mvord: Multivariate Ordinal Regression Models.

Description

Multivariate ordinal regression models in the R package mvord can be fitted using the function mvord(). Two different data structures can be passed on to mvord() through the use of two different multiple measurement objects MMO and MMO2 in the left-hand side of the model formula. MMO uses a long data format, which has the advantage that it allows for varying covariates across multiple measurements. This flexibility requires the specification a subject index as well as a multiple measurement index. In contrast to MMO, the function MMO2 has a simplified data structure, but is only applicable in settings where the covariates do not vary between the multiple measurements. In this case, the multiple ordinal observations as well as the covariates are stored in different columns of a data.frame. We refer to this data structure as wide data format.

Usage

mvord(formula, data, error.structure = cor_general(~1), link = mvprobit(),
  response.levels = NULL, coef.constraints = NULL, coef.values = NULL,
  threshold.constraints = NULL, threshold.values = NULL,
  weights.name = NULL, offset = NULL, PL.lag = NULL, contrasts = NULL,
  control = mvord.control())

Arguments

formula

an object of class formula of the form y ~ X1 + ... + Xp.

data

data.frame containing a subject index, an index for the multiple measurements, an ordinal response y and covariates X1, ..., Xp.

error.structure

different error.structures: general correlation structure (default) cor_general(~1), general covariance structure cov_general(~ 1), factor dependent correlation structure cov_general(~ f), factor dependent covariance structure cov_general(~ f), a constant cor_equi(~ 1) or a covariate dependent equicorrelation structure cor_equi(~ S), AR(1) correlation structure cor_ar1(~ 1) or a covariate dependent AR(1) correlation structure cor_ar1(~ S). See error_struct or 'Details'.

link

specifies the link function by mvprobit() (multivariate normally distributed errors - default) or mvlogit(df = 8) (multivariate logistically distributed errors), where df specifies the degrees of freedom of the t copula.

response.levels

(optional) list of length equal to the number of multiple measurements to specify the category labels in case of varying categories across multiple measurements

coef.constraints

(optional) vector or matrix of constraints on the regression coefficients. See 'Details'.

coef.values

(optional) matrix setting fixed values on the regression coefficients. See 'Details'.

threshold.constraints

(optional) vector of constraints on the threshold parameters. See 'Details'.

threshold.values

(optional) list of (optional) fixed values for the threshold parameters. See 'Details'.

weights.name

(optional) character string with the column name of subject-specific weights in data which need to be constant across multiple measurements. Negative weights are not allowed.

offset

(optional) this can be used to specify an a priori known component to be included in the linear predictor during fitting. This should be NULL or a numeric vector of length equal to the number of cases. One or more offset terms can be included in the formula instead or as well, and if more than one is specified their sum is used. See model.offset.

PL.lag

(optional) specifies the time lag of the pairs in the pairwise likelihood approach to be optimized (can be used with cor_ar1).

contrasts

(optional) an optional list. See the contrasts.arg of model.matrix.default.

control

(optional) a list of parameters for controlling the fitting process. See mvord.control for details.

Value

The function mvord returns an object of class "mvord".

The functions summary and print are used to display the results. The function coef extracts the regression coefficients, a function thresholds the threshold coefficients and the function error_structure returns the estimated parameters of the corresponding error structure.

An object of class "mvord" is a list containing the following components:

beta
a named matrix of regression coefficients
theta

a named list of threshold parameters
error.struct
an object of class error_struct containing the parameters of the error structure
sebeta
a named matrix of the standard errors of the regression coefficients
setheta
a named list of the standard errors of the threshold parameters
seerror.struct
a vector of standard errors for the parameters of the error structure
rho
a list of all objects that are used in mvord()

Details

Implementation MMO:

data: In MMO we use a long format for the input of data, where each row contains a subject index (i), a multiple measurement index (j), an ordinal observation (Y) and all the covariates (X1 to Xp). This long format data structure is internally transformed to a matrix of responses which contains NA in the case of missing entries and a list of covariate matrices. This is performed by the multiple measurement object MMO(Y, i, j) specifying the column names of the subject index and the multiple measurement index in data. The column containing the ordinal observations can contain integer or character values or can be of class (ordered) 'factor'. When using the long data structure, this column is basically a concatenated vector of each of the multiple ordinal responses. Internally, this vector is then split according to the measurement index. Then the ordinal variable corresponding to each measurement index is transformed into an ordered factor. For an integer or a character vector the natural ordering is used (ascending, or alphabetical). If for character vectors the alphabetical order does not correspond to the ordering of the categories, the optional argument response.levels allows to specify the levels for each response explicitly. This is performed by a list of length q, where each element contains the names of the levels of the ordered categories in ascending (or if desired descending) order. If all the multiple measurements use the same number of classes and same labelling of the classes, the column Y can be stored as an ordered 'factor' (as it is often the case in longitudinal studies). The order of the multiple measurements is needed when specifying constraints on the thresh- old or regression parameters. This order is based on the type of the multiple measurement index column in data. For 'integer', 'character' or 'factor' the natural ordering is used (ascending, or alphabetical). If a different order of the multiple responses is desired, the multiple measurement index column should be an ordered factor with a corresponding ordering of the levels.
If the categories differ across multiple measurements (either the number of categories or the category labels) one needs to specify the response.levels explicitly. This is performed by a list of length $J$ (number of multiple measurements), where each element contains the names of the levels of the ordered categories in ascending or descending order.
```
response.levels = list(c("G","F","E", "D", "C", "B", "A"),
                       c("G","F","E", "D", "C", "B", "A"),
                       c("O","N","M","L", "K", "J", "I", "H"))
```
formula The ordinal responses (e.g., rating) are passed by a formula object. Intercepts can be included or excluded in the model depending on the model paramterization:
- Model without intercept: If the intercept should be removed the formula for a given response (rating) and covariates (X1 to Xp) has the following form:
  
  formula = MMO(rating, firm_id, rater_id) ~ 0 + X1 + ... + Xp.
- Model with intercept: If one wants to include an intercept in the model, there are two equivalent possibilities to set the model formula. Either one includes the intercept explicitly by:
  
  formula = MMO(rating, firm_id, rater_id) ~ 1 + X1 + ... + Xp,
  
  or by
  
  formula = MMO(rating, firm_id, rater_id) ~ X1 + ... + Xp.

Implementation MMO2:

data:The data structure applied by MMO2 is slightly simplified, where the multiple ordinal observations as well as the covariates are stored as columns in a data.frame. Each subject $i$ corresponds to one row of the data frame, where all outcomes (with missing observations set to NA) and all the covariates are stored in different columns. Ideally each outcome column is of type ordered factor. For column types like 'integer', 'character' or 'factor' a warning is given and the natural ordering is used (ascending, or alphabetical).
formula The ordinal responses (e.g., rating) are passed by a formula object. Intercepts can be included or excluded in the model depending on the model parameterization:
formula = MMO2(rater1, rater2, rater3) ~ X1 + ... + Xp.

error.structure

We allow for different error structures depending on the model parameterization:

Correlation:
- cor_general The most common parameterization is the general correlation matrix.
  
  error.structure = cor_general(~ 1)
  
  This parameterization can be extended by allowing a factor dependent correlation structure, where the correlation of each subject $i$ depends on a given subject-specific factor f. This factor f is not allowed to vary across multiple measurements $j$ for the same subject $i$ and due to numerical constraints only up to maximum 30 levels are allowed.
  
  error.structure = cor_general(~ f)
- cor_equi A covariate dependent equicorrelation structure, where the correlations are equal across all $J$ dimensions and depend on subject-specific covariates S1, ..., Sm. It has to be noted that these covariates S1, ..., Sm are not allowed to vary across multiple measurements $j$ for the same subject $i$.
  
  error.structure = cor_equi(~ S1 + ... + Sm)
- cor_ar1 In order to account for some heterogeneity the $AR(1)$ error structure is allowed to depend on covariates X1, ..., Xp that are constant over time for each subject $i$.
  
  error.structure = cor_ar1(~ S1 + ... + Sm)
Covariance:
- cov_general
  
  In case of a full variance-covariance parameterization the standard parameterization with a full variance-covariance is obtained by:
  
  error.structure = cov_general(~ 1)
  
  This parameterization can be extended to the factor dependent covariance structure, where the covariance of each subject depends on a given factor f:
  
  error.structure = cov_general(~ f)

coef.constraints

The package supports constraints on the regression coefficients. Firstly, the user can specify whether the regression coefficients should be equal across some or all response dimensions. Secondly, the values of some of the regression coefficients can be fixed.

As there is no unanimous way to specify such constraints, we offer two options. The first option is similar to the specification of constraints on the thresholds. The constraints can be specified in this case as a vector or matrix of integers, where coefficients getting same integer value are set equal. Values of the regression coefficients can be fixed through a matrix. Alternatively constraints on the regression coefficients can be specified by using the design employed by the VGAM package. The constraints in this setting are set through a named list, where each element of the list contains a matrix full-column rank. If the values of some regression coefficients should be fixed, offsets can be used. This design has the advantage that it supports constraints on outcome-specific as well as category-specific regression coefficients. While the first option has the advantage of requiring a more concise input, it does not support category-specific coefficients. The second option offers a more flexible design in this respect. For further information on the second option we refer to the vignette and to the documentation of vglm.

Using the first option, constraints can be specified by a vector or a matrix coef.constraints. First, a simple and less flexible way by specifying a vector coef.constraints of dimension $J$. This vector is allocated in the following way: The first element of the vector coef.constraints gets a value of 1. If the coefficients of the multiple measurement $j = 2$ should be equal to the coefficients of the first dimension ($j=1$) again a value of 1 is set. If the coefficients should be different to the coefficients of the first dimension a value of 2 is set. In analogy, if the coefficients of dimensions two and three should be the same one sets both values to 2 and if they should be different, a value of 3 is set. Constraints on the regression coefficients of the remaining multiple measurements are set analogously.

coef.constraints <- c(1,1,2,3)

This vector coef.constraints sets the coefficients of the first two raters equal $$\beta_{1\cdot} = \beta_{2\cdot}$$ A more flexible way to specify constraints on the regression coefficients is a matrix with $J$ rows and $p$ columns, where each column specifies constraints on one of the $p$ coefficients in the same way as above. In addition, a value of NA excludes a corresponding coefficient (meaning it should be fixed to zero).

coef.constraints <- cbind(c(1,2,3,4), c(1,1,1,2), c(NA,NA,NA,1),
                          c(1,1,1,NA), c(1,2,3,4), c(1,2,3,4))

This matrix coef.constraints gives the following constraints:

$\beta_{12} = \beta_{22} = \beta_{32}$
$\beta_{13} = 0$
$\beta_{23} = 0$
$\beta_{33} = 0$
$\beta_{44} = 0$
$\beta_{14} = \beta_{24} = \beta_{34}$

coef.values

In addition, specific values on regression coefficients can be set in the matrix coef.values. Parameters are removed if the value is set to zero (default for NA's in coef.constraints) or to some fixed value. If constraints on parameters are set, these dimensions need to have the same value in coef.values. Again each column corresponds to one regression coefficient.

Together with the coef.constraints from above we impose:

coef.constraints <- cbind(c(1,2,2), c(1,1,2), c(NA,1,2),
                          c(NA,NA,NA), c(1,1,2))

coef.values <- cbind(c(NA,NA,NA), c(NA,NA,NA), c(0,NA,NA),
                     c(1,1,1), c(NA,NA,NA))

Interaction terms: When constraints on the regression coefficient should be specified in models with interaction terms, the coef.constraints matrix has to be expanded manually. In case of interaction terms (specified either by X1 + X2 + X1:X2 or equivalently by X1*X2), one additional column at the end of coef.constraints for the interaction term has to be specified for numerical variables. For interaction terms including factor variables suitably more columns have to be added to the coef.constraints matrix.

threshold.constraints

Similarly, constraints on the threshold parameters can be imposed by a vector of positive integers, where dimensions with equal threshold parameters get the same integer. When restricting the thresholds of two outcome dimensions to be the same, one has to be careful that the number of categories in the two outcome dimensions must be the same. In our example with $J=4$ different outcomes we impose:

threshold.constraints <- c(1,1,2)

gives the following restrictions:

$\bm\theta_{1} = \bm\theta_{2}$
$\bm\theta_{3}$ arbitrary.

threshold.values

In addition, threshold parameter values can be specified by threshold.values in accordance with identifiability constraints. For this purpose we use a list with $J$ elements, where each element specifies the constraints of the particular dimension by a vector of length of the number of threshold parameters (number of categories - 1). A number specifies a threshold parameter to a specific value and NA leaves the parameter flexible. For data_mvord we have

threshold.constraints <- NULL

threshold.values <- list(c(-4,NA,NA,NA,NA,4.5),
                         c(-4,NA,NA,NA,NA,4.5),
                         c(-5,NA,NA,NA,NA,NA,4.5))

References

Hirk R, Hornik K, Vana L (2020). <U+201C>mvord: An R Package for Fitting Multivariate Ordinal Regression Models.<U+201D> Journal of Statistical Software, 93(4), 1<U+2013>41, 10.18637/jss.v093.i04.

Examples

Run this code

# NOT RUN {
library(mvord)

#toy example
data(data_mvord_toy)

#wide data format with MMO2
res <- mvord(formula = MMO2(Y1, Y2) ~ 0 + X1 + X2,
             data = data_mvord_toy)
print(res)
summary(res)
thresholds(res)
coefficients(res)
head(error_structure(res))

# convert data_mvord_toy into long format
df <- cbind.data.frame("i" = rep(1:100,2), "j" = rep(1:2,each = 100),
                       "Y" = c(data_mvord_toy$Y1,data_mvord_toy$Y2),
                       "X1" = rep(data_mvord_toy$X1,2),
                       "X2" = rep(data_mvord_toy$X2,2))

#for long format data, use MMO instead of MMO2
res <- mvord(formula = MMO(Y, i, j) ~ 0 + X1 + X2, #or formula = MMO(Y) ~ 0 + X1 + X2
               data = df)
print(res)
summary(res)
thresholds(res)
coefficients(res)
head(error_structure(res))
# }
# NOT RUN {
res2 <- mvord(formula = MMO(Y) ~ 0 + X1 + X2,
               data = df,
               control = mvord.control(solver = "BFGS"),
               threshold.constraints = c(1,1),
               coef.constraints = c(1,1))
print(res2)
summary(res2)
thresholds(res2)
coefficients(res2)
head(error_structure(res2))

## examples
#load data
data(data_mvord)
head(data_mvord)

#-------------
# cor_general
#-------------
# approx 2 min
res_cor <- mvord(formula = MMO(rating) ~ 0 + X1 + X2 + X3 + X4 + X5,
                 data = data_mvord,
                 coef.constraints = cbind(c(1,2,2),
                                          c(1,1,2),
                                          c(NA,1,2),
                                          c(NA,NA,NA),
                                          c(1,1,2)),
                 coef.values = cbind(c(NA,NA,NA),
                                     c(NA,NA,NA),
                                     c(0,NA,NA),
                                     c(1,1,1),
                                     c(NA,NA,NA)),
                 threshold.constraints = c(1,1,2),
                 control = mvord.control(solver = "newuoa"))
print(res_cor)
summary(res_cor)
thresholds(res_cor)
coefficients(res_cor)
head(error_structure(res_cor))

#-------------
# cov_general
#-------------
#approx 4 min
res_cov <- mvord(formula = MMO(rating) ~ 1 + X1 + X2 + X3 + X4 + X5,
                 data = data_mvord,
                 error.structure = cov_general(~1),
                 threshold.values = list(c(-4,NA,NA,NA,NA,4.5),
                                         c(-4,NA,NA,NA,NA,4),
                                         c(-5,NA,NA,NA,NA,NA,4.5))
) #does not converge with BFGS

print(res_cov)
summary(res_cov)
thresholds(res_cov)
coefficients(res_cov)
head(error_structure(res_cov))


#-------------
# cor_ar1
#-------------
#approx 4min
data(data_mvord_panel)
head(data_mvord_panel)

#select subset of data
subset_dat <- data_mvord_panel$year %in% c("year3", "year4", "year5", "year6", "year7")
data_mvord_panel <- data_mvord_panel[subset_dat,]

mult.obs <- 5
res_AR1 <- mvord(formula = MMO(rating) ~ 0 + X1 + X2 + X3 + X4 + X5,
                 data = data_mvord_panel,
                 error.structure = cor_ar1(~1),
                 threshold.constraints = c(1,1,1,2,2),
                 coef.constraints = c(1,1,1,2,2),
                 control = mvord.control(solver = "BFGS"))
print(res_AR1)
summary(res_AR1)
thresholds(res_AR1)
coefficients(res_AR1)
head(error_structure(res_AR1))
head(error_structure(res_AR1, type = "corr"))

data(data_mvord2)
# approx 2 min
res_cor <- mvord(formula = MMO2(rater1, rater2, rater3) ~ 0 + X1 + X2 + X3 + X4 + X5,
                 data = data_mvord2,
                 coef.constraints = cbind(c(1,2,2),
                                          c(1,1,2),
                                          c(NA,1,2),
                                          c(NA,NA,NA),
                                          c(1,1,2)),
                 coef.values = cbind(c(NA,NA,NA),
                                     c(NA,NA,NA),
                                     c(0,NA,NA),
                                     c(1,1,1),
                                     c(NA,NA,NA)),
                 threshold.constraints = c(1,1,2),
                 control = mvord.control(solver = "newuoa"))
print(res_cor)
summary(res_cor)
thresholds(res_cor)
coefficients(res_cor)
head(error_structure(res_cor))
# }
# NOT RUN {
# }

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