com.imsl.stat
Class FactorAnalysis

java.lang.Object
  com.imsl.stat.FactorAnalysis

All Implemented Interfaces:

Serializable, Cloneable

public class FactorAnalysis
extends Object
implements Serializable, Cloneable

Performs Principal Component Analysis or Factor Analysis on a covariance or correlation matrix.

Class FactorAnalysis computes principal components or initial factor loading estimates for a variance-covariance or correlation matrix using exploratory factor analysis models.

Models available are the principal component model for factor analysis and the common factor model with additions to the common factor model in alpha factor analysis and image analysis. Methods of estimation include principal components, principal factor, image analysis, unweighted least squares, generalized least squares, and maximum likelihood.

For the principal component model there are methods to compute the characteristic roots, characteristic vectors, standard errors for the characteristic roots, and the correlations of the principal component scores with the original variables. Principal components obtained from correlation matrices are the same as principal components obtained from standardized (to unit variance) variables.

The principal component scores are the elements of the vector where is the matrix whose columns are the characteristic vectors (eigenvectors) of the sample covariance (or correlation) matrix and x is the vector of observed (or standardized) random variables. The variances of the principal component scores are the characteristic roots (eigenvalues) of the covariance (correlation) matrix.

Asymptotic variances for the characteristic roots were first obtained by Girshick (1939) and are given more recently by Kendall, Stuart, and Ord (1983, page 331). These variances are computed either for variance-covariance matrices or for correlation matrices.

The correlations of the principal components with the observed (or standardized) variables are the same as the unrotated factor loadings obtained for the principal components model for factor analysis when a correlation matrix is input.

In the factor analysis model used for factor extraction, the basic model is given as where is the population covariance matrix. is the matrix of factor loadings relating the factors f to the observed variables x, and is the matrix of covariances of the unique errors e. Here, p represents the number of variables and k is the number of factors. The relationship between the factors, the unique errors, and the observed variables is given as where, in addition, it is assumed that the expected values of e, f, and x are zero. (The sample means can be subtracted from x if the expected value of x is not zero.) It is also assumed that each factor has unit variance, the factors are independent of each other, and that the factors and the unique errors are mutually independent. In the common factor model, the elements of the vector of unique errors e are also assumed to be independent of one another so that the matrix is diagonal. This is not the case in the principal component model in which the errors may be correlated.

Further differences between the various methods concern the criterion that is optimized and the amount of computer effort required to obtain estimates. Generally speaking, the least-squares and maximum likelihood methods, which use iterative algorithms, require the most computer time with the principal factor, principal component, and the image methods requiring much less time since the algorithms in these methods are not iterative. The algorithm in alpha factor analysis is also iterative, but the estimates in this method generally require somewhat less computer effort than the least-squares and maximum likelihood estimates. In all algorithms one eigensystem analysis is required on each iteration.

See Also:

Principal Component Example, Maximum Likelihood Example, Serialized Form

Nested Class Summary

static class	FactorAnalysis.BadVarianceException Bad variance error.
static class	FactorAnalysis.EigenvalueException Eigenvalue error.
static class	FactorAnalysis.NoDegreesOfFreedomException No degrees of freedom error.
static class	FactorAnalysis.NonPositiveEigenvalueException Non positive eigenvalue error.
static class	FactorAnalysis.NotPositiveDefiniteException Covariance matrix not positive definite.
static class	FactorAnalysis.NotPositiveSemiDefiniteException Covariance matrix not positive semi-definite.
static class	FactorAnalysis.NotSemiDefiniteException Hessian matrix not semi-definite.
static class	FactorAnalysis.RankException Rank of covariance matrix error.
static class	FactorAnalysis.SingularException Covariance matrix singular error.

Field Summary

static int	ALPHA_FACTOR_ANALYSIS Indicates alpha factor analysis.
static int	CORRELATION_MATRIX Indicates correlation matrix.
static int	GENERALIZED_LEAST_SQUARES Indicates generalized least squares method.
static int	IMAGE_FACTOR_ANALYSIS Indicates image factor analysis.
static int	MAXIMUM_LIKELIHOOD Indicates maximum likelihood method.
static int	PRINCIPAL_COMPONENT_MODEL Indicates principal component model.
static int	PRINCIPAL_FACTOR_MODEL Indicates principal factor model.
static int	UNWEIGHTED_LEAST_SQUARES Indicates unweighted least squares method.
static int	VARIANCE_COVARIANCE_MATRIX Indicates variance-covariance matrix.

Constructor Summary

FactorAnalysis(double[][] cov, int matrixType, int nf)
Constructor for FactorAnalysis.

Method Summary

double[][]	getCorrelations() Returns the correlations of the principal components.
double[][]	getFactorLoadings() Returns the unrotated factor loadings.
double[]	getParameterUpdates() Returns the parameter updates.
double[]	getPercents() Returns the cumulative percent of the total variance explained by each principal component.
double[]	getStandardErrors() Returns the estimated asymptotic standard errors of the eigenvalues.
double[]	getStatistics() Returns statistics.
double[]	getValues() Returns the eigenvalues.
double[]	getVariances() Gets the unique variances.
double[][]	getVectors() Returns the eigenvectors.
void	setConvergenceCriterion1(double eps) Sets the convergence criterion used to terminate the iterations.
void	setConvergenceCriterion2(double epse) Sets the convergence criterion used to switch to exact second derivatives.
void	setDegreesOfFreedom(int ndf) Sets the number of degrees of freedom.
void	setFactorLoadingEstimationMethod(int methodType) Sets the factor loading estimation method.
void	setMaxIterations(int maxit) Sets the maximum number of iterations in the iterative procedure.
void	setMaxStep(int maxstp) Sets the maximum number of step halvings allowed during an iteration.
void	setVarianceEstimationMethod(int init) Sets the variance estimation method.
void	setVariances(double[] uniq) Sets the variances.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

ALPHA_FACTOR_ANALYSIS

public static final int ALPHA_FACTOR_ANALYSIS

Indicates alpha factor analysis.

See Also:

Constant Field Values

CORRELATION_MATRIX

public static final int CORRELATION_MATRIX

Indicates correlation matrix.

See Also:

Constant Field Values

GENERALIZED_LEAST_SQUARES

public static final int GENERALIZED_LEAST_SQUARES

Indicates generalized least squares method.

See Also:

Constant Field Values

IMAGE_FACTOR_ANALYSIS

public static final int IMAGE_FACTOR_ANALYSIS

Indicates image factor analysis.

See Also:

Constant Field Values

MAXIMUM_LIKELIHOOD

public static final int MAXIMUM_LIKELIHOOD

Indicates maximum likelihood method.

See Also:

Constant Field Values

PRINCIPAL_COMPONENT_MODEL

public static final int PRINCIPAL_COMPONENT_MODEL

Indicates principal component model.

See Also:

Constant Field Values

PRINCIPAL_FACTOR_MODEL

public static final int PRINCIPAL_FACTOR_MODEL

Indicates principal factor model.

See Also:

Constant Field Values

UNWEIGHTED_LEAST_SQUARES

public static final int UNWEIGHTED_LEAST_SQUARES

Indicates unweighted least squares method.

See Also:

Constant Field Values

VARIANCE_COVARIANCE_MATRIX

public static final int VARIANCE_COVARIANCE_MATRIX

Indicates variance-covariance matrix.

See Also:

Constant Field Values

Constructor Detail

FactorAnalysis

public FactorAnalysis(double[][] cov,
                      int matrixType,
                      int nf)

Constructor for FactorAnalysis.

Parameters:

cov - A double matrix containing the covariance or correlation matrix.

matrixType - An int scalar indicating the type of matrix that is input. Uses class member VARIANCE_COVARIANCE_MATRIX, CORRELATION_MATRIX for matrixType.

nf - An int scalar indicating the number of factors in the model. If nf is not known in advance, several different values of nf should be used, and the most reasonable value kept in the final solution. Since, in practice, the non-iterative methods often lead to solutions which differ little from the iterative methods, it is usually suggested that a non-iterative method be used in the initial stages of the factor analysis, and that the iterative methods be used once issues such as the number of factors have been resolved.

Throws:

IllegalArgumentException - is thrown if x.length, and x[0].length are equal to 0.

Method Detail

getCorrelations

public double[][] getCorrelations()
                           throws FactorAnalysis.RankException,
                                  FactorAnalysis.NoDegreesOfFreedomException,
                                  FactorAnalysis.NotSemiDefiniteException,
                                  FactorAnalysis.NotPositiveSemiDefiniteException,
                                  FactorAnalysis.NotPositiveDefiniteException,
                                  FactorAnalysis.SingularException,
                                  FactorAnalysis.BadVarianceException,
                                  FactorAnalysis.EigenvalueException,
                                  FactorAnalysis.NonPositiveEigenvalueException

Returns the correlations of the principal components.

Returns:

An double matrix containing the correlations of the principal components with the observed/standardized variables. If a covariance matrix is input to the constructor, then the correlations are with the observed variables. Otherwise, the correlations are with the standardized (to a variance of 1.0) variables. Only valid for the principal components model.

Throws:

FactorAnalysis.RankException

FactorAnalysis.NoDegreesOfFreedomException

FactorAnalysis.NotSemiDefiniteException

FactorAnalysis.NotPositiveSemiDefiniteException

FactorAnalysis.NotPositiveDefiniteException

FactorAnalysis.SingularException

FactorAnalysis.BadVarianceException

FactorAnalysis.EigenvalueException

FactorAnalysis.NonPositiveEigenvalueException

getFactorLoadings

public double[][] getFactorLoadings()
                             throws FactorAnalysis.RankException,
                                    FactorAnalysis.NoDegreesOfFreedomException,
                                    FactorAnalysis.NotSemiDefiniteException,
                                    FactorAnalysis.NotPositiveSemiDefiniteException,
                                    FactorAnalysis.NotPositiveDefiniteException,
                                    FactorAnalysis.SingularException,
                                    FactorAnalysis.BadVarianceException,
                                    FactorAnalysis.EigenvalueException,
                                    FactorAnalysis.NonPositiveEigenvalueException

Returns the unrotated factor loadings.

Returns:

A double matrix containing the unrotated factor loadings.

Throws:

FactorAnalysis.RankException

FactorAnalysis.NoDegreesOfFreedomException

FactorAnalysis.NotSemiDefiniteException

FactorAnalysis.NotPositiveSemiDefiniteException

FactorAnalysis.NotPositiveDefiniteException

FactorAnalysis.SingularException

FactorAnalysis.BadVarianceException

FactorAnalysis.EigenvalueException

FactorAnalysis.NonPositiveEigenvalueException

getParameterUpdates

public double[] getParameterUpdates()
                             throws FactorAnalysis.RankException,
                                    FactorAnalysis.NoDegreesOfFreedomException,
                                    FactorAnalysis.NotSemiDefiniteException,
                                    FactorAnalysis.NotPositiveSemiDefiniteException,
                                    FactorAnalysis.NotPositiveDefiniteException,
                                    FactorAnalysis.SingularException,
                                    FactorAnalysis.BadVarianceException,
                                    FactorAnalysis.EigenvalueException,
                                    FactorAnalysis.NonPositiveEigenvalueException

Returns the parameter updates.

Returns:

A double array containing the parameter updates when convergence was reached (or the iterations terminated). The parameter updates are only meaningful for the common factor model. The parameter updates are set to 0.0 for the principal component model.

Throws:

FactorAnalysis.RankException

FactorAnalysis.NoDegreesOfFreedomException

FactorAnalysis.NotSemiDefiniteException

FactorAnalysis.NotPositiveSemiDefiniteException

FactorAnalysis.NotPositiveDefiniteException

FactorAnalysis.SingularException

FactorAnalysis.BadVarianceException

FactorAnalysis.EigenvalueException

FactorAnalysis.NonPositiveEigenvalueException

getPercents

public double[] getPercents()
                     throws FactorAnalysis.RankException,
                            FactorAnalysis.NoDegreesOfFreedomException,
                            FactorAnalysis.NotSemiDefiniteException,
                            FactorAnalysis.NotPositiveSemiDefiniteException,
                            FactorAnalysis.NotPositiveDefiniteException,
                            FactorAnalysis.SingularException,
                            FactorAnalysis.BadVarianceException,
                            FactorAnalysis.EigenvalueException,
                            FactorAnalysis.NonPositiveEigenvalueException

Returns the cumulative percent of the total variance explained by each principal component. Valid for the principal component model.

Returns:

An double array containing the total variance explained by each principal component.

Throws:

FactorAnalysis.RankException

FactorAnalysis.NoDegreesOfFreedomException

FactorAnalysis.NotSemiDefiniteException

FactorAnalysis.NotPositiveSemiDefiniteException

FactorAnalysis.NotPositiveDefiniteException

FactorAnalysis.SingularException

FactorAnalysis.BadVarianceException

FactorAnalysis.EigenvalueException

FactorAnalysis.NonPositiveEigenvalueException

getStandardErrors

public double[] getStandardErrors()
                           throws FactorAnalysis.RankException,
                                  FactorAnalysis.NoDegreesOfFreedomException,
                                  FactorAnalysis.NotSemiDefiniteException,
                                  FactorAnalysis.NotPositiveSemiDefiniteException,
                                  FactorAnalysis.NotPositiveDefiniteException,
                                  FactorAnalysis.SingularException,
                                  FactorAnalysis.BadVarianceException,
                                  FactorAnalysis.EigenvalueException,
                                  FactorAnalysis.NonPositiveEigenvalueException

Returns the estimated asymptotic standard errors of the eigenvalues.

Returns:

An double array containing the estimated asymptotic standard errors of the eigenvalues.

Throws:

FactorAnalysis.RankException

FactorAnalysis.NoDegreesOfFreedomException

FactorAnalysis.NotSemiDefiniteException

FactorAnalysis.NotPositiveSemiDefiniteException

FactorAnalysis.NotPositiveDefiniteException

FactorAnalysis.SingularException

FactorAnalysis.BadVarianceException

FactorAnalysis.EigenvalueException

FactorAnalysis.NonPositiveEigenvalueException

getStatistics

public double[] getStatistics()
                       throws FactorAnalysis.RankException,
                              FactorAnalysis.NoDegreesOfFreedomException,
                              FactorAnalysis.NotSemiDefiniteException,
                              FactorAnalysis.NotPositiveSemiDefiniteException,
                              FactorAnalysis.NotPositiveDefiniteException,
                              FactorAnalysis.SingularException,
                              FactorAnalysis.BadVarianceException,
                              FactorAnalysis.EigenvalueException,
                              FactorAnalysis.NonPositiveEigenvalueException

Returns statistics.

Returns:

A double array (Stat) containing output statistics. Stat is not defined and is set to NaN when the method used to obtain the estimates, is the principal component method, principal factor method, image factor analysis method, or alpha analysis method.

i	Stat[i]
0	Value of the function minimum.
1	Tucker reliability coefficient.
2	Chi-squared test statistic for testing that the number of factors in the model are adequate for the data.
3	Degrees of freedom in chi-squared. This is computed as where nvar is the number of variables and nf is the number of factors in the model.
4	Probability of a greater chi-squared statistic.
5	Number of iterations.

Throws:

FactorAnalysis.RankException

FactorAnalysis.NoDegreesOfFreedomException

FactorAnalysis.NotSemiDefiniteException

FactorAnalysis.NotPositiveSemiDefiniteException

FactorAnalysis.NotPositiveDefiniteException

FactorAnalysis.SingularException

FactorAnalysis.BadVarianceException

FactorAnalysis.EigenvalueException

FactorAnalysis.NonPositiveEigenvalueException

getValues

public double[] getValues()
                   throws FactorAnalysis.RankException,
                          FactorAnalysis.NoDegreesOfFreedomException,
                          FactorAnalysis.NotSemiDefiniteException,
                          FactorAnalysis.NotPositiveSemiDefiniteException,
                          FactorAnalysis.NotPositiveDefiniteException,
                          FactorAnalysis.SingularException,
                          FactorAnalysis.BadVarianceException,
                          FactorAnalysis.EigenvalueException,
                          FactorAnalysis.NonPositiveEigenvalueException

Returns the eigenvalues.

Returns:

A double array containing the eigenvalues of the matrix from which the factors were extracted ordered from largest to smallest. If Alpha Factor analysis is used, then the first nf positions of the array contain the Alpha coefficients. Here, nf is the number of factors in the model. If the algorithm fails to converge for a particular eigenvalue, that eigenvalue is set to NaN. Note that the eigenvalues are usually not the eigenvalues of the input matrix cov. They are the eigenvalues of the input matrix cov when the principal component method is used.

Throws:

FactorAnalysis.RankException

FactorAnalysis.NoDegreesOfFreedomException

FactorAnalysis.NotSemiDefiniteException

FactorAnalysis.NotPositiveSemiDefiniteException

FactorAnalysis.NotPositiveDefiniteException

FactorAnalysis.SingularException

FactorAnalysis.BadVarianceException

FactorAnalysis.EigenvalueException

FactorAnalysis.NonPositiveEigenvalueException

getVariances

public double[] getVariances()
                      throws FactorAnalysis.RankException,
                             FactorAnalysis.NoDegreesOfFreedomException,
                             FactorAnalysis.NotSemiDefiniteException,
                             FactorAnalysis.NotPositiveSemiDefiniteException,
                             FactorAnalysis.NotPositiveDefiniteException,
                             FactorAnalysis.SingularException,
                             FactorAnalysis.BadVarianceException,
                             FactorAnalysis.EigenvalueException,
                             FactorAnalysis.NonPositiveEigenvalueException

Gets the unique variances.

Returns:

A double array of length nvar containing the unique variances, where nvar is the number of variables.

Throws:

FactorAnalysis.RankException

FactorAnalysis.NoDegreesOfFreedomException

FactorAnalysis.NotSemiDefiniteException

FactorAnalysis.NotPositiveSemiDefiniteException

FactorAnalysis.NotPositiveDefiniteException

FactorAnalysis.SingularException

FactorAnalysis.BadVarianceException

FactorAnalysis.EigenvalueException

FactorAnalysis.NonPositiveEigenvalueException

getVectors

public double[][] getVectors()
                      throws FactorAnalysis.RankException,
                             FactorAnalysis.NoDegreesOfFreedomException,
                             FactorAnalysis.NotSemiDefiniteException,
                             FactorAnalysis.NotPositiveSemiDefiniteException,
                             FactorAnalysis.NotPositiveDefiniteException,
                             FactorAnalysis.SingularException,
                             FactorAnalysis.BadVarianceException,
                             FactorAnalysis.EigenvalueException,
                             FactorAnalysis.NonPositiveEigenvalueException

Returns the eigenvectors.

Returns:

A double matrix containing the eigenvectors of the matrix from which the factors were extracted. The j-th column of the eigenvector matrix corresponds to the j-th eigenvalue. The eigenvectors are normalized to each have Euclidean length equal to one. Also, the sign of each vector is set so that the largest component in magnitude (the first of the largest if there are ties) is made positive. Note that the eigenvectors are usually not the eigenvectors of the input matrix cov. They are the eigenvectors of the input matrix cov when the principal component method is used.

Throws:

FactorAnalysis.RankException

FactorAnalysis.NoDegreesOfFreedomException

FactorAnalysis.NotSemiDefiniteException

FactorAnalysis.NotPositiveSemiDefiniteException

FactorAnalysis.NotPositiveDefiniteException

FactorAnalysis.SingularException

FactorAnalysis.BadVarianceException

FactorAnalysis.EigenvalueException

FactorAnalysis.NonPositiveEigenvalueException

setConvergenceCriterion1

public void setConvergenceCriterion1(double eps)

Sets the convergence criterion used to terminate the iterations.

Parameters:

eps - A double used to terminate the iterations. For the least squares and and maximum likelihood methods convergence is assumed when the relative change in the criterion is less than eps. For alpha factor analysis, convergence is assumed when the maximum change (relative to the variance) of a uniqueness is less than eps. eps is not referenced for the other estimation methods. If this member function is not called, eps is set to 0.0001.

setConvergenceCriterion2

public void setConvergenceCriterion2(double epse)

Sets the convergence criterion used to switch to exact second derivatives.

Parameters:

epse - A double used to switch to exact second derivatives. When the largest relative change in the unique standard deviation vector is less than epse exact second derivative vectors are used. If this member function is not called, epse is set to 0.1. Not referenced for principal component, principal factor, image factor, or alpha factor methods.

setDegreesOfFreedom

public void setDegreesOfFreedom(int ndf)

Sets the number of degrees of freedom.

Parameters:

ndf - An int value specifying the number of degrees of freedom in the input matrix. If this member function is not called 100 degrees of freedom are assumed.

setFactorLoadingEstimationMethod

public void setFactorLoadingEstimationMethod(int methodType)

Sets the factor loading estimation method.

Parameters:

methodType - An int scalar indicating the method to be applied for obtaining the factor loadings. Use class member PRINCIPAL_COMPONENT_MODEL, PRINCIPAL_FACTOR_MODEL, UNWEIGHTED_LEAST_SQUARES, GENERALIZED_LEAST_SQUARES, MAXIMUM_LIKELIHOOD, IMAGE_FACTOR_ANALYSIS, or ALPHA_FACTOR_ANALYSIS for methodType. If this member function is not called, the PRINCIPAL_COMPONENT_MODEL is used.

For the principal component and principal factor methods, the factor loading estimates are computed as

where and the diagonal matrix are the eigenvalues and eigenvectors of a matrix. In the principal component model, the eigensystem analysis is performed on the sample covariance (correlation) matrix while in the principal factor model the matrix is used. If the unique error variances are not known in the principal factor model, then they are estimated. This is achieved by calling the member function setVarianceEstimationMethod and setting init to 0. If the principal component model is used, the error variances are set to 0.0 automatically.

The basic idea in the principal component method is to find factors that maximize the variance in the original data that is explained by the factors. Because this method allows the unique errors to be correlated, some factor analysts insist that the principal component method is not a factor analytic method. Usually however, the estimates obtained via the principal component model and other models in factor analysis will be quite similar.

It should be noted that both the principal component and the principal factor methods give different results when the correlation matrix is used in place of the covariance matrix. Indeed, any rescaling of the sample covariance matrix can lead to different estimates with either of these methods. A further difficulty with the principal factor method is the problem of estimating the unique error variances. Theoretically, these must be known in advance and passed in through member function setVariances. In practice, the estimates of these parameters produced by calling the member function setVarianceEstimationMethod and setting init to 0 are often used. In either case, the resulting adjusted covariance (correlation) matrix

may not yield the nf positive eigenvalues required for nf factors to be obtained. If this occurs, the user must either lower the number of factors to be estimated or give new unique error variance values.

For the least-squares and maximum likelihood methods an iterative algorithm is used to obtain the estimates (see Joreskog 1977). As with the principal factor model, the user may either input the initial unique error variances or allow the algorithm to compute initial estimates. Unlike the principal factor method, the code then optimizes the criterion function with respect to both and . (In the principal factor method, is assumed to be known. Given , estimates for may be obtained.)

The major differences between the estimation methods described in this member function are in the criterion function that is optimized. Let denote the sample covariance (correlation) matrix, and let denote the covariance matrix that is to be estimated by the factor model. In the unweighted least-squares method, also called the iterated principal factor method or the minres method (see Harman 1976, page 177), the function minimized is the sum of the squared differences between and . This is written as .

Generalized least-squares and maximum likelihood estimates are asymptotically equivalent methods. Maximum likelihood estimates maximize the (normal theory) likelihood while generalized least squares optimizes the function .

In all three methods, a two-stage optimization procedure is used. This proceeds by first solving the likelihood equations for in terms of and substituting the solution into the likelihood. This gives a criterion , which is optimized with respect to . In the second stage, the estimates

are obtained from the estimates for .

The generalized least-squares and the maximum likelihood methods allow for the computation of a statistic for testing that nf common factors are adequate to fit the model. This is a chi-squared test that all remaining parameters associated with additional factors are zero. If the probability of a larger chi-squared is small (see stat[4] under getStatistics) so that the null hypothesis is rejected, then additional factors are needed (although these factors may not be of any practical importance). Failure to reject does not legitimize the model. The statistic stat[2] is a likelihood ratio statistic in maximum likelihood estimates. As such, it asymptotically follows a chi-squared distribution with degrees of freedom given in stat[3].

The Tucker and Lewis (1973) reliability coefficient, , is returned in stat[1] when the maximum likelihood or generalized least-squares methods are used. This coefficient is an estimate of the ratio of explained to the total variation in the data. It is computed as follows:

where is the determinant of cov, p is the number of variables, k is the number of factors, is the optimized criterion, and d is the number of degrees of freedom.

The term "image analysis" is used here to denote the noniterative image method of Kaiser (1963). It is not the image factor analysis discussed by Harman (1976, page 226). The image method (as well as the alpha factor analysis method) begins with the notion that only a finite number from an infinite number of possible variables have been measured. The image factor pattern is calculated under the assumption that the ratio of the number of factors to the number of observed variables is near zero so that a very good estimate for the unique error variances (for standardized variables) is given as one minus the squared multiple correlation of the variable under consideration with all variables in the covariance matrix.

First, the matrix is computed where the operator "diag" results in a matrix consisting of the diagonal elements of its argument, and is the sample covariance (correlation) matrix. Then, the eigenvalues and eigenvectors of the matrix are computed. Finally, the unrotated image factor pattern matrix is computed as .

The alpha factor analysis method of Kaiser and Caffrey (1965) finds factor-loading estimates to maximize the correlation between the factors and the complete universe of variables of interest. The basic idea in this method is as follows: only a finite number of variables out of a much larger set of possible variables is observed. The population factors are linearly related to this larger set while the observed factors are linearly related to the observed variables. Let f denote the factors obtainable from a finite set of observed random variables, and let denote the factors obtainable from the universe of observable variables. Then, the alpha method attempts to find factor-loading estimates so as to maximize the correlation between f and . In order to obtain these estimates, the iterative algorithm of Kaiser and Caffrey (1965) is used.

setMaxIterations

public void setMaxIterations(int maxit)

Sets the maximum number of iterations in the iterative procedure.

Parameters:

maxit - An int used as the maximum number of iterations allowed during the iterative portion of the algorithm. If this member function is not called, maxit is set to 60. Not referenced for factor loading methods principal component, principal factor, or image factor methods.

setMaxStep

public void setMaxStep(int maxstp)

Sets the maximum number of step halvings allowed during an iteration.

Parameters:

maxstp - An int used as the maximum number of step halvings allowed during an iteration. If this member function is not called, maxstp is set to 8. Not referenced for principal component, principal factor, image factor, or alpha factor methods.

setVarianceEstimationMethod

public void setVarianceEstimationMethod(int init)

Sets the variance estimation method.

Parameters:

init - An int used to designate the method to be applied for obtaining the initial estimates of the unique variances. If this member function is not called, init is set to 1.

init	Method
0	Initial estimates are taken as the constant 1-nf/(2*nvar) divided by the diagonal elements of the inverse of input matrix cov, where nvar is the number of variables.
1	Initial estimates are input by the user in vector uniq (setVariances).

Note that when the factor loading estimation method is PRINCIPAL_COMPONENT_MODEL, the initial estimates in uniq are reset to 0.0.

setVariances

public void setVariances(double[] uniq)

Sets the variances.

Parameters:

uniq - A double array of length nvar containing the unique variances, where nvar is the number of variables. If this member function is not called, the elements of uniq are set to 0.0. If the iterative methods fail for the unique variances used, new initial estimates should be tried. These may be obtained by use of another factoring method (use the final estimates from the new method as initial estimates in the old method). Another alternative is to call member function setVarianceEstimationMethod and set the input argument to 0. This will cause the initial unique variances to be estimated by the code.