Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s)

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4.2. The Elderly at Home 69Table 4.3. Variables used in the PCA for the elderly at home.1. Age 11. Separate kitchen2. Sex 12. Hot water3. Marital status 13. Car or van ownership4. Employed 14. Number of elderly in household5. Birthplace 15. Owner occupier6. Father’s birthplace 16. Council tenant7. Length of residence in 17. Private tenantpresent household 18. Lives alone8. Density: persons per room 19. Lives with spouse or sibling9. Lavatory 20. Lives with younger generation10. Bathroominferential techniques that depend on assumptions such as multivariatenormality (see Section 3.7) are not invoked, there is no real necessity forthe variables to have any particular distribution. Admittedly, correlationsor covariances, on which PCs are based, have particular relevance for normalrandom variables, but they are still valid for discrete variables providedthat the possible values of the discrete variables have a genuine interpretation.Variables should not be defined with more than two possible values,unless the values have a valid meaning relative to each other. If 0, 1, 3 arepossible values for a variable, then the values 1 and 3 must really be twiceas far apart as the values 0 and 1. Further discussion of PCA and relatedtechniques for discrete variables is given in Section 13.1.It is widely accepted that old people who have only just passed retirementage are different from the ‘very old,’ so that it might be misleading to dealwith all 2622 individuals together. Hunt (1978), too, recognized possibledifferences between age groups by taking a larger proportion of elderlywhose age was 75 or over in her sample—compared to those between 65and 74—than is present in the population as a whole. It was thereforedecided to analyse the two age groups 65–74 and 75+ separately, and partof each analysis consisted of a PCA on the correlation matrices for the20 variables listed in Table 4.3. It would certainly not be appropriate touse the covariance matrix here, where the variables are of several differenttypes.It turned out that for both age groups as many as 11 PCs could bereasonably well interpreted, in the sense that not too many coefficientswere far from zero. Because there are relatively few strong correlationsamong the 20 variables, the effective dimensionality of the 20 variablesis around 10 or 11, a much less substantial reduction than occurs whenthere are large correlations between most of the variables (see Sections 4.3and 6.4, for example). Eleven PCs accounted for 85.0% and 86.6% of thetotal variation for the 65–74 and 75+ age groups, respectively.
70 4. Interpreting Principal Components: ExamplesTable 4.4. Interpretations for the first 11 PCs for the ‘elderly at home.’65–74 75+Contrasts single elderly living alone withothers.Contrasts those lacking basic amenities(lavatory, bathroom, hot water)in private rented accommodation withothers.Contrasts council tenants, living incrowded conditions with others.Contrasts immigrants living with nextgeneration with others. There are elementshere of overcrowding and possessionof a car.Component 1 (16.0%; 17.8%) ∗Component 2 (13.0%; 12.9%)Component 3 (9.5%; 10.1%)Component 4 (9.2%; 9.2%)Contrasts single elderly, particularly female,living alone with others.Contrasts those lacking basic amenities(lavatory, bathroom, hot water), whoalso mainly lack a car and are in private,rented accommodation, not living withthe next generation with others.Contraststhosewhohaveacar,donotlive in council housing (and tend to livein own accommodation) and tend to livewith the next generation with others.Contrasts council tenants, mainly immigrant,living in crowded conditions withothers.Component 5 (7.3%; 8.3%)Contrasts immigrants not living with Contrasts immigrants with others.next generation, with others. They tendto be older, fewer employed, fewer witha car, than in component 4.Component 6 (6.7%; 5.6%)Contrasts the younger employed people(tendency to be male), in fairly male) employed with others.Contrasts younger (to a certain extent,crowded conditions, often living withnext generation with others.Component 7 (5.6%; 5.1%)Contrasts long-stay people with a kitchenwith others.Contrasts those lacking kitchen facilitieswith others. (NB: 1243 out of 1268 havekitchen facilities)Component 8 (5.0%; 4.9%)Contrasts women living in private Contrasts private tenants with others.accommodation with others.Component 9 (4.6%; 4.5%)Contrasts old with others.Contrasts long-stay, mainly unemployed,individuals with others.Component 10 (4.4%; 4.4%)Contrasts long-stay individuals, without Contrasts very old with others.a kitchen, with others.Component 11 (3.7%; 3.8,%)Contrasts employed (mainly female) Contrasts men with women.with others.∗ The two percentages are the percentages of variation accounted for by the relevant PCfor the 65–74 and 75+ age groups, respectively.
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Principal ComponentAnalysis,Second
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viPreface to the Second Editionerty
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viiiPreface to the Second EditionA
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xPreface to the First Editionand in
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xiiPreface to the First EditionIn m
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xviAcknowledgmentsthese institution
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xviiiContents3.4.1 Example ........
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xxContents10 Outlier Detection, Inf
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xxivList of Figures5.2 Artistic qua
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xxviiiList of Tables6.1 First six e
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2 1. IntroductionFigure 1.1. Plot o
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4 1. IntroductionFigure 1.3. Studen
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1.2. A Brief History of Principal C
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1.2. A Brief History of Principal C
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2.1. Optimal Algebraic Properties o
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Page 94 and 95: 4Principal Components as a SmallNum
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Page 142 and 143: 6Choosing a Subset of PrincipalComp
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6.1. How Many Principal Components?
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6.2. Choosing m, the Number of Comp
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6.2. Choosing m, the Number of Comp
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6.3. Selecting a Subset of Variable
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6.4. Examples Illustrating Variable
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7.1. Models for Factor Analysis 151
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7.2. Estimation of the Factor Model
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7.3. Comparisons Between Factor and
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7.4. An Example of Factor Analysis
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7.4. An Example of Factor Analysis
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7.5. Concluding Remarks 165To illus
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8Principal Components in Regression
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8.1. Principal Component Regression
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8.1. Principal Component Regression
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8.2. Selecting Components in Princi
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8.2. Selecting Components in Princi
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8.3. Connections Between PC Regress
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8.4. Variations on Principal Compon
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8.5. Variable Selection in Regressi
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8.5. Variable Selection in Regressi
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8.6. Functional and Structural Rela
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8.7. Examples of Principal Componen
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Table 8.3. Principal component regr
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9Principal Components Used withOthe
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9.1. Discriminant Analysis 201on th
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9.1. Discriminant Analysis 203Figur
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9.1. Discriminant Analysis 205Corbi
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9.1. Discriminant Analysis 207that
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9.1. Discriminant Analysis 209betwe
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9.2. Cluster Analysis 211dimensiona
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9.2. Cluster Analysis 213Before loo
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9.2. Cluster Analysis 215Figure 9.3
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9.2. Cluster Analysis 217demographi
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9.2. Cluster Analysis 219county clu
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9.2. Cluster Analysis 221choosing a
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9.3. Canonical Correlation Analysis
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10.1. Detection of Outliers Using P
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10.2. Influential Observations in a
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10.3. Sensitivity and Stability 259
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10.3. Sensitivity and Stability 261
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10.4. Robust Estimation of Principa
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11Rotation and Interpretation ofPri
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11.1. Rotation of Principal Compone
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oot of the corresponding eigenvalue
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11.2. Alternatives to Rotation 279w
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11.2. Alternatives to Rotation 281F
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11.2. Alternatives to Rotation 283F
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11.2. Alternatives to Rotation 285T
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11.2. Alternatives to Rotation 287T
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11.2. Alternatives to Rotation 289A
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11.2. Alternatives to Rotation 291
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11.3. Simplified Approximations to
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11.3. Simplified Approximations to
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11.4. Physical Interpretation of Pr
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12Principal Component Analysis forT
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12.1. Introduction 301series is alm
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12.2. PCA and Atmospheric Time Seri
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and a typical row of the matrix is1
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12.3. Functional PCA 317A key refer
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12.3. Functional PCA 319The sample
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12.3. Functional PCA 321speed (mete
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12.3. Functional PCA 323of the data
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12.3. Functional PCA 325subject to
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12.3. Functional PCA 327series than
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12.4. PCA and Non-Independent Data
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13.1. Principal Component Analysis
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13.2. Analysis of Size and Shape 34
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13.2. Analysis of Size and Shape 34
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13.5. Common Principal Components 3
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13.7. PCA in Statistical Process Co
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13.8. Some Other Types of Data 369A
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13.8. Some Other Types of Data 371d
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14Generalizations and Adaptations o
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14.1. Non-Linear Extensions of Prin
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14.1. Additive Principal Components
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14.2. Weights, Metrics, Transformat
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14.3. PCs in the Presence of Second
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14.4. PCA for Non-Normal Distributi
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14.5. Three-Mode, Multiway and Mult
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14.5. Three-Mode, Multiway and Mult
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14.6. Miscellanea 401• Linear App
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14.6. Miscellanea 40314.6.3 Regress
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14.7. Concluding Remarks 405space o
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Appendix AComputation of Principal
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A.1. Numerical Calculation of Princ
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ReferencesAguilera, A.M., Gutiérre
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References 417Apley, D.W. and Shi,
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References 419Benasseni, J. (1986b)
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References 421Boik, R.J. (1986). Te
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References 423Castro, P.E., Lawton,
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References 425Cook, R.D. (1986). As
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References 427Dempster, A.P., Laird
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References 429Feeney, G.J. and Hest
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References 431in Descriptive Multiv
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References 433Gunst, R.F. and Mason
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References 435Hocking, R.R., Speed,
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References 437Jeffers, J.N.R. (1978
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References 439Kazi-Aoual, F., Sabat
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References 441Krzanowski, W.J. (200
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References 443Mann, M.E. and Park,
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References 445Monahan, A.H., Tangan
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References 447Pack, P., Jolliffe, I
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References 449Richman M.B. (1993).
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References 451Soofi, E.S. (1988). P
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References 453Tenenbaum, J.B., de S
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References 455Vong, R., Geladi, P.,
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References 457regularities in multi
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Index 459116, 127-130, 133, 270, 27
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Index 461computationin (PC) regress
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Index 463discriminant principal com
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Index 465of correlations between va
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Index 467see also hypothesis testin
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Index 469PC algorithms with noise 4
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Index 471in functional and structur
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Index 473variance ellipsoids,S-esti
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Index 475spatial correlation/covari
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Index 477for covariance matrices 26
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Author Index 479Belsley, D.A. 169Be
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Author Index 481Fowlkes, E.B. 377Fr
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Author Index 483Krzanowski, W.J. 46
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Author Index 485Rencher, A.C. 64, 1
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Author Index 487Yaguchi, H. 371Yana
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