Editing Factor analysis (section)

===Terminology===
{{glossary}}
{{term|Factor loadings}}
{{defn|1=Communality is the square of the standardized outer loading of an item. Analogous to [[Pearson product-moment correlation coefficient|Pearson's r]]-squared, 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. (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. {{pb}}When interpreting, by one rule of thumb in confirmatory factor analysis, factor loadings should be .7 or higher to confirm that independent variables identified a priori are represented by a particular factor, on the rationale that the .7 level corresponds to about half of the variance in the indicator being explained by the factor. However, the .7 standard is a high one and real-life data may well not meet this criterion, which is why some researchers, particularly for exploratory purposes, will use a lower level such as .4 for the central factor and .25 for other factors. In any event, factor loadings must be interpreted in the light of theory, not by arbitrary cutoff levels. {{pb}}In [[Angle#Types of angles|oblique]] rotation, one may examine 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 [[coefficient]]s 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.<ref>{{cite journal | last1=Liou | first1=C.-Y. | last2=Musicus | first2=B.R. | title=Cross Entropy Approximation of Structured Gaussian Covariance Matrices |journal=IEEE Transactions on Signal Processing |volume=56 |issue=7 |pages=3362–3367 |year=2008 |doi=10.1109/TSP.2008.917878 | bibcode=2008ITSP...56.3362L | s2cid=15255630 | url=http://ntur.lib.ntu.edu.tw/bitstream/246246/155199/1/23.pdf }}</ref>}}
{{term|Communality}}
{{defn|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. 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 in the context of the factors being posited.}}
{{term|Spurious solutions}}
{{defn|If the communality exceeds 1.0, there is a spurious solution, which may reflect too small a sample or the choice to extract too many or too few factors.}}
{{term|Uniqueness of a variable}}
{{defn|The variability of a variable minus its communality.}}
{{term|Eigenvalues/characteristic roots}}
{{defn|Eigenvalues measure the amount of variation in the total sample accounted for by each 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 less important than the factors with higher eigenvalues.}}
{{term|Extraction sums of squared loadings}}
{{defn|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. SPSS also prints "Rotation Sums of Squared Loadings" and even for PCA, these eigenvalues will differ from initial and extraction eigenvalues, though their total will be the same.}}
{{term|Factor scores}}
{{term|Component scores (in PCA)|multi=yes}}
{{defn|1={{ghat|Explained from PCA perspective, not from Factor Analysis perspective.}} The scores of each case (row) on each factor (column). 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.}}
{{glossary end}}