36.0%.0% 5.0% 0.0% 2.0% 2.0% 00.0% % 22.0% 24.0%.0% 6.0% 5.0%.0% 00.0% Average 44.4% 2.6%.% 7.7% 6.7% 5.3% 3.3% 00.0% Fgure. Percentage of varaton n each assessor s confguraton explaned by the two frst prncpal coponents; (*) refers to the average percentage of varaton over assessors PC2 (%) (*) PC (%) 2 9

10 Concluson The paraount features of PCAMDS are, on the one hand, ts splcty and, on the other hand, ts ablty to provde perforance ndces assocated wth the assessors. These ndces ght be a sple tool for a panel leader to consder ndvdual dfferences between the panelsts wth respect to ther percepton of dfferent densons. Moreover, the ethod s supported by a statstcal odel that assues the exstence of latent varables whch are related to the orgnal data n the sense that the attrbutes can be retreved as lnear cobnatons of these underlyng latent varables. The ethod of analyss bascally nvolves perforng PCA on the erged data sets. Therefore, further developents of the ethod ay rely on the wde statstcal lterature devoted to PCA whch s by far the ost popular ethod n ultvarate analyss (Jollffe, 986). For nstance, t s possble to extend the ethod of analyss to encopass the case where there are ssng data (soe assessors ay be unavodably retaned fro attendng all sessons). Indeed, ths s a trcky proble when usng such ethod as GPA whereas solutons to ths proble exst wthn PCA fraework. Ths ssue wll be nvestgated n further research. Acknowledgeent The second author s grateful to the Deutsche Forschungsgeenschaft (SFB 475, "Reducton of coplexty for ultvarate data structures") for the fnancal support of ths work. 0

11 References ARNOLD, G. M. and WILLIAMS, A. A. (986). The use of Generalzed Procrustes Analyss n sensory analyss. In: Statstcal Procedures n Food Research, ed. J. R. Pggott. Elsever BROCKHOFF, P. M., HIRST, D. and NÆS, T. (996). Analysng ndvdual profles by three way factor analyss. In: Multvarate analyss of data n sensory scence, ed. T. Næs and E. Rsvk. Elsever DIJKSTERHUIS, G. B. and GOWER, J. C. (99). The nterpretaton of Generalzed Procrustes Analyss and alled ethods. Food Qualty and Preference, 3, DIJKSTERHUIS, G. B. and PUNTER, P. H. (990). Interpretng Generalzed Procrustes Analyss 'analyss of varance' tables. Food Qualty and Preference, 2, GOWER, J. C. (975). Generalzed Procrustes Analyss. Psychoetrka, 40, JOLLIFFE, I. T. (986). Prncpal Coponent Analyss. New York: Sprnger-Verlag. JONG, J. C. and KOTZ, S. (999). On a relaton between Prncpal Coponent Analyss and regresson analyss. The Aercan Statstcan, 53 (4), KUNERT, J. (998). Sensory experents as crossover studes. Food Qualty and Preference, 9, KUNERT, J. and QANNARI, E. M. (999). A sple alternatve to Generalzed Procrustes Analyss: Applcaton to sensory proflng data. Journal of Sensory Studes, 4, TUCKER, L. R. (966). Soe atheatcal notes on three-ode factor analyss. Psychoetrka, 3,

12 Appendx The a of ths appendx s to suarze the varous steps of the analyss. Step : Copute the sotropc scalng factors α ( =,, ) accordng to () and set Y =α X. Step 2 : Perfor a PCA on the erged data tables Y = (Y Y 2 Y ); relatonshps aong products can be depcted on the bass of the unstandardzed prncpal coponents. Step 3 : Denote by c, c 2, c p the standardzed prncpal coponents assocated wth Y. For each data table Y ( =,, ) and for each prncpal coponent c j (j =,, p) copute the percentage of total varance n Y explaned by c j accordng to SSQ( Yˆ ( ) j ) λ j =, nt ( ) where SSQ( ˆ ) j Y s coputed as the su of the squared covarances between the attrbutes n Y and c j, n s the nuber of products and T s defned accordng to (2). λ j reflects the portance that assessor attaches to denson c j. These ndces can be copared to each other and to ther average, whch s as a atter of fact the percentage of varaton explaned by c j (step 2). 2