In this article
Once variables have been added, and any adjustments to the advanced settings have been made, click Run factor analysis to produce a calculation output as in the example below .
Figure 1 - Example of a Factor Analysis output
Several variations of the factor analysis can be made, with each calculation run saved separately within the factor analysis object. The most recent calculation run will display first. Calculation runs can be renamed or deleted if required.
Click the Send to Excel button to export the tables to Excel.
Correlation Matrix
This table .contains the correlation matrix that is used as a basis for the factor analysis.
Figure 2 - The Correlation Matrix
The rows and columns of the correlation matrix correspond to the seven variables included in the analysis: Row 1 (and column 1) represents variable 1, row 2 (and column 2) represents variable 2, and so forth. Where a given row and column intersect, you will find the correlation between the two corresponding variables.
For example, where the row for variable 2 intersects with the column for variable 1, you find a correlation of -0.05; this means that the correlation between variables 1 and 2 is -0.05.
The Eigenvalues Table
This table contains information about the eigenvalues of the Correlation/Covariance matrix .
Figure 3 - The Eigenvalues table
Each eigenvalue will correspond to a factor in the model, and the percentage of total variance of the data that is explained by each factor is also shown. If rotation of factors is chosen in the model, the table will also show the sum of squared loadings and the percentage of total variance explained by the rotated factors
How Factors are Determined
When the factors are created, the first component extracted in a principal component analysis accounts for a maximal amount of total variance in the observed variables. Under typical conditions, this means that the first component will be correlated with at least some of the observed variables. It may be correlated with many.
The second component extracted will have two important characteristics. First, this component will account for a maximal amount of variance in the data set that was not accounted for by the first component. Again under typical conditions, this means that the second component will be correlated with some of the observed variables that did not display strong correlations with component 1. The second characteristic of the second component is that it will be uncorrelated with the first component. Literally, if you were to compute the correlation between components 1 and 2, that correlation would be zero.
The remaining components that are extracted in the analysis display the same two characteristics: each component accounts for a maximal amount of variance in the observed variables that was not accounted for by the preceding components, and each is uncorrelated with all of the preceding components. A factor analysis proceeds in this fashion, with each new component accounting for progressively smaller and smaller amounts of variance (this is why only the first few components are usually retained and interpreted). When the analysis is complete, the resulting components will display varying degrees of correlation with the observed variables, but are completely uncorrelated with one another.
Factors Table
This table contains further information about the factors as calculated by the Factor Analysis .
Figure 4 - The Factors table
The loadings of each variable on each factor are shown, as well as the communalities which show the amount of variance in the specific variable that is explained by the model. If rotation of factors is chosen in the model, the table will also show the loadings for the rotated factors.
When a variable is given a great deal of weight in constructing a principal component, we say that the variable loads on that component. For example, if the question Q1_1 (“Amenities”) is given a lot of weight in creating the first factor, we say that this item loads on factor 1.
These factor loadings are equivalent to bivariate correlations between the variables and the factors. For example, where the rows for the variables intersect with the column for factor 1, you can see that the correlation between Q1_1 and the first component is 0.75; the correlation between Q1_2 and the first component is -0.065, and so forth.
Communality refers to the percent of variance in a variable that is accounted for by the retained components (or factors). A given variable will display a large communality if it loads heavily on at least one of the study’s retained components.
Looking at the results from the example data, we see in the rotated factors (from the cell highlighting) that the strong correlations associated with the variables could be grouped into three underlying factors we could classify as:
- Factor 1: “Facilities” (accounting for the variables Amenities, Parking, Room Comfort)
- Factor 2: “People” (accounting for the variables Friendly and Welcoming, Check-in procedure)
- Factor 3: “Food and Drink” (accounting for the variables Breakfast, Minibar / in-room snacks)
The article “Creating Factor Score Variables in the Data Source” explains how we can create these factor variables in the data source for use in subsequent analyses (go to Creating Factor Scores Variables in the Data Source for more information). You could then use these three new artificial variables (rather than the seven original variables) as predictor variables in multiple regressions, or in any other type of analysis. In essence, this is what is accomplished by factor analysis: it allows you to reduce a set of observed variables into a smaller set of artificial variables called factors. The resulting factors may then be used in subsequent analyses.
Scree Plot
A scree plot is a visual tool plotting the eigenvalues associated with each component. The component numbers are listed on the horizontal axis, while eigenvalues are listed on the vertical axis .
Figure 5 - Example of a Scree Plot
This plot can be used in determining the number of “Meaningful” factors to retain by looking for a “break” between the components with relatively large eigenvalues and those with small eigenvalues. The components that appear before the break are assumed to be meaningful; those appearing after the break are assumed to be unimportant and are not retained.
Sometimes a scree plot will display several large breaks. When this is the case, you should look for the last big break before the eigenvalues begin to level off. Only the components that appear before this last large break should be retained.
With the example, notice that there is a relatively large break following component 3. The breaks between components , 4-7 are all relatively small, so a three-factor solution seems correct.