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Factor Analysis
Factor analysis is a multivariate statistical technique in which there is no distinction between dependent and independent variables. In factor analysis, all variables under investigation are analysed together to extract the underlined factors. Factor analysis is a data reduction method. It is a very useful method to reduce a large number of variables resulting in data complexity to a few manageable factors. These factors explain most part of the variations of the original set of data.
For example, A market researcher might have collect data on say, more than 50 attributes (or items) of a product which may become very difficult to analyse. Factor analysis could help to reduce the data on 50 odd attributes to a few manageable factors. It helps in identifying the underlying structure of the data.
Factor: A factor is a linear combination of variables. It is a construct that is not directly observable but that needs to be inferred from the input variables. The factors are statistically independent. We will show you their application in a regression analysis as a factor scores, when used as independent variables in regression analysis, help to solve the problem of multicollinearity. The factor scores could also used in other multivariate techniques.
Uses of Factor Analysis
The technique of factor analysis has multiple uses as discussed in the following situations:
- Scale Construction: Factor analysis could be used to develop concise multiple item scales for measuring various constructs. Factor analysis can reduce the set of statements to a concise instrument and at the same time, ensure that the retained statements adequately represent the critical aspects of the constructs being measured.
- Establish antecedents: The factor analysis method reduces multiple input variables into grouped factors. Thus, the independent variables can be grouped into broad factors. For examples, all the variables that measure the safety clauses in a mutual fund could be reduced to a factor called safety clause.
- Psychographic profiling: Different independent variables are grouped to measure independent factors. These are then used for identifying personality types. One of the most well-known inventories based on this technique is called the 16 PF inventory.
- Segmentation Analysis: Factor analysis could also be used for segmentation. For example, there could be different sets of two-wheelers-customers owning two wheelers because of different importance they give to factors like prestige, economy consideration and functional features.
- Marketing Studies: The technique has extensive use in the field of marketing and can be successfully used for new product development; product acceptance research, developing of advertising copy, pricing studies and for branding studies. For example, to identify the attributes of brands that influence consumers’ choice, to identify the characteristics of price sensitive customers.
Types of Factor Analysis
There are two main types of factor analysis:
- Exploratory factor analysis
- Confirmatory factor analysis
1. Exploratory Factor Analysis: Exploratory factor analysis (EFA) attempts to
discover the nature of the constructs inuencing a set of responses. Exploratory factor analysis is used to measure the underlying factors that affect the variables in a data structure without setting any predefined structure to the outcome. EFA is a technique within factor analysis whose overarching goal is to identify the underlying relationships between measured variables. It is commonly used by researchers when developing a scale (a scale is a collection of questions used to measure a particular research topic) and serves to identify a set of latent constructs underlying a battery of measured variables. It should be used when the researcher has no a priori hypothesis about factors or patterns of measured variables.
Objectives • The primary objectives of an EFA are to determine
- The number of common factors inuencing a set of measures.
- The strength of the relationship between each factor and each observed measure. - Some common uses of EFA are to
- Identify the nature of the constructs underlying responses in a specic content area.
- (^) Determine what sets of items “hang together” in a questionnaire.
of these denitions, however, are easier to interpret theoretically than others. By rotating your factors, you attempt to nd a factor solution that is equal to that obtained in the initial extraction, but which has the simplest interpretation. There are many different types of rotation, but they all try make your factors each highly responsive to a small subset of your items (as opposed to being moderately responsive to a broad set). There are two major categories of rotations, orthogonal rotations, which produce uncorrelated factors, and oblique rotations, which produce correlated factors. The best orthogonal rotation is widely believed to be Varimax. Oblique rotations are less distinguishable, with the three most commonly used being Direct Quartimin, Promax, and Harris-Kaiser Orthoblique.
vi. Interpret your factor structure: Each of your measures will be linearly related to each of your factors. The strength of this relationship is contained in the respective factor loading, produced by your rotation. This loading can be interpreted as a standardized regression coefficient, regressing the factor on the measures. You dene a factor by considering the possible theoretical constructs that could be responsible for the observed pattern of positive and negative loadings. To ease interpretation, you have the option of multiplying all of the loadings for a given factor by -1. This essentially reverses the scale of the factor, allowing you, for example, to turn an “unfriendliness” factor into a “friendliness” factor. vii. Construct factor scores for further analysis: If you wish to perform additional analyses using the factors as variables you will need to construct factor scores. The score for a given factor is a linear combination of all the measures, weighted by the corresponding factor loading. Sometimes factor scores are idealized, assigning a value of 1 to strongly positive loadings, a value of -1 to strongly negative loadings, and a value of 0 to intermediate loadings. These factor scores can then be used in analyses just like any other variable, although you should remember that they will be strongly collinear with the measures used to generate them.
2. Confirmatory Factor Analysis
CFA allows the researcher to test the hypothesis that a relationship between the observed variables and their underlying latent construct(s) exists. The researcher uses knowledge of the theory, empirical research, or both, postulates the relationship pattern a priori and then
tests the hypothesis statistically. The primary objective of a CFA is to determine the ability of a predened factor model to t an observed set of data.
Some common uses of CFA are to
- Establish the validity of a single factor model.
- (^) Compare the ability of two different models to account for the same set of data.
- Test the signcance of a specic factor loading.
- Test the relationship between two or more factor loadings.
- Test whether a set of factors are correlated or uncorrelated.
- Assess the convergent and discriminant validity of a set of measures.
Performing CFA
There are six basic steps to performing an CFA:
- Dene the factor model : The rst thing you need to do is to precisely dene the model you wish to test. This involves selecting the number of factors, and dening the nature of the loadings between the factors and the measures. These loadings can be xed at zero, xed at another constant value, allowed to vary freely, or be allowed to vary under specied constraints (such as being equal to another loading in the model).
- Collect measurements : You need to measure your variables on the same (or matched) experimental units.
- Obtain the correlation matrix: You need to obtain the correlations (or covariances) between each of your variables.
- Fit the model to the data: You will need to choose a method to obtain the estimates of factor loadings that were free to vary. The most common model-tting procedure is Maximum likelihood estimation, which should probably be used unless your measures seriously lack multivariate normality. In this case you might wish to try using Asymptotically distribution free estimation.
- Evaluate model adequacy: When the factor model is t to the data, the factor loadings are chosen to minimize the discrepancy between the correlation matrix
3. Size of Sample: The size of the sample respondents should be at least four to five
times more than the number of variables (number of statements).
4. Establishing the strength of the factor analysis solution: In order to
establish the strength of the factor analysis solution it is essential to establish the reliability and validity of the obtained reduction. It is done with the help of KMO and Bartlett’s test of Sphericity.
- The basic principle behind the application of factor analysis is that the initial set of variables should be highly correlated. If the correlation coefficient between all the variables are small, factor analysis may not be an appropriate technique. A correlation matrix of the variables could be computed and tested for its statistical significance. The hypothesis to be tested may be written as:
H 0 : Correlation matrix is insignificant, i.e. correlation matrix is an identity matrix where diagonal elements are one and off diagonal elements are zero.
H 1 : Correlation matrix is significant.
The test is carried out by using a Barttlet test of Sphericity, which takes the determinants of the correlation matrix into consideration. The significance of the correlation matrix ensure that a factor analysis exercise could be carried out.
- Another condition which needs to be fulfilled before a factor analysis could be carried out is the value of Kaiser-Meyer-Olkin (KMO) statistics which takes a value of between 0 and 1. For the application of factor analysis, the value of KMO statistics should be greater than 0.5. The KMO statistics compares the magnitude of observed correlation coefficient with the magnitudes of partial correlation coefficients. A small value of KMO shows that correlation between variables cannot be explained by other variables.
