Qualitative and Quantitative Research Methods

Quantitative Research Methods Dr Nripendra P Rana Professor in Digital Marketing Head of International Business, Marketing & Branding Associate Editor : International Journal of Information Management 18 th September 2019 Lecture – Structural Equation Modelling (SEM)  What is SEM?  What are alternatives to SEM?  Why is it important?  What are main components of a SEM model?  What different types of “ Model Fit Indices ” do you know?  What is the role of “ Model Fit Indices ” in SEM? SEM (Opening Discussion for 10 Minutes) • It is a statistical technique that takes a confirmatory (i.e., hypothesis testing) approach to the analysis of a structural theory bearing on some phenomenon (Byrne, 2010). • As per Gefen et al. (2000), it is a multivariate technique combining the following two aspects to estimate a series of interrelated dependence relationships simultaneously: – multiple regression - dependence relationships – factor analysis - unmeasured concepts with multiple variables • As per Kline (2011), the following falls within the scope of SEM: – Confirmatory factor analysis – Path analysis – Partial least squares path modelling – Latent growth modelling Structural Equation Modelling (SEM) • First generation regression models: – Linear regression, LOGIT, ANOVA, MANOVA – Limitation: can analyze only one layer of linkage between independent and dependent variables at a time. • Second generation data analysis techniques: – LISREL (covariance -based SEM), Partial Least Squares (PLS) – Facilitates modelling the relationships among multiple independent and dependent constructs simultaneously – It can be used to test the extent to which research meets recognized standards for high quality statistical analysis • Test for statistical conclusion validity Why SEM? (Gefen et al . 2000) • In the same analysis SEM: – assesses the structural model – relationship among a set of dependent and independent constructs, and – evaluates the measurement model – loadings of observed items (measurements) on their expected latent variables. • Integrated analysis of the measurement and the structural model enables: – measurement errors of the observed variables to be analyzed as an integral part of the model, and – factor analysis to be combined in one operation with the hypotheses testing. • Fuller information about the extent to which the research model is supported by the data than in regression techniques Why SEM? (Gefen et al . 2000) Understanding SEM Model (Gefen et al . 2000) SEM models can include two types of variables: observed and latent • Observed variables (X or Y) are shown as squares [or rectangles] – Observed value used as an indirect measure of a concept or latent variable that cannot be measured or observed directly – They have empirical data, like the numeric responses to a rating (i.e. Likert) scale item on a questionnaire – Also known as indicator, item or measure – Empirical data are assumed to contain measurement error – Observed variables either reflect (hence termed as reflective) or form (hence termed as formative) the latent constructs Understanding SEM Model (Gefen et al . 2000) • Latent variables (A, B, C, D & E) are commonly depicted as ovals, circles or ellipses – Construct that is not observable or measured directly , but is measured indirectly through observable variables that reflect or form the construct – Two types: exogenous and endogenous – also have an error element that is typically not drawn in the diagram but is always part of the complete statistical model • Arrows between the latent constructs represent the hypothesized causation paths – the extent to which the latent variables vary linearly with other latent variables in the model – Coefficients estimating the strength of the relationships are either βs or γs • They may also correlate through error variance, which is shown as a double headed curved arrow Understanding SEM Model (Gefen et al . 2000) Reflective vs Formative variables • Reflective variables – Observed variables that “reflect” the latent variable – They should be unidimensional and correlated – Arrows connecting latent variables to the measurement variables point away from the latent variables – LISREL/AMOS assumes that the measurement variables reflect the construct represented by the latent variable • Formative variables – Observed variables that “ cause ” the latent variable – Represent different dimensions of latent variables – In PLS, arrows may also point to (rather than from ) a latent variable Understanding SEM Model (Gefen et al . 2000) • Exogenous latent constructs called Xi or Ksi (ξ) – Acts only as a predictor or "cause" for other constructs in the model – Exogenous constructs have only causal arrows leading out of them and are not predicted by any other constructs in the model – In Figure 1, latent constructs A and B are considered as exogenous in nature as no other variable in this model predicts them – The following causal paths were estimated as a γ coefficient: • A ⇒ C (γ CA ) • B ⇒ C ( γCB ) – X variables represent the measure of the exogenous constructs • Each X should load onto one exogenous – Lambda X ( λ Χ) - the path between an observed variable X and an exogenous construct • the item loading on its latent variable Understanding SEM Model (Gefen et al . 2000) • Endogenous latent constructs termed as Eta ( η) – Construct that is the dependent or outcome variable in at least one causal relationship – There are one or more arrows leading into the endogenous construct – In Figure 1, latent constructs C, D and are endogenous variables as these three are predicted by one or more other latent constructs – The following causal paths between endogenous variables were estimated as a β coefficient : • C ⇒ D (β DC ); C ⇒ E (β EC ) – Y variables represent the measure of the endogenous constructs • Each Y should load onto one endogenous construct – Lambda Y ( λ Y ) - the path between an observed variable Y and an endogenous construct • Item loading on its latent variable Understanding SEM Model (Gefen et al . 2000) • Gamma (γ) coefficients – paths connecting exogenous to endogenous variables • Beta ( β) – paths between two endogenous variables • Phi ( φ) – shared correlation matrix among exogenous variables • Psi (ψ) – shared correlation matrix among the error terms of the η • The error terms are termed as ζ (Zeta) Understanding SEM Model (Gefen et al . 2000) • Theta Delta ( Θ δ) – the error variance associated with X items – the variance not reflecting its exogenous latent variable • Theta Epsilon ( Θε ) – the error variance associated with Y items – the variance not reflecting its endogenous latent variable • The Θ δ and Θ ε error terms are supposed to load only on their corresponding items. • The λ Χ and λ Y matrixes are full and fixed – Mandatory to connect each item to its latent construct Understanding SEM Model (Gefen et al . 2000) • Measures for assessing how well the proposed theory fits the data • A variety of model fit indices are available • Lack of consensus in terms of: – which indices to report – the cut -offs for various indices • Two main categories – Absolute Fit Indices : Chi -Squared test, RMSEA, GFI, AGFI, the RMR and the SRMR – Incremental Fit Indices : NFI, CFI, PNFI Model Fit Indices (Hooper et al. 2008 ) • Model chi -square (χ 2) – In order to demonstrate a good model fit, this should be insignificant with a p -value above .05 – This criterion is generally not satisfied as chi -square is sensitive to larger sample size • Relative/normed ( the ratio of χ 2 to degrees of freedom ) chi -square (χ 2/df ) – An alternative to chi -square that minimizes the impact of sample size on the model chi -square – The ratio should not be higher than 5.0 and lower than 2.0 Absolute Fit Indices (Hooper et al. 2008 ) • Root mean square error of approximation (RMSEA) – Considered as an informative fit index due to its sensitivity to the number of estimated parameters in the model • RMSEA favors parsimony in that it will choose the model with the lesser number of parameters – Should n o t be more than .07 • In a well -fitting model the lower limit is close to 0 while the upper limit should be less than 0.07 Absolute Fit Indices (Hooper et al. 2008 ) • GFI/AGFI – A cut -off point of 0.90 has been recommended for the GFI but when factor loadings and sample sizes are low a higher cut -off of 0.95 is more appropriate – Values for the AGFI also range between 0 and 1 and it is generally accepted that values of 0.80 or greater indicate well fitting models Absolute Fit Indices (Hooper et al. 2008 ) • Standardised Root Mean Square Residual (SRMR) – Values for the SRMR should range b/w zero to 1.0 with well fitting models obtaining values less than .05 – But values as high as 0.08 are deemed acceptable – An SRMR of 0 indicates perfect fit • but it must be noted that SRMR will be lower when there is a high number of parameters in the model and in models based on large sample sizes Absolute Fit Indices (Hooper et al. 2008 ) • Normed -fit index (NFI) – Values for this statistic range between 0 and 1 – Values greater than 0.90 suggest a good fit. – More recent suggestion: the cut -off criteria should be NFI ≥ .95 – A major drawback to this index is that it is sensitive to sample size, underestimating fit for samples less than 200 Incremental Fit Indices (Hooper et al. 2008 ) • Comparative Fit Index (CFI) – A revised form of the NFI which takes into account sample size – Performs well even when sample size is small – One of the most popularly reported fit indices due to being one of the measures least effected by sample size – Values for this statistic range between 0.0 and 1.0 with values closer to 1.0 indicating good fit – A value of CFI ≥ 0.95 is considered as an indicative of good fit Incremental Fit Indices (Hooper et al. 2008 ) • Parsimony Goodness -of -Fit Index (PGFI) and Parsimonious Normed Fit Index (PNFI) • No threshold levels for these statistics have been recommended, which has made them more difficult to interpret • It is possible to obtain parsimony fit indices within the .50 region Parsimony Fit Indices (Hooper et al. 2008 ) • Avoid including all indices in order not to burden both a reader and a reviewer • A variety of indices (one from each category) should be reported because different indices reflect a different aspect of model fit: – Chi -Square statistic along with its degrees of freedom and p value – RMSEA and its associated confidence interval – SRMR – CFI – One parsimony fit index such as the PNFI • The aforementioned indices have been found to be the most insensitive to sample size, model misspecification and parameter estimates Reporting Fit Indices (Hooper et al. 2008 ) • It is not uncommon to experience the poor fit of a proposed model/theory specified with the data • Some modifications are acceptable for achieving improvement in results – Weak items with low multiple r 2 should be removed from the analysis as they likely to contain very high levels of error – Check discriminant validity – If Phi value is greater than 1.0 discriminant validity has not been achieved • Further inspections of item cross -loadings need to be made • Items with high Lambda -Y modification indices are possible candidates for deletion • This would not cause any major theoretical repercussions Improving Model Fit (Hooper et al. 2008 ) • Correlation of error terms can also help but its widely acceptable – A step that should be taken with caution – This suggests that there is some other issue that is not specified within the model that is causing the covariation – Decision to correlate error terms should accompany with a strong theoretical justification – Correlating within -factor error terms is easier to justify and more acceptable than attempting to correlate error terms across latent variables Improving Model Fit (Hooper et al. 2008 ) • Model Specification • Measurement Model • Structural Model SEM Steps Model Specification: Example (Dwivedi et al. 2017) Measurement Model: Example Structural Model: Example Structural Model: Example • Byrne, B . M . (2010 ). Structural equation modeling with AMOS : Basic concepts, applications, and programming . 2nd Edition, Routledge , New York, USA • Dwivedi , Y.K ., Rana, N .P., Janssen, M ., Lal, B ., Williams, M .D . & Clement, R .M . (2017 ). An Empirical Validation of a Unified Model of Electronic Government Adoption (UMEGA) . Government Information Quarterly, 34 (2), 211 -230 . • Gefen , D ., Straub, D ., & Boudreau, M . C . (2000 ). Structural equation modeling and regression : Guidelines for research practice . Communications of the Association for Information Systems , 4(1), 7. Available at : http ://aisel .aisnet .org/cgi/viewcontent .cgi?article= 2531 &context=cais • Hooper, D ., Coughlan, J., & Mullen, M . (2008 ). Structural equation modelling : Guidelines for determining model fit . Articles , 2. Available at http ://arrow .dit .ie/cgi/viewcontent .cgi?article= 1001 &context=buschmanart • Kaplan, D . (2007 ). Structural Equation Modeling . Sage . pp . 1089 –1093 . ISBN 9781412950589 . • Kline, R . (2011 ). Principles and Practice of Structural Equation Modeling (Third ed .). Guilford . ISBN 9781606238769 . • Schumacker , R . E ., & Lomax, R . G . (2010 ). A beginner's guide to structural equation modeling . 3rd Edition, Routledge , New York, USA . • Wikipedia page on “Structural equation modelling”, which also available at https ://en .wikipedia .org/wiki/Structural_equation_modeling References ( Recommended Readings) Thank you! Questions?