FACTOR ANALYSIS. Examples. Identification of factors of intelligence out of a variety of tests.

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1 FACTOR ANALYSIS Factor Analysis is an explorative statistical technique to identify a relatively small number of factors that can be used to represent relationships among sets of many interrelated variables. Examples Identification of factors of intelligence out of a variety of tests. Calculation of touristic attractiveness from single touristic opportunities. Identification of factors of urban renewal out of single renewal measures. Steps in Factor Analysis 1. Standardization of the variables For the comparison of variables measured on different scales it is useful to perform the socalled z- transformation (mean: 0, variance: 1): 1

2 z ik = (x ik - x k ) / s k x ik x k s k z ik Value of variable k on observation i Mean of variable k Standard deviation of variable k Standardized value of variable x ik 2. Correlation analysis The correlation matrix is computed from the matrix of the standardized variables: Z Transposed of Z K Number of variables R = (1/(K-1))*ZZ Variables that do not appear to be related to other variables can be identified from the matrix. The appropriateness of the factor model can be evaluated. Factor analysis assumes, that the correlations between the (standardized) variables can be described by certain factors and tries to uncover these factors by mathematical and statististical techniques. 2

3 3. The basic assumptions of factor analysis Assumption: Each variable x ik (or the standardized z ik ) can be described as a linear combination of certain unobservable factors: z ik = a i1 *p 1k + a i2 *p 2k a iq *p qk Z = A * P (K*I) = (K*Q)*(Q*I) K I Q A P Number of variables Number of observations Number of factors K*Q matrix of factor-loadings; describing the relationship between variables and factors Q*I matrix of factor-scores; describing the relationship between factors and observations Substituting the last relationship into the first equation yields: R = 1/(K-1) ZZ = 1/(K-1) (AP)(AP) = 1/(K-1) APP A = ACA C Correlation matrix of the factor scores As the factors are supposed to be uncorrelated, C turns out to be the unity matrix and we can derive 3

4 the socalled fundamental theorem of factor analysis: R = AA The fundamental theorem implies, that the correlation matrix of the data to be analysed can be represented by the product of the matrix of the factor loadings and its transposed. If the extracted factors explain the variance of the basic data completely, then the squared sum of the factor loadings equals 1. But usually the basic variables are not only explained by the extracted factors (common factors), but also by factors which are unique to the dependent variable (unique factors). The squared sum of the loadings of common factors is called communality: R Reduced correlation matrix U Matrix of unique factors R = AA + UU R = AA 4

5 R differs from the correlation matrix, that in the main diagonal we find the communalities (and not values of 1). 4. The problem of the estimation of the communalities The exact estimation of the communalities can be performed only after factor extraction, on the other hand they are necessary for solving the fundamental theorem: this fact leads to a central problem of factor analysis, namely to the necessity of an a- priori estimation of the communalities: This can be done in different ways: As an estimated value for the communality the highest correlation coeffiecient of the variable under consideration with any other variable is used. This method has to be considered a very rough estimation, but practical applications have turned out to be quite meaningful. Another method would be to estimate the communalities iteratively with the highest correlation coefficient as the starting value. Both methods are known as Principal Axis Factoring (Hauptachsenanalyse). 5

6 No explicit estimation of the communalities is performed, but the original correlation matrix (with one s in the main diagonal) is used. It is assumed that the total variance can be explained by the common factors. This method is called Principal Component Analysis (Hauptkomponentenanalyse). In principal components analysis analysis linear combinations of the observed variables are formed. The first component is the combination that accounts for the largest ammount of variance. Succesive components explain progressively smaller portions of the total sample variance, and are all uncorrelated with each other. It is possible to compute as many principal components as there are variables. If all principal components are used, each variable can be exactly represented by them, but nothing has been gained, since there are as many factors as variables. In general, principal components analysis is a separate technique from factor analysis. That is, it can be used whenever uncorrelated linear combinations of the observed variables are desired. 5. The phase of factor extraction Several different methods can be used to obtain estimates of the common factors. These methods differ in the criterion used to define "good fit". The unweighted leastsquares method produces, for example, for a fixed 6

7 number of factors, a factor pattern matrix that minimizes the sum of the squared differences between the observed and reproduced correlation matrices (ignoring the diagonals). The generalized least squares method minimizes the same criterion; however, correlations are weighted inversely by the uniqueness of the variables. That is, correlations involving variables with high uniqueness are given less weight than correlations involving variables with low uniqueness. The second problem in factor extraction phase is to determine the number of factors needed to describe adequately the data. Similar to factor extraction different methods may be applied to that problem. Two examples are presented in detail: Kaiser Criterion: The number of factors to be extracted equals the number of factors for which the eigenvalue is greater than 1 (the eigenvalue represents the percentage of total variance explained by each factor and equals the sum of the squared factor loadings). Factors with a variance less than 1 are no better than a single variable, since each variable has a variance of 1. Although this a very commonly used criterion (also default criterion in the SPSS Factor Analysis), it is not always a good solution. 7

8 Bartlett-Test II: Checks, when k factors have already been extracted, if factor k+1 is still significant due to a Chi-Square distribution. 6. The Rotation Phase Although the factor matrix obtained in the extraction phase indicates the relation between the factors and the individual variables, it is usually difficult to identify meaningful factors based on this matrix. Often the variables and factors do not appear to be correlated in any interpretable pattern. Most factors are correlated with many variables. Since one of the goals of factor analysis is to identify factors, that are substantively meaningful (in the sense that they summarize sets of closely related variables), the rotation phase of factor analysis aims to transform the initial matrix into one that is easier to interpret. Consider the following plot with two factors and four hypothetical variables. From the factor loadings, it is difficult to interpret any of the factors, since the variables and factors are intertwined. That is, all factor loadings are quite high, and both factors explain all of the variables. But this changes dramatically, if the axes are rotated (according to the dotted line). The rotation is called orthogonal, when the axes are maintained at right angles, otherwise the rotation is called oblique. 8

9 The purpose of rotation is to achieve a simple structure. This means that we would like each fac- 9

10 tor to have nonzero loadings for only some of the variables, which helps us to interpret the factors. We would also like each variable to have nonzero loadings for only a few factors, preferebly one. This permits the factors to differ from each other. If several factors have high loadings on the same variables, it is difficult to ascertain how the factors differ. A variety of algorithms is used for orthogonal rotation to a simple structure. The most commonly used method is the varimax method, which attempts to minimize the number of variables that have high loadings on a factor. This should enhance the interpretability of the factors. The quartimax method emphasizes simple interpretation of variables, since the solution minimizes the number of factors needed to explain a variable. A quartimax rotation often results in a general factor with high-to-moderate loadings on most variables. This is one of the main shortcomings of the quartimax method. The equamax method is a combination of the varimax method, which simplifies the factors, and the quartimax method, which simplifies variables. 10

11 7. Estimation of Factor Scores Since one of the goals of factor analysis is to reduce a large number of variables to a smaller number of factors, it is often desirable to estimate factor scores for each case. The factor scores can be used in subsequent analyses to represent the values of the factors. Plots of factor scores for pairs of factors are useful for detecting unusual observations. Recall that a factor can be estimated as a linear combination of the original variables. That is, for case k, the score for the jth factor is estimated as: p F = jk W ji Xik i= 1 where X ik is the standardized value of the ith variable for case k and W ji is the factor score coefficient for the jth factor and the ith variable. There are several methods of estimating factor score coefficients. Each has different properties and results in different scores. The most common one (and the default in the SPSS-Procedure Factor Analysis) is the regression method. 11

12 Recommendations for Factor Analysis Data: The variables should be quantitative at the interval or ratio level. Categorical data (such as religion or country of origin) are not suitable for factor analysis. Data for which Pearson correlation coefficients can sensibly be calculated should be suitable for factor analysis. Assumptions: The data should have a bivariate normal distribution for each pair of variables, and observations should be independent. The factor analysis model specifies that variables are determined by common factors (the factors estimated by the model) and unique factors (which do not overlap between observed variables); the computed estimates are based on the assumption that all unique factors are uncorrelated with each other and with the common factors. Three main fields of applications of factor analysis can be identified: Initial structuring of a relatively little known new field of scientific interest (classical example of intelligence research) but also: Identification of 12

13 characteristical "constructs" (e.g economic-, family-, ethnic status) within the "Social Area Analysis". Estimation of variables which cannot be measured or observed directly. Simple data reduction (Reduction of dimensions to describe certain variables) Applying factor analysis requires a very careful selection of the variables involved. Forgetting important variables could lead to severe mistakes interpreting the results. Before an interpretation the significances of the factors have to be examined. Significant factors, which additionally approach a simple structure should be interpreted on the basis of scientific knowledge and experience. Factors which obviously represent pure statistical artefacts have to be dropped from the analysis. Applying factor analysis involves two kinds of statistical procedures: 13

14 straight forward algorithms (e.g. standardizing the variables or calculating the correlations), procedures depending on the subjective judgements of the scientist (number of factors, estimation of communalities). Especially decisions of the second type require a high ammount of personal skill and responsability. 14

15 SPSS PROCEDURES FOR FACTOR ANALYSIS 15

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21 SPSS-OUTPUT FROM FACTOR- ANALYSIS 1. Principal Component Analysis Descriptive Statistics ABIPRO ARL_RATE BFZ_20EW BFZ_50EW BFZ_BEZ BFZ_LHS BFZ_NBEZ BFZ_NLHS BFZ_WIEN KKPOTIV KKPOTOV LANDP SFZ_20EW SFZ_50EW SFZ_BEZ SFZ_LHS SFZ_NBEZ SFZ_NLHS SFZ_WIEN SO WI Std. Mean Deviation Analysis N 8,8828 4, ,57E-02 1,5062E ,23 35, ,70 36, ,06 21, ,23 49, ,13 19, ,41 40, ,95 107, , , , , , , ,20 17, ,64 17, ,85 9, ,74 23, ,55 8, ,35 19, ,12 81, ,3024, ,2058,

22 Component ABIPRO ARL_RATE BFZ_20EW BFZ_50EW BFZ_BEZ BFZ_LHS BFZ_NBEZ BFZ_NLHS BFZ_WIEN KKPOTIV KKPOTOV LANDP SFZ_20EW SFZ_50EW SFZ_BEZ SFZ_LHS SFZ_NBEZ SFZ_NLHS SFZ_WIEN SO WI Communalities Initial Extraction Extraction Method: Principal Component Analysis. Total Variance Explained Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings % of Cumulative % of Cumulative % of Cumulative Total Variance % Total Variance % Total Variance % 9,135 43,500 43,500 9,135 43,500 43,500 6,718 31,989 31,989 2,971 14,147 57,647 2,971 14,147 57,647 4,103 19,536 51,525 2,849 13,565 71,212 2,849 13,565 71,212 2,122 10,104 61,629 1,576 7,505 78,717 1,576 7,505 78,717 1,791 8,530 70,159,947 4,509 83,226,947 4,509 83,226 1,120 5,332 75,491,734 3,494 86,720,734 3,494 86,720 1,014 4,831 80,322,635 3,024 89,744,635 3,024 89,744 1,009 4,803 85,125,449 2,139 91,882,449 2,139 91,882,996 4,742 89,867,377 1,797 93,679,377 1,797 93,679,536 2,554 92,421,277 1,319 94,998,277 1,319 94,998,362 1,722 94,143,247 1,176 96,174,247 1,176 96,174,305 1,452 95,595,211 1,003 97,176,211 1,003 97,176,215 1,023 96,618,194,925 98,102,194,925 98,102,211 1,004 97,622,140,668 98,770,140,668 98,770,165,785 98,407,120,573 99,343,120,573 99,343 8,980E-02,428 98,835 5,003E-02,238 99,581 5,003E-02,238 99,581 8,960E-02,427 99,261 3,551E-02,169 99,750 3,551E-02,169 99,750 8,665E-02,413 99,674 2,446E-02,116 99,867 2,446E-02,116 99,867 2,826E-02,135 99,809 1,385E-02 6,593E-02 99,933 1,385E-02 6,593E-02 99,933 2,051E-02 9,765E-02 99, ,505E-03 3,574E-02 99,968 7,505E-03 3,574E-02 99,968 1,276E-02 6,078E-02 99, ,621E-03 3,153E ,000 6,621E-03 3,153E ,000 6,937E-03 3,303E ,000 Extraction Method: Principal Component Analysis. 22

23 10 Scree Plot Eigenvalue Component Number Component Matrix a Component ABIPRO ARL_RATE BFZ_20EW BFZ_50EW BFZ_BEZ BFZ_LHS BFZ_NBEZ BFZ_NLHS BFZ_WIEN KKPOTIV KKPOTOV LANDP SFZ_20EW SFZ_50EW SFZ_BEZ SFZ_LHS SFZ_NBEZ SFZ_NLHS SFZ_WIEN SO WI ,498,337 2,645E-02,488 4,664E-02,145 -,422,414,134-3,24E-02 8,161E-02-2,14E-02-3,49E-02 5,847E-03 1,408E-02 7,695E-03 3,082E-03-1,19E-03-9,97E-04 8,854E-04 2,105E-04,262,445-4,71E-02,176,569 -,588,103 5,361E-02 7,803E-02 6,697E-02-1,46E-02 6,979E-02-3,96E-02 1,571E-02 1,177E-02-4,54E-03 6,620E-03 3,608E-03 1,546E-03-2,80E-04 8,148E-05,867 9,643E-02 -,225-5,32E-02 -,175,127-1,59E-02-5,46E-02,221,206 5,069E-02,151 -,103-2,68E-02 2,280E-02-2,48E-02 5,761E-02-2,75E-03 4,203E-02 2,896E-02 1,209E-02,882,124 -,243-8,54E-03 3,805E-02 4,827E-02 3,755E-02-8,69E-02,158 -,221,200 3,462E-02-3,41E-03 -,110-4,52E-02-3,45E-02 5,491E-02-1,05E-02-2,37E-02-3,45E-02-1,62E-02,648 -,340,591,240 7,979E-02 4,837E-02 3,789E-02-3,54E-02-8,85E-03-2,03E-02 3,366E-02 2,908E-02 8,762E-03-5,94E-02,170,101 1,482E-02 1,870E-03 1,636E-03-2,15E-02 3,571E-02,784 3,331E-02 -,440,274 1,303E-02 5,196E-02-5,28E-02 1,067E-02 -,273 1,965E-02 5,831E-02 9,442E-02-6,65E-02-8,55E-02-1,35E-02 1,913E-03-8,12E-02 4,819E-02 4,367E-02-1,61E-02-1,26E-02,670 -,311,589,221 5,526E-02 4,171E-02 3,053E-02-1,31E-02 2,286E-03 2,198E-02 5,064E-02 5,119E-02-3,33E-02 6,911E-02 -,165,112 1,535E-02 1,783E-03-5,97E-03 1,891E-02-3,27E-02,875,112 -,368 7,176E-02-1,80E-02 6,633E-02 3,633E-03-5,27E-02 2,562E-03 -,124 8,528E-02 9,232E-02-3,65E-02,173 5,424E-02-8,07E-03-5,77E-02 1,811E-02-5,57E-02 2,700E-02 1,792E-02,386,559,368 -,539,210,190-1,05E-02 8,396E-02-6,44E-02 3,331E-03 4,027E-02 7,339E-02-3,18E-02-1,63E-02 5,479E-03 1,188E-02-5,96E-02-9,58E-02 8,786E-03-1,71E-03-1,60E-03 -,725,260-9,93E-02,279,158,220,318-1,56E-02-4,49E-02,183,265-1,93E-02,191 4,272E-02 9,064E-03-7,15E-03 2,763E-03-1,20E-04 1,975E-03 6,657E-04 3,317E-04 -,541,414-2,47E-02,397 2,775E-02,284,456 5,687E-02 6,822E-02 -,117 -,199 4,385E-02 -,156-3,10E-02-8,96E-03 3,924E-03-2,62E-04-5,07E-04-1,22E-03-2,29E-04-2,05E-04,257 -,615 -,175 -,358 -,184 -,153,331,470-3,31E-02-1,82E-02 8,331E-02 2,801E-02-2,15E-02 6,927E-04 1,279E-02 3,637E-03 7,154E-03 4,564E-03-6,60E-04 4,507E-04 2,654E-04,877,145 -,190-5,95E-02-8,73E-02 9,000E-02 1,999E-02 6,872E-02,199,281 -,135-6,40E-02 4,618E-02-3,05E-02-3,29E-03 2,512E-02-4,73E-02 1,617E-02-4,54E-02-2,88E-02-1,09E-02,884,171 -,218-1,34E-02,105 6,160E-03 7,441E-02 3,998E-02,133 -,140-2,97E-02 -,207,156 -,110-5,03E-02 3,477E-02-3,47E-02 8,497E-03 2,727E-02 3,498E-02 1,421E-02,648 -,334,603,224 8,394E-02 5,015E-02 3,868E-02 1,993E-02 4,392E-03-2,86E-03-3,00E-02-3,94E-02 3,346E-02-4,09E-02,158 -,103-1,37E-02-3,15E-03-4,65E-03 2,176E-02-3,85E-02,790 7,583E-02 -,372,270 4,333E-02 3,531E-02-4,29E-02 7,322E-02 -,340 8,439E-02-8,49E-02-4,64E-02 1,242E-02-7,12E-02-3,09E-02-9,99E-03 7,783E-02-4,78E-02-3,61E-02 1,366E-02 1,017E-02,676 -,312,588,210 6,778E-02 5,075E-02 2,937E-02 4,870E-02 1,735E-02 2,460E-02-1,88E-02-2,40E-02 2,914E-03 7,082E-02 -,158 -,110-2,01E-02-2,06E-03 1,171E-02-1,94E-02 3,539E-02,874,154 -,342 4,385E-02-6,77E-05 6,498E-02 1,847E-02 3,336E-02-1,38E-02-6,29E-02-9,20E-02-9,19E-02 5,980E-02,231 6,528E-02 1,525E-02 4,302E-02-2,05E-02 4,549E-02-2,39E-02-1,50E-02,286,584,439 -,510,216,203-2,42E-02 8,230E-02 -,121-3,12E-03-1,01E-02-3,03E-02-1,79E-02 1,093E-02 3,707E-03-1,04E-02 5,547E-02 9,669E-02-6,94E-03 2,161E-03 1,810E-03,201,661,392,134 -,456 -,184 1,961E-02 6,261E-02-3,48E-02-6,54E-02-8,02E-02,205,224-1,47E-02-1,22E-02-4,40E-03 4,393E-03 5,205E-03 8,890E-04-5,95E-04-5,92E-04,266,643,375,116 -,433 -,235 9,709E-02-4,76E-02-5,96E-02 4,383E-02,148 -,223 -,173 1,236E-02 1,713E-02 1,646E-03-9,62E-03-6,46E-03-1,26E-04-3,91E-04 6,007E-04 Extraction Method: Principal Component Analysis. a. 21 components extracted. Rotated Component Matrix a Component ABIPRO ARL_RATE BFZ_20EW BFZ_50EW BFZ_BEZ BFZ_LHS BFZ_NBEZ BFZ_NLHS BFZ_WIEN KKPOTIV KKPOTOV LANDP SFZ_20EW SFZ_50EW SFZ_BEZ SFZ_LHS SFZ_NBEZ SFZ_NLHS SFZ_WIEN SO WI ,248 -,189-9,43E-02 8,955E-02,172 3,546E-02 -,225,893 7,450E-02-2,32E-02 3,405E-03-9,60E-03-5,46E-03-6,98E-03-6,90E-04 5,879E-04 1,369E-03-3,50E-04-6,28E-04 1,439E-04 8,100E-06,215 4,948E-03,126,105 2,444E-02,957-9,82E-02 3,032E-02 1,311E-02-5,62E-04 1,111E-02 1,223E-02 1,817E-03 2,825E-03 5,432E-06 1,580E-04 5,796E-04 1,299E-04-7,57E-04 3,862E-05-3,35E-06,831,195,119,113 -,136-5,46E-02 5,938E-02 -,124-8,69E-02,363 -,112 -,143 7,831E-03,118 1,094E-02-1,40E-02 1,567E-02 4,094E-03,128 6,583E-03 9,731E-05,867,210,140 4,826E-02 -,102 8,771E-02 3,310E-02 -,128-6,80E-02-4,51E-02 -,168 9,349E-02 1,308E-02,321 7,820E-03-5,29E-03-4,64E-04 2,188E-03 3,384E-03 9,507E-03 3,215E-04,187,948 4,281E-02 4,267E-02-8,54E-02 4,005E-03 2,166E-02-8,02E-02-4,23E-02-6,99E-03-2,34E-03-1,52E-03-3,92E-03 2,061E-02-4,73E-02 -,204 1,807E-02 2,553E-03 3,129E-03 2,766E-03 4,263E-02,911,139 -,106-1,21E-02-6,66E-02 6,988E-02 1,279E-02-4,67E-03-2,91E-03-9,00E-02,237 -,110 2,863E-03 6,410E-04 9,409E-03-1,60E-02,230 3,477E-03-9,60E-04 2,315E-02 6,609E-04,206,936 6,508E-02 7,178E-02-9,83E-02 6,599E-03 3,534E-02-7,31E-02-4,49E-02 1,846E-02-4,78E-03-2,89E-02 5,242E-03 2,055E-02,220 4,409E-02 8,631E-03 1,543E-03 2,189E-03-2,83E-03-3,11E-02,951,145 5,207E-02 4,054E-02-8,61E-02 5,726E-02 3,411E-02-9,81E-02-6,40E-02-8,12E-02 -,136 -,105-6,85E-03-1,85E-02 8,267E-03 2,168E-03-9,23E-03 1,298E-02-2,65E-02-8,83E-02-1,06E-03,159 9,121E-02,950,176-4,84E-02 8,063E-02-3,19E-02-6,08E-02-3,02E-02 3,890E-02-3,70E-02-9,14E-03-1,82E-02 3,133E-02 7,391E-03-7,48E-03 1,978E-02,120 3,138E-03-4,04E-03 1,174E-04 -,401 -,366 -,102-4,15E-02,420 3,482E-02 -,170,164,677-3,66E-02 3,017E-04-1,47E-02 5,056E-03-6,87E-03-1,65E-04-1,06E-03 2,170E-03-1,82E-04-7,80E-04 6,867E-05 9,284E-06 -,245 -,255-5,59E-02,120,884 2,657E-02 -,164,167,138-2,44E-02-6,47E-03-5,26E-03-2,41E-03-5,63E-03-1,09E-03-1,36E-04-1,09E-03-4,35E-04-6,65E-04 8,011E-05-1,07E-05,120,112 -,126 -,198 -,160 -,113,911 -,210-7,21E-02 1,692E-02 5,122E-04 6,023E-03 1,199E-04 1,371E-03 6,924E-05 1,026E-03-6,24E-04 4,044E-04 2,348E-04 5,696E-05-7,96E-06,810,211,180,111 -,110 3,118E-02 9,035E-02-9,42E-02-7,60E-02,449 4,584E-03 9,538E-02-1,95E-03-7,19E-02-4,94E-03 1,348E-02-2,08E-02-6,14E-04-5,62E-02-2,38E-03-1,69E-04,843,214,186 5,472E-02-6,96E-02,163 6,566E-02 -,103-7,62E-02 4,025E-02-4,74E-02,384 2,879E-03 4,956E-02-1,10E-02 3,929E-03-2,61E-02-1,74E-03-2,31E-03 1,999E-03-2,40E-05,178,951 6,433E-02 4,309E-02-7,81E-02 6,922E-03 4,635E-02-5,91E-02-5,93E-02 2,200E-02 1,756E-02 3,645E-02 3,895E-04-1,19E-02 -,192-3,76E-02-1,13E-02-2,03E-03-1,09E-03-2,48E-03-5,07E-02,874,172-4,83E-02 2,047E-02-5,32E-02,102 1,128E-02 3,296E-03-8,08E-03-4,57E-02,429-1,66E-02 1,488E-03-5,75E-02-3,70E-03 4,495E-03 8,353E-03-2,62E-03-3,36E-03-2,71E-03-1,60E-04,209,938 8,136E-02 5,776E-02-8,74E-02 7,490E-03 6,254E-02-4,57E-02-6,48E-02 3,986E-02 1,481E-02 2,459E-02 2,381E-03-9,07E-03 3,744E-02,209-1,38E-02-1,55E-03-1,91E-03 2,957E-03 3,979E-02,932,139,105 5,528E-02-6,43E-02 7,230E-02 5,378E-02-8,06E-02-8,20E-02-1,89E-02-3,47E-02 3,900E-02-5,25E-04 -,185-1,02E-02 1,523E-02 -,174-1,02E-02-1,25E-02 6,490E-02 8,572E-04 5,310E-02 9,108E-02,959,198-1,69E-02 6,298E-02-8,41E-02-3,12E-02-2,15E-02 6,009E-04 1,898E-02 2,964E-02 1,793E-02-2,13E-02-6,63E-03 9,032E-03-2,68E-02 -,116-2,22E-03 4,059E-03-1,49E-04 7,314E-02 8,416E-02,228,895 7,310E-02 5,334E-02 -,115 9,214E-02-3,15E-02 1,315E-02-1,86E-03 1,248E-03 -,323-5,05E-03-1,60E-04-5,08E-04-1,86E-03 1,614E-03 6,873E-05-1,89E-04 2,723E-05,118,105,220,894 4,472E-02 8,897E-02 -,110 5,954E-03 8,152E-03 1,817E-02 9,700E-04 5,675E-03,325 8,664E-03 1,723E-03 1,483E-03-6,51E-04-1,46E-03 9,612E-04 2,395E-04-6,17E-05 Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization. a. Rotation converged in 22 iterations. 23

24 2. Principal Axis Factoring Communalities Initial Extraction ABIPRO,446,460 ARL_RATE,304,192 BFZ_20EW,969,791 BFZ_50EW,972,835 BFZ_BEZ,977,926 BFZ_LHS,965,861 BFZ_NBEZ,974,927 BFZ_NLHS,976,921 BFZ_WIEN,948,896 KKPOTIV,682,625 KKPOTOV,581,531 LANDP,445,470 SFZ_20EW,969,809 SFZ_50EW,971,841 SFZ_BEZ,980,934 SFZ_LHS,956,804 SFZ_NBEZ,977,932 SFZ_NLHS,969,904 SFZ_WIEN,947,861 SO,661,542 WI,657,529 Extraction Method: Principal Axis Factoring. 24

25 Factor Total Variance Explained Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings % of Cumulative % of Cumulative % of Cumulative Total Variance % Total Variance % Total Variance % 9,135 43,500 43,500 8,955 42,645 42,645 6,856 32,648 32,648 2,971 14,147 57,647 2,760 13,141 55,786 4,140 19,715 52,363 2,849 13,565 71,212 2,594 12,351 68,137 2,547 12,129 64,492 1,576 7,505 78,717 1,282 6,104 74,241 2,047 9,749 74,241,947 4,509 83,226,734 3,494 86,720,635 3,024 89,744,449 2,139 91,882,377 1,797 93,679,277 1,319 94,998,247 1,176 96,174,211 1,003 97,176,194,925 98,102,140,668 98,770,120,573 99,343 5,003E-02,238 99,581 3,551E-02,169 99,750 2,446E-02,116 99,867 1,385E-02 6,593E-02 99,933 7,505E-03 3,574E-02 99,968 6,621E-03 3,153E ,000 Extraction Method: Principal Axis Factoring. 10 Scree Plot Eigenvalue Factor Number 25

26 ABIPRO ARL_RATE BFZ_20EW BFZ_50EW BFZ_BEZ BFZ_LHS BFZ_NBEZ BFZ_NLHS BFZ_WIEN KKPOTIV KKPOTOV LANDP SFZ_20EW SFZ_50EW SFZ_BEZ SFZ_LHS SFZ_NBEZ SFZ_NLHS SFZ_WIEN SO WI Factor Matrix a Factor ,472,123,230,412,242,210,262,145,855,236-3,85E-02-4,95E-02,874,264-2,85E-02-1,24E-02,650 -,684 1,784E-02,186,780,366 -,234,254,673 -,666 4,349E-02,172,876,367 -,116 6,954E-02,385 2,533E-02,730 -,464 -,701,188,139,281 -,516,201,277,383,243 -,154 -,512 -,355,867,234 2,124E-02-5,22E-02,877,268 2,492E-02-1,22E-02,652 -,692 3,238E-02,171,782,327 -,157,247,679 -,666 4,406E-02,159,873,367-6,21E-02 4,693E-02,284-1,94E-02,779 -,416,194 3,856E-02,671,228,255 4,271E-02,647,209 Extraction Method: Principal Axis Factoring. a. 4 factors extracted. 7 iterations required. 26

27 ABIPRO ARL_RATE BFZ_20EW BFZ_50EW BFZ_BEZ BFZ_LHS BFZ_NBEZ BFZ_NLHS BFZ_WIEN KKPOTIV KKPOTOV LANDP SFZ_20EW SFZ_50EW SFZ_BEZ SFZ_LHS SFZ_NBEZ SFZ_NLHS SFZ_WIEN SO WI Rotated Factor Matrix a Factor ,279 -,205 -,140,566,302 1,127E-03,192,253,823,210,192 -,184,858,207,184 -,148,193,935 5,869E-02-9,97E-02,905,140 -,146-1,96E-02,216,929 9,211E-02-9,68E-02,939,142 7,045E-02 -,115,161 7,763E-02,930-1,82E-02 -,442 -,413 -,196,469 -,284 -,297-9,64E-02,594,120,119 -,171 -,642,823,220,244 -,155,856,208,228 -,117,185,940 7,867E-02 -,105,875,176-8,02E-02 1,164E-02,218,929,101 -,108,927,137,126 -,101 5,640E-02 8,101E-02,921 5,846E-02,134,164,468,527,187,184,472,487 Extraction Method: Principal Axis Factoring. Rotation Method: Varimax with Kaiser Normalization. a. Rotation converged in 7 iterations. The first factor could be interpreted as "peripheral". All distance variables (BFZs and SFZs take on high factor loadings; also there is little tourism). The second factor could be denominated as "intermediate". In many variables (tourism, purchasing power) it is close to the first factor, but accessibility is generally better. The third factor resembles the second factor concerning accessibility, except that the factor loadings for distances to Vienna are very high. Moreover the loadings for tourism (SO, WI) are quite high. Factor 3 could therefore be characterized as "touristic-west austrian". Finally the fourth factor is clearly the "central place" factor (negative 27

28 loadings with the distance variables; highest with purchasing power and share of graduates). In this example you can see the factor scores for some communities of the federal state of Burgenland. Obviously all the scores for factor 3 (touristic-west Austria) are negative. Looking at factor 4 the federal capital Eisenstadt takes on the highest value. 28