Fit the log-multiplicative uniform difference model (UNIDIFF, see Erikson & Goldthorpe, 1992), also called the log-multiplicative layer effect model (Xie, 1992). For square tables, diagonal cells can be handled separately.
unidiff(tab, diagonal = c("included", "excluded", "only"),
constrain = "auto",
weighting = c("marginal", "uniform", "none"), norm = 2,
family = poisson,
tolerance = 1e-8, iterMax = 5000, eliminate=NULL,
trace = FALSE, verbose = TRUE,
checkEstimability = TRUE, ...)a three-way table, or an object (such as a matrix) that can be coerced into a table; if present, dimensions above three will be collapsed as appropriate.
included fits the standard model with full two-way interaction;
excluded adds to this model diagonal-specific parameters for each years, effectively
removing the influence of diagonal cells on the layer coefficients; only fits a model
without the full two-way interaction, where only diagonal parameters are affected by the layer
effect (see “Details” below).
(non-eliminated) coefficients to constrain, specified by a regular expression, a numeric vector of indices, a logical vector, a character vector of names, or "[?]" to select from a Tk dialog. The default constrains to 0 the first layer parameter and interaction coefficients for the first row and column of the table.
what weights should be used when normalizing coefficients. This does not affect
layer coefficients, which are set to 1 for the first layer, but only two-way interaction
coefficients and layer association levels, which are layer coefficients times the intrinsic
association coefficient (see maor) for the first layer.
the norm to use to compute the mean absolute odds ratio (see maor).
a specification of the error distribution and link function to be used in the model.
This can be a character string naming a family function; a family function, or the result of
a call to a family function. See family details of family functions.
a positive numeric value specifying the tolerance level for convergence; higher values will speed up the fitting process, but beware of numerical instability of estimated scores!
a positive integer specifying the maximum number of main iterations to perform; consider raising this value if your model does not converge.
either NULL (the default) to estimate all parameters, NA
to skip the estimation of some parameters for increased efficiency, or the name of a
factor to be passed as gnm's corresponding argument.
a logical value indicating whether the deviance should be printed after each iteration.
a logical value indicating whether progress indicators should be printed, including a diagnostic error message if the algorithm restarts.
a logical value indicating whether the estimability of the contrasts should
be checked via checkEstimable. Disabling this check can improve performance for large models.
more arguments to be passed to gnm
A unidiff object, with all the components of a gnm object, plus an
unidiff component holding the most relevant information:
a qvcalc object holding the (log) layer coefficients, their standard
errors and quasi-standard errors.
the value of the intrinsic association coefficient (see maor) for
each layer.
the value of the Mean absolute odds ratio (see maor) for
each layer.
a data frame object holding the two-way interaction coefficients, and their standard errors.
the value of the diagonal argument above.
the value of the weighting argument above.
The equation of the fitted model is:
$$ log F_{ijk} = \lambda + \lambda^I_i + \lambda^J_j + \lambda^K_k
+ \lambda^{IK}_{ik} + \lambda^{JK}_{jk}
+ \phi_k \psi^{IJ}_{ij} $$
where \(F_{ijk}\) is the expected frequency for the cell at the intersection of row i, column j and
layer k of tab. When diagonal = "excluded", \(\lambda^{IJK}_{ijk}\) parameters are added
but set to 0 when \(i \neq j\) (off-diagonal). When diagonal = "only", \(\psi^{IJ}_{ij}\) is set
to 0 when \(i \neq j\).
Note that by default weighting="marginal", meaning that reported
interaction coefficients do not correspond to what is usually expected
in log-linear modeling. Use weighting="none" or weighting="uniform"
to use more classic identification constraints (effects coding).
Layer coefficients \(\phi_k\) are internally exponentiated in the gnm formula, which means the reported
values are in log scale, with reference 0 for the first year. Interaction coefficients use the
“sum” contrast, also known as “effect” coding, except when diagonal is different from
included, in which case “treatment” constrast (a.k.a “reference” or “dummy”
coding) is used.
Actual model fitting is performed using gnm, which implements the Newton-Raphson algorithm.
This function simply allows for direct identification of the log-multiplicative parameters by setting the
appropriate constraints, and improves performance by eliminating less interesting coefficients.
Erikson, R., and Goldthorpe, J.H. (1992). The Constant Flux: A Study of Class Mobility in Industrial Societies. Oxford: Clarendon Press. Ch. 3.
Xie, Yu (1992). The Log-Multiplicative Layer Effect Model for Comparing Mobility Tables. Am. Sociol. Rev. 57(3):380-395.
Yaish, M. (1998). Opportunities, Little Change. Class Mobility in Israeli Society, 1974-1991. Ph.D. thesis, Nuffield College, University of Oxford.
Yaish, M. (2004). Class Mobility Trends in Israeli Society, 1974-1991. Lewiston: Edwin Mellen Press.
# NOT RUN {
## Yaish (1998, 2004)
data(yaish)
# Last layer omitted because of low frequencies
yaish <- yaish[,,-7]
# Layer (education) must be the third dimension
yaish <- aperm(yaish, 3:1)
model <- unidiff(yaish)
model
summary(model)
plot(model)
# }
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