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Association log-linear analysis and canonical correlation analysis Chapter 9 Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 1 Association between qualitative variables • Association is a generic term referring to the relationship of two variables • Correlation measures strictly refer to quantitative variables • Thus, association, generally refers to qualitative variables • Two qualitative variables are said to be associated when changes in one variable lead to changes in the other variable (i.e. they are not independent. For example, education is generally associated with job position. • Association measures for categorical variables are based on tables of frequencies, also termed contingency tables Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 2 Contingency tables • Frequency tables show the joint frequencies of two categorical variables • The marginal totals, that is the row and column totals of the contingency table, represent the univariate frequency distribution for each of the two variables • If these variables are independent one would expect that the distribution of frequencies across the internal cells of the contingency table only depends on the marginal totals and the sample size Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 3 Contingency table (frequencies) Food & non-alcoholic beverage (Binned) * Anonymised hhold inc + allowances (Banded) Crosstabulation Food & non-alcoholic beverage (Binned) £ 20 or less From £ 20 to £ 40 From £ 40 to £ 60 More than £ 60 Total Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi Count % of Total Count % of Total Count % of Total Count % of Total Count % of Total Anonymised hhold inc + allowances (Banded) Medium-low Medium-high Low income income income High income 47 19 18 4 9.4% 3.8% 3.6% .8% 57 48 24 22 11.4% 9.6% 4.8% 4.4% 17 31 45 40 3.4% 6.2% 9.0% 8.0% 4 27 38 59 .8% 5.4% 7.6% 11.8% 125 125 125 125 25.0% 25.0% 25.0% 25.0% Total 88 17.6% 151 30.2% 133 26.6% 128 25.6% 500 100.0% 4 Independent variables • In probability terms, two events are regarded as independent when their joint probability is the product of the probabilities of the two individual events Prob(X=a,Y=b)=Prob(X=a)Prob(Y=b) • Similarly, two categorical variables are independent when the joint probability of two categorical outcomes is equal to the product of the probabilities of the individual outcomes for each variable Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 5 Expected frequencies under independence • Thus, the frequencies within the contingency table should not be too different from these expected values: ni 0 n0 j fi 0 f0 j * fij fi 0 f 0 j n00 f00 where • nij and fij are the absolute and relative frequencies, respectively • ni0 and n0j (or fi0 and f0j) are the marginal totals for row i and column j, respectively • n00 is the sample size (hence the total relative frequency f00 equals one). Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 6 Independence and association – testing • The more that empirical frequencies are at a distance from the expected frequency under independence, the more the two categorical variables are associated. • Thus, a synthetic measure of association is given by ( f ij f ij* ) 2 2 i, j Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi f ij* 7 The Chi-square statistic • The more distant the actual joint frequencies are from the expected ones, the larger is the Chi-square statistics • Under the independence assumption, the chi-square statistic has a known probability distribution, so that its empirical values can be associated with a probability value to test independence • The observed frequency values may differ from the expected values fij* because of random errors, so that the discrepancy can be tested using a statistical tool, the Chi-square distribution • As usual, the basic principle is also to measure the probability that the discrepancy between the expected and observed value is due to randomness only • If this probability value (from the Chi-square theoretical distribution) is very low (below the significance threshold), then one rejects the null hypothesis of independence between the two variable and proceed assuming some association Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 8 Other association measures • contingency coefficient: ranges from zero (independence) to values close to one for strong association, but its value depends on the shape (number of rows and columns) of the contingency table • Cramers V, bound between zero and one does not suffer from the above shortcoming (but strong associations may translate in relatively low - below 0.5 – values • Goodman and Kruskal's Lambda for strictly nominal variables, compares predictions obtained for one of the variables using two different methods, one which only considers the marginal frequency distribution for that variable, the other which picks up the most likely values after considering the distribution of the other variable. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 9 Other association measures • Uncertainty coefficient as the Goodman and Kruskals lambda, but considers the reduction in the prediction error rather than the rate of correct predictions. • Ordinal variables: • Gamma statistic (between minus one and one, zero indicates independence) • Somers d statistic, adjustment of the Gamma statistics to account for the direction of the relationshop • Kendall’s Tau b and Tau c statistics, for square and rectangular tables, respectively • These statistics check all pairs of values assumed by the two variables to see if (a) a category increase in one variable leads to a category increase in the second one (positive association); or (b) whether the opposite happens (negative association); or (c) the ordering of one variable is independent from the ordering of the other. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 10 Directional vs. symmetric measures • Directional measures (e.g. Somer’s d) assume that the change in one variable depends on the change in the other variable (there is a direction) • Symmetric measures assume no direction Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 11 Association measures in SPSS Click here to see the list of available statistics Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 12 Chi-square test in SPSS In a typical week, what type of fresh or frozen chicken do you buy for your household's home consumption? * Marital status Crosstabulation Count In a typical week, what type of fresh or frozen chicken do you buy for your household's home consumption? Marital status single married 14 31 'Value' chicken 133 36 233 8 35 5 48 15 59 9 83 101 258 55 414 'Organic' chicken Chi-square test Chi-Square Tests Pearson Chi-Square Likelihood Ratio Linear-by-Linear Association N of Valid Cases Value 8.517a 8.739 1.120 6 6 Asymp. Sig. (2-sided) .203 .189 1 .290 df 414 a. 0 cells (.0%) have expected count less than 5. The minimum expected count is 6.38. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi Total 50 64 Total other 5 'Standard' chicken 'Luxury' chicken Contingency Table As the p-value is above 0.05, the hypothesis of independence cannot be rejected at the 95% confidence level 13 Other association measures (symmetric) In a typical week, what type of fresh or frozen chicken do you buyfor your household's home consumption? * How would you describe the financial situation of your household? Crosstabulation Count In a typical week, what type of fresh or frozen chicken do you buy for your household's home consumption? 'Value' chicken How would you describe the financial situation of your household? Not very well off Difficult Modest Reasonable Well off 8 7 20 11 4 Total 50 'Standard' chicken 14 20 75 81 40 230 'Organic' chicken 0 6 12 19 11 48 'Luxury' chicken Total 1 7 21 40 14 83 23 40 128 151 69 411 Symmetric Measures Nominal by Nominal Phi Cramer's V Contingency Coefficient N of Valid Cases Value .268 .155 .259 411 Approx. Sig. .003 .003 .003 a. Not assuming the null hypothesis. b. Using the asymptotic standard error assuming the null hypothesis. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 14 Directional association measures Directional Measures Nominal by Nominal Lambda Goodman and Kruskal tau Uncertainty Coefficient Symmetric In a typical week, what type of fresh or frozen chicken do you buy for your household's home consumption? Dependent How would you describe the financial situation of your household? Dependent In a typical week, what type of fresh or frozen chicken do you buy for your household's home consumption? Dependent How would you describe the financial situation of your household? Dependent Symmetric In a typical week, what type of fresh or frozen chicken do you buy for your household's home consumption? Dependent How would you describe the financial situation of your household? Dependent Value .020 Asymp. a Std. Error .012 Approx. T 1.622 .000 .000 . . .035 .021 1.622 .105 .017 .006 .044 .016 .007 .010 .029 .009 3.104 .002e .033 .010 3.104 .002 .026 .008 3.104 .002 b c Approx. Sig. .105 c d d e e a. Not assuming the null hypothesis. b. Using the asymptotic standard error assuming the null hypothesis. c. Cannot be computed because the asymptotic standard error equals zero. Statistics for Marketing & Consumer Research d. Based on chi-square approximation Copyright © 2008 - Mario Mazzocchi e. Likelihood ratio chi-square probability. 15 More than two variables: three-way contingency tables Government Completely distrust Count Female Male Total UK Italy Germany Netherlands France Total UK Italy Germany Netherlands France Total 8 9 5 2 11 35 1 7 0 0 9 17 52 Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 2.00 Count 6 4 3 3 4 20 1 3 4 0 2 10 30 3.00 Count 11 9 3 4 6 33 1 3 1 1 4 10 43 Neither Count 22 8 17 13 16 76 1 6 7 5 6 25 101 5.00 Count 12 8 14 12 13 59 1 11 4 6 6 28 87 6.00 Count 15 8 20 21 4 68 1 9 6 8 2 26 94 Completely trust Count 7 5 9 18 9 48 2 8 6 6 5 27 75 Total Count 81 51 71 73 63 339 8 47 28 26 34 143 482 16 Log-linear analysis • The objective of log-linear analysis is to explore the association between more than two categorical variables check whether associations are significant and explore how the variables are associated • Log-linear analysis can be applied by considering a general log-linear model Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 17 Log-linear analysis: saturated model • Consider the three variables of the three-way contingency table: T(in government) Gender and Country • The frequency of each cell in the table can be rewritten as: nijk q uGuCuTuGCuGTuCTuGCT • where: nijk is the frequency for trust-level i, gender j and country k uG is the main effect of Gender (Trust, Country) uGT is the interaction effect of Gender and Trust uGCT is the interaction effect of Gender Trust and Country q is scale parameter which depends on the total number of obs. and similarly for uT, uC, uGC,... The frequency of each cell is fully explained when considering all of the main and interaction effect (the model is saturated) Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 18 Interpretation of the model nijk q uGuCuTuGCuGTuCTuGCT The u terms represent the main and interaction effects and can be interpreted as the expected relative frequencies • For example, in a two by two contingency table with no interaction, one would have nij=Nfi0f0j • Instead, if there is dependence (relevant interaction), the frequencies of a two by two contingency table are exactly explained (this is in fact a saturated model) by f ij nij Nf i 0 f 0 j f f i0 0 j where the term between brackets reflects the frequency explained by the interaction term and is one under independence Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 19 The log-linear model • By taking the logarithms one moves to a linear rather than multiplicative form log(nijk ) log(q ) log(uG ) + log(uC ) + log(uT ) + log(uGC ) + log(uGT ) + log(uCT ) + log(uGCT ) • The saturated model is not very useful, as it fits the data perfectly and does not tell much about the relevance of each of the effects • Thus, log-linear analysis check whether simplified log-linear models are as good as the saturated model in predicting the table frequencies Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 20 1. What log-linear analysis does 1. Computes the main and interaction effects for the saturated model 2. Simplifies the saturated model by deleting (according to a given rule) some of the main and interaction effects and obtains estimates for all of the main and interaction effects left in the simplified model 3. Compares the simplified model with the benchmark model • • If the simplified model performs well, it goes back to No. 2 and proceeds with attempts for further simplification Otherwise it stops Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 21 Simplified models • For example, suppose that the three-way interaction term is omitted, the log-linear model becomes: log(nijk ) log(q ) log(uG ) + log(uC ) + log(uT ) + log(uGC ) + log(uGT ) + log(uCT ) + ijt • Now there is an error term • The effects cannot be computed exactly, but can be estimated through a regression-like model, where • the dependent variable is the (logarithm of) cell frequency • the explanatory variables are a set of dummy variables with value one when a main effect or interaction is relevant to that cell of the contingency table and zero otherwise • the estimated coefficients are the (logarithm of) the corresponding main or interaction effects Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 22 Hierarchical log-linear analysis Proceeds hierarchically backward • Delete the highest-order interaction first (uGCT) • Delete lower-order interactions (uGC, uCT , uGT), one by one, two together, three altogether • Delete main effects (uG, uT, uC), one by one, two together, three altogether Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 23 Hierarchical LLA in SPSS Select the categorical variables and define their range of values Select backward elimination for hierarchical LLA Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 24 Options Provides the (exact) estimates for the saturated model Provides the association table (useful for deleting terms) Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 25 K-way effects: Output K=1 MAIN EFFECTS K=2 2-WAY INTERACTION K=3 3-WAY INTERACTION The 3-way interaction can be omitted, but other effects seem to be relevant K-W ay and Higher-Order Effects K-way and Higher a Order Effects K-way Effects b K 1 2 3 1 2 3 df 69 58 24 11 34 24 Likelihood Ratio Chi-Square Sig. 277.095 .000 124.416 .000 17.155 .842 152.678 .000 107.262 .000 17.155 .842 a. Tests that k-way and higher order effects are zero. b. Tests that k-way effects are zero. Test the effect of deleting that kway order effect ONLY Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi Pearson Chi-Square Sig. 276.963 .000 115.059 .000 14.784 .927 161.904 .000 100.275 .000 14.784 .927 Number of Iterations 0 2 4 0 0 0 Test the effect of deleting that k-way order effect AND ALL EFFECTS OF AN HIGHER ORDER 26 Deletion of terms • Now it is possible to look within a given kway class (partial association table) Partial Associations Effect q50*q64 q50*q43j q64*q43j q50 q64 q43j df 4 6 24 1 4 6 Partial Chi-Square 37.915 3.225 63.880 82.057 .752 69.869 Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi Sig. .000 .780 .000 .000 .945 .000 Number of Iterations 2 2 2 2 2 2 Deletion of these terms does not make the prediction of the contingency table cells worse compared to the model with the term 27 Specification search Step Summary Step 0 b Generating Class c Deleted Effect 1 Generating Class 1 c Deleted Effect 2 Generating Class 1 2 3 c Deleted Effect 3 Generating Class 1 2 c a Effects q50*q64* q43j q50*q64* q43j q50*q64, q50*q43j, q64*q43j q50*q64 q50*q43j q64*q43j q50*q64, q64*q43j q50*q64 Chi-Square df Number of Iterations Sig. .000 0 . 17.155 24 .842 17.155 24 .842 37.915 3.225 63.880 4 6 24 .000 .780 .000 20.379 30 .906 39.036 4 q64*q43j 65.001 24 No more terms can be 2 .000 eliminated .000 2 q50*q64, q64*q43j 20.379 30 .906 4 This term can be eliminated 2 2 2 a. For 'Deleted Effect', this is the change in the Chi-Square after the effect is deleted from the model. b. At each step, the effect with the largest significance level for the Likelihood Ratio Change is deleted, provided the significance level is larger than .050. c. Statistics are displayed for the best model at each step after step 0. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 28 Further steps • The hierarchical procedure stops when it cannot eliminate all effects for a given order • However, the partial association table showed that the main effect for country might be non-relevant • It may be desirable to test another model where that main effect is eliminated Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 29 Further steps Select the variables here Click here to define the model Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 30 The model Specify the model by inserting those 2-way interaction terms retained from hierarchical analysis and deleting the main effect for q64 Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 31 Output Goodness-of-Fit Testsa,b Likelihood Ratio Pearson Chi-Square Value 20.379 18.410 df 30 30 Sig. .906 .952 The model is still acceptable after deleting the country main effect a. Model: Poisson b. Design: Constant + q43j + q50 + q43j * q64 + q50 * q64 Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 32 And the winner is... log(nijk ) log(q ) log(uG ) + log(uC ) + log(uT ) + log(uGC ) + log(uTC ) + ijt • This model explains the contingency table cells almost as well as the saturated (exact) model • Thus, (a) the interaction among country, trust level and gender; and (b) the interaction between trust level and gender are not relevant Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 33 Parameter estimates • These can be regarded as size effects (how much are these terms relevant? comparisons are allowed!) – check the Z statistic • Check the SPSS output (click on OPTIONS to ask for estimates) • Odds-ratio (the ratio between estimates of the Z for different cells) indicate the ratio of the probabilities of ending up in a cell compared to the one chosen as a benchmark Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 34 Odds-ratio example • compare UK females (Z=3.97) with German females (Z=1.02) • The ratio is about four, which means that: • the interaction between being female and from the UK is about four times more important than the interaction between being female and from Germany in explaining departure from a flat distribution • the effect is positive (it increase frequencies) • This would suggest that in contingency tables it is more likely to find UK females than German females after accounting for all other effects Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 35 Canonical Correlation Analysis (CCA) (1) • This technique allows one to explore the relationship between a set of dependent variables and a set of explanatory variables. • Multiple regression analysis can be seen as a special case of canonical correlation analysis where there is a single dependent variable • CCA is applicable to both metric and nonmetric variables. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 36 CCA(2) • Link with correlation analysis: canonical correlation is the method which maximizes the correlation between two sets of variables rather than individual variables Example: relation between attitudes towards chicken and general food lifestyles in the Trust data-set • Attitudes towards chicken are measured through a set of variables which include taste, perceived safety, value for money, safety, etc. (items in q12) • Lifestyle measurement is based on agreement with statements like “I purchase the best quality food I can afford” or “I am afraid of things that I have never eaten before” (items in q25) Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 37 Canonical variates and canonical correlation • CCA relates two sets of variables • This technique also needs to combine variables within each set to obtain two composite measures which can be correlated • In standard correlation analysis this synthesis consists in a linear combination of the original variables for each set leading to the estimation of canonical variates or linear composites • The bivariate correlation between the two canonical variates is the canonical correlation Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 38 Canonical correlation equations 1)m dependent variables, y1, y2, …, ym 2)k independent variables, x1, x2, …, xk The objective is to estimate several (say c) canonical variates as follows: Y S1 11 y1 12 y2 ... 1m ym X S1 11 x1 12 x2 ... 1k xk Y S 2 21 y1 22 y2 ... 2 m ym X S 2 21 x1 22 x2 ... 2 k xk ... Y Sc c1 y1 c 2 y2 ... cm ym X Sc c1 x1 c 2 x2 ... ck xk the (canonical) correlation between the canonical variables YS1 and XS1 is the highest, followed by the correlation between YS2 and XS2 and so on Furthermore, the extracted canonical variates are not correlated between each other, so that CORR(YSi, YSj)=0 and CORR(XSi, XSj)=0 for any i≠j, which also implies CORR(YSi, XSj)=0. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 39 Canonical functions • The bivariate linear relationship between variates, YSi=f(XSi) is the i-th canonical function • The maximum number of canonical functions c (canonical variates) is equal to m or k, whichever the smaller. • CCA estimates the canonical coefficients and in a way that they maximize the canonical correlation between the two covariates • The coefficients are usually normalized in a way that each canonical variable has a variance of one. • The method can be generalized to deal with partial canonical correlation (controlling for other sets of variables) and nonlinear canonical correlation (where the canonical variates show a non-linear relationship). Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 40 Output elements • Canonical loadings – linear correlations between each of the original variable and their respective canonical variate • Cross-loadings – correlations with the opposite canonical variate. • Eigenvalues (or canonical roots) – squared canonical correlations, they represent how much of the original variability is shared by the two canonical variables of each canonical correlation • Canonical scores – value of the canonical function for each of the observations, based on the canonical variates • Canonical redundancy index – it measures how much of the variance in one of the canonical variates is explained by the other canonical variate Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 41 Canonical correlation analysis in SPSS • There is no menu-driven routine for CCA • A macro routine written through the command (syntax) editor is necessary Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 42 Canonical correlation macro Indicate here the path to the SPSS directory Run the program List the variables of the two sets Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 43 Output Values of the canonical correlation The first 3 correlations are different from 0 at a 95% confidence level Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 44 Canonical coefficients for the 1st set of canonical variates Y Set 1 ( coefficients) S1 11 y1 12 y2 ... 1m ym Canonical variate 1 2 3 4 5 6 7 8 9 10 q12a -0.293 0.445 0.392 -0.261 -0.379 0.354 -0.226 0.048 -0.264 0.748 q12b 0.305 -0.362 -0.071 -0.591 0.237 0.267 -0.778 0.276 0.215 -0.25 q12c -0.365 0.431 -0.267 -0.24 -0.187 0.209 -0.075 0.13 0.639 0.023 q12d 0.244 -0.211 -0.5 0.301 0.026 0.194 0.183 0.039 -0.201 0.026 q12e 0.217 -0.436 0.025 0.708 -0.453 0.398 0.257 -0.464 0.191 0.395 q12f -0.407 -0.292 -0.005 -0.76 0.354 -0.041 0.706 -0.056 0.304 -0.704 q12g -0.246 0.343 -0.049 -0.018 -0.452 -0.176 -0.363 -0.505 -0.957 -0.245 q12h 0.633 0.11 0.322 -0.083 0.406 -0.103 0.692 0.639 0.521 0.558 q12i 0.338 0.156 -0.211 -0.252 0.179 0.164 0.045 -0.837 0.072 0.437 q12j 0.349 0.469 0.286 0.099 -0.438 -0.2 -0.183 -0.211 0.313 -0.526 q12k 0.187 0.327 -0.039 0.079 0.23 0.721 0.306 0.117 -0.377 -0.346 q12l -0.245 -0.23 0.607 0.278 0.344 0.055 -0.341 -0.141 0.165 0.071 Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 45 Canonical coefficients for the 2nd set of canonical variates X Set 2 ( coefficients) S1 11x1 12 x2 ... 1k xk Canonical variate 1 2 3 4 5 6 7 8 9 10 q25a -0.55 0.283 -0.018 -0.215 -0.183 0.651 -0.759 0.116 0.431 0.526 q25b 0.14 0.222 0.038 0.135 -0.889 -0.084 -0.277 0.224 -0.533 0.59 q25c 0.323 0.103 -0.127 0.087 0.674 -0.465 -0.09 -0.536 -0.619 0.484 q25d 0.277 0.154 0.614 -0.315 -0.062 -0.258 0.212 -0.219 0.467 0.42 q25e -0.142 -0.673 0.769 0.665 -0.089 -0.047 -0.487 0.15 -0.217 -0.417 q25f 0.295 0.682 -0.168 -0.425 -0.38 0.391 0.232 -0.258 -0.207 -0.796 q25g 0.322 0.057 -0.12 -0.438 0.39 -0.306 -0.979 0.058 0.421 -0.495 q25h 0.087 -0.504 -0.12 -0.725 0.022 0.326 0.083 0.315 -0.308 0.148 q25i -0.175 0.329 0.444 0.135 0.476 0.284 0.092 0.766 -0.375 -0.032 q25j 0.5 -0.305 -0.115 0.428 0.025 0.847 0.206 -0.449 0.187 0.09 Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 46 Loadings (correlations) Set 1 q12a q12b q12c q12d q12e q12f q12g q12h q12i q12j q12k q12l 1 -0.06 0.443 -0.354 0.373 0.118 -0.085 0.216 0.516 0.217 0.494 0.025 -0.175 2 -0.1 -0.404 0.496 -0.209 -0.499 -0.372 0.096 0.119 0.258 0.219 0.405 0.038 3 0.382 0.096 -0.363 -0.284 0.232 0.308 0.258 0.375 -0.217 0.362 0.069 0.592 4 -0.384 -0.512 0.034 0.08 -0.007 -0.528 -0.357 -0.281 -0.174 0.031 0.216 0.325 5 -0.564 -0.213 -0.018 -0.326 -0.604 -0.305 -0.277 -0.042 0.465 -0.494 0.41 0.506 6 0.371 0.339 0.262 0.265 0.404 0.168 -0.06 -0.09 -0.017 -0.026 0.672 0.211 7 0.073 -0.371 -0.049 0.188 0.24 0.502 0.167 0.372 -0.061 -0.011 0.078 -0.157 8 0.126 0.177 0.115 0.215 -0.143 -0.136 -0.013 0.294 -0.714 -0.134 0.03 -0.169 9 -0.174 0.001 0.505 -0.143 0.132 0 -0.452 -0.055 0.113 0.248 -0.18 0.115 10 0.305 -0.077 0.056 0.001 0.04 -0.234 -0.011 0.228 0.246 -0.432 -0.316 -0.037 1 -0.616 0.577 -0.108 0.286 -0.268 -0.1 0.678 0.038 0.305 0.719 2 0.164 0.148 0.114 0.109 -0.317 0.346 0.055 -0.56 0.395 -0.09 3 0.178 -0.126 0.153 0.77 0.624 0.305 -0.183 0.087 0.229 -0.113 4 -0.204 0.205 -0.119 -0.338 0.001 -0.328 -0.005 -0.7 0.184 0.367 5 0.046 -0.494 0.38 -0.048 -0.102 -0.16 0.128 0.009 0.395 0.089 6 0.402 -0.111 0.003 -0.074 0.09 0.271 -0.129 0.234 0.235 0.503 7 -0.372 -0.233 -0.221 0.154 -0.278 -0.029 -0.478 0.026 0.071 -0.02 8 -0.421 0.409 -0.621 -0.181 -0.397 -0.492 0.392 0.059 0.663 0.071 9 -0.038 -0.172 -0.567 0.283 -0.33 -0.382 0.271 -0.348 -0.081 0.205 10 0.197 0.284 0.193 0.233 -0.277 -0.426 -0.122 0.079 0.059 0.141 Set 2 q25a q25b q25c q25d q25e q25f q25g q25h q25i q25j Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 47 Cross-loadings Cross 1-2 q12a q12b q12c q12d q12e q12f q12g q12h q12i q12j q12k q12l 1 -0.027 0.195 -0.156 0.164 0.052 -0.037 0.095 0.227 0.095 0.217 0.011 -0.077 2 -0.036 -0.145 0.178 -0.075 -0.179 -0.134 0.035 0.043 0.093 0.079 0.146 0.014 3 0.123 0.031 -0.116 -0.091 0.074 0.099 0.083 0.12 -0.07 0.116 0.022 0.19 4 -0.106 -0.142 0.01 0.022 -0.002 -0.146 -0.099 -0.078 -0.048 0.008 0.06 0.09 5 -0.131 -0.05 -0.004 -0.076 -0.14 -0.071 -0.064 -0.01 0.108 -0.115 0.095 0.117 6 0.078 0.071 0.055 0.056 0.085 0.035 -0.013 -0.019 -0.004 -0.005 0.141 0.044 7 0.011 -0.055 -0.007 0.028 0.036 0.075 0.025 0.055 -0.009 -0.002 0.012 -0.023 8 0.016 0.023 0.015 0.028 -0.019 -0.018 -0.002 0.038 -0.093 -0.017 0.004 -0.022 9 -0.014 0 0.041 -0.012 0.011 0 -0.037 -0.004 0.009 0.02 -0.015 0.009 10 0.012 -0.003 0.002 0 0.002 -0.009 0 0.009 0.01 -0.017 -0.012 -0.001 Cross 2-1 q25a q25b q25c q25d q25e q25f q25g q25h q25i q25j -0.271 0.253 -0.047 0.126 -0.118 -0.044 0.298 0.017 0.134 0.316 0.059 0.053 0.041 0.039 -0.114 0.124 0.02 -0.201 0.142 -0.032 0.057 -0.041 0.049 0.247 0.2 0.098 -0.059 0.028 0.073 -0.036 -0.056 0.057 -0.033 -0.093 0 -0.091 -0.001 -0.194 0.051 0.101 0.011 -0.115 0.088 -0.011 -0.024 -0.037 0.03 0.002 0.092 0.021 0.085 -0.023 0.001 -0.016 0.019 0.057 -0.027 0.049 0.049 0.106 -0.055 -0.035 -0.033 0.023 -0.041 -0.004 -0.071 0.004 0.011 -0.003 -0.055 0.053 -0.081 -0.024 -0.052 -0.064 0.051 0.008 0.086 0.009 -0.003 -0.014 -0.046 0.023 -0.027 -0.031 0.022 -0.028 -0.007 0.017 0.008 0.011 0.007 0.009 -0.011 -0.017 -0.005 0.003 0.002 0.005 Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 48 Redundancy analysis % OF VARIANCE EXPLAINED BY: Set 1 by Set 1 Set 2 by Set 2 Set 1 by Set 2 Set 2 by Set 1 CV1-1 0.093 0.018 0.196 0.038 CV1-2 0.096 0.012 0.077 0.01 CV1-3 0.105 0.011 0.125 0.013 CV1-4 0.091 0.007 0.098 0.007 CV1-5 0.158 0.008 0.061 0.003 CV1-6 0.091 0.004 0.064 0.003 CV1-7 0.058 0.001 0.058 0.001 CV1-8 0.066 0.001 0.176 0.003 CV1-9 0.054 0 0.093 0.001 CV1-10 0.047 0 0.051 0 Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 49