make_restrictions and Factanal. Although it is possible to simply estimate and use the unbiased sample
covariance matrix, there are many other ways to estimate a covariance
that can be superior, particularly when the traditional maximum likelihood
discrepancy function is not chosen in the call to make_restrictions.
In technical terms, make_manifest is the constructor for objects of
manifest-class, which houses the sample covariance estimate
and some ancillary information in its slots. The three arguments in the signature
of the S4 generic function are:
x, data, and covmat
"make_manifest"(covmat, n.obs = NA_integer_, shrink = FALSE)
"make_manifest"(covmat, shrink = FALSE)
"make_manifest"(covmat, n.obs = NA_integer_, shrink = FALSE, sds = NULL)
"make_manifest"(covmat)
# Use the methods above when only the covariance matrix is available
# Use the methods below when the raw data are available (preferable)
"make_manifest"(x, subset, shrink = FALSE,
bootstrap = 0, how = "default", seed = 12345, wt = NULL, ...)
"make_manifest"(data, subset, shrink = FALSE,
bootstrap = 0, how = "default", seed = 12345, wt = NULL, ...)
"make_manifest"(data, subset, shrink = FALSE,
bootstrap = 0, how = "default", seed = 12345, wt = NULL, ...)
"make_manifest"(x, subset, shrink = FALSE,
bootstrap = 0, how = "default", seed = 12345, wt = NULL, ...)
"make_manifest"(x, data, subset, shrink = FALSE, na.action = "na.pass",
bootstrap = 0, how = "default", seed = 12345, wt = NULL, ...)data must be a data.frame and the formula should not have a
response. If a data.frame or a matrix of data, then all its columns are used.formula is not specified, then all its columns are used and similarly
if it is a matrix of data.CovMcd-class, an
object of S4 class "hetcor" from the polycor package, or missing. If a list, it
must contain an element named "cov" and may contain the following named elements:
"cov" element
make_restrictions-methods can calculate various
weight matrices if the raw data are passed to make_manifest, so this mechanism
should only be used if those options are inadequate
"cov" is really
a correlation matrix
covmat is a covariance
matrix or if covmat is a list with no element named n.obs. It is possible
to obtain maximum likelihood estimates without knowing the number of observations but nothing
elseTRUE, then the “minimax shrinkage” estimator
discussed in theorem 3.1 of Dey and Srinivasan (1985) is invoked on the sample covariance
matrix as calculated according to the other arguments. In some circumstances, shrink
is inappropriate and ignored with a warningNULL or a numeric vector that contains the standard deviations of
the manifest variables, which is used when covmat is a correlation matrixhow = "mcd" or bootstrap > 0. If NULL, then the
current seed is used. This argument defaults to 12345.x is specified. By default, the observations are weighted equally.
The wt argument can be used in two ways. First, it is passed to the
the corresponding argument of cov.wt if appropriate (see below).
Second, it is passed to the prob argument of sample when
bootstrap > 0.na.action to be used if x is a formula.covmat
is unspecified, implying that the raw data are being used to estimate the sample covariance.manifest-class.
how argument dictates how the covariance matrix is estimated.
There is much to be said in favor the Minimum Covariance Determinant (CovMcd)
estimator (see Pison et. al. 2003) and it is used as the default when there are no missing
data, although it can subtly affect the sampling distributions of estimates that subsequently
derived from it. The same could probably be said for the shrinkage estimators
(either via how = "lambda" or shrink = TRUE). The Dey and Srinivasan (1985)
shrinkage estimator preserves the eigenvectors of the preliminarily-calculated covariance
matrix but deterministically compresses the eigenvalues. The cov.shrink
estimator in the corpcor package is based on the idea that the amount shrinkage should be
proportional to the variance of the covariance estimates. Use how = "mle" or
how = "unbiased" to obtain either the maximum likelihood or unbiased sample covariance
estimator, the latter of which is the one used in virtually all factor applications whether
appropriate or not.Next, consider the case where at least one manifest variable is an ordered factor. If
how = "ranks", Spearman correlations are estimated from the integer codes
underlying the ordered factors. This mechanism is recommended only if there are at least
five levels of each ordered factor and no missing data. In that case, one would presumably
want to specify method = "ADF" in the subsequent call to make_restrictions).
If how != "ranks" all pairwise correlations are estimated under bivariate
normality assumptions via hetcor in the suggested polycor package,
which will allow pairwise-deletion when there are missing data. If how != "ranks" and
bootstrap > 0 (recommended), then there must not be any missing data because
the bootstrapping utilizes fast Spearman correlations and then tries to correct the bias
by rescaling the bootstrapped means to equal to point estimates calculated with the call
to hetcor.
In general, bootstrapping is good for estimating the uncertainty of the estimated sample covariances and this uncertainty estimate is needed for the ADF discrepancy function and its special cases. In some cases, bootstrapping is the only way to obtain such an uncertainty estimate.
Pison, G., Rousseeuw, P.J., Filzmoser, P. and Croux, C. (2003) Robust factor analysis. Journal of Multivariate Analysis, 84, 145--172.
Factanal, make_restrictions, manifest-class,
covMcd, cov.wt, hetcor,
mlest, cov.shrink, and cov. man <- make_manifest(covmat = Harman23.cor)
show(man) # some basic info
if(require(nFactors)) screeplot(man) # advanced Scree plot
cormat(man) # sample correlation matrix
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