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Tilburg University the Use of Canonical Analysis Kuylen, A.A.A Tilburg University The use of canonical analysis Kuylen, A.A.A.; Verhallen, T.M.M. Published in: Journal of Economic Psychology Publication date: 1981 Link to publication in Tilburg University Research Portal Citation for published version (APA): Kuylen, A. A. A., & Verhallen, T. M. M. (1981). The use of canonical analysis. Journal of Economic Psychology, 1, 217-237. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 28. sep. 2021 7 21 Journal f Economio , c Psycholog (19811 y ) 217-237 North-Holland Publishing Company THE USE OF CANONICAL ANALYSIS Anton A.A. KUYLE Thed Nan o M.M. VERHALLEN* Tilburg University, NetherlandsThe Received August 25, 1980; accepted April 20, 1981 e firsTh t par f thio t s non-technical revie f canonicawo l analysi concernes si d wit principlee hth e ,th data requirements interpretatione th , evaluatioe , th applicatioe th d nan consumen ni r researcf ho canonical analysis. In this context, concepts as canonical correlation, weights, loadings and scores e explainedar . Their interrelationship discussede sar . The characteristice nth f specifiso c formf o s canonical analysis: canonical correlation, canonical regression, redundancy analysis and partial canonical analysis are discussed. Their different application properties are emphasized. The last part consist application a f o s f canonicano l correlation analysi r branfo s d positioning. Here th e relationship with discriminant analysi illustrateds i s . Furthermor usage eth f canonicaeo l analysis for optimal scaling purposes is illustrated for the same example. 1. Introduction This article aims at making the technique of canonical analysis accessi- researchee th consumee o t th e n i bl r r behavio rfirse areath t n parI . a t general revie essentialswe gives i th f no date th ,a requiremente th d san interpretation and evaluation of problems associated with canonical analysis. Thi s followei s a shor y db t discussio f specifio n c formf o s canonical analysi applicationd an s consumen i s r research lase th t n I . part exampln a , brann eo d positionin gives gi demonstrato nt e bote hth relationshi f canonicapo l analysis with discriminant analysi wels sa s a l optimar fo e us l s scalinit g purposes. e authorTh * s wis o thant h k Meeuwi . GeleijnJ s r programminfo s a packagg n canonicao e l analysis and his useful comments on this paper. Mailing address: A.A.A. Kuylen, Katholieke Hogeschool, G 712, Postbus 90153, 5000 LE Tilburg Netherlandse Th , . - 0167-4870/81/0000-0000/S02.5 1980© 1 North-Holland 218 A.A.A. Kuylen, T.M.M. Verhallen / The use of canonical analysis 1.1. What is canonical analysis? e maiTh n characteristi f canonicaco l analysi investigatioe th s si e th f no relationship between two sets of variables. One set is the predictor set , analyticallyor independenf o t se e th , t variables secone Th . d consistf so the criteria or the dependent variables. A frequently occurring research problem with this structure is when the relationship between attitudes, the predictors, and product-usage characteristics, the criteria, is to be investigated. In a canonical analysis, variates are computed from both sets of variables. A canonical variate is similar to a factor in a principal component analysis, with the difference that a variate consists of a maximally correlated predictor and a criterion part. Analogous to factor analysis, a maximum of N variates (factors) can be extracted, whic independene har f eaco t h other numbee . th T Vs i f variablero s from the smallest set. To get acquainted with canonical analysis an example as we would obtain from many computer programs (e.g. SPSS) is given in table 1. Table 1 exampln A f canonicaeo l analysis. Variables Variate 1a Variate 2 a can. weights can. weights Predictor set Extensive 0.21 0.17 Sober -0.04 0.13 Imaginative 0.10 0.01 Varied 0.30 -0.17 With family -0.07 0.09 With care -0.09 0.22 Fast -0.29 -0.45 Neat -0.02 0.41 Good looking -0.18 -0.19 Criterion set Bread 0.23 -0.32 Meats 0.65 -0.33 Cheese 0.08 0.05 Eggs 0.15 0.03 Table cloth 0.15 0.69 Dishes -0.10 0.33 Canonical correlation 0.95 0.80 _————————————————————————^—————————————————— canonicae Th a l variate standardizede sar . A.A.A. Kuylen, T.M.M. Verhallen / The use of canonical analysis 219 In this exampl relationshie eth p between breakfast evaluatione th d san usag f breakfaso e t product s beeha s n investigated using canonical analysis canonicafirse o tw Th .t l variate givee sar n here. Canonical correlations (0.9 d 0.80 e analogouan 5 ar ) o ordinart s y correlation coefficients. It is important to keep in mind that a canonical correlation expresses the association between two underlying con- structs relationshipe Th . s betwee originae nth l observed variables can- not be derived directly from this. The implication of this will be discusse followine th n di g section. 1.2. The interpretation of canonical analysis results usee canonicaf th ro r Fo l analysi numbesa problemf ro arisy sema with regard to the interpretation and evaluation of results obtained from canonical analysis. 7.2.7. Canonical weights The canonical weights are comparable with ^-weights or, when the variates have been standardized, as usually is done in computer pro- grams,with beta-weights fro mmultipla e regression analysis. They serve o transfort e originamth lo obtai t variable s a a n y n suci wa s a h maximum correlation between predicto criteriod an r n set f variateso s . magnitude Th weigha f eo t expresse importance sth variabla f eo e from one set with regard to the other set in obtaining a maximum correlation between sets. We could start from these weights to interpret the results. Howeve problemso r thertw e ear : (1) In the first place these weights may be unstable due to multicol- linearity [1]. Some variable y obtaima s a nsmal l weigh r eveo t a n negative weight becaus face th t f thaeo variance th t variabla n ei s eha already been explaine othey db r variables thin I . s typ situatiof eo e nth weight t giv no cleae a o sd r pictur relevance th variablese f eo th f eo . (2) Canonical analysis maximizes the correlation between predictor and criterion parts of significant canonical variates. The correlation, however computes i , constructedn do t observeno , d variables. Techni- cally, a canonical solution is the maximum correlation between pairs of [1] Multicollinearity exists when variables from the same set are mutually highly correlated. When e multicollinearity th a picture siz f t th e o ewant ge f on o eo t s e coefficienth , f multiplo t e determinatio e computenb (Rn ca r eac) fo d h variable wit l e samotheral hth en i se setSe . 2 Johnston(1972). I 220 A.A.A. Kuylen, T.M.M. Verhallen / The use of iwwnu-ut analysis. linearly transformed variables. As many new orthogonal variates may be computed from the residual variance as the number of variables from the smallest set. In table 1, with six criterion variables a maximum of six orthogonal canonical variates may be.computed. Every canonical variat regressioe th constructeds e i th f no , non-observed variate th n eo observed variables. As these are constructed variables it is not necessary that the relationship between actual and observed variables has any significance. In order to investigate how much of the variance.of the observed variable s retainesi canonicaa n di l variate necessars i t i , computo yt e canonical loadings. A canonical loading, analogous to factor loadings in a factor analysis, expresse e associatioth s n betwee a nvariabl a d an e canonical variate. Thu t offersi opportunitn a s interpretatioe th r yfo f no a variate. 1.2.2. Canonical loadings canonicae Th l loading foune b correlatiny n db s ca variabl w ra e gth e scores wit variate hth e scores. These canonical variate score analoe sar - gou o factot s r score n factoi s r analysis. They expres scoree th s f o s respondent r objects(o whicn o s variablee hth measurede ar s e th n o ) canonical variates exampln .A computationa e th f eo l procedure involved is given below. Thus the correlation of variable X- with /v" forms the canonical loadin f variablgo variatn o i eX e /).canonicatabln e I . th e2 l variate loading r examplou r presentede sfo ear . One interpretation of the results based on the canonical loadings is as follows: persons who evaluate their breakfast as varied, extensive d imaginativan t emeats ea ten o t d, brea d cheesan d t breakfasa e t (variate 1), and people who consider their breakfast as neat, good looking, prepared with car table tenus clote a o dishe d t han s when having breakfast.
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