| Title: | Extended RC Models for Contingency Tables |
|---|---|
| Description: | Maximum likelihood estimation of an extended class of row-column (RC) association models for two-dimensional contingency tables, which are formulated by a condition of reduced rank on a matrix of extended association parameters; see Forcina (2019) <arXiv:1910.13848>. These parameters are defined by choosing the logit type for the row and column variables among four different options and a transformation derived from suitable divergence measures. |
| Authors: | Francesco Bartolucci, Antonio Forcina |
| Maintainer: | Francesco Bartolucci <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 1.2 |
| Built: | 2025-01-14 05:52:29 UTC |
| Source: | https://github.com/cran/extRC |
Estimation of extended RC models, which are formulate by constraining different types of association parameters to have a reduced rank.
The package contains functions for maximum likelihood (ML) estimation of an extended class of row-column (RC) association models for two-dimensional contingency tables, as described in Forcina (2019). These models are formulated by a condition of reduced rank on a matrix of extended association parameters, which are defined by choosing the logit type for the row and column variables among four different options and a transformation derived from Cressie and Read (1984). Among the available alternatives, it is possible to use log-odds ratio based on different types of aggregation of the joint probabilities. The class of models generalizes that proposed in Kateri and Papaioannou (1994), Bartolucci and Forcina (2002), and Espendiller (2017), and includes the original RC association models of Goodman (1979) and the correspondence analysis model, as formulated in Goodman (1981) and Gilula et al. (1988). Maximum likelihood estimation is based on an algorithm that is an adaptation of the Aitchison and Silvey (1958) algorithm for constrained ML estimation and is related to the algorithm described in Evans and Forcina (2013) for fitting constrained marginal models.
The main function in the package is extRC that provides an output that may be shown by usual R commands print, summary, and plot.
Francesco Bartolucci, Antonio Forcina
Maintainer: Francesco Bartolucci <[email protected]>
Aitchison, J. and Silvey (1958). Maximum-likelihood estimation of parameters subject to restraints. The Annals of Mathematical Statistics, 29, 813-828.
Bartolucci, F. and Forcina, A. (2002). Extended RC association models allowing for order restrictions and marginal modeling. Journal of the American Statistical Association, 97, 1192-1199.
Cressie, N. and Read, T.R., 1984. Multinomial goodness-of-fit tests. Journal of the Royal Statistical Society: Series B, 46, 440-464.
Espendiller, M., 2017. Association in contingency tables. Ph.D. thesis.
Evans, R.J. and Forcina, A. (2013). Two algorithms for fitting constrained marginal models. Computational statistics & Data analysis, 66, 1-7.
Forcina (2019), An extended class of RC association models: definition and estimation, arXiv:1910.13848.
Gilula, Z., Krieger, A.M., and Ritov, Y., 1988. Ordinal association in contingency tables: Some interpretive aspects. Journal of the American Statistical Association, 83, 540-545.
Goodman, L.A. (1979). Simple models for the analysis of association in cross-classifications having ordered categories. Journal of the American Statistical Association, 74, 537-552.
Goodman, L.A., 1981. Association models and canonical correlation in the analysis of cross-classifications having ordered categories. Journal of the American Statistical Association, 76, 320-334.
Kateri, M. and Papaioannou, T. (1994). f-divergence Association Models. University of Ioannina.
# load data data(mobility) # fit model for a single la out = extRC(mobility,mod=c("l","l"),k=1,la=0.6) summary(out)# load data data(mobility) # fit model for a single la out = extRC(mobility,mod=c("l","l"),k=1,la=0.6) summary(out)
Internal function that computes step length of the estimation algorithm in extRC by fitting a cubic polynomial.
cuby(g)cuby(g)
g |
vector of likelihood values at different step lengths |
comp1 |
optimal length |
Francesco Bartolucci, Antonio Forcina
Given a vector of canonical parameters coding distribution for an IxJ contingency table and the RC model specification in list Model, it computes vector of marginal and joint parameters and matrix of its derivatives with respect the canonical parameters.
Deta(the, Model, der = FALSE)Deta(the, Model, der = FALSE)
the |
vector of canonical parameters |
Model |
list specifying all model components |
der |
to require derivative computation (optional) |
eta |
vector of marginal parameters |
Der |
derivative matrix with respect to canonical parameters |
Francesco Bartolucci, Antonio Forcina
It creates a matrix of first differences of order k.
dfm(k)dfm(k)
k |
size of the matrix |
D |
first difference matrix |
Francesco Bartolucci, Antonio Forcina
D = dfm(5) x = runif(5) (D%*%x)D = dfm(5) x = runif(5) (D%*%x)
Given the row vectorized matrix, it computes the vector of discrepancies with respect to a certain rank and its derivative.
Drank(ga, lev, k, der = FALSE)Drank(ga, lev, k, der = FALSE)
ga |
row vectorized matrix of interaction |
lev |
vector of the number of row and column categories in the original table (the numbers of rows and columns of the input matrix must be increased by 1) |
k |
matrix rank |
der |
to require derivative |
fr |
vector of discrepancies with respect to the rank |
Dfr |
derivative of fr |
Bartolucci, F. and Forcina, A. (2002). Extended RC association models allowing for order restrictions and marginal modeling. Journal of the American Statistical Association, 97, 1192-1199.
A = matrix(rnorm(12),4) # matrix the rank of which must be checked a = as.vector(t(A)) out = Drank(a,c(5,4),1,der=TRUE) (out$fr) (out$Dfr)A = matrix(rnorm(12),4) # matrix the rank of which must be checked a = as.vector(t(A)) out = Drank(a,c(5,4),1,der=TRUE) (out$fr) (out$Dfr)
Main function that fits extended RC models based on different types of aggregation (continuation, local, global) and different divergence functions defined by a suitable value of lambda.
extRC(N, mod, k, la, marg.cons = c("free","equal","shift"))extRC(N, mod, k, la, marg.cons = c("free","equal","shift"))
N |
observed contingency table |
mod |
vector indicating the types of aggregation for row and column variables ("c" for continuation, "l" for local, "g" for global) |
k |
rank required for the matrix of interaction parameters |
la |
value of lambda parameter |
marg.cons |
type of constraint on the marginal distributions |
la |
vector of lambda values (when a vector is in input) |
dev |
deviance of the fitted model (when only one lambda value is in input) or vector of deviances (when a vector of lambda values is in input) |
df |
degrees of freedom (when only one lambda value is in input) |
it |
number of iterations (when only one lambda value is in input) |
dis |
final discrepancy (when only one lambda value is in input) |
pj |
vector of joint probabilities under the fitted model (when only one lambda value is in input) |
eta |
full vector of marginal parameters (when only one lambda value is in input) |
etaX |
vector of row marginal parameters (when only one lambda value is in input) |
etaY |
vector of column marginal parameters (whenonly one lambda value is in input) |
Eta |
matrix of association parameters (when only one lambda value is in input) |
la |
vector of lambda values (when more lambda values are in input) |
dev |
vector of deviance values (when more lambda values are in input) |
Francesco Bartolucci, Antonio Forcina
# load data data(mobility) # for a single value of lambda, fit model with constraints of rank 1 on # local-local logits and without constraints on the marginal distributions out = extRC(mobility,mod=c("l","l"),k=1,la=0.6) summary(out) # for a single value of lambda, fit model with constraints of rank 1 on # local-local logits and under constrain of equal marginal distributions out = extRC(mobility,mod=c("l","l"),k=1,la=0.6,marg.cons="equal") summary(out) # for a single value of lambda, fit model with constraints of rank 2 on # global-global logits and under constraint that marginal distributions # are equal up to a constant shift out = extRC(mobility,mod=c("g","g"),k=2,la=0.6,marg.cons="shift") summary(out) # fit model for a vector of lambdas la = seq(-1.8,0.6,length.out=10) out1 = extRC(mobility,mod=c("l","l"),k=1,la=la) plot(out1)# load data data(mobility) # for a single value of lambda, fit model with constraints of rank 1 on # local-local logits and without constraints on the marginal distributions out = extRC(mobility,mod=c("l","l"),k=1,la=0.6) summary(out) # for a single value of lambda, fit model with constraints of rank 1 on # local-local logits and under constrain of equal marginal distributions out = extRC(mobility,mod=c("l","l"),k=1,la=0.6,marg.cons="equal") summary(out) # for a single value of lambda, fit model with constraints of rank 2 on # global-global logits and under constraint that marginal distributions # are equal up to a constant shift out = extRC(mobility,mod=c("g","g"),k=2,la=0.6,marg.cons="shift") summary(out) # fit model for a vector of lambdas la = seq(-1.8,0.6,length.out=10) out1 = extRC(mobility,mod=c("l","l"),k=1,la=la) plot(out1)
Internal function that performs a matrix algebra transformation that is used for estimation in extRC.
Hmat(G)Hmat(G)
G |
input matrix |
H |
transformed matrix |
Francesco Bartolucci, Antonio Forcina
Internal function that implements the Aitchinson-Silvey algorithm to estimate extended RC models.
MainRC(y, Model, the0 = NULL, output = FALSE)MainRC(y, Model, the0 = NULL, output = FALSE)
y |
row vectorized vector of frequencies of the contingency table |
Model |
list of model components |
the0 |
initial vector of canonical parameters (optional) |
output |
to require full output (optional) |
dev |
final deviance |
df |
degrees of freedom |
pj |
vector of joint probabilities |
it |
number of iterations |
dis |
final discrepancy |
Francesco Bartolucci, Antonio Forcina
Computation of aggregation matrices for generalized interactions that are used in codeextRC to estimate extended RC models.
MatIn(lev, mod)MatIn(lev, mod)
lev |
vector number of rows and columns |
mod |
type of logit for each dimension |
R0 |
aggregation matrix for the row margin upper level |
R1 |
aggregation matrix for the row margin lower level |
C0 |
aggregation matrix for the column margin upper level |
C1 |
aggregation matrix for the column margin lower level |
J00 |
aggregation matrix for the left upper quadrant |
J01 |
aggregation matrix for the rigth upper quadrant |
J10 |
aggregation matrix for the left lower quadrant |
J11 |
aggregation matrix for the right lower quadrant |
Francesco Bartolucci, Antonio Forcina
Social mobility table of 3,500 British individuals, who are cross-classified according to their occupational status and the occupation status of their fathers.
data("mobility")data("mobility")
The format is: num [1:5, 1:5] 50 28 11 14 3 45 174 78 150 42 ... - attr(*, "dimnames")=List of 2 ..$ : chr [1:5] "F1" "F2" "F3" "F4" ... ..$ : chr [1:5] "S1" "S2" "S3" "S4" ...
Mosteller, F. (1968). Association and estimation in contingency tables. Journal of the American Statistical Association, 63, 1-28.
Bartolucci, F. and Forcina, A. (2002). Extended RC association models allowing for order restrictions and marginal modeling. Journal of the American Statistical Association, 97, 1192-1199.
It plots the output of codeextRC function for a vector of lambda values.
## S3 method for class 'extRC' plot(x, ...)## S3 method for class 'extRC' plot(x, ...)
x |
output from |
... |
further arguments passed to or from other methods |
None |
Francesco Bartolucci, Antonio Forcina
Internal function that, given a vector of canonical parameters for an IxJ table (vectorized by row) and the RC model specification in list Model, computes vector of discrepancies and matrix of its derivatives with respect to the canonical parameters.
PraD(the, Model, der = FALSE)PraD(the, Model, der = FALSE)
the |
vector of canonical parameters |
Model |
list of model components |
der |
to require the derivative (optional) |
eta |
vector of parameters (logits, interactions) |
hdis |
vector of discrepancies |
Hdis |
matrix of derivatives of discrepancies with respect to the canonical parameter (optional) |
Francesco Bartolucci, Antonio Forcina
Given the output of codeextRC function, it is written in a readable form.
## S3 method for class 'extRC' print(x, ...)## S3 method for class 'extRC' print(x, ...)
x |
output from |
... |
further arguments passed to or from other methods |
None |
Francesco Bartolucci, Antonio Forcina
Summary method for the output of codeextRC function.
## S3 method for class 'extRC' summary(object, ...)## S3 method for class 'extRC' summary(object, ...)
object |
output from |
... |
further arguments passed to or from other methods |
None |
Francesco Bartolucci, Antonio Forcina
Given a square matrix, it provides the lower triangular part, including the main diagonal.
tril(M)tril(M)
M |
square matrix |
N |
transformed matrix |
Francesco Bartolucci, Antonio Forcina
M = matrix(1:9,3) N = tril(M)M = matrix(1:9,3) N = tril(M)