Overview www.wiley.com/wires/compstats Bootstrapped Confidence Intervals variable tends to be associated with one (or a small After the number of components to keep has been number)of the components,and each component determined,we can compute confidence intervals represents only a small number of variables.In for the eigenvalues of using the bootstrap.34-39 addition,the components can often be interpreted To use the bootstrap,we draw a large number of from the opposition of few variables with positive samples (e.g.,1000 or 10,000)with replacement loadings to few variables with negative loadings. from the learning set.Each sample produces a set Formally varimax searches for a linear combination of eigenvalues.The whole set of eigenvalues can then of the original factors such that the variance of the be used to compute confidence intervals. squared loadings is maximized,which amounts to maximizing ROTATION v=∑q呢-2 (23) After the number of components has been determined, and in order to facilitate the interpretation,the with being the squared loading of the j-th variable analysis often involves a rotation of the components of matrix Q on component e and being the mean that were retained [see,e.g.,Ref 40 and 67,for of the squared loadings. more details].Two main types of rotation are used: orthogonal when the new axes are also orthogonal to each other,and oblique when the new axes are Oblique Rotations not required to be orthogonal.Because the rotations With oblique rotations,the new axes are free to are always performed in a subspace,the new axes take any position in the component space,but the will always explain less inertia than the original degree of correlation allowed among factors is small components (which are computed to be optimal). because two highly correlated components are better However,the part of the inertia explained by the interpreted as only one factor.Oblique rotations, total subspace after rotation is the same as it was therefore,relax the orthogonality constraint in order before rotation (only the partition of the inertia has to gain simplicity in the interpretation.They were changed).It is also important to note that because strongly recommended by Thurstone,42 but are used rotation always takes place in a subspace (i.e.,the more rarely than their orthogonal counterparts. space of the retained components),the choice of this For oblique rotations,the promax rotation has subspace strongly influences the result of the rotation. the advantage of being fast and conceptually simple. Therefore,it is strongly recommended to try several The first step in promax rotation defines the target sizes for the subspace of the retained components in matrix,almost always obtained as the result of a order to assess the robustness of the interpretation of varimax rotation whose entries are raised to some the rotation.When performing a rotation,the term power(typically between 2 and 4)in order to force loadings almost always refer to the elements of matrix the structure of the loadings to become bipolar. Q.We will follow this tradition in this section. The second step is obtained by computing a least square fit from the varimax solution to the target matrix.Promax rotations are interpreted by looking Orthogonal Rotation at the correlations-regarded as loadings-between An orthogonal rotation is specified by a rotation the rotated axes and the original variables.An matrix,denoted R,where the rows stand for the interesting recent development of the concept of original factors and the columns for the new(rotated) oblique rotation corresponds to the technique of factors.At the intersection of row m and column n we independent component analysis (ICA)where the have the cosine of the angle between the original axis axes are computed in order to replace the notion and the new one:rm,n=cos em.A rotation matrix of orthogonality by statistical independence [see Ref has the important property of being orthonormal 43,for a tutorial]. because it corresponds to a matrix of direction cosines and therefore RR =I. Varimax rotation,developed by Kaiser,41 is the When and Why Using Rotations most popular rotation method.For varimax a simple The main reason for using rotation is to facilitate the solution means that each component has a small interpretation.When the data follow a model(such number of large loadings and a large number of zero as the psychometric model)stipulating(1)that each (or small)loadings.This simplifies the interpretation variable load on only one factor and (2)that there because,after a varimax rotation,each original is a clear difference in intensity between the relevant 442 2010 John Wiley Sons,Inc. Volume 2,July/August 2010Overview www.wiley.com/wires/compstats Bootstrapped Confidence Intervals After the number of components to keep has been determined, we can compute confidence intervals for the eigenvalues of X. using the bootstrap.34–39 To use the bootstrap, we draw a large number of samples (e.g., 1000 or 10,000) with replacement from the learning set. Each sample produces a set of eigenvalues. The whole set of eigenvalues can then be used to compute confidence intervals. ROTATION After the number of components has been determined, and in order to facilitate the interpretation, the analysis often involves a rotation of the components that were retained [see, e.g., Ref 40 and 67, for more details]. Two main types of rotation are used: orthogonal when the new axes are also orthogonal to each other, and oblique when the new axes are not required to be orthogonal. Because the rotations are always performed in a subspace, the new axes will always explain less inertia than the original components (which are computed to be optimal). However, the part of the inertia explained by the total subspace after rotation is the same as it was before rotation (only the partition of the inertia has changed). It is also important to note that because rotation always takes place in a subspace (i.e., the space of the retained components), the choice of this subspace strongly influences the result of the rotation. Therefore, it is strongly recommended to try several sizes for the subspace of the retained components in order to assess the robustness of the interpretation of the rotation. When performing a rotation, the term loadings almost always refer to the elements of matrix Q. We will follow this tradition in this section. Orthogonal Rotation An orthogonal rotation is specified by a rotation matrix, denoted R, where the rows stand for the original factors and the columns for the new (rotated) factors. At the intersection of row m and column n we have the cosine of the angle between the original axis and the new one: rm,n = cos θm,n. A rotation matrix has the important property of being orthonormal because it corresponds to a matrix of direction cosines and therefore RTR = I. Varimax rotation, developed by Kaiser,41 is the most popular rotation method. For varimax a simple solution means that each component has a small number of large loadings and a large number of zero (or small) loadings. This simplifies the interpretation because, after a varimax rotation, each original variable tends to be associated with one (or a small number) of the components, and each component represents only a small number of variables. In addition, the components can often be interpreted from the opposition of few variables with positive loadings to few variables with negative loadings. Formally varimax searches for a linear combination of the original factors such that the variance of the squared loadings is maximized, which amounts to maximizing ν = #(q2 j,# − q2 # ) 2 (23) with q2 j,# being the squared loading of the j-th variable of matrix Q on component # and q2 # being the mean of the squared loadings. Oblique Rotations With oblique rotations, the new axes are free to take any position in the component space, but the degree of correlation allowed among factors is small because two highly correlated components are better interpreted as only one factor. Oblique rotations, therefore, relax the orthogonality constraint in order to gain simplicity in the interpretation. They were strongly recommended by Thurstone,42 but are used more rarely than their orthogonal counterparts. For oblique rotations, the promax rotation has the advantage of being fast and conceptually simple. The first step in promax rotation defines the target matrix, almost always obtained as the result of a varimax rotation whose entries are raised to some power (typically between 2 and 4) in order to force the structure of the loadings to become bipolar. The second step is obtained by computing a least square fit from the varimax solution to the target matrix. Promax rotations are interpreted by looking at the correlations—regarded as loadings—between the rotated axes and the original variables. An interesting recent development of the concept of oblique rotation corresponds to the technique of independent component analysis (ica) where the axes are computed in order to replace the notion of orthogonality by statistical independence [see Ref 43,for a tutorial]. When and Why Using Rotations The main reason for using rotation is to facilitate the interpretation. When the data follow a model (such as the psychometric model) stipulating (1) that each variable load on only one factor and (2) that there is a clear difference in intensity between the relevant 442  2010 John Wiley & Son s, In c. Volume 2, July/Augu st 2010