Introduction to data analysis
- 1. P a g e 1 | 8 DESCRIPTION OF THE TOPIC Items Description of the Topic Course Data Analysis for Social Science Teachers Topic Introduction to Data Analysis Module Id 1.1 Introduction In the recent past, quite a bit of importance has been given to data analysis in research. One of the possible reasons is that empirical evidence establishes a firm grounding to either accept or reject the proposed hypotheses. The choice of the statistical technique depends on the nature of the research problem or question and also on the nature of the data set. The research questions to solve a research gap or problem may be related to identifying the degree of relationships among variables, checking for the significance of group differences, predicting of group memberships or structure, or it could be time-related. In order to identify associations between two or more variables, depending on whether their nature of being parametric or non-parametric, correlation, and regression or chi- square techniques may be adopted. This can be done as a Bi-variate correlation and regression, multiple correlation and regression, Canonical correlation, Multiple Discriminant Analysis, and Log-it regression. The bi-variate correlation is a good starting point to identify the degree of relationship between two continuous variables, such as job and family satisfaction where either of them can be treated as a DV and IV as the research question may be. But bi-variate regression would require one of them to be defined as the DV and the other as the IV. Although these are not multivariate techniques, they form the basis of the Multivariate Analysis (MVA). 1. Importance of Multivariate Analysis (MVA) If watching a movie needs to be a pleasant experience, the lighting, the projected film light and sound effects in the theatre must be optimum. The other factors that may contribute to a pleasant viewing experience may include, but not limited to, seating arrangements, air- conditioning and hall odour. If one has to study or measure the pleasantness of watching a movie in a theatre, all of the above factors must be studied together and not in isolation. There is a possibility of an unpleasant parking experience that may negatively impact the pleasantness of watching a movie. So, the real value of measuring the pleasantness of a movie-watching experience lies in measuring all the influencing factors together. This is exactly what Multivariate analysis is all about. So, analysis of multiple variables simultaneously would result in a better picture to arrive at inferences instead of multiple uni-variate analyses done with the individual variables. Statistical Techniques that simultaneously analyse multiple measurements of the observed variables are known as Multivariate Analysis (MVA). We may perform MVA by using multiple variables in a single relationship or in multiple relationships.
- 2. P a g e 2 | 8 In a truly multivariate scenario, all variables must be: i. Random in nature, ii. Inter-related, and iii. Interpreted in unison. Reading the paper related to testing the Greenhaus and Allen model by Pattusamy and Jacob (2015) will help in understanding our forthcoming discussions and answering a few questions in the end. The theoretical model is shown in Figure 1. Figure 1 - Theoretical Model From Figure 1, it is seen that family-work conflict (FWC) will have a negative effect on job satisfaction (JS) while family-work facilitation (FWF) will have a positive effect on job satisfaction. Similarly, work-family facilitation (WFF) will have a positive effect on family satisfaction (FS) while work-family conflict (WFC) will have a negative effect on family satisfaction. Both job and family satisfaction will influence feelings of work-family balance positively which in turn will positively influence life satisfaction (LS). All the above statements have been hypothesized and can be stated conclusively if we have empirical data to establish the stated hypotheses. The use of appropriate statistical methods will facilitate the data analysis to arrive at well-grounded inferences and conclusions. Univariate statistical tests involve one dependent variable. Examples include, but are not limited to, t-tests of means, analysis of variance (ANOVA), analysis of covariance and simple linear regression (with one dependent and one independent variable). Having said so much about the importance of data analysis, let us have a quick look at a few multivariate techniques that we are likely to study in detail during the course of this study. The next section leads us to the classification of MVA.
- 3. P a g e 3 | 8 2. Classification of MVA MVA can be classified as Dependence techniques and Interdependence techniques. 2.1 Dependence techniques (used when there are one or more dependent variables and independent variables. Eg. Multiple regression analysis) i. Multiple regression and multiple correlation ii. Multiple Discriminant Analysis (MDA) and Logistic Regression iii. Canonical Correlation Analysis iv. Multivariate Analysis of Variance and Covariance v. Conjoint Analysis vi. Structural Equation Modelling (SEM) and Confirmatory Factor Analysis (CFA) 2.1.1 Multiple Regression Let us presume that some previous research has established that cars with higher engine capacity and higher unladen weight offer lesser fuel efficiency (possibly validated using a correlation analysis). If a researcher wants to predict the fuel efficiency based on engine capacity and unladen weight, then fuel efficiency is treated as the dependent variable while engine capacity and unladen weight are treated as the independent variables. The researcher collects data on fuel efficiency, engine capacity and an unladen weight of about 100 cars or more (that run on the same type of fuel) and would possibly use the multiple regression (MR) method to predict fuel efficiency. In order to use the MR method the dependent and the independent variables (two or more) must be metric data. 2.1.2 Multiple Discriminant Analysis (MDA) and Logit Analysis If the dependent variable is dichotomous (Yes/No, Men / Women) type, then MDA is an appropriate technique. The independent variables need to be metric data. MDA helps to understand group differences and to predict the possibility that an observation or object would belong to a specific group. An example that we had discussed in MR in the previous section, suppose we had data on the engine capacity and unladen weight of about 100 plus cars (that run on the same type of fuel) and if we want to classify them as Big and Small cars, then MDA would be a relevant technique. Logit Analysis also is known as Logistics regression is a combination of MR and MDA. Although the regression principle is similar to that of MR, the DV in Logit regression need not be metric as in the case of MR but can be a dichotomous variable as in MDA. Another distinguishing fact of Logit regression is that it can accommodate both metric and non-metric IVs and overlook the multivariate normality assumption.
- 4. P a g e 4 | 8 2.1.3 Canonical Correlation Analysis If there are multiple metric dependent and metric independent variables to be correlated and regressed, then the right tool is Canonical Correlation Analysis. We actually try to determine the associations between two sets of variables. For example, we might study the relationship between a number of indices of fuel efficiency (the DVs such as Indicated Horse Power (IHP) and Brake Horse Power (BHP)) and the IVs (such as engine capacity, unladen weight of the car, and age of the car). 2.1.4 Multivariate Analysis of Variance and Covariance In-order to simultaneously explore the relationship between multiple categorical independent variables, which are also called treatments, and two or more metric dependent variables, an ideal technique would be the Multivariate Analysis of Variance and Covariance (MANOVA). If the analysis requires the elimination of the effect of the uncontrolled metric independent variables, which are known as covariates, on the dependent variables, then the multivariate analysis of covariance (MANCOVA) is used. Both MANOVA and MANCOVA may be done as one way or factorial. In our car example with fuel efficiency as the DV, age of the car can be treated as a covariate. 2.1.5 Conjoint Analysis Conjoint Analysis is a contemporary dependence technique that would help a decision-maker (product design head) evaluate the importance of attributes (typically product attributes) along with its levels. Let us say we have three attributes of a car, namely, airbags (2, 4, or 6 airbags), speakers for infotainment (2, 4 or 6 speakers) and steering wheel height adjustment (low, medium and high). If we want to know popular combinations preferred by car enthusiasts, we may have to ask them to rate all of the 27 combinations. For example a car enthusiast may prefer 6 airbags, 4 speakers and medium height for his steering wheel. Likewise, there are 27 possible combinations. However, using conjoint analysis it is possible to capture the ratings of the prospective car buyer with just 9 or more combinations. The conjoint analysis helps a great deal is product design simulation studies. 2.1.6 Structural Equation Modelling (SEM) and Confirmatory Factor Analysis (CFA) While multiple regression examines a single relationship between a DV and multiple IVs in an SEM, it is possible to examine multiple relationships simultaneously. Generally, a CFA is done prior to the SEM. The SEM consists of the structural and the measurement model. The structural model may have one or more DVs and one or more IVs with all relationships defined. Each of the DVs and IVS may be either uni- or multi-dimensional and each of the dimensions may be measured using scale items for indicators. The CFA will show the contribution of each scale item to its dimension and the extent to which it measures the same. By this the measurement model is evaluated. After the validity and reliability of the
- 5. P a g e 5 | 8 measurement model are established, the structural model is evaluated to establish and prove or disprove hypotheses. Hence, SEM supports simultaneous assessment of relationships and accommodates multi-item scales. 2. 2 Interdependence techniques (absence of dependent or independent variables but involves techniques to simultaneously analyze all variables together in the set. Eg. Factor Analysis). a) Factor Analysis (both Principal Component Analysis and Common Factor Analysis) b) Cluster Analysis c) Perceptual Mapping (also called as Multidimensional Scaling) d) Correspondence Analysis 2.2.1 Factor Analysis The objective of factor analysis is to reduce the number of measured variables into meaningful factors (or variates) with minimal loss of information. This can either be done by the PCA method or by common factor analysis. Suppose a prospective car buyer is considering the color of the car, the aerodynamic design, body-colored bumpers, height- adjustable steering column, driver seat height adjustment, touch screen for infotainment, ABS and Airbags. If the opinion of the car buyer is captured using a 7 point Likert scale, either PCA or common factor analysis may group these eight variables in three groups, namely, external features (colour of the car, the aerodynamic design, body-colored bumpers), internal features (height-adjustable steering column, driver seat height adjustment, touch screen for infotainment) and safety features (ABS and Airbags). So factor analysis helps us to reduce eight variables into three meaningful factors (variates). 2.2.2 Cluster Analysis In the car example that we have been discussing so far, suppose we have the data on engine capacities of about 130 cars with the engine capacities ranging from a minimum of 799cc to 2399cc and we want these 130 cars to be placed in three groups, namely, small, medium and large cars, cluster analysis would be a recommended technique. The Cluster analysis algorithm places the objects in homogeneous groups depending on the characteristics specified by the researcher. In our example, the cars would be placed in groups based on engine capacity. Clustering can be done based on multiple characteristics too. Either hierarchical or non-hierarchical clustering procedures may be adopted. Basically hierarchical methods could be either agglomerative or divisive. The algorithms followed in the hierarchical methods are single, complete and average linkage methods. The other methods are the Centroid and Ward methods. Alternatively the non-hierarchical clustering popularly follows the k-means algorithm and places objects in cluster groups once the number of clusters is specified. The decision on whether to adopt the hierarchical or non-hierarchical procedure depends on the choice of the researcher and the problem defined.
- 6. P a g e 6 | 8 2.2.3 Perceptual Mapping If we consider two dimensions of the car, namely, fuel efficiency and driving comfort and we want to know how the brands of cars currently available in the market are positioned in the minds of the car enthusiasts and perceived by the car enthusiasts, the right technique is Perceptual Mapping (PM) also known as Multi-dimensional Scaling (MDS). MDS typically helps a researcher to determine the perceived relative image of the cars (in this case) considering the two dimensions. In MDS, unlike in factor or cluster analysis, a solution can be obtained for each respondent and there is no variate. The researcher makes choices between similarity and preference data, disaggregate and aggregate analysis and on whether to use the Compositional or decompositional methods. Although earlier MDS programs were predominantly non-metric in output, the contemporary programs provide metric output. 2.2.4 Correspondence Analysis If we have non-metric data such as colors of the cars, classification of car size such as small, medium and large and we want to position the cars in a perceptual map, then the technique to be adopted is the Correspondence Analysis (CA). It starts with a cross-tabulation of the two attributes, namely, colors and car size; after that it carries out a non-metric to metric conversion, and then leads to dimension reduction and finally the perceptual map is prepared. CA is the best option for a multivariate representation of interdependence for non- metric data. 3. Nature of Data The following table gives a summary of the nature of data: Name of the Multivariate Technique Nature of the Data DV IV Canonical Correlation Metric, Non-metric Metric, Non-metric MANOVA Metric Non-metric ANOVA Metric Non-metric MDA Non-metric Metric Multiple Regression Metric Metric, Non-metric Conjoint Analysis Non-metric, Metric Non-metric SEM Metric Metric, Non-metric 4 Some Generic Tips to Perform Multivariate Analysis
- 7. P a g e 7 | 8 While performing MVA on the research problem, it would help if the researcher observes the following tips: 1. Ensure that both statistical and practical significance exists in the research being done. 2. The sample size should be adequate but neither under sized nor over sized. 3. Clearly, understand the nature of the data. 4. Use a minimum number of variables in the model to obtain the desired results. 5. Identify and eliminate errors. 6. Ensure a fool-proof validation of the results. I hope the above content gives you a fair idea of the existing multivariate techniques that we would be covering in our course and a snapshot of their applications. For further learning, may I also suggest the open courseware by Cynthia et al., (2011), titled “Statistical Thinking and Data Analysis”. Although at the beginning of this discussion, I had suggested the reading of the paper by Pattusamy and Jacob (2015), throughout the discussion I used examples relating to cars. If you have understood the application of the discussed MVA tests with the variables in the car example, you should be able to answer a few fundamental questions relating to data analysis with respect to the variables in the paper. Here are your challenges. Self-Assessment: You could suggest appropriate statistical tests to answer the following research questions. It does help if you could also justify your choice of the technique. 1. Are men more satisfied with their jobs than women? 2. Does life satisfaction vary with age? 3. Will feelings of work-life balance influence the relationship between job satisfaction and life satisfaction? 4. Would there be a difference in the strength of the relationship between family satisfaction and life satisfaction between men and women? 5. Would it be possible to categorize men who are highly and moderately satisfied in their lives?
- 8. P a g e 8 | 8 References 1. Barbara G.T and Linda S.F, Using Multivariate Statistics, 6th Edition, Pearson Education Inc, pp. 612-680. 2. Cynthia Rudin, Allison Chang, and Dimitrios Bisias. 15.075J Statistical Thinking and Data Analysis. Fall 2011. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA. 3. Hair J.F, Black W.C, Babin B.J and Anderson R.E, Multivariate Data Analysis, 7th Edition, Pearson Education (South Asia), pp. 89-149. 4. Murugan Pattusamy and Jayanth Jacob, A test of Greenhaus and Allen (2011) model on Work Family Balance, Current Psychology, Springer, 2015. 5. Zumbo B.D. (2014) Univariate Tests. In: Michalos A.C. (eds) Encyclopedia of Quality of Life and Well-Being Research. Springer, Dordrecht ***************************************************************************