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# ROTATION IN FACTOR ANALYSIS Research Methodology

❶The "reduced correlation matrix" is defined as.

## Illustration and Benefits of Factor Analysis

This is the most common factor analysis used by researchers and it is not based on any prior theory. Confirmatory factor analysis CFA: Used to determine the factor and factor loading of measured variables, and to confirm what is expected on the basic or pre-established theory.

CFA assumes that each factor is associated with a specified subset of measured variables. It commonly uses two approaches:. Principal components analysis and exploratory and confirmatory factor analysis. Evaluating the use of exploratory factor analysis in psychological research. Psychological Methods, 4 3 , Multivariate data analysis with readings 4th ed. Upper Saddle River, NJ: A step-by-step approach to using the SAS system for factor analysis and structural equation modeling. The multivariate social scientist: Introductory statistics using generalized linear models.

Introduction to factor analysis: What it is and how to do it. Statistical methods and practical issues. Factor analysis as a statistical method. The Statistician, 12 3 , Canonical analysis and factor comparison. Making sense of factor analysis: The use of factor analysis for instrument development in health care research. Factor analysis of Gulf War illness: What does it add to our understanding of possible health effects of deployment, American Journal of Epidemiology, , Construct explication through factor or component analysis: A review and evaluation of alternative procedures for determining the number of factors or components.

Honoring Douglas Jackson at seventy. Common factor analysis versus principal component analysis: Differential bias in representing model parameters, Multivariate Behavioral Research, 28 , This was then used to estimate the factors and the loadings. With the advent of high-speed computers, the minimization problem can be solved iteratively with adequate speed, and the communalities are calculated in the process, rather than being needed beforehand. The MinRes algorithm is particularly suited to this problem, but is hardly the only iterative means of finding a solution.

If the solution factors are allowed to be correlated as in oblimin rotation, for example , then the corresponding mathematical model uses skew coordinates rather than orthogonal coordinates. The parameters and variables of factor analysis can be given a geometrical interpretation.

This follows from the model equation. In the above example, the hyperplane is just a 2-dimensional plane defined by the two factor vectors. The projection of the data vectors onto the hyperplane is given by. The goal of factor analysis is to find a hyperplane which is a "best fit" to the data in some sense, so it doesn't matter how the factor vectors which define this hyperplane are chosen, as long as they are independent and lie in the hyperplane.

After a suitable set of factors are found, they may also be arbitrarily rotated within the hyperplane, so that any rotation of the factor vectors will define the same hyperplane, and also be a solution. As a result, in the above example, in which the fitting hyperplane is two dimensional, if we do not know beforehand that the two types of intelligence are uncorrelated, then we cannot interpret the two factors as the two different types of intelligence. Even if they are uncorrelated, we cannot tell which factor corresponds to verbal intelligence and which corresponds to mathematical intelligence, or whether the factors are linear combinations of both, without an outside argument.

The diagonal elements will clearly be 1's and the off diagonal elements will have absolute values less than or equal to unity. The "reduced correlation matrix" is defined as. The goal of factor analysis is to choose the fitting hyperplane such that the reduced correlation matrix reproduces the correlation matrix as nearly as possible, except for the diagonal elements of the correlation matrix which are known to have unit value.

In other words, the goal is to reproduce as accurately as possible the cross-correlations in the data. Specifically, for the fitting hyperplane, the mean square error in the off-diagonal components. It can be seen that. The term on the right is just the covariance of the errors. In the model, the error covariance is stated to be a diagonal matrix and so the above minimization problem will in fact yield a "best fit" to the model: It will yield a sample estimate of the error covariance which has its off-diagonal components minimized in the mean square sense.

The square of these lengths are just the diagonal elements of the reduced correlation matrix. These diagonal elements of the reduced correlation matrix are known as "communalities":. Large values of the communalities will indicate that the fitting hyperplane is rather accurately reproducing the correlation matrix.

It should be noted that the mean values of the factors must also be constrained to be zero, from which it follows that the mean values of the errors will also be zero. Exploratory factor analysis EFA is used to identify complex interrelationships among items and group items that are part of unified concepts.

Confirmatory factor analysis CFA is a more complex approach that tests the hypothesis that the items are associated with specific factors. Canonical factor analysis, also called Rao's canonical factoring, is a different method of computing the same model as PCA, which uses the principal axis method. Canonical factor analysis seeks factors which have the highest canonical correlation with the observed variables.

Canonical factor analysis is unaffected by arbitrary rescaling of the data. Common factor analysis, also called principal factor analysis PFA or principal axis factoring PAF , seeks the least number of factors which can account for the common variance correlation of a set of variables.

Image factoring is based on the correlation matrix of predicted variables rather than actual variables, where each variable is predicted from the others using multiple regression. Alpha factoring is based on maximizing the reliability of factors, assuming variables are randomly sampled from a universe of variables. All other methods assume cases to be sampled and variables fixed. Factor regression model is a combinatorial model of factor model and regression model; or alternatively, it can be viewed as the hybrid factor model, [11] whose factors are partially known.

Commonality is the square of standardized outer loading of an item. Analogous to Pearson's r , the squared factor loading is the percent of variance in that indicator variable explained by the factor. To get the percent of variance in all the variables accounted for by each factor, add the sum of the squared factor loadings for that factor column and divide by the number of variables. Note the number of variables equals the sum of their variances as the variance of a standardized variable is 1.

This is the same as dividing the factor's eigenvalue by the number of variables. By one rule of thumb in confirmatory factor analysis, loadings should be. In any event, factor loadings must be interpreted in the light of theory, not by arbitrary cutoff levels.

In oblique rotation, one gets both a pattern matrix and a structure matrix. The structure matrix is simply the factor loading matrix as in orthogonal rotation, representing the variance in a measured variable explained by a factor on both a unique and common contributions basis. The pattern matrix, in contrast, contains coefficients which just represent unique contributions.

The more factors, the lower the pattern coefficients as a rule since there will be more common contributions to variance explained. For oblique rotation, the researcher looks at both the structure and pattern coefficients when attributing a label to a factor.

Principles of oblique rotation can be derived from both cross entropy and its dual entropy. The sum of the squared factor loadings for all factors for a given variable row is the variance in that variable accounted for by all the factors, and this is called the communality. The communality measures the percent of variance in a given variable explained by all the factors jointly and may be interpreted as the reliability of the indicator.

If the communality exceeds 1. Uniqueness of a variable: That is, uniqueness is the variability of a variable minus its communality. The eigenvalue for a given factor measures the variance in all the variables which is accounted for by that factor. The ratio of eigenvalues is the ratio of explanatory importance of the factors with respect to the variables.

If a factor has a low eigenvalue, then it is contributing little to the explanation of variances in the variables and may be ignored as redundant with more important factors. Eigenvalues measure the amount of variation in the total sample accounted for by each factor. Extraction sums of squared loadings: Initial eigenvalues and eigenvalues after extraction listed by SPSS as "Extraction Sums of Squared Loadings" are the same for PCA extraction, but for other extraction methods, eigenvalues after extraction will be lower than their initial counterparts.

Factor scores also called component scores in PCA: To compute the factor score for a given case for a given factor, one takes the case's standardized score on each variable, multiplies by the corresponding loadings of the variable for the given factor, and sums these products.

Computing factor scores allows one to look for factor outliers. Also, factor scores may be used as variables in subsequent modeling. Researchers wish to avoid such subjective or arbitrary criteria for factor retention as "it made sense to me".

A number of objective methods have been developed to solve this problem, allowing users to determine an appropriate range of solutions to investigate. Methods may not agree. For instance, the parallel analysis may suggest 5 factors while Velicer's MAP suggests 6, so the researcher may request both 5 and 6-factor solutions and discuss each in terms of their relation to external data and theory.

Horn 's parallel analysis PA: A Monte-Carlo based simulation method that compares the observed eigenvalues with those obtained from uncorrelated normal variables. A factor or component is retained if the associated eigenvalue is bigger than the 95th percentile of the distribution of eigenvalues derived from the random data.

PA is one of the most recommended rules for determining the number of components to retain, [ citation needed ] but many programs fail to include this option a notable exception being R. On Step 1, the first principal component and its associated items are partialed out. Thereafter, the average squared off-diagonal correlation for the subsequent correlation matrix is then computed for Step 1.

On Step 2, the first two principal components are partialed out and the resultant average squared off-diagonal correlation is again computed. The computations are carried out for k minus one step k representing the total number of variables in the matrix. Thereafter, all of the average squared correlations for each step are lined up and the step number in the analyses that resulted in the lowest average squared partial correlation determines the number of components or factors to retain.

Although methodologically akin to principal components analysis, the MAP technique has been shown to perform quite well in determining the number of factors to retain in multiple simulation studies. See Courtney [19] for guidance. The Kaiser rule is to drop all components with eigenvalues under 1. The Kaiser criterion is the default in SPSS and most statistical software but is not recommended when used as the sole cut-off criterion for estimating the number of factors as it tends to over-extract factors.

As one moves to the right, toward later components, the eigenvalues drop. When the drop ceases and the curve makes an elbow toward less steep decline, Cattell's scree test says to drop all further components after the one starting the elbow.

This rule is sometimes criticised for being amenable to researcher-controlled " fudging ". That is, as picking the "elbow" can be subjective because the curve has multiple elbows or is a smooth curve, the researcher may be tempted to set the cut-off at the number of factors desired by their research agenda.

The unrotated output maximizes variance accounted for by the first and subsequent factors, and forces the factors to be orthogonal. This data-compression comes at the cost of having most items load on the early factors, and usually, of having many items load substantially on more than one factor.

Rotation serves to make the output more understandable, by seeking so-called "Simple Structure": Varimax rotation is one such method that maximizes simultaneously for all factors the variance of the loadings within each factor. The variance of a factor is largest when its smallest loadings tend towards zero and its largest loadings tend towards unity. In essence, the solution obtained through varimax rotation produces factors that are characterized by large loadings on relatively few variables.

The other method of rotation is known as quartimax rotation wherein the factor loadings are transformed until the variance of the squared factor loadings throughout the matrix is maximized. As a result, the solution obtained through this method permits a general factor to emerge, whereas in case of varimax solution such a thing is not possible. But both solutions produce orthogonal factors i. It should, however, be emphasized that right rotation must be selected for making sense of the results of factor analysis.

Measurement And Scaling Techniques. Analysis Of Variance And Co-variance. Interpretation And Report Writing. Its Role In Research. Research Methodology Interview Questions. Research Methodology Practice Tests.

## Main Topics

Factor analysis decision process Objectives of factor analysis Designing a factor analysis Assumptions in factor analysis Deriving factors and assessing overall fit Interpreting the factors Validation of factor analysis Additional use of factor analysis research 9 10

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Factor Analysis is also extensively used in the field of marketing and market research related to product attributes and perceptions. The construction of ‘Perceptual Maps’ and product positioning studies are some crucial areas where Factor Analysis is widely used along with other quantitative research and analysis tools.