FAfitStats: Summary Statistics for a TSFA Models

Description

FAfitStats calculates various statistics for a TSFestModel or all possible (unrotated factanal) models for a data matrix. This function is also used by the summary method for a TSFestModel.

Usage

FAfitStats(object, ...)
    # S3 method for default
FAfitStats(object, diff. = TRUE,
               N=(nrow(object) - diff.), 
	       control=list(lower = 0.0001, opt=list(maxit=1000)), ...)
    # S3 method for TSFmodel
FAfitStats(object, diff. = TRUE,
               N=(nrow(object$data) - diff.), ...)

Arguments

object

a time series matrix or TSFestModel.

diff.

logical indicating if data should be differenced.

sample size.

control

a list of arguments passed to factanal.

...

further arguments passed to other methods.

Value

a vector or list of various fit statistics. See details.

Details

In the case of the method for a TSFmodel the model parameters are extracted from the model and the result is a vector of various fit statistics (see below). (Calculations are done by the internal function FAmodelFitStats.)

Most of these statistics are described in Wansbeek and Meijer (2000, WM below). The sample size \(N\) is used in the calculation of these statistics. The default is the number of number of observations, as in WM. That is, the number of rows in the data matrix, minus one if the data is differenced. Many authors use \(N - 1\), which would be \(N-2\) if the data is differenced. The exact calculations can be determined by examining the code: print(tsfa:::FAmodelFitStats). The vector of statistics is:

chisq: Chi-square statistic (see, for example, WM p298).
df: degrees of freedom, which takes the rotational freedom into account (WM p169).
pval: p-value
delta: delta
RMSEA: Root mean square error of approximation (WM p309).
RNI: Relative noncentrality index (WM p307).
CFI: Comparative fit index (WM p307).
MCI: McDonald's centrality index.
GFI: Goodness of fit index ( J<U+001B2973>kog and S<U+001B28AF>m, 1981, 1986, WM p305).
AGFI: Adjusted GFI (J<U+001B2973>kog and S<U+001B28AF>m, 1981, 1986).
AIC: Akaike's information criterion (WM p309).
CAIC: Consistent AIC(WM p310).
SIC: Schwarz's Bayesian information criterion.
CAK: Cudeck & Browne's rescaled AIC.
CK: Cudeck & Browne's cross-validation index.

The information criteria account for rotational freedom. Some of these goodness of fit statistics should be used with caution, because they are not yet based on sound statistical theory. Future versions of tsfa will probably provide improved versions of these goodness-of-fit statistics.

In the case of the default method, which expects a matrix of data with columns for each indicator series, models are calculated with factanal for factors up to the Ledermann bound. No rotation is needed, since rotation does not affect the fit statistics. Values for the saturated model are also appended to facilitate a sequential comparison.

If factanal does not obtain a satisfactory solution it may produce an error "unable to optimize from these starting value(s)." This can sometimes be fixed by increasing the opt, maxit value in the control list.

The result for the default method is a list with elements

fitStats: a matrix with rows as for a single model above, and a column for each possible number of factors.
seqfitStats: a matrix with rows chisq, df, and pval, and columns indicating the comparative fit for an additional factor starting with the null (zero factor) model. (See also independence model, WM, p305)

The largest model can correspond to the saturated model, but will not if the Ledermann bound is not an integer, or even in the case of an integer bound but implicit contraints resulting in a Heywood case (see Dijkstra, 1992). In these situations it might make sense to remove the model corresponding to the largest integer, and make the last sequential comparison between the second to largest integer and the saturated solution. The code does not do this automatically.

References

Dijkstra, T. K. (1992) On Statistical Inference with Parameter Estimates on the Boundary of the Parameter Space, British Journal of Mathematical and Statistical Psychology, 45, 289--309.

Hu, L.-t., and Bentler, P. (1995) Evaluating model fit. In R. H. Hoyle (Ed.), Structural equation modeling: Concepts, issues, and applications (pp. 76--99). Thousand Oaks, CA: Sage.

J<U+001B2973>kog, K. G., and S<U+001B28AF>m, D. (1981) LISREL V user's guide. Chicago: National Educational Resources.

J<U+001B2973>kog, K. G., and S<U+001B28AF>m, D. (1986) LISREL VI: Analysis of linear structural relationships by maximum likelihood, instrumental variables, and least squares methods (User's Guide, 4th ed.). Mooresville, IN: Scientific Software.

Ogasawara, Haruhiko. (2001). Approximations to the Distributions of Fit Indexes for Misspecified Structural Equation Models. Structural Equation Modeling, 8, 556--574.

Wansbeek, Tom and Meijer, Erik (2000) Measurement Error and Latent Variables in Econometrics, Amsterdam: North-Holland.

Examples

Run this code

# NOT RUN {
  if (require("CDNmoney")){
    data("CanadianMoneyData.asof.28Jan2005", package="CDNmoney")
    data("CanadianCreditData.asof.28Jan2005", package="CDNmoney")

    z <- tframed(tbind(
        MB2001,
        MB486 + MB452 + MB453 ,
        NonbankCheq,
        MB472 + MB473 + MB487p,
        MB475,
        NonbankNonCheq + MB454 + NonbankTerm + MB2046 + MB2047 + MB2048 +
        MB2057 + MB2058 + MB482),
        names=c("currency", "personal cheq.", "NonbankCheq",
        "N-P demand & notice", "N-P term", "Investment" )
      )

    z <- tfwindow(tbind (z, ConsumerCredit, ResidentialMortgage,
                        ShortTermBusinessCredit, OtherBusinessCredit),
         start=c(1981,11), end=c(2004,11))

    cpi <- 100 * M1total / M1real
    popm <- M1total / M1PerCapita
    scale <- tfwindow(1e8 /(popm * cpi), tf=tframe(z))

    MBandCredit <- sweep(z, 1, scale, "*")

    FAfitStats(MBandCredit)

    c4withML  <- estTSF.ML(MBandCredit, 4)
    FAfitStats(c4withML)
  }
# }

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