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AIM:TO START FROM THE COMPANY NEEDS IN QUALITY CONTROL, TO GAIN AN APPROPRIATE SENSORY METHODOLOGY. According the expressed needs, this program could make up and focus on the following points:
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AIM:TO GIVE THEORICAL BASIS IN STATISTICS TO ASSIST THE STATISTICAL RESULT INTERPRETATION IN SENSORY ANALYSIS According the expressed needs, this program could make up and focus on the following points:
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This annex has for aim to present the main TASTEL statistics in order to show how to apply or to validate on examples. These examples, that will have been taken from norms, or in the statistical literature, can present to the user a part of results obtained by TASTEL, and to allow, thus, a validation of calculations and possibly a deepening of theoretical elements. Thus, these examples on general methods have not only as support sensory analysis, but also will be able to calculate as well medicine, or economy fields, or other subjects.DISCRIMINATION TESTSCode design
General calculation
A no-A Test
RANKING TESTSFriedman Statistics
It can be taken as validation example: Norm application described in Annex A. Results of the ranking test with TASTEL:
Assessor number: 8 PRODUCT A: Sum of ranks: 17 PRODUCT B: Sum of ranks: 31 PRODUCT C: Sum of ranks: 32 PRODUCT D: Sum of ranks: 23 PRODUCT E: Sum of ranks: 17 SIGNIFICANT TEST : YES PRODUCT POSITION : PREFERRED REJECTED GLOBAL . ... ... ... .. : . ... ... ..AE...D.. ..BC.. ... . SIGNIF. DISTANCE : -----------.. ... ... ... ... . This test evidences samples differences in a general manner without hypothesis, In case or pre-determined order, the Page Statistics can be possible conducted. Page Statistics
INTENSITY SCALE VARIABLESForeword: In case of missing values in the extracted data, following calculation choices are suggested automatically:
Balance Incomplete Blocks
Couple comparison testsThese comparisons intervene in the TASTEL system in the product couple graphs.t of Student (EPS) calculation
Not matched calculation
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SENSORY ANALYSIS AND CONTROL QUALITY TRAINING
To give an overview of the Sensory Analysis techniques in Quality Insurance
Discussion about the products and the Company control organisation
Aim of the setting up of Sensory Analysis in the framework of Quality Insurance
Alpha risk (1st hand): Test level (risk to declare products as different, even if they are not it)
| - Mean calculation of the difference between the product 1 and the product 2 by variable (MOY) |
| - Standard deviation calculation of the difference between the product 1 and the product 2 by variable (ECA) |
| - Calculation of the t observed: SIGNIP(I%)=MOY*((N%(PRO%))^(1/2))/ECA |
| Significance: |
| Significant, if the t observed value is higher or equal than the t theoretical value: |
| If Tobs=ABS(SIGNIP(I%))>=EPS then the comparison is significantly different. |
| Values displayed in TASTEL couple graphs for matched calculation: |
| . Product means |
| . EPS (theoretical t), and Tobs (observed t), and significance of the asked threshold (*,if yes), |
| . Counting of signs (+) of the product 1, by report to the product 2. |
It will be able to refer to works dealing with such calculations in: Méthodes statistiques, B.Grais - DUNOD 1992, ou Probabilités analyse de données et statistiques, G.Saporta - TECHNIP 1990.
Analysis of variance
| This analysis identifies the effects of the possibly four main factors intervening in an evaluation, with or without repetition, these four factors are the following: |
| . PRODUCT factor |
| . ASSESSOR factor |
| . TEST factor |
| . DATE factor |
| This analysis examines the information variability to identify: |
| . What are the significant factors for each attribute, and, |
| . Within each factor, what are the elements that different some by report to others by signification level. |
| TASTEL suggests to display effects as well as tests of mean comparisons. |
| Three tests can be issued: |
| Least Significant Difference test, |
| Newmann & Keuls test (or Tukey), and |
| Duncan test. |
| The order 2, or order 3 interaction effects can be displayed, if need be. |
| Also, the graph representation displays the following information, by attribute, or in summary: |
| . DF : Degrees of Freedom (n - 1) for the main factors |
| . SS : Sum of squares |
| . MS : Mean Square |
| . F Ratio: Mean square of the considered factor/Mean square of the residual (or interaction) |
| . F : Table theoretical value for the 1%, or 5% threshold (defined in calculation parameters) |
| . Signif.: Positive (*) when Fratio > Ftheorical |
| . LSD : Least Significant Difference, mean smallest difference between two elements in order that these last are considered as significantly different ; The calculation is the following: |
| LSD = t(1or 5%) X square root(2XMS_err/nb_observ_fact) for the LSD test |
| . SSA : Smaller Significant Amplitude, same interpretation as LSD |
| SSA = Q(1or 5%) X square root(MS_ERR/nb_observ_fact) |
| . Signification segments, according to the chosen options |
| . Layout of the factors representing the MS values. |
| It will be able to refer to tables published by ITCF – “Comparaisons de moyennes et de variances - application à l'agronomie” - J.P. GOUET, concerning these calculations. |
| Analysis of the variance of evaluation panel - H. Van Hove - Belgian Wine Review and Spirit, 2/1983 |
| It is displayed furthermore an example of application (in French language, due to the publication origin): |
Principal Component Analysis (PCA)
| The principle of the principal component analysis is to research the best data representation with the less possible dimensions, so that to reduce the variable number, or the initial space dimension number. This allows consequently to explain and to display data with a reduced axis number to facilitate interpretation in a way of synthetic results. The graphic proximity of products and/or attributes leads characteristics linked in term of data and therefore of behaviour. Thus, the problem consists to find orthogonal main axes (A1, A2,. .) such that the variance of A be maximum, constituted of eigenvectors a1,a2,. .. aq, each of these main axes representing a part of the data information (inertia). It will be able to refer to the following book, concerning the theoretical basis of these calculations. : Analyse de données multidimensionnelles P.Bertier et JM Bouroche - PRESSES UNIVERSITAIRES DE FRANCE, 1981. |
| The application example displayed in the next pages is also extracted from this book. |
| Analysis in principal components - Example of EDCO countries |
| One can observe that starting from 18 original variables, such synthetic comparison appears with 3 axes representing 69 % of raw data information. Then, one can gather consequently similar country behaviour as Spain, Portugal, or Greece, in opposition with United States on a development and equipment axis... A simple analogy between countries and products in a first hand, and variables and sensory attributes in the other hand, allows the interpretation of sensory product profile data. |
Correspondence Factorial Analysis (CFA)
| Principles of the Correspondence Factorial Analysis, in similar manner to the PCA, is to research the best data representation with the less possible dimensions, so that to reduce the variable number of the initial space. This analysis is applied on data constituted of a frequency tables, a criterion of distinction from simple matrix can be the meaning and the possible interpretation given on the sum of rows and columns. It will be able to refer to the same following book as PCA, concerning the theoretical basis of these calculations: Analyse de données multidimensionnelles P.Bertier et JM Bouroche - PRESSES UNIVERSITAIRES DE FRANCE, 1981. The application example of the next pages is extracted from the following book: Statistique textuelle - L.Lebart, A. Salem DUNOD,1994. |
| L'exemple d'application traité dans les pages suivantes est tiré d'une utilisation similaire : Statistique textuelle - L.Lebart, A. Salem DUNOD,1994. |
| Analyse factorielle des correspondances Réponses à une question libre : Enfant, selon les niveaux d'éducation |
| One can observe that from totality of terms, a synthetic comparison is representing about 80 % (57.04%+21.13%) of the data information. Then, it is possible to link consequently in interpretation terms as 'Chômage' (unemployment) and 'Difficultés' (difficulties) with people 'sans dipl.' (without diploma), and other questions for academic people as questions of 'Avenir' (Future), and 'Conjoncture économique' (economic conjuncture), . ... |
| In similar manner, an analogy between education and products, and terms in response to a question for example of qualities or defaults on products, allows to analyse product studies. |
Discriminant Analysis
| After having taken knowledge on multidimensional methods, descriptive on quantitative data, or qualitative, the discriminant analysis allows to better surround the membership of a product group to its origin family by one or several quantitative variables. |
| Thus, this method will be able to answer to next questions: |
| . Are the identified groups distinguished by one or several variables? |
| . And with the knowledge of these variables, will be able to put back individuals in their origin groups, and with what error? And finally |
| . What would be variables that discriminate the better the different groups? |
| A simple analogy allows the replacement of, for example, variables by sensory data on samples of wines, and groups, by the vintages. |
| The used method implemented in TASTEL is a stepwise discriminant analysis. It will be able to refer to the following book for added information on the theoretical basis: J.M. ROMEDER - Méthodes et programmes d'analyse discriminante Dunod, Paris 1973. |
| The application example is extracted from the following reference: Le modèle euclidien en analyse de données - J.Pontier A.B. Dufour M. Normand - Ed. UNIVERSITE DE BRUXELLES, 1990. |
| Discriminant analysis - Analysis of the handball player morphology |
| This analysis checks and gathers individuals from data on the handball player morphology, according to the different location on the handball field, and thus to show the possible links between these physical characteristics and the ability to take one location or one other. TASTEL numerical results are displayed in next page. |
Analysis STATIS
| This analysis is probably the most complex, but also the most recent in comparison to the different previous techniques. Indeed, schematically, principles of calculations are the following: |
| In a first time, mathematical operations will adjust data groups between them with rotations, transfers, similarities. This allows consequently a solution called 'Compromise' or 'inter-structure' between analysed elements (products, assessors) more robust than from raw data, then to represent individual differences or trajectories (coming initially from different data in the time). |
| In application to sensory analysis, and with TASTEL, this analysis displays: |
| * The product study: |
| . With the compromise between each product (multidimensional representation of links between the different products between them), with the possibility or not to represent in a similar manner raw data using a PCA. |
| . Trajectories are using an envelope graph of the smallest convex. Each point is the product quotations by the assessors. |
| . Attributes correlation allowing to analyse inter or intra-structure difference by the envelope graphs, but also to observe the sensory attribute distribution, and therefore the coherence, or consensus of the panel to this product, for example. |
| * The assessor study: |
| . With the compromise between each assessor (multidimensional representation of links between the different assessors between them), with the possibility or not to represent in a similar manner raw data using a PCA. |
| . Trajectories are using an envelope graph of the smallest convex. Each point is the assessor quotations of the different products. |
| . Attribute correlation allowing to analyse inter or intra-structure difference by the envelope graphs, but also to observe the sensory attribute distribution, and therefore the coherence, or repeatability between standards to this assessor, for example. |
| The used method implemented in TASTEL is an analysis STATIS with an allocation of a same weight for each of individuals without additional individuals neither tables. It will be able to refer to the following book for theoretical basis and for the application example: * Ch. LAVIT - Analyse conjointe de tableaux quantitatifs - Masson, Paris 1988 as well as to the articles of P. SCHLICH comparing the interest of STATIS as compared to a GPA |
| (Generalised Procrustean Analysis): * P. SCHLICH. 1993. RV Coefficient and STATIS : Useful multivariate statistical methods for sensory evaluation - Food 2000 Preservation. Boston * P. SCHLICH. 1992. GPA ou STATIS, Consensus ou compromis. Evaluation sensorielle - Bruxelles 1992/ COMETT |
| STATIS analysis example of the 'auxologique' french file : data of child growth 4 to 15 years. |
| The E* are the results of 30 children. For the analysis, there exists an opposition between small children and strong on the axis 1, and small and great on the axis 2. In analogy, as compared to products, the interpretation undertakes in similar manner to a PCA. |
| This trajectory representation is totally fitted to this case depending of the time. This graph displays the children position evolution according to their growth more or less rapid, or average as compared to the others. Concerning by analogy, products, and so rather of the quotation 'envelopes', the interpretation been made according to the envelope more or less narrow (intra-structure: consensus or coherence of assessors on the considered product), and the location as compared to the average of products according to the interpretation of axis in relation with the sensory attributes (inter-structure). |
| Correlation display analysed variables, or the attributes so as to possible interpret the constitution of main axis. The attribute envelope spread can give information on the correct use (for the discrimination of products) or on the assessor training concerning these attributes. (A great spread would indicate different comprehension or different recognition by assessors). The dual (on assessors) use and interpretation undertakes in similar manner. |
Transformation of intensity scales in distribution/ranking
| Two complementary transformations can be undertaken on the intensity scales: . Quotation distribution . Transformation of intensities in ranks. For these transformations, two tests of significance are suggested: |
Calculation of the Cochran Q test
| This is a dichotomic test, that allocates accepted values as (1), and rejected as (0), then the hypothesis consists to determine if the k products are identical or no, with a significance threshold of 5%. In the case of more of two products: products are significantly different if at least one combination is significant. Combinations are formed by the complementary of each of possible cases (if three modes: three tests are undertaken, for example). |
| The statistic is the following: |
| k : product number to compare |
| Gi : Sum of values (0 or 1) by product |
| Gm : Mean of sums of values by product |
| Di : Sum of values (0 or 1) by assessor |
| Q = [k(k - 1) * sum(Gi - Gm)2] / [k*sumDi - sumDi2] |
| This value is compared to a khi2 distribution statistic with k - 1 degrees of freedom. |
| It will be able to refer to the following publication: No parametric statistics for the behavioural sciences - International student publishing - Mac Graw Hill 1956. |
Friedman test
| Cf. Test of ranking in the previous points. |
Mean comparisons - non balance factors
| Two data groups with unknown and unequal variances are considered. On the two unknown means m1 and m2 will be carried out appropriate hypothesis test: ASPIN WELCH test. This test is referenced in NF X 06-065 norm, and can also be consulted in: “Aide-mémoire statistique - CISIA/CERESTA Editor, 1995”. In case of not balanced factors in analysis of variance in TASTEL, this test is automatically calculated and printed. The main steps of this calculation are the following ones: |
| Identification of pair comparisons according to the related factors: PRODUCT, ASSESSOR, DATE, for example. |
| Then, for each comparison: |
| .Counting of the two sample number, n1, n2 |
| .Computation of each pair mean, for the two samples m1, m2 |
| .Computation of each pair variance, for the two samples s²1, s²2 |
| .Computation of a mu value: |
| 1/mu = 1/(n1-1) [s²1/n1/s²d]² + 1/(n2-1) [s²2/n2/s²d]² , with s²d = s²1/n1+s²2/n2 |
| Test is then the following: H0 (m1 = m2) against H1 (m1¹m2) with a threshold for a Student distribution m1-m2 < -t1-a/2 (mu) * sd -> hypothesis H0 rejected |
| Remark: Real risks are given by interpolation line/column of the t table values. |
| Result printing for all comparisons |
Willcoxon test
| Willcoxon test is used to identify a significant difference between two products with a not Gaussian statistics. This statistics can then be used in the case of not normal data distribution. In a first step, data are sorted in ranks, then R+ et R- values are calculated in order to be compared with R(n) Willcoxon critical values, n being the comparison number with a non zero difference. It is advised to use bilateral test results for the comparison of two products. We can refer to the following publication « Aide-mémoire statistique – CISIA/CERESTA –p.134-136 ». |
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| Prod. | Ass.1 | Ass.2 | Ass.3 | Ass.4 | Ass.5 | Ass.6 | Ass.7 | Ass.8 | Ass.9 |
|-------|-------|-------|-------|-------|-------|-------|-------|-------|-------|
| a=pA | 47 | 100 | 33 | 70 | 94 | 85 | 39 | 52 | 47 |
| b=pB | 41 | 98 | 46 | 61 | 84 | 87 | 36 | 52 | 51 |
| d=a-b | 6 | 2 | -13 | 9 | 10 | -2 | 3 | 0 | -4 |
| r | 5 | 1.5 | 8 | 6 | 7 | 1.5 | 3 | | 4 |
| r+ | 5 | 1.5 | | 6 | 7 | | 3 | | |
| r- | | | 8 | | | 1.5 | | | 4 |
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R+ = 22.5 R- = 13.5 n = 8
Bilateral test
H0[(a)=(b)] against H1[(a)<>(b)]
min(R-,R+) = 13.5 > R_alpha(n) = 3
->We accept H0
Right unilateral test
H0[(a)=(b)] against H1[(a)<(b)]
R+ = 22.5 > R_alpha(n) = 5
->We accept H0
Left unilateral test
H0[(a)=(b)] against H1[(a)>(b)]
R- = 13.5 > R_alpha(n) = 5
->We accept H0
Internal Preference Mapping
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This analysis details the consumer preferences in order to assess the behaviour preference diversity |
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Thus, this will allow representation of consumer segments based on 5 or more products. Graphs will display names, or panellist criteria, but also, it will be possible to preview the consumer densities, and so, the most interesting points to study. |
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Consumer counting options are possible to evidence particular densities in order to express market segments.
Probabilistic Analysis
| This analysis is especially dedicated for protocols in Balance Incomplete Blocks, to find from personal ranking in block: "panellist", a collective order for the totality of the panel. These models are developed for pairs and trios, extensions for quartets, ... are under way. These calculations are based on frequencies to be first, second, or third. |
| In case of trios, two models are suggested: |
| . Dependent model: Choice twice of the first product (frequency of the p product to be first to the square) by the choice once to be second (frequency of the p product to be second) - The whole, divided by the totality of terms, by forbidding the equal places. |
| . Independent model: Choice of the first product among the three, then the second one among the two remainders. |
External Preference Mapping - 'Descriptive' / ‘Preference’ data links
| Multidimensional techniques as “Joint PCA” can represent simultaneously consumer groups according to their appreciation on products, with expert evaluation on these same products. Indeed, an automatic link exists with our complementary TASTEL system dedicated for sensory analysis assistance. A preference optimum is calculated by the system, calculations are based on consumer answer densities. |
| Then, with the help of this optimum, and with techniques of 'REVERSE INGENEERING', it is possible to find the sensory co-ordinates that a potential product would have had being placed in this place. Checking the model is easily conducted by placing the sensory co-ordinate of this supplementary individual in the joint PCA: This position is the point 'PRODUIT IDEAL' (Ideal product). These modelled co-ordinates are displayed in the last figure. |
| These different statistical tools allow thus to bring rules of decision on sensory improvements to |