QCI WORKSHOP- Factor analysis-
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Factor analysis was conducted on 8 variables related to reasons for choosing a university. The analysis revealed 2 key factors: 1) Academic Quality captured variables like quality of education, experts/labs, and international recognition. 2) Campus Life captured variables like campus/security, years operating, graduates employed, and accommodations. The analysis showed that the reasons for university choice could be understood in terms of these two underlying dimensions or factors of Academic Quality and Campus Life experience. This provides useful insight into how to segment and target prospective students.
QCI WORKSHOP- Factor analysis-
- 1. QCI- 5 Days Online Workshop on Enhancing Research Capability Dr. Jai Singh FACTOR ANALYSIS
- 2. Impact of Covid-19 in Health Sector • Insurance Industry • Pharmaceutical Companies • Physicians • Patients • Government • Nurse/Carers, staff, unions, • Voluntary organisations • Social services, • Local health authority • Primary care groups • Local health groups
- 3. Factor Analysis • Factor Analysis (FA) is an exploratory technique applied to a set of large number of observed variables to find basic small number of factors (subsets of variables) from which the observed variables were generated. Ex.- An individual’s response to the questions on a college entrance test is influenced by underlying variables such as intelligence, years in school, age, emotional state on the day of the test, amount of practice taking tests, and so on. Ex.- The analyst hopes to reduce the interpretation of a 200- question test to the study of 4 or 5 factors.
- 4. Continue- • Factor analysis is a procedure used to reduce a large amount of questions (condense the data) into few variables (Factors) according to their relevance. • Factors are assumed to represent dimensions within data. • The answers to the questions are the observed variables. The underlying, influential variables are the factors. • The most common technique is known as Principal Component Analysis (PCA).
- 6. Continue- • Factor analysis is a useful tool for investigating variable relationships for complex concepts such as socioeconomic status, dietary patterns, or psychological scales. • It allows researchers to investigate concepts that are not easily measured directly by collapsing a large number of variables into a few interpretable underlying factors. • Factor analysis is commonly used in: -Scale development -The evaluation of the psychometric quality of a measure, and -The assessment of the dimensionality of a set of variables.
- 7. Factor Analysis –Structural Aspect • Factor analysis provides a tool for analyzing the structure of interrelationships (Correlations) among variables by defining a set of variables which are highly correlated known as Factors.
- 8. What is a factor? • Multiple observed variables have similar patterns of responses because they are all associated with a latent (i.e. not directly measured) variable. - Ex.- People may respond similarly to questions about income, education, and occupation, which are all associated with the latent variable socioeconomic status. • In every factor analysis, there are the same number of factors as there are variables. Each factor captures a certain amount of the overall variance in the observed variables. • Factors are always listed in order of how much variation they explain.
- 9. Exploratory Factor Analysis (EFA) Exploratory - When the dimensions/factors are theoretically unknown - Exploratory factor analysis is a statistical approach that can be used to analyze interrelationships (correlation) among a large number of variables in a data set and to explain these variables in terms of a smaller number of common underlying dimensions. - This involves finding a way of condensing the information contained in some of the original variables into a smaller set of implicit variables (called factors) with a minimum loss of information. - This type of analysis provides a factor structure (a grouping of variables based on strong correlations).
- 10. Confirmatory Factor Analysis • Confirmatory – When researcher has preconceived thoughts about the actual structure of data based on theoretical support or prior research. • Researcher may wish to test hypothesis involving issues as which variables should be grouped together on a factor. • Example - Retail firm identified 80 characteristics of retail stores and their services that consumers mentioned as affecting their patronage choice among stores. Retailer want to find the broader dimensions on which he can conduct a survey
- 11. Exploratory Factor Analysis • Assumptions: Metric Data (Interval) Multicollinearity must be present i.e. Correlation among variables Adequate sample size • Purpose Obtaining independent factors Data Reduction
- 12. Extract Factor (Example-1) • Several questions closely related to aspects of customer satisfaction -How satisfied are you with our product? -Would you recommend our product to a friend or family member? -How likely are to you purchase our product in the future
- 13. Continue- • We want one variable to represent a customer satisfaction score. - One option would be to average the three question responses. - Another option would be to create a factor dependent variable. • This can be done by PCA and keeping the first Principal Component (also known as a factor). - The advantage of PCA over an average is that it automatically weights each of the variables in the calculation.
- 14. Example-2 • Purchase barriers of potential customers • Possible barriers to purchase - Factor analysis can uncover the trends of how these questions will move together. - Loadings for 3 factors for each of the variables.
- 16. Principal components weights for the variables • The first component heavily weights variables related to cost, • The second weights variables related to IT, and • The third weights variables related to organizational factors. - We can give our new super variables appropriate names.
- 17. If we were to cluster the customers based on these three components, we can see some trends. Customers tend to be high in Cost barriers or Org barriers, but not both.
- 19. Scope of Factor analysis Psychographics (Agree/Disagree): • I value family • I believe brand represents value Behavioral (Agree/Disagree): • I purchase the cheapest option • I am a bargain shopper Attitudinal (Agree/Disagree): • The economy is not improving • I am pleased with the product Activity-Based (Agree/Disagree): • I love sports • I sometimes shop online during work hours Behavioral and psychographic questions are especially suited for factor analysis.
- 20. Eigenvalue • The eigenvalue is a measure of how much of the variance of the observed variables a factor explains. Eigenvalues represent the total amount of variance that can be explained by a given principal component. • Any factor with an eigenvalue ≥1 explains more variance than a single observed variable. • Factor for socioeconomic status had an eigenvalue of 2.3 it would explain as much variance as 2.3 of the three variables. • If eigenvalues are greater than zero, then it’s a good sign. • Since variance cannot be negative, negative eigenvalues imply the model is ill-conditioned. • Eigenvalues close to zero imply there is item multicollinearity, since all the variance can be taken up by the first component. • The factors that explain the least amount of variance are generally discarded.
- 22. What are factor loadings? Variables Factor 1 Factor 2 Income 0.65 0.11 Education 0.59 0.25 Occupation 0.48 0.19 House value 0.38 0.60 Number of public parks in neighborhood 0.13 0.57 Number of violent crimes per year in neighborhood 0.23 0.55 The relationship of each variable to the underlying factor is expressed by the so-called factor loading. Example- indicators of wealth with six variables and two resulting factors.
- 23. Interpretation • The variable income has the strongest association to the underlying latent variable. Factor 1, with a factor loading of 0.65. • Two other variables, education and occupation, are also associated with Factor 1. Based on the variables loading highly onto Factor 1, we could call it “Individual socioeconomic status.” • House value, number of public parks, and number of violent crimes per year, however, have high factor loadings on the other factor, Factor 2. They seem to indicate the overall wealth within the neighborhood, so we may want to call Factor 2 “Neighborhood socioeconomic status.” • Since factor loadings can be interpreted like standardized regression coefficients, one could also say that the variable income has a correlation of 0.65 with Factor 1.
- 24. Continue- • Variable house value also is marginally important in Factor 1 (loading = 0.38). • This makes sense, since the value of a person’s house should be associated with his or her income. • .50 loading denotes that 25% of variance is accounted for by the factor. • Loading must exceed .70 for factor to account for .50 variance. • Loadings .5 are considered practically significant and .7 are indicative of well defined structure.
- 25. Communalities • Communalities indicate the amount of variance in each variable that is accounted for by the components. • Initial communalities are estimates of the variance in each variable accounted for by all components or factors. • For principal components extraction, this is always equal to 1.0 for correlation analyses. • Extraction communalities are estimates of the variance in each variable accounted for by the components. • The high communalities indicates that the extracted components represent the variables well. • If any communalities are very low in a principal components extraction, you may need to extract another component.
- 26. Variance in factor analysis • Two types of variance- common and unique • Common variance is the amount of variance that is shared among a set of items. Items that are highly correlated will share a lot of variance. – Communality (also called h2) is a definition of common variance that ranges between 00 and 11. Values closer to 1 suggest that extracted factors explain more of the variance of an individual item. • Unique variance is any portion of variance that’s not common. There are two types: – Specific variance: is variance that is specific to a particular item – Error variance: comes from errors of measurement and basically anything unexplained by common or specific variance.
- 27. Kaiser-Meyer-Olkin (KMO) Test - Measure of how suited data is for Factor Analysis. - Sampling adequacy for each variable in the model and for the complete model. - Measure of the proportion of variance among variables that might be common variance. - The lower the proportion, the more suited data to Factor Analysis.
- 28. KMO Values • KMO values between 0 and 1. • A rule of thumb for interpreting the statistic - KMO values between 0.8 and 1 indicate the sampling is adequate. - KMO values less than 0.6 indicate the sampling is not adequate and that remedial action should be taken. - Some times we accept KMO values between 0.5 and 0.6.
- 29. Bartlett test of sphericity Checks statistical significance that correlation matrix has significant correlation among at least some variables • Bartlett test should be significant i.e. less than 0.05 this means that the variables are correlated highly enough to provide a reasonable basis for factor analysis. • Correlation matrix is significantly different from an identity matrix in which correlations between variables are all zero. • Test compares an observed correlation matrix to the identity matrix. Null hypothesis of the test –variables are orthogonal, i.e. not correlated. Alternative hypothesis - variables are not orthogonal, i.e. they are correlated enough - correlation matrix diverges significantly from identity matrix.
- 31. How many number of factors should be extracted? There are some criteria, but no 100% foolproof statistical test exists. • Drawing screeplot: Connect eigenvalues (as representing variances explained by each factor) for many possible factors from maximum to minimum. The adequate number of factors is before the sudden downward inflextion of the plot. • Parallel analysis: Compare actual screeplot with the possible screeplot based on randomly resampled data. The adequate number of factors is at the crossing point of the two plots. • Eigenvalues > 1: Eigenvalues sum to the number of items, so an eigenvalue more than 1 is more informative than a single average item.
- 32. Factor rotation • Un rotated factor don’t provide the required information. • Axis of the factors can be rotated within the multidimensional variable space while determining the best fit between the variables and the latent factors. • Factor rotation improves interpretation of data by reducing ambiguities. • Rotation - reference axes of factors are turned about the origin - some best position has been achieved. • Un rotated factor solution extract factors in order of their variances extracted. • Effect of rotation - redistribute the variance from earlier factors to later ones.
- 33. Orthogonal and oblique factor rotation • Orthogonal factor rotation - Axes are maintained at 90 degree. - More suitable when research goal is data reduction - Rotations that assume the factors are not correlated are called orthogonal rotations. • Oblique factor rotation - Axes are rotated but they don’t retain the 90 degree angle between reference axes. - Oblique is more flexible. - Best suited to the goal of obtaining several theoretically meaningful factors. - Rotations that allow for correlation are called oblique rotations.
- 34. Correlation between factors • High Individual socioeconomic status (Factor 1) lives also in an area that has a high Neighborhood socioeconomic status (Factor 2). • That means the factors should be correlated. • the two axes of the two factors are probably closer together than an orthogonal rotation axis. • the angle between the two factors is now smaller than 90 degrees. • meaning the factors are now correlated. • In this example, an oblique rotation accommodates the data better than an orthogonal rotation.
- 35. Sample Example A researcher is interested to investigate the reasons of choosing a university for education. Several variables were identified which influence the individuals (guardians/students) to choose a university. • Variables- 1. Cost of Education 2. Quality of Education 3. Availability of experts and modern laboratories 4. Having own campus and security 5. Numbers of years operating 6. Number of graduates in job and abroad 7. International recognition and 8. Accommodations and food Seven Point Scale; 1=Not Important to 7=Very Important