Perceptual Mapping Based on Three-Way Binary Data

Transcription

1 Perceptual Mapping Based on Three-Way Binary Data B. Kemal Büyükkurt, Concordia University, Montreal, Canada & Pieter M. Kroonenberg, Leiden University, The Netherlands

2 Perceptual Mapping - Perceptual mapping: Graphical display summarizing consumers perceptions of multi-attribute objects. Example: Displaying brands in a product class together with their attributes e.g. brands for treating stomach problems. Brunswik s (955) Lens Model: Theoretical foundation for understanding the importance of perceptions in consumer purchases Perceptions Preferences Choice

3 Perceptual Mapping - 2 Goals perceptual mapping Aid for strategic marketing decisions Summarizing nature and degree of competition among a set brands via key product attributes. Common application areas product positioning identification of market gaps for new product development.

4 Perceptual Mapping - 3 Basic data Brands are scored on a number of attributes by several individuals Scores averaged over individuals Result: Brand by Attribute matrix Common data analysis techniques correspondence analysis Brands principal component analysis, multidimensional analysis discriminant analysis Attributes factor analysis Brands Average Attributes Individuals

5 Basic data Perceptual Mapping - 4 Doctor thinks a brand posesses an attribute => score =, if not: score = 0 Three-way binary data: Brands Attributes Doctors Why average over doctors? Different doctors may be sensitive to different attributes Three-way data analysis techniques Three-mode binary hierarchical cluster analysis Three-mode principal component analysis (numerical)

6 The Binary Data Cube j=,...,j Variables (Attributes) MODE B i=,...,i Objects (Brands) MODE A k=,...,k Subjects (Doctors) MODE C Fibers Slices

7 Stacked Two-Way Data Columns: Attributes through J Doctor (k=) Doctor 2 (k=2) Rows: Brands through I Rows: Brands through I Doctor K (k=k) Rows: Brands through I

8 HICLAS3: Algebraic Representation (Tucker3-HICLAS) Hiclas3 model (uses Boolean algebra) P Q R ~ x ˆ = m = a~ b c~ g~ ijk ijk p = q = r = ip jq kr pqr m ijk = iffã ip = and b jq = and c kr = and g pqr = for at least one combination of p, q, and r; ã ip, b jq, c kr : elements binary component matrices A, B, and C, respectively (brands, attributes, doctors). : element of the P Q R three-way binary core array G, indicates links between binary components of the three modes g pqr

9 HICLAS3 Pictorial Representation A B G C G m 2 = as a 22 b 2 c g 22 = (all other 7 combinations contain a zero) brands attributes doctors core array c2 b b2 a a2 a a2 b b2 c

10 Three-Mode Component Analysis Tucker3 model (numerical) xˆ ijk = m ijk = P Q R p = q = r = a ip b jq c kr g pqr i=,...,i (brands); j=,...,j (attributes); k=,...,k (doctors); m ijk is the model matrix or structural image a ip, b jq, c kr : elements loading matrices A, B, and C, respectively (brands, attributes, doctors). g pqr : element of the P Q R three-way core array G; indicates strength of the link between the components of the three modes

11 Three-Mode Binary Analysis in Action Perceptions of Medical Doctors w.r.t. Gastro-Intestinal Drugs

12 Perceptions of Medical Doctors Gastro-Intestinal Drugs Tagamet Zantac Pepcid Axid Sulcrate Cytotec Losec

13 Attributes Adjectives [Binary answers no (0) or yes ()] Relieves Pain RelPain Does not have serious side effects NoSideEf Relatively safe w.r.t. potential interactions with other drugs Safe Flexible in terms of dosage FlexDose Not too costly for the patient LowCost Relieves symptoms RelSymptoms Promotes healing Heals Prophylactic Prophylactic

14 Data: Brands Attributes Doctors ( ) j=,...,8 Variables (Attributes) MODE B i=,...,7 Objects (Brands) MODE A k=,...,283 Subjects (Doctors) MODE C

15 Perceptions of Medical Doctors Central questions What is the position of brands w.r.t. each other? Which attributes are related to this positioning? Do doctors differ in their perceptions in which brands have which attributes?

16 HiClas3 Model Tucker3 hierarchical classes model Basic elements Binary components for all three modes (doctors, brands and attributes) Plus linkage information about the components Basic literature Papers by Ceulemans, Van Mechelen in Psychometrika (Catholic University Leuven, Belgium)

17 Number of discrepancies HiClas3 Choosing a Model,, 2,2, 3,3, Brands Attributes Doctors 2,,2,2,2 2,2,2 3,2,2 2,3,2 3,3, Model complexity: (3,3,2) = (Brands = 3 components ; Attr = 3; Docs = 2) Discrepancy : Data have a, model matrix a 0 and vice versa 3,,3 2,2,3,3,3 3,2,3 2,3,3 3,3,3 Degrees of freedom

18 Binary Component Matrices (brands; attributes) B = Cytoprotective agent B2 = Tagamet (Oldest) B3 = Histamines; H-2 blocker Brand Discre- pancies Fit B B2 B Sulcrate Cytotec Zantac Pepcid Axid Losec Tagamet Attribute Discre- pancies Fit A A2 A Relieves Pain Relieves Symptoms Promotes Health No Side Effects Relatively Safe Flexible Dose Prophylatic Low Cost A = Primary medical A2 = Use in practice A3 = Secondary medical Low Cost had no relations with other attributes

19 Binary Component Matrices (doctors) Doctors MD MD2 f Prop. s Doctor Type Doctor Type Doctor Type Doctor Type Average sd =.09 Doctor Type 4 (0,0) has no links with other doctors

20 Dr2 ( 0) Binary Core Array A A2 A3 Prim. Prac Secon. Med. tice Med B (Cytoprotective Cytoprotective) 0 0 B2 (Tagamet ) 0 B3 (Histamines ) Dr3 (0 ) B (Cytoprotective Cytoprotective) 0 B2 (Tagamet ) 0 0 B3 (Histamines ) : a link exists between components of the three modes Dr = Dr2 + Dr3

21 Doctor Type 2 Sulcrate Zantac, Axid, Pepcid, Losec Cytotec Tagamet n = 69 No side effects Safe Prophylactic Flexible dose Low cost Relieves pain and symptoms, Promotes health A

22 Doctor Type 3 Sulcrate Zantac, Axid, Pepcid, Losec Cytotec Tagamet n = 98 No side effects Safe Prophylactic Flexible dose Low cost Relieves pain and symptoms, Promotes health

23 Doctor Type Sulcrate Zantac, Axid, Pepcid, Losec Cytotec Tagamet n = 70 No side effects Safe Prophylactic Flexible dose Low cost Relieves pain and symptoms, Promotes health A

24 Doctor Types Sulcrate Zantac, Axid, Pepcid, Losec Cytotec Tagamet Dr2 Low cost Dr3 No side effects Safe Prophylactic Flexible dose Dr A Relieves pain and symptoms, Promotes health

25 Characterisation of Doctor Types Brand Name Dr 2 (n=69) Dr 3 (n=98) Dr (n=70) Zantac a0, a a0,,a2, a3 a0, a, a2, a3 Axid a0, a a0,,a2, a3 a0, a, a2, a3 Pepcid a0, a a0,,a2, a3 a0, a, a2, a3 Losec a0, a a0,,a2, a3 a0, a, a2, a3 Cytotec a0, a a0, a,,a3 a0, a, XX,a3 Sulcrate a0, a a0, a, a2, a3 a0, a, a2, a3 Tagamet a0, a, a2 a0,,a2 a0, a, a2,xx a0={relieves Pain, Relieves Symptoms, Promotes Healing} a={prophylactic}, a2={flexible Dosage}, a3={no Side Effects, Safe} Doctor Type 4 has no links; Low Cost has no links

26 Further Considerations No information on Low Cost (more complex HiClas models can model a separate component for Low Cost) Tagamet is relatively inexpensive, while the others are not Don t the doctors see this? HiClas3 suggest they do not.

27 Proportions of Ones across Doctors Tagamet Zantac PepCid Axid Losec Sulcrate Cytotec RelievePain RelieveSymptoms PromotesHealth NoSideEffect RelativeSafe FlexbileDose Prophylactic LowCost

28 Further Considerations Surprise Tagamet is the only relatively inexpensive brand Possible reason: Doctors from all groups say Tagamet is not expensive. Thus unrelated to the present groups. Possible solution: More groups for attributes (we are working on this) Question Other variability not present in HiClas solution?

29 Further Analyses Treat the binary data as numerical and analyse with Tucker3. Handle the data such that emphasis is on: relative differences between brands relative differences between attributes.

30 Three-Mode Component Analysis Concentrate on consensus and individual differences between doctors in the relationships between brands and attributes. Absolute differences between brand and between attributes are ignored. 0 M 5 A B C D E 5 M 0 A B C D E -2 M +2 A B C D E -2 M +2 A B C D E

31 Component Scores * * * Consensus among doctors Tucker3_332 Subject Component Individual differences between doctors Tucker3_332 Component 2

32 Joint Biplot (Consensus among doctors - Mean) Second Component 2 0 Tagamet LowCost Losec RelPain RelSymptoms Heals Pepcid Axid Zantac NoSideEff FlexDose Safe - -2 Mean of each brand and each attribute -3 Cytotec Prophylactic Sulcrate First Component

33 Joint Biplot (Individual differences between doctors - Deviations from mean) Second Component Cytotec Safe Losec Sulcrate Heals Relieves Relieves Pain Symptoms NoSideEff FlexDose Pepcid Zantac Axid Prophylatic LowCost -2 Tagamet First Component

34 Conclusions - HiClas model Given the data are binary, the binary hierarchical classes model is an obvious analysis method and has a relatively straightforward interpretation. Effective graphics to display results Many components might be necessary to model all systematic variability present.

35 Tucker3 model Conclusions - 2 By using a numerical model variance can be portrayed in a different and also insightful manner Differential weighting may simplify model description Enlightning graphics are available (joint biplots), but it requires some training to understand them

36 Conclusions - 3 Substantive conclusions concern the perceptual mappings of the brands with respect to the attributes as seen by the doctors. The main patterns have been discussed during the presentation and will not be repeated.

37 Thank You

38 Tucker3 Model in Matrix Notation X = AG ( B ' C ' ) + ε A, (I P) loadings matrix for brands B, (J Q) loadings matrix for attributes C, (K R) loadings matrix for subjects G, (P Q R) core array with links between the components

39 PARAFAC/CANDECOMP Model: xˆ ijk = m ijk = s = (Harshman 970, 976; Harshman and Lundy 984, 994; Carroll and Chang 970) Based on the principle of Parallel Proportional Profiles (Cattell 944). S g sss m ijk is the model matrix or structural image A is the (I S) loadings matrix for brands B is the (J S) loadings matrix for attributes C is the (K S) loadings matrix for subjects G is the (S S S) superdiagonal core array exclusive links between the components s of the three modes a is b js c ks

40 Three-Mode Components Analysis: Model Comparison MODELS NUMBER OF COMPONENTS STANDARDIZED Number St.Fit/#Param A B C SS of (x000) Param. TUCKALS TUCKALS TUCKALS TUCKALS TUCKALS TUCKALS TUCKALS TUCKALS TRILIN TUCKALS TUCKALS TRILIN TUCKALS COMPUTATION OF NUMBER OF PARAMETERS: A + B + C + core - transformational freedom TUCKALS2: I*P + J*Q + + P*Q*K - P**2 - Q**2 TUCKALS3: I*P + J*Q + K*R + P*Q*R - P**2 - Q**2 - R**2 PARAFAC : I*S + J*S + K*S + S - S - S - S

41 Varimax Rotation: Deciding On Weights Relative Weights Varimax Value A B C Core A B C unrotated

42 Joint biplot for Brands and Attrubutes First versus Third Component for Second Component of Doctors.5 Cytotc PromHl Sulcrt RlvPn RlvSym Third Component Losec RelSaf Axid PepCid Zantac NoSiEf Tagamt NotCst FlxDoz Prophy /05/08 0:48:7 First Component

43 Components for Brands and Attributes Mode Unrotated Components (Orthonormal) Components After Varimax Rotation of the Core Matrix Components After Joint Varimax Rotation of Components and the Core Brands: A G D E B C F Attributes: Inexp NoSiEf Safe Prophy RelPain FlexDo RelSym Heals

44 Core Array Components for Brands: Frontal Slice : Unrotated Varimax Rotation of the Core Only Components for Attributes Joint Varimax Rotation of the Components and the Core Frontal Slice 2:

45 Assessment of Goodness of Model Fit: Kroonenberg and De Leeuw (980), and Kroonenberg (983) show that SS(Residual) = SS (Total) - SS(Fit) SS Accounted For = SS(Fit) / SS (Total) Also, as it has been shown by Ten Berge, De Leeuw, and Kroonenberg (987), when the ALS algorithm has converged, SS (Residual m ) = SS (Total m ) - SS(Fit m ) where m stands for any level of any mode of the data matrix. Using the last relationship, the relative fit of individual levels of a mode can be established. Also, whether a given level fits the model well or badly can be determined.

46 Model selection Tucker3 model Deviance versus Sum of Numbers of Components (Three-Mode Scree Plot) 400 Deviance versus Sum of Numbers of Components Deviance (SS(Residual)) xx 2xx2 x2x2 2x2x 2x2x2 3xx3 3x3x x3x3 2x3x2 3x2x2 2x2x3 3x3x2 3x2x3 2x3x3 3x3x /05/08 09:58:28 Sum of Numbers of Components (S = P + Q + R)

47 Tucker 3 Solutions Raw SS Standardized SS SS(Total) A.EST.SS(Fit) B.EST.SS(Fit) C.EST.SS(Fit) SS(Fit) SS(Residual) DF = Number of data points (minus loss of information due to preprocessing or missing data) minus the number of independent parameters Number of independent parameters = (I*P) + (J*Q) + (K*R) + (P*Q*R) - P**2 - Q**2 - R**2 with I, J, K the numbers of levels of st, 2nd, and 3rd modes, respectively, and P, Q, R the numbers of components of st, 2nd, and 3rd modes, respectively.

48 Relating Subject Components to External Variables Y: Number of years of experience as a medical doctor (standardized) X : First component score for subjects mode, X 2 : Second component score for subjects mode, Linear Model: Y = X B + X 2 B 2 Estimates: B = -.054, std. error = 0.997, t=.058, p-value=0.29 B 2 =.350, std. error = 0.997, t =.345, p-value=0.8 R 2 =0.0, F-value=.477, df=(2, 282), p-value=0.23. Conclusion: Subject components are not related to number of years of experience as a medical doctor.

49 Relating Residuals to External Variables: Y: Number of years of experience as a medical doctor (standardized) X: Sum of squares of residuals for each subject Linear Regression: Y = B X Estimated B = , R 2 = , F-value = 0.22, d.f. = (, 282) p-value = Conclusion : Residuals are not related to number of years of experience.

50 Result HiClas3-model (2D 3A 3B) Sulcrate Zantac, Axid, Pepcid, Losec Cytotec Tagamet D,D3 D,D3 D,D3 D,D2 D,D2 D4 No side effects Prophylactic Flexible dose Safe Relieves pain and symptoms, Promotes health D ( ) = 70 D2 ( 0) = 69 D3 (0 ) = 98 D4 (0 0) = 46 Low cost

51 HICLAS3: Example (Tucker3-HICLAS) Stimuli (0) Ignored in restaurant; Disconnecting operator; Closing store; Missing page in book; Accused by instructor; People tell lies about you; Persistently contradicted; Unfairly blamed for error () P P3 P2 P3 P P3 Subjects Response Grimace (00) Hands tremble; (00) Perspire; Want to strike Turn away; Lose patience; Feel irritated; Curse () Become enraged; Become tense; Heart beats faster (0) Based on example Leuven group

52 Number of discrepancies HiClas3 Three-mode scree plot Doctors Attributes Brands,, 2,2, 2,,2,2,2 3,3, ,,3,3,3 2,2,2 2,2,3 2,3,2 3,2,2 3,3, 2,3,3 3,2,3 3,3,2 Sum of Numbers of Components (S = P + Q + R) **

53 Preprocessing: Double Centring (in three-mode component analysis) Double centring: doctors may not use the attrubutes uniformly across the brands and across the attributes. Double centring: Scores in deviations from brand means attribute means. x. i x ḳ jk Origin = zero point for both brands and attributes of each subject s scores.i.i Brands Doctors k.k...j...j Attributes * x ijk = x ijk x. jk x i. k + x.. k jk ik I x K matrix of means removed J x K matrix of means removed

54 Preprocessing: Double centring Three-way factorial design without replacement ( observation per cell): Dependent variable: Brand possesses attribute (score = ) x = m + a + b + c + ac + bc + ab + ijk i j k ik jk ij abc ijk After centring: x = m + a + b + c + ac + bc + ab + ijk i j k ik jk ij abc ijk Analysed with Three-mode PCA ab ij = consensus of doctors about attributes of brands abc ijk = differences between doctors about attributes of brands

55 Model selection Tucker3 model Deviance versus Degrees-of-Freedom 400 Deviance (SS(Residual)) x3x3 3xx3 3x3x 3x2x2 3x2x3 3x3x2 2x2x32x2x2 2x3x2 2x3x3 2x2x 2xx2 x2x2 x3x3 xx Degrees-of-Freedom Model complexity: (Docs = 2; Attr = 3; Brands = 3) or (Docs = 3; Attr = 3; Brands = 3)

56 Third Component LowCost Tagamet Joint biplot (Consensus) Cytotec Heals RelSymptoms RelPain Prophylactic Losec Axid FlexDose Sulcrate NoSideEff Pepcid Zantac Safe First Component

57 Joint biplot (Consensus) Second Component Tagamet LowCost Losec RelPain RelSymptoms Heals PepCid AxidZantac NoSideEff FlexDose Safe Sulcrate Cytotec Third Component LowCost Tagamet Prophylactic Cytotec Losec Heals Safe RelSymptoms RelPain Sulcrate NoSideEff Pepcid AxidZantac FlexDose -3 Prophylactic

58 Conclusions Where to go from here Irregular patterns in some doctors combined with low number of ones were excluded from the HiClas analysis while these doctors were scattered all over the plot of the doctors components. Thus Tucker analysis picked up some information which was not available to the HiClas analysis. Similarly for the LowCost attribute. Is the numerical information such as the variance somewhere to be found in the HiClas results and if so can it be used? Construct exactly fitting hierarchical classes models and run a Tucker3 analysis on them. Construct doctors/attributes/brands artificially according to a specific pattern and include them in the analysis to facilitate interpretation. Sort out the mathematics of the comparison between models.

59 HiClas3 Three-Mode Deviance Plot Number of discrepancies ,,,2,2,3,3 Doctors Attributes Brands 2,,2 2,2, 2,2,2 2,2,3 2,3, Model complexity: (Docs = 2; Attr = 3; Brands = 3) or (Docs = 3; Attr = 3; Brands = 3) 2,3,3 3,,3 3,2,2 3,2,3 3,3, 3,3,2 3,3,3 Degrees of freedom **

60 Component Scores Individual differences between doctors Tucker3_332 Component HiClas3_332 Consensus among doctors = Dr 0 = Dr2 0 = Dr3 00 = Dr Tucker3_332 Subject Component.2

61 Gastro SVD-Biplot (means across doctors - attributes centred) Second Component NoSiEf RelSaf RlvPn RlvSym PromHl Losec Zantac PepCid FlxDoz Axid Tagamet NotCst Sulcrt Prophy Cytotc First Component Third Component Losec Cytotc RelSaf PepCid Axid PromHl NoSiEf Sulcrt Zantac Prophy RlvSym RlvPn FlxDoz Tagamet NotCst First Component 24/06/08 3:4:59