## Stata 15 help for pca

```
[MV] pca -- Principal component analysis

Syntax

Principal component analysis of data

pca varlist [if] [in] [weight] [, options]

Principal component analysis of a correlation or covariance matrix

pcamat matname , n(#) [options pcamat_options]

matname is a k x k symmetric matrix or a k(k+1)/2 long row or column
vector containing the upper or lower triangle of the correlation or
covariance matrix.

options              Description
-------------------------------------------------------------------------
Model 2
components(#)      retain maximum of # principal components; factors()
is a synonym
mineigen(#)        retain eigenvalues larger than #; default is 1e-5
correlation        perform PCA of the correlation matrix; the default
covariance         perform PCA of the covariance matrix
vce(none)          do not compute VCE of the eigenvalues and vectors;
the default
vce(normal)        compute VCE of the eigenvalues and vectors assuming
multivariate normality

Reporting
level(#)           set confidence level; default is level(95)
novce              suppress display of SEs even though calculated
# means              display summary statistics of variables

tol(#)             advanced option; see Options for details
ignore             advanced option; see Options for details

norotated          display unrotated results, even if rotated results
are available (replay only)
-------------------------------------------------------------------------
# means is not allowed with pcamat.
norotated is not available in the dialog box.

pcamat_options       Description
-------------------------------------------------------------------------
Model
shape(full)        matname is a square symmetric matrix; the default
shape(lower)       matname is a vector with the rowwise lower triangle
(with diagonal)
shape(upper)       matname is a vector with the rowwise upper triangle
(with diagonal)
names(namelist)    variable names; required if matname is triangular
forcepsd           modifies matname to be positive semidefinite
* n(#)               number of observations
sds(matname2)      vector with standard deviations of variables
means(matname3)    vector with means of variables
-------------------------------------------------------------------------
* n() is required for pcamat.

bootstrap, by, jackknife, rolling, statsby, and xi are allowed with pca;
see prefix.  However, bootstrap and jackknife results should be
interpreted with caution; identification of the pca parameters involves
overdispersed estimates (Milan and Whittaker 1995).
Weights are not allowed with the bootstrap prefix.
aweights are not allowed with the jackknife prefix.
aweights and fweights are allowed with pca; see weight.
See [MV] pca postestimation for features available after estimation.

pca

Statistics > Multivariate analysis > Factor and principal component
analysis > Principal component analysis (PCA)

pcamat

Statistics > Multivariate analysis > Factor and principal component
analysis > PCA of a correlation or covariance matrix

Description

pca and pcamat display the eigenvalues and eigenvectors from the
principal component analysis (PCA) eigen decomposition.  The eigenvectors
are returned in orthonormal form, that is, uncorrelated and normalized.

pca can be used to reduce the number of variables or to learn about the
underlying structure of the data.  pcamat provides the correlation or
covariance matrix directly.  For pca, the correlation or covariance
matrix is computed from the variables in varlist.

Options

+---------+
----+ Model 2 +----------------------------------------------------------

components(#) and mineigen(#) specify the maximum number of components
(eigenvectors or factors) to be retained.  components() specifies the
number directly, and mineigen() specifies it indirectly, keeping all
components with eigenvalues greater than the indicated value.  The
options can be specified individually, together, or not at all.
factors() is a synonym for components().

components(#) sets the maximum number of components (factors) to be
retained.  pca and pcamat always display the full set of eigenvalues
but display eigenvectors only for retained components.  Specifying a
number larger than the number of variables in varlist is equivalent
to specifying the number of variables in varlist and is the default.

mineigen(#) sets the minimum value of eigenvalues to be retained.
The default is 1e-5 or the value of tol() if specified.

Specifying components() and mineigen() affects only the number of
components to be displayed and stored in e(); it does not enforce the
assumption that the other eigenvalues are 0.  In particular, the
standard errors reported when vce(normal) is specified do not depend
on the number of retained components.

correlation and covariance specify that principal components be
calculated for the correlation matrix and covariance matrix,
respectively.  The default is correlation.  Unlike factor analysis,
PCA is not scale invariant; the eigenvalues and eigenvectors of a
covariance matrix differ from those of the associated correlation
matrix.  Usually, a PCA of a covariance matrix is meaningful only if
the variables are expressed in the same units.

For pcamat, do not confuse the type of the matrix to be analyzed with
the type of matname.  Obviously, if matname is a correlation matrix
and the option sds() is not specified, it is not possible to perform
a PCA of the covariance matrix.

vce(none|normal) specifies whether standard errors are to be computed for
the eigenvalues, the eigenvectors, and the (cumulative) percentage of
explained variance (confirmatory PCA). These standard errors are
obtained assuming multivariate normality of the data and are valid
only for a PCA of a covariance matrix.  Be cautious if applying these
to correlation matrices.

+-----------+
----+ Reporting +--------------------------------------------------------

level(#) specifies the confidence level, as a percentage, for confidence
intervals.  The default is level(95) or as set by set level.  level()
is allowed only with vce(normal).

This option is ignored when specified with vce(normal).

novce suppresses the display of standard errors, even though they are
computed, and displays the PCA results in a matrix/table style.  You
can specify novce during estimation in combination with vce(normal).
More likely, you will want to use novce during replay.

means displays summary statistics of the variables over the estimation
sample.  This option is not available with pcamat.

+----------+

tol(#) is an advanced, rarely used option and is available only with
vce(normal).  An eigenvalue, ev_i, is classified as being close to
zero if ev_i < tol * max(ev).  Two eigenvalues, ev_1 and ev_2, are
"close" if abs(ev_1-ev_2) < tol*max(ev).  The default is tol(1e-5).
See option ignore and Technical note below.

ignore is an advanced, rarely used option and is available only with
vce(normal).  It continues the computation of standard errors and
tests, even if some eigenvalues are suspiciously close to zero or
suspiciously close to other eigenvalues, violating crucial
assumptions of the asymptotic theory used to estimate standard errors
and tests.  See Technical note below.

The following option is available with pca and pcamat but is not shown in
the dialog box:

norotated displays the unrotated principal components, even if rotated
components are available.  This option may be specified only when
replaying results.

Options unique to pcamat

+-------+
----+ Model +------------------------------------------------------------

shape(shape_arg) specifies the shape (storage mode) for the covariance or
correlation matrix matname.  The following shapes are supported:

full specifies that the correlation or covariance structure of k
variables is stored as a symmetric k x k matrix.  Specifying
shape(full) is optional in this case.

lower specifies that the correlation or covariance structure of k
variables is stored as a vector with k(k+1)/2 elements in rowwise
lower-triangular order:

C(11) C(21) C(22) C(31) C(32) C(33) ... C(k1) C(k2) ... C(kk)

upper specifies that the correlation or covariance structure of k
variables is stored as a vector with k(k+1)/2 elements in rowwise
upper-triangular order:

C(11) C(12) C(13) ... C(1k) C(22) C(23) ... C(2k) ...  C(k-1
k-1) C(k-1 k) C(kk)

names(namelist) specifies a list of k different names, which are used to
document output and to label estimation results and are used as
variable names by predict.  By default, pcamat verifies that the row
and column names of matname and the column or row names of matname2
and matname3 from the sds() and means() options are in agreement.
Using the names() option turns off this check.

forcepsd modifies the matrix matname to be positive semidefinite (psd)
and so to be a proper covariance matrix.  If matname is not positive
semidefinite, it will have negative eigenvalues.  By setting negative
eigenvalues to 0 and reconstructing, we obtain the least-squares
positive-semidefinite approximation to matname.  This approximation
is a singular covariance matrix.

n(#) is required and specifies the number of observations.

sds(matname2) specifies a k x 1 or 1 x k matrix with the standard
deviations of the variables.  The row or column names should match
the variable names, unless the names() option is specified.  sds()
may be specified only if matname is a correlation matrix.

means(matname3) specifies a k x 1 or 1 x k matrix with the means of the
variables.  The row or column names should match the variable names,
unless the names() option is specified.  Specify means() if you have
variables in your dataset and want to use predict after pcamat.

Technical note

pca and pcamat with the vce(normal) option assume that

(A1) the variables are multivariate normal distributed and

(A2) the variance-covariance matrix of the observations has all
distinct and strictly positive eigenvalues.

Under assumptions A1 and A2, the eigenvalues and eigenvectors of the
sample covariance matrix can be seen as maximum likelihood estimates for
the population analogues that are asymptotically (multivariate) normally
distributed (Anderson 1963; Jackson 2003).  See Tyler (1981) for related
results for elliptic distributions.  Be cautious when interpreting
because the asymptotic variances are rather sensitive to violations of
assumption A1 (and A2).  Wald tests of hypotheses that are in conflict
with assumption A2 (for example, testing that the first and second
eigenvalue are the same) produce incorrect p-values.

Because the statistical theory for a PCA of a correlation matrix is much
more complicated, pca and pcamat compute standard errors and tests of a
correlation matrix as if it were a covariance matrix.  This practice is
in line with the application of asymptotic theory in Jackson (2003).
This will usually lead to some underestimation of standard errors, but we
believe that this problem is smaller than the consequences of deviations
from normality.

You may conduct tests for multivariate normality using the mvtest
normality command (see [MV] mvtest normality).

Examples

Standard PCA for descriptive use
. sysuse auto
. pca trunk weight length headroom
. pca trunk weight length headroom, comp(2) covariance

PCA assuming multivariate normality of the data
. webuse bg2
. pca bg2cost*, vce(normal)

PCA of a covariance or correlation matrix
. matrix S = ( 10.167, 22.690,  2.040  \ ///
22.690, 56.949,  3.788  \ ///
2.040,  3.788,  0.688  )
. matrix rownames S = visual hearing taste
. matrix colnames S = visual hearing taste
. pcamat S, n(979) comp(2)

Same as above
. matrix S = ( 10.167, 22.690, 2.040, ///
56.949, 3.788, ///
0.688 )
. pcamat S, n(979) shape(upper) comp(2) names(visual hearing taste)

Stored results

pca and pcamat without the vce(normal) option store the following in e():

Scalars
e(N)                number of observations
e(f)                number of retained components
e(rho)              fraction of explained variance
e(trace)            trace of e(C)
e(lndet)            ln of the determinant of e(C)
e(cond)             condition number of e(C)

Macros
e(cmd)              pca (even for pcamat)
e(cmdline)          command as typed
e(Ctype)            correlation or covariance
e(wtype)            weight type
e(wexp)             weight expression
e(title)            title in output
e(properties)       nob noV eigen
e(rotate_cmd)       program used to implement rotate
e(estat_cmd)        program used to implement estat
e(predict)          program used to implement predict
e(marginsnotok)     predictions disallowed by margins

Matrices
e(C)                p x p correlation or covariance matrix
e(means)            1 x p matrix of means
e(sds)              1 x p matrix of standard deviations
e(Ev)               1 x p matrix of eigenvalues (sorted)
e(L)                p x f matrix of eigenvectors = components
e(Psi)              1 x p matrix of unexplained variance

Functions
e(sample)           marks estimation sample

pca and pcamat with the vce(normal) option store the above, as well as
the following:

Scalars
e(v_rho)            variance of e(rho)
e(chi2_i)           chi-squared statistic for test of independence
e(df_i)             degrees of freedom for test of independence
e(p_i)              p-value for test of independence
e(chi2_s)           chi-squared statistic for test of sphericity
e(df_s)             degrees of freedom for test of sphericity
e(p_s)              p-value for test of sphericity
e(rank)             rank of e(V)

Macros
e(vce)              multivariate normality
e(properties)       b V eigen

Matrices
e(b)                1 x p+fp coefficient vector (all eigenvalues and
retained eigenvectors)
e(Ev_bias)          1 x p matrix: bias of eigenvalues
e(Ev_stats)         p x 5 matrix with statistics on explained variance
e(V)                variance-covariance matrix of the estimates e(b)

References

Anderson, T. W. 1963. Asymptotic theory for principal component analysis.
Annals of Mathematical Statistics 34: 122-148.

Jackson, J. E. 2003. A User's Guide to Principal Components.  New York:
Wiley.

Milan, L., and J. Whittaker. 1995. Application of the parametric
bootstrap to models that incorporate a singular value decomposition.
Applied Statistics 44: 31-49.

Tyler, D. E. 1981. Asymptotic inference for eigenvectors.  Annals of
Statistics 9: 725-736.

```