ordFusion: Fusion and selection of dummy coefficients of ordinal predictors

Description

Fits dummy coefficients of ordinally scaled independent variables with a fused lasso penalty on differences of adjacent dummy coefficients. Using the ordinalNet algorithm if cumulative logit model is fitted, otherwise glmpath algorithm is used.

Usage

ordFusion(x, y, u = NULL, z = NULL, offset = rep(0,length(y)), lambda,  
  model = c("linear", "logit", "poisson", "cumulative"), 
  restriction = c("refcat", "effect"), scalex = TRUE, nonpenx = NULL, 
  frac.arclength = NULL, ...)

Value

An ordPen object, which is a list containing:

fitted: the matrix of fitted response values of the training data. Columns correspond to different lambda values.
coefficients: the matrix of fitted coefficients with respect to dummy-coded (ordinal or nominal) categorical input variables (including the reference category) as well as metric predictors. Columns correspond to different lambda values.
model: the type of the fitted model: "linear", "logit", "poisson", or "cumulative".
restriction: the type of restriction used for identifiability.
lambda: the used lambda values.
xlevels: a vector giving the number of levels of the ordinal predictors.
ulevels: a vector giving the number of levels of the nominal predictors (if any).
zcovars: the number of metric covariates (if any).

Arguments

x: the matrix of ordinal predictors, with each column corresponding to one predictor and containing numeric values from {1,2,...}; for each covariate, category 1 is taken as reference category with zero dummy coefficient.
y: the response vector.
u: a matrix (or data.frame) of additional categorical (nominal) predictors, with each column corresponding to one (additional) predictor and containing numeric values from {1,2,...}; corresponding dummy coefficients will not be penalized, and for each covariate category 1 is taken as reference category. Curretnly not supported if model == "cumulative".
z: a matrix (or data.frame) of additional metric predictors, with each column corresponding to one (additional) predictor; corresponding coefficients will not be penalized. Curretnly not supported if model == "cumulative".
offset: vector of offset values.
lambda: vector of penalty parameters, i.e., lambda values.
model: the model which is to be fitted. Possible choices are "linear" (default), "logit", "poisson" or "cumulative". See details below.
restriction: identifiability restriction for dummy coding. "reference" takes category 1 is as reference category (default), while with "effect" dummy coefficients sum up to 0 (known as effect coding).
scalex: logical. Should (split-coded) design matrix corresponding to x be scaled to have unit variance over columns before fitting? See details below.
nonpenx: vectors of indices indicating columns of x whose regression coefficients are not penalized. Curretnly not supported if model == "cumulative".
frac.arclength: just in case the corresponding glmpath argument is to be modified; default is 1 for model == "linear", and 0.1 otherwise.
...: additional arguments to ordinalNet (if model == "cumulative") or glmpath.

Author

Jan Gertheiss, Aisouda Hoshiyar

Details

The method assumes that categorical covariates (contained in x and u) take values 1,2,...,max, where max denotes the (columnwise) highest level observed in the data. If any level between 1 and max is not observed for an ordinal predictor, a corresponding (dummy) coefficient is fitted anyway (by linear interpolation, due to some additional but small quadratic penalty, see glmpath for details). If any level > max is not observed but possible in principle, and a corresponding coefficient is to be fitted, the easiest way is to add a corresponding row to x (and u,z) with corresponding y value being NA.

If a linear regression model is fitted, response vector y may contain any numeric values; if a logit model is fitted, y has to be 0/1 coded; if a poisson model is fitted, y has to contain count data. If a cumulative logit model is fitted, y takes values 1,2,...,max.

If scalex is TRUE, (split-coded) design matrix constructed from x is scaled to have unit variance over columns (see standardize argument of glmpath or/and ordinalNet).

References

Gertheiss, J. and G. Tutz (2010). Sparse modeling of categorial explanatory variables. The Annals of Applied Statistics, 4, 2150-2180.

Hoshiyar, A., Gertheiss, L.H., and Gertheiss, J. (2023). Regularization and Model Selection for Item-on-Items Regression with Applications to Food Products' Survey Data. Preprint, available from https://arxiv.org/abs/2309.16373.

Park, M.Y. and T. Hastie (2007). L1 regularization path algorithm for generalized linear models. Journal of the Royal Statistical Society B, 69, 659-677.

Tutz, G. and J. Gertheiss (2014). Rating scales as predictors -- the old question of scale level and some answers. Psychometrica, 79, 357-376.

Tutz, G. and J. Gertheiss (2016). Regularized regression for categorical data. Statistical Modelling, 16, 161-200.

Examples

Run this code

# fusion and selection of ordinal covariates on a simulated dataset
set.seed(123)

# generate (ordinal) predictors
x1 <- sample(1:8,100,replace=TRUE)
x2 <- sample(1:6,100,replace=TRUE)
x3 <- sample(1:7,100,replace=TRUE)

# the response
y <- -1 + log(x1) + sin(3*(x2-1)/pi) + rnorm(100)

# x matrix
x <- cbind(x1,x2,x3)

# lambda values
lambda <- c(80,70,60,50,40,30,20,10,5,1) 

# fusion and selection
ofu <- ordFusion(x = x, y = y, lambda = lambda)

# results
round(ofu$coef,digits=3)
plot(ofu)

# If for a certain plot the x-axis should be annotated in a different way,
# this can (for example) be done as follows:
plot(ofu, whx = 1, xlim = c(0,9), xaxt = "n")
axis(side = 1, at = c(1,8), labels = c("no agreement","total agreement"))

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