Description Usage Arguments Details Value Note Author(s) References See Also Examples

This function is intended for users and estimates a factor analysis model that has been
set up previously with a call to `make_manifest`

and a call to
`make_restrictions`

.

1 2 3 |

`manifest` |
An object that inherits from | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

`restrictions` |
An object that inherits from | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

`scores` |
Type of factor scores to produce, if any. The default is | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

`seeds` |
A vector of length one or two to be used as the random
number generator seeds corresponding to the | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

`lower` |
A lower bound. In exploratory factor analysis, | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

`analytic` |
A logical (default to | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

`reject` |
Logical indicating whether to reject starting values that fail the
constraints required by the model; see | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

`NelderMead` |
Logical indicating whether to call | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

`impatient` |
Logical that defaults to | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

`...` |
Further arguments that are passed to
The following arguments to
Other arguments to |

The call to `Factanal`

is somewhat of a formality in the sense that most of the
difficult decisions were already made in the call to `make_restrictions`

and the call to `make_manifest`

. The most important remaining detail is
the specification of the values for the starting population in the genetic algorithm.

It is not necessary to provide starting values, since there are methods for this
purpose; see `create_start`

. Also, if `starting.values = NA`

, then
a population of starting values will be created using the typical mechanism in
`genoud`

, namely random uniform draws from the domain of the
parameter.

Otherwise, if `reject = TRUE`

, starting values that fail one or more constraints
are rejected and new vectors of starting values are generated until the population is
filled with admissable starting values. In some cases, the constraints are quite difficult
to satisfy by chance, and it may be more practical to specify `reject = FALSE`

or to
supply starting values explicitly. If starting values are supplied, it is helpful if at
least one member of the genetic population satisfies all the constraints imposed on the
model. Note the rownames of `restrictions@Domains`

, which indicate the proper order
of the free parameters.

A matrix (or vector) of starting values can be passed as `starting.values`

.
(Also, it is possible to pass an object of `FA-class`

to
`starting.values`

, in which case the estimates from the previous call to
`Factanal`

are used as the starting values.) If a matrix, it should have
columns equal to the number of rows in `restrictions@Domains`

in the specified
order and one or more rows up to the number of genetic individuals in the population.

If `starting.values`

is a vector, its length can be equal to the number of rows
in `restrictions@Domains`

in which case it is treated as a one-row matrix, or its
length can be equal to the number of manifest variables, in which case it is passed
to the `start`

argument of `create_start`

as a vector of initial
communality estimes, thus avoiding the sometimes time-consuming process of generating
good initial communality estimates. This process can also be accelerated by specifying
`impatient = TRUE`

.

An object of that inherits from `FA-class`

.

The underlying genetic algorithm can print a variety of output as it progresses. On Windows, you either have to move the scrollbar periodically to flush the output to the screen or disable buffering by either going to the Misc menu or by clicking Control+W. The output will, by default, look something like this

Generation | First | Second | ... | Last | Discrepancy |

number | constraint | constraint | constraint | function | |

0 | -1.0 | -1.0 | ... | -1.0 | double |

1 | -1.0 | -1.0 | ... | -1.0 | double |

... | ... | ... | ... | ... | ... |

42 | -1.0 | -1.0 | ... | -1.0 | double |

The integer on the far left indicates the generation number. If it appears to
skip one or more generations, that signifies that the best individual in the
“missing” generation was no better than the best individual in the
previous generation. The sequence of *-1.0* indicates that various constraints
are being satisfied by the best individual in the generation. Some of these
constraints are hard-coded, some are added by the choices the user makes in the call
to `make_restrictions`

. The curious are referred to the source code,
but for the most part users need not worry about them provided they are *-1.0*.
If any but the last are not *-1.0* after the first few generations, there is a
**major** problem because no individual is satisfying all the constraints.
The last number is a double-precision number indicating the value of the discrepancy
function. This number will decrease, sometimes painfully slowly, sometimes intermittently,
over the generations since the discrepancy function is being minimized, subject to the
aforementioned constraints.

Ben Goodrich

Barthlomew, D. J. and Knott, M. (1990) *Latent Variable Analysis
and Factor Analysis.* Second Edition, Arnold.

Beauducel, A. (2007) In spite of indeterminancy, many common factor score
estimates yield an identical reproduced covariance matrix.
*Psychometrika*, **72**, 437–441.

Smith, G. A. and Stanley G. (1983)
Clocking *g*: relating intelligence and measures of timed
performance. *Intelligence*, **7**, 353–368.

Venables, W. N. and Ripley, B. D. (2002)
*Modern Applied Statistics with S.* Fourth edition. Springer.

`make_manifest`

, `make_restrictions`

, and
`Rotate`

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | ```
## Example from Venables and Ripley (2002, p. 323)
## Previously from Bartholomew and Knott (1999, p. 68--72)
## Originally from Smith and Stanley (1983)
## Replicated from example(ability.cov)
man <- make_manifest(covmat = ability.cov)
## Not run:
## Here is the easy way to set up a SEFA model, which uses pop-up menus
res <- make_restrictions(manifest = man, factors = 2, model = "SEFA")
## End(Not run)
## This is the hard way to set up a restrictions object without pop-up menus
beta <- matrix(NA_real_, nrow = nrow(cormat(man)), ncol = 2)
rownames(beta) <- rownames(cormat(man))
free <- is.na(beta)
beta <- new("parameter.coef.SEFA", x = beta, free = free, num_free = sum(free))
Phi <- diag(2)
free <- lower.tri(Phi)
Phi <- new("parameter.cormat", x = Phi, free = free, num_free = sum(free))
res <- make_restrictions(manifest = man, beta = beta, Phi = Phi)
# This is how to make starting values where Phi is the correlation matrix
# among factors, beta is the matrix of coefficients, and the scales are
# the logarithm of the sample standard deviations. It is also the MLE.
starts <- c( 4.46294498156615e-01, # Phi_{21}
4.67036349420035e-01, # beta_{11}
6.42220238211291e-01, # beta_{21}
8.88564379236454e-01, # beta_{31}
4.77779639176941e-01, # beta_{41}
-7.13405536379741e-02, # beta_{51}
-9.47782525342137e-08, # beta_{61}
4.04993872375487e-01, # beta_{12}
-1.04604290549591e-08, # beta_{22}
-9.44950629176182e-03, # beta_{32}
2.63078925240678e-04, # beta_{42}
9.38038168787216e-01, # beta_{52}
8.43618801925473e-01, # beta_{62}
log(man@sds)) # log manifest standard deviations
sefa <- Factanal(manifest = man, restrictions = res,
# NOTE: Do NOT specify any of the following tiny values in a
# real research situation; it is done here solely for speed
starting.values = starts, pop.size = 2, max.generations = 6,
wait.generations = 1)
nsim <- 101 # number of simulations, also too small for real work
show(sefa)
summary(sefa, nsim = nsim)
model_comparison(sefa, nsim = nsim)
stuff <- list() # output list for various methods
stuff$model.matrix <- model.matrix(sefa) # sample correlation matrix
stuff$fitted <- fitted(sefa, reduced = TRUE) # reduced covariance matrix
stuff$residuals <- residuals(sefa) # difference between model.matrix and fitted
stuff$rstandard <- rstandard(sefa) # normalized residual matrix
stuff$weights <- weights(sefa) # (scaled) approximate weights for residuals
stuff$influence <- influence(sefa) # weights * residuals
stuff$cormat <- cormat(sefa, matrix = "RF") # reference factor correlations
stuff$uniquenesses <- uniquenesses(sefa, standardized = FALSE) # uniquenesses
stuff$FC <- loadings(sefa, matrix = "FC") # factor contribution matrix
stuff$draws <- FA2draws(sefa, nsim = nsim) # draws from sampling distribution
if(require(nFactors)) screeplot(sefa) # Enhanced scree plot
profile(sefa) # profile plots of non-free parameters
pairs(sefa) # Thurstone-style plot
if(require(Rgraphviz)) plot(sefa) # DAG
``` |

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