Multivariate Data Analysis - Assignment Example

Multivariate Data Analysis Table of Contents Structural Equations Model (SEM) 2 Confirmatory Factor Analysis (CFA) 2 Case Study of a Structural Measurement Model 3 Bibliography 7 Structural Equations Model (SEM) Structural Equations Models are a group of multivariate equations, involving dependent and independent variables and depicting different forms of equation through which the components are related to one another. The components or constructs are of two kinds – endogenous and exogenous, depending upon the way that their values are attained. The purpose of an SME is to estimate the values of all such endogenous variables in terms of the exogenous ones, through assessing the interrelationships being defined. Confirmatory Factor Analysis (CFA) There might be cases where the constructs (both endogenous and exogenous) are each defined by a few latent or unobservable variables. These variables need to be specified prior to setting up a structural model. The process through which these latent variables are decided is known as Confirmatory Factor Analysis (CFA). Thus, the measurement model for a CFA also comes in the form of a multivariate regression equation. However, CFA precedes SME since the exogenous variables included in an SME are determined through CFA. CFA and SME together form a measurement model, and help in evaluating the underlying relationship between variables, with least measurement errors. An SEM generally consists of a number of multivariate equations which often leads to errors in recording the inputs. Hence the preferred form of input representation in an SEM should be through a covariance matrix with defined row and column names, so as to avoid confusion and errors in providing inputs. Post estimation, there remains the task of assessing the fitness of the predicted model. Model fit implies the degree to which the estimated model can resemble the observed population model. Hence, the more that the observed covariance matrix corresponds to the estimated one, the better is the model fitness. Generally, model fit could be of two types – goodness of fit and badness of fit; in the former case, the estimated model is considered to be a good representation as the value of the statistic rises in contrast to what the defining factor should be in case of badness of fit. Examples of goodness of fit are GFI, CFI and TLI and those of badness of fit are RMSEA and SRMR. However, there is no benchmark to evaluate the validity of the model based on the value of goodness of fit of the same. The only way-out would be to assess the goodness of fit through figuring out the same via multiple indices. Case Study of a Structural Measurement Model A numeric example of a hypothesized Measurement Model has been provided below, which says that the factors ‘Supervision’, ‘Work Environment’ and ‘Coworkers’ bear a causal relation between them as is evident from the arrow heads. In fact, the directions of the arrows indicate the direction of relationship between the constructs. The figures adjoining the arrows are the proportions of the dependence existing between the variables. Figure 1: An example of a Hypothesized Structural Measurement Model There are five variables involved in the model, which means that there should be (5 x 5 =) 25 covariances and variances involving them. These 25 values could be represented in the form of a correlation matrix, with the diagonal values representing the variances and the non-diagonal ones representing the covariances between the same. These 25 values are the observed ones, which are hypothetical in the present case in the sense that they have been calculated on the basis of hypothetical sample data. One point to be kept in mind over here is that even though Job Search and Job Satisfaction are the endogenous variables, with the remaining being exogenous in nature, the correlation matrix provides information about the association existing between any two variables irrespective of their nature. The observed covariance matrix is followed by an estimated covariance matrix which again is followed by the covariance matrix for the estimated residual terms, i.e., the differences between the observed and estimated covariances. Most of the estimated residuals are found to be equal to 0.00, implying that the estimated model is probably a good fit, as will be seen later. Observed Covariance Matrix (S) Supervision (S) Work Environment (WE) Coworkers (CW) Job Satisfaction (JS) Job Search (JSh) Supervision Var (SP) Work Environment 0.2 Var (WE) Coworkers 0.2 0.15 Var (CW) Job Satisfaction 0.2 0.3 0.5 Var (JS) Job Search -0.05 0.25 0.4 0.5 Var (JSh) Table 1: Observed covariance Matrices The observed covariance matrix intimidates about the extent to which the three exogenous variables, namely, ‘Supervision’, ‘Work Environment’ and ‘Coworkers’ are related to each other. This could be summarized with the help of the following path diagram, where the arrow heads between the exogenous variables indicate the relationship that each of them hold with the other. Figure 2: Path diagram representing the observed impact of the exogenous constructs on the endogenous ones and the observed covariances between the exogenous constructs On the other hand, the arrow heads pointing towards the endogenous variables clearly suggest that out of the three exogenous variables, the factor ‘Coworkers’ impose the greatest impact on the dependent variable ‘Job Satisfaction’, while the least effect is bestowed by the factor ‘Supervision’. The numerical values of these impacts are the result of a regression model being estimated with the sample data (not presented over here). Furthermore, the variable ‘Job Satisfaction’ which turns up to be an exogenous variable in determining the degree of ‘Job Search’, imposes a 50% impact on the latter. Thus, the Structural Model, including the SEM and CFA, could be presented as follows, Job Satisfaction = 0.065 Supervision + 0.219 Work Environment + 0.454 Coworkers … (1) Job Search = 0.500 Job Satisfaction … (2) Equation (2) is the Structural Equation Model in this case and equation (1) is the Confirmatory factor Analysis model, which precedes the SEM. The above two equations provide the following modified equation to figure out the value of ‘Job Search’ in a way such as to take care of all the latent variables as well. Thus, equation (2) could be re-written as, Job Search = 0.500 (0.065 Supervision + 0.219 Work Environment + 0.454 Coworkers) … (3) The final task is that of assessing the model fitness, which could be done through predicting the estimated covariance matrix and figuring out the predicted residuals. The estimated covariance matrix could be computed through figuring out the values of each of the paths in the structural model. The values would be a sum of the direct and indirect paths, which when subtracted from the observed covariances evolved from sample observations, would yield the predicted residuals. The following paragraph shows how the estimated covariances are evolved, with the help of an example. Suppose the covariance value between Work Environment and Job Satisfaction needs to be estimated. Hence, the process will be to figure out the value of the direct path, which is the value of the impact that the former has on the latter. Hence, value of direct path = 0.219 The next step will be to figure out the value of the indirect path which means that the impact of the variables which are related to ‘Work Environment’ and have an impact upon ‘Job Satisfaction’ needs to be calculated as follows – Value of indirect path from Work Environment to Job Satisfaction through Supervision = 0.200 x 0.065 = 0.013 Value of indirect path from Work Environment to Job Satisfaction through Coworkers = 0.150 x 0.454 = 0.068 Thus, the total value of the path, which is equal to the estimated covariance = 0.219 + 0.013 + 0.068 = 0.300. In this way, the estimated covariance matrix could be attained for every pair of variable as the following table would show. Estimated Covariance Matrix (E) Supervision Work Environment Coworkers Job Satisfaction Job Search Supervision - Work Environment 0.2 - Coworkers 0.2 0.15 - Job Satisfaction 0.2 0.3 0.5 - Job Search 0.1 0.15 0.25 0.5 - Table 2: Estimated Covariance Matrix Finally, after the estimated covariance matrix has been obtained, the final step will be to evaluate the predicted residual matrix which will figure out the model fitness. Residual Covariance Matrix (S - E) Supervision Work Environment Coworkers Job Satisfaction Job Search Supervision - - - - - Work Environment 0 - - - - Coworkers 0 0 - - - Job Satisfaction 0 0 0 - - Job Search 0.15 -0.1 -0.15 0 - Table 3: Residual Covariance Matrix The residual covariance matrix above shows that the difference between the observed and estimated covariances for most of the pairs is 0.00, other than for three pairs out of 25, which is a clear indication of the fact that the estimated model is a rather good fit. Bibliography Hair et al. (2010), Multivariate Data Analysis, 7th Edition, Pearson Global Edition Read More