Goals of Factor Analysis (1) (1)to reduce the number of variables and (2) to detect structure in the relationships between variables, that is to classify variables. Factor can be thought of as manifestation of an abstract underlying dimension

Basis of Factor Analysis Combining Two Variables into a Single Factor. One can summarize the correlation between two variables in a scatterplot. A regression line represents the "best" summary of the linear relationship between the variables. A variable that would approximate the regression line would capture most of the "essence" of the two items. In a sense we have reduced the two variables to one factor.

Extracting Principal Components (most common kind of Factor Analysis) Principal Components extends the previous logic of expressing two or more variables with a single factor. In the case of multiple variables, the computations become more involved, but the basic principle of expressing two or more variables by a single factor remains the same.

Extracting Principal Components Extracts as many factors as there are variables. Tries to extract factors that are independent of each other The first factor explains the highest percentage of variance, the second factor next highest.

Two variable (A, B) situation Refers to common variance or extracted factor Three variable (A, B & C) situation Refers to common variance or extracted factor

Refers to common variance or extracted factor 1 Multiple variable (A, B, C, D, & E) situation, More than one Factor Refers to common variance or extracted factor 2

Applications of Factor Analysis Screening of Variables for use in further statistical analysis: In the previous example, suppose the company wants to identify the factors that affect satisfaction with company. This can be done by predicting satisfaction from the predictor variable. By identifying the natural groupings of variables, we can use only one of them for further predictive analysis (regressions). Another option is to use the Factor Score itself.

Applications of Factor Analysis Clustering of People: By identifying the natural groupings of variables, one can also identify divide people who score high on one factor or the other. In the previous example, suppose you identified three factors that affect satisfaction with company: price, quality, image of product. People might rate higher on one factor or the other.

Interpreting SPSS output Eigenvalues: Percentage of variance explained by extracted factor. Scree Plot Look at total cumulative percentage explained to decide how many factors to include.

Number of Factors: Should be a good summary solution Total Percentage Explained Are factors comprehensible How many factors to retain?

In the language of factor analysis, the proportion of variance of a particular item that is due to common factors (shared with other items) is called communality. Communality If a factor is high on this, then the variable is mostly independent

The factor loadings could be plot in a scatterplot, with each variable represented as a point. The axis of this plot could be rotated in any direction without changing the relative locations of the points to each other; however, the actual coordinates of the points, that is, the factor loadings would change. Sometimes such rotations allow a clearer view of the factors Rotation

The goal of all of rotational strategies is to obtain a clear pattern of loadings, that is, factors that are somehow clearly marked by high loadings for some variables and low loadings for others. Typical rotational strategies are varimax (variance maximizing), quartimax, and equamax. Rotational Strategies