Structural equation modeling Chapter 15 Statistics for Marketing

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Structural equation modeling • Structural Equation Modelling (SEM) is a powerful method to estimate multiple and simultaneous relationships involving several dependent variables and explanatory variables, and allows for the inclusion of latent variables which cannot be directly measured but can be expressed as a function of other measurable variables Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 2

SEM extends the regression assumptions as • Several dependent variables can be considered at the same time • Explanatory variables can be assumed to be measured with a random error • Endogenous variables can be used to explain dependent variables • Correlation between explanatory variables is allowed for • And this is not all. Another key feature in SEM is the possibility of including in the model, as endogenous or exogenous variables, some latent (unobservable) variables. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 3

Latent variables • A latent variable cannot be observed or directly measured as it quantifies an object or construct which cannot be defined and measured in an unequivocal and verifiable way • Latent variables are not directly measured but can be approximated by a set of observable variables • Factors or principal components are latent variables Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 4

SEM and latent variables • SEM allows one to incorporate FA into a regression model, where latent factors appear in the SEM as explanatory and/or dependent variables. • SEM can be seen as a broad model which embodies regression and FA and canonical correlation analysis (the relationship between sets of dependent and independent variables are estimated) • However, many issues need to be considered to estimate SEMs Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 5

SEM as confirmatory FA • SEM is so flexible that it may lead to hundreds of potential alternative models • This is the reason why SEM is classified as a confirmatory rather than exploratory technique (like FA) and is sometime referred to as confirmatory factor analysis Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 6

Exploratory FA • Exploratory techniques • • • Get the data Specify a parametric structure Use an estimation procedure to quantify the relationship among variables • Exploratory FA • • • Assumes a linear structure and defines the factors as unobservable variables Factors are expressed as linear combinations of the observed variables Factor loadings and the factor scores are estimated, together with some of the diagnostics (mainly the eigenvalues) which allow ne to decide on the number of factors Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 7

Confirmatory FA Confirmatory techniques • The specific relationship among the variables need to be specified prior to the analysis, based on some pre-existing theory • The technique is used to fit this pre-specified model to the data and diagnostics are used to assess whether the model is good or not. SEM as confirmatory FA • It requires the model and all of the relationships between the variables to be chosen prior to estimation • With confirmatory analysis it is still possible to compare statistically alternative models through a competing model strategy using some statistical criterion to choose the winning mode Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 8

Latent constructs and indicators • Latent constructs cannot be directly and univocally measured • E. g. attitudes, passion, trust, risk aversion • There is no unique measure because they have many dimensions • Researchers usually exploit several items in a questionnaire to obtain some indirect measurement • E. g. measure risk aversion by asking respondents to rate how dangerous they consider driving at 100 mph, diving from ten meters, swimming in the ocean, jumping off a plane with a parachute, etc. • These are indicators (or manifest variables) which help measuring the construct, although with some measurement error Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 9

Causality • Constructs are the foundation of structural equation modeling, because they represent the real partners in a causal relationship • Causation: when two constructs are significantly associated (correlated) to one another but one occurs before the other then the construct occurring earlier is said to cause the construct provided there are no other reasons for such result – E. g. a high risk aversion precedes the decision to avoid risky actions, so risk aversion causes behaviour. • It is debatable whether statistics are able to identify causal relationships • In practise causation from the explanatory variable to the dependent one is simply hypothesized by the researcher and statistical models merely assess whether the assumed relationships fits the data appropriately Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 10

From variables to constructs • • • If the true risk aversion were measurable then it would be possible to explore its role in determining some specific behavior like purchasing chicken when newspapers report stories about bird flu It would be a nonsense to relate the fear of flying to chicken consumption But it is no nonsense to: • • • use the fear of flying and ten other risk aversion measures to quantify the construct of risk aversion relate the construct to chicken consumption SEM moves from variables to constructs because the latter can be seen as the outcome of several measurable variables. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 11

Variables in SEM • Manifest variables that can be directly measured and serve as indicators • Latent variables are not observable and are the actual components of the causal relationship • Just like variables in regression analysis one can distinguish between exogenous constructs and endogenous constructs in a causal relationship • Exogenous constructs only act as explanatory factors and do not depend on any other construct • Endogenous constructs play the role of the dependent variables and they appear on the left-hand side in at least one of the SEM causal relationships Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 12

Examples of marketing applications for SEM • Scarcity and willingness to buy How perceived scarcity influences willingness to buy through a set of mediating variables, like perceived quality, perceived monetary sacrifice and perceived symbolic benefits (Wu and Hsing (2006)) • E-shopping On-line shopping as the interaction of four variables: education, technological know-how, economic condition and trust (Mahmood et al. (2005)) • Customer relationship Customer satisfaction for credit cards considering two latent constructs, customer service and card member rewards as measured by some manifest variables, like being courteous, accurate in answering questions, correcting errors (for customer service) or ease of obtaining rewards (Dillon et al. (1997)) • Human brand. Why people become so attached to famous personalities who are the subject of marketing communication? Three latent determinants are considered, referring to the extent that the personality is perceived to fulfil a consumer’s (a) autonomy (or self-determination), (b) relatedness (sense of closeness with others) and (c) competence (the tendency to seek for achievement (Thomson (2006)) Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 13

SEM & related techniques • Structural Equation Modelling is a comprehensive method which includes as special cases: – confirmatory factor analysis – path analysis – multivariate regression (simultaneous equation systems) Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 14

Confirmatory factor analysis • In CFA the number of factors and the loadings of the original variable are assumed to follow some prior theory • The researcher runs the factor analysis on the basis of these assumptions on the number of factors and the loadings (constraining to 0 the loadings for those variables that are not expected to load on a specific factor) • Results are then evaluated with goodness-of-fit diagnostic Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 15

Path analysis • It generalizes the regression model to deal with the causality concept • Path Analysis is based on the path diagram • In Path Analysis all variables are directly measured, which marks the distinction with SEM, which includes latent constructs • The path diagram represents the relationship between the variables through arrows and boxes: – boxes are the variables – straight arrows leave the boxes containing predictors and point towards the boxes containing the dependent variables – It is also possible that two variables are correlated without implying causation, in which case the arrows are curved. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 16

Example of path diagram • A model where x 1 and x 2 cause y and there is nonzero correlation (there is collinearity) between x 1 and x 2 These are correlations between the variables linked by the arrows This is the direct effect of x 1 on y The indirect effect of x 1 on y is given by 0. 3 x 0. 5 = 0. 15 The total correlation of x 1 and y is 0. 85 Similarly, the total correlation of x 2 and y is 0. 71 (0. 5+0. 3 x 0. 7) Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 17

Multivariate regression • Multivariate regression analysis or simultaneous equation system • a system of regression equation • the dependent variable in one equation can appear on the right hand side of other equations • Some of the key assumption of standard regression analysis are violated: • the endogenous variables which appear on the right hand side are correlated with the residuals • Least squares estimates are inconsistent – alternative estimation methods are used • SEM as a system of structural equations • The equations of a system can be defined as structural equations when their specification depends on some theoretical basis • The structural form can generally be simplified into a reduced form by solving the system analytically • Differences • SEM allows for the inclusion of latent constructs (unobserved factors) and uses a confirmatory approach • Multivariate regression is based on observed (manifest) variables only and is generally exploited for exploratory rather than confirmatory analysis Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 18

Structural equation models • A structural equation model is composed of • a measurement model which links the latent construct to the manifest indicators; and • a structural model which summarizes the relationships linking the endogenous and exogenous constructs • The measurement model corresponds to confirmatory factor analysis: it can be tested whether the measurement of the latent variables through the manifest indicators is satisfactory • A structural equation model can be represented through a path diagram • manifest variables are shown in square or rectangular boxes • latent variables (and measurement errors) are shown through ovals or circles • causality relationships are indicated through straight arrows • correlation without causality is shown through a curved arrow Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 19

Identification • The path diagram is drawn according to some theory • Quantities to be estimated • Latent constructs • Parameters (causal relationships and correlations) • Before estimation it is important to check for identification issues • A model can be • Under-identified when the unknown parameters are too many compared to the available observations • just-identified when the model allows a single solution since there is a single set of estimates compatible with the available observations • over-identified when a single structural model is compatible with various sets of estimates (thus estimates might be not unique nor optimal) Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 20

Over-identification and SEM • Signs of identification problems • standard errors look too large • indicators are too highly correlated between each other • some of the estimates are unacceptable (like negative variance) • If the number of relationships exceeds what is needed for just-identification there is over-identification • SEM allows one to test over-identified models and find a solution which is statistically acceptable and test the validity of a theory (or compare competing theories) • To check for identification and over-identification, one may look at the degrees of freedom Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 21

Degrees of freedom (Do. F) • Do. F count the number of independent observations available for estimation • They are a measure of the discrepancy between the available number of observations and the constraints associated with the estimation of unknown pardegrees of freedomameters • Under-identified models: negative degrees of freedom • Just-identified models: degrees of freedom equal to zero • Over-identified models: positive • Rule of thumb for identification. At least three manifest indicators for each latent variable. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 22

Estimation • Maximum likelihood estimation (MLE) • the manifest indicators must follow a multivariate normal distribution (i. e. they are normally distributed for any value of the other indicators) • the latent constructs are also assumed to be normally distributed • Key aspects of SEM estimation • Individual cases only enter the estimation process to obtain the covariance matrix • SEM does not use the individual observations (cases) for the estimation of the parameters • Estimation is based on the covariance matrix not on the individual cases • Thus, degrees of freedom refer to the elements in the covariance matrix • An adequate sample size is still needed • Identification problems may emerge when many of the elements of the covariance matrix are close to zero unless the sample size is large enough • A simple rule of thumb requires at least fifteen observations per measured variable or indicator Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 23

Testing • When there are more elements in the covariance matrix than parameters to be estimated, then the model is over-identified and it is possible to test its theoretical foundation • How is theory tested? • First, parameter estimates should be reasonable, both in terms of the founding theory and for statistical acceptability • Goodness-of-fit tests Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 24

Goodness-of-fit tests • Chi-square statistic: tests whether the observed covariance matrix is equal to the estimated one (which is what one hopes) • The number of degrees of freedom indicates whether the model is just-identified (zero Do. F) or over-identified (more than zero Do. F) • If the p-value of the Chi-square test is larger than 0. 05 (0. 01), then the observed covariance matrix is not different from the estimated one at a 95% (99%) level of confidence • Non-rejection of the null hypothesis suggests that theory is acceptable although this does not rule out better models • The output from the tested model is usually compared to two boundaries • the independence model (no correlation between the endogenous and exogenous variables) • the saturated model (no constraints at all, perfect fit with the data, just like in log-linear analysis). • The tested model lies between these two extremes Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 25

Goodness-of-fit indicators • Minimum sample discrepancy (CMIN) checks whether the model perfectly fits the data (very unlikely and not really useful as a test). • When this measure is divided by the degree of freedom (CMIN/DF), one obtains the above mentioned Chi-square test • Root mean square residual (RMR) refer to the residuals between the estimated and sample covariance matrices. It can be used to compare alternative models, where a smaller RMR indicates better fit • Goodness-of-fit index (GFI), should be above 0. 90 for acceptable theories. The adjusted version (AGFI) has similar interpretation • Other indices which are expected to be as close as possible to one (and not below 0. 90): • • • normed fit index (NFI) relative fit index (RFI) incremental fit index (IFI) Tucker-Lewis coefficient (TLI) comparative fit index (CFI) Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 26

More indicators and tests • The non-centrality parameter (NCP) and the root mean square of approximation (RMSEA) consider both the discrepancy criterion (like the CMIN) and some parsimony criteria accounting for degrees of freedom – The RMSEA should be less than 0. 05 for a good model – The hypothesis that RMSEA<0. 05 is tested through the PCLOSE test. • Other measures for comparing alternative models are the AIC and BIC information criteria and similar information indices • The Hoelter’s critical N shows the largest sample size which still allows to accept the model – It is a useful complement to the Chi-square test, as the latter tends to reject the model when the sample size is large. – Better models require larger sample sizes to be rejected and generally one would expect a critical N of at least 200 for a good model. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 27

The SPSS package for SEM – AMOS • Graphical interface • Alternative softwares • LAMOS reads SPSS data sets • ISREL, the first computer program which dates back to 1973 after the pioneering work of the statistician Karl Jöreskog (1967 and 1969) which has evolved together with the method. • SAS also provides a procedure for estimating structural equation models proc CALIS. • Other packages designed for structural equation models are EQS and Mplus Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 28

The example • Behavioral model from the Trust data-set • Theory of planned behaviour (Ajzen): human action is determined by a combination of three dimensions • Behavioural beliefs (beliefs about the outcome of the action) produce either a positive or a negative attitude (A) toward the behaviour • normative beliefs refer to subjective norms (SN) or perceived social forces • control beliefs lead to perceived behavioural control (PBC) • All these produce intentions to behave (ITB) Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 29

Trust example • The Trust data-set is based to some extent on theory of planned behaviour • Intentions to purchase chicken (ITP) - question q 7 • eleven behavioural beliefs (bba to bbk in the Trust data -set) which act as the manifest indicators for the latent construct attitude (A) • For simplicity the subjective norm (SN) and perceived behavioural control (PBC) are assumed to be measured directly (although with error) through individual measures PBC and SN in the Trust data-set Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 31

The AMOS graphic interface The path diagram can be drawn directly here These are elements and functions to draw the diagrams easily Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 33

Some useful functions Draw a latent variable or adds indicators to a latent variable Draw a square (manifest variable) Draw a circle (latent variable) When clicking on an indicator, associate it with a latent unobserved variable (generally an error) Otherwise it creates the latent variable Draw a single arrow (causation) Draw a double arrow (correlation) Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi Copies objects Moves objects Delete objects 34

The data • To include the observed variables it is now time to open the SPSS file: or simply click on this button Click on FILE NAME to open the data, then click on OK Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 36

Adding names of variables from the data-set Click here to see the list of available variables Simply drag the desired variables on to the desired square Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 37

Unobservable variables and errors • It is also necessary to give names to the latent variables and to the errors (circles) • Latent variables: Right-click on the desired circle, select PROPERTIES and assign a a name and a label • Errors: all remaining variables can be assigned an automated name as follows Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 38

Before estimation Intercept is required for estimation if there are missing data Choose estimation method Decide whether results for the independence and saturated should be shown Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 40

Analysis properties Set estimation options Ask for additional output (e. g. factor Convergenceweights for the criteria measurement model) Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 41

Estimation output As the estimation procedure terminates this button becomes available to show estimates directly in the path diagram This pane shows the final Chi-square degrees of freedom and signals potential convergence problems As expected, the model is over-identified Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 43

Output Computation of Do. F(Default model) Number of distinct sample moments: 119 Number of distinct parameters to be estimated: 45 Degrees of freedom (119 - 45): 74 Do. F show that the model is overidentified The Chi-square statistic is high and significant: this means that there is a discrepancy between the observed covariance Result (Default model) matrix and the estimated one Minimum was achieved Chi-square = 619. 428 Degrees of freedom = 74 Probability level =. 000 Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi This is a frequent result but the Chi-square statistic Do. Ftends to be inflated when there are many Do. F(or large sample sizes) and is very sensitive to the assumption of multivariate normality 47

Estimates Estimate S. E. C. R. P . 074 13. 298 *** This coefficient is constrained to one in order to ensure identification bba <--- A 1. 000 bbb <--- A . 985 bbc <--- A -. 097 . 067 -1. 461 . 144 bbd <--- A . 836 . 067 12. 496 *** bbe <--- A 1. 045 bbf <--- A . 853 bbg <--- A . 887 bbh <--- A . 965 . 079 12. 296 *** bbi <--- A -. 297 . 064 -4. 628 *** bbj <--- A . 674 . 077 8. 807 *** bbk <--- A -. 028 . 084 -. 327 . 744 q 7 <--- A . 073 . 012 6. 307 *** q 7 <--- sn . 008 1. 051 . 293 q 7 <--- pbc . 016 . 010 1. 581 . 114 Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi These are the loadings of the. 078 13. 485 *** manifest 13. 872 indicators on the latent. 061 *** variable attitude (A). 074 11. 953 *** Attitudes have a positive and significant (albeit small) impact on SN & PBC do not intentions have a significant impact on intentions 48

Standardized regression weights Standardized weights can be interpreted like correlations, but they assume causation Estimate bba <--- A . 735 bbb <--- A . 656 bbc <--- A -. 072 bbd <--- A . 616 bbe <--- A . 661 bbf <--- A . 682 bbg <--- A . 594 bbh <--- A . 633 bbi <--- A -. 228 bbj <--- A . 447 bbk <--- A -. 017 q 7 <--- A . 324 q 7 <--- sn . 048 q 7 <--- pbc . 071 The behavioral belief which is most important to measure the latent construct (attitude) is bba, i. e. chicken taste These weights are negatively related to attitudes. They measure the following beliefs: bbc: difficulty of preparation bbi: the agreement with the statement that chicken lacks flavour A standardized weight of 0. 32 indicates a positive but bbk: animal welfare concern small relation between attitudes and intentions to purchase chicken Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 49

Correlations (no causation) Covariances: (Group number 1 - Default model) S. E. C. R. P 3. 712 4. 219 . 880 . 379 Estimate sn <--> pbc <--> A -2. 880 3. 548 -. 812 . 417 sn <--> A 12. 017 4. 461 2. 694 . 007 Correlations: (Group number 1 - Default model) Estimate sn <--> pbc . 040 pbc <--> A -. 040 sn <--> A . 134 Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi The only bidirectional relation which emerges as significant is between subjective norm (referent beliefs) and attitudes 50

Goodness-of-fit (1) Model Values above fve suggest rejection of the model In a good model all these indicators should be above 0. 9 Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 51

A competing model • The theoretical model is rejected by the coefficients are significant • Competing model strategy • try and remove all the non-significant components of the model • add some other explanatory variables • Problem: measurement of the latent construct for attitude because the presence of items with negative wording might lead to the identification of more than one latent factor • Additional explanatory variable • some variable explaining habit, which could be correlated with attitude and influence ITP • For example, variable q 2 b measures the frequency of chicken purchases and we label it as habit Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 54