Fits ordinal regression models with elastic net penalty by coordinate descent. Supported model families include cumulative probability, stopping ratio, continuation ratio, and adjacent category. These families are a subset of vector glm's which belong to a model class we call the elementwise link multinomial-ordinal (ELMO) class. Each family in this class links a vector of covariates to a vector of class probabilities. Each of these families has a parallel form, which is appropriate for ordinal response data, as well as a nonparallel form that is appropriate for an unordered categorical response, or as a more flexible model for ordinal data. The parallel model has a single set of coefficients, whereas the nonparallel model has a set of coefficients for each response category except the baseline category. It is also possible to fit a model with both parallel and nonparallel terms, which we call the semi-parallel model. The semi-parallel model has the flexibility of the nonparallel model, but the elastic net penalty shrinks it toward the parallel model.
ordinalNet(
x,
y,
alpha = 1,
standardize = TRUE,
penaltyFactors = NULL,
positiveID = NULL,
family = c("cumulative", "sratio", "cratio", "acat"),
reverse = FALSE,
link = c("logit", "probit", "cloglog", "cauchit"),
customLink = NULL,
parallelTerms = TRUE,
nonparallelTerms = FALSE,
parallelPenaltyFactor = 1,
lambdaVals = NULL,
nLambda = 20,
lambdaMinRatio = 0.01,
includeLambda0 = FALSE,
alphaMin = 0.01,
pMin = 1e-08,
stopThresh = 1e-08,
threshOut = 1e-08,
threshIn = 1e-08,
maxiterOut = 100,
maxiterIn = 100,
printIter = FALSE,
printBeta = FALSE,
warn = TRUE,
keepTrainingData = TRUE
)Covariate matrix. It is recommended that categorical covariates are converted to a set of indicator variables with a variable for each category (i.e. no baseline category); otherwise the choice of baseline category will affect the model fit.
Response variable. Can be a factor, ordered factor, or a matrix where each row is a multinomial vector of counts. A weighted fit can be obtained using the matrix option, since the row sums are essentially observation weights. Non-integer matrix entries are allowed.
The elastic net mixing parameter, with 0 <= alpha <= 1.
alpha=1 corresponds to the lasso penalty, and alpha=0 corresponds
to the ridge penalty.
If standardize=TRUE, the predictor variables are
scaled to have unit variance. Coefficient estimates are returned on the
original scale.
Optional nonnegative vector of penalty factors with
length equal to the number of columns in x. If this argument is used,
then the penalty for each variable is scaled by its corresponding factor.
If NULL, the penalty factor is one for each coefficient.
Logical vector indicating whether each coefficient should
be constrained to be non-negative. If NULL, the default value is FALSE
for all coefficients.
Specifies the type of model family. Options are "cumulative" for cumulative probability, "sratio" for stopping ratio, "cratio" for continuation ratio, and "acat" for adjacent category.
Logical. If TRUE, then the "backward" form of the model
is fit, i.e. the model is defined with response categories in reverse order.
For example, the reverse cumulative model with \(K+1\) response categories
applies the link function to the cumulative probabilities \(P(Y \ge 2),
\ldots, P(Y \ge K+1)\), rather then \(P(Y \le 1), \ldots, P(Y \le K)\).
Specifies the link function. The options supported are logit,
probit, complementary log-log, and cauchit. Only used if customLink=NULL.
Optional list containing a vectorized link function g,
a vectorized inverse link h, and the Jacobian function of the inverse link
getQ. The Jacobian should be defined as \(\partial h(\eta) / \partial \eta^T\)
(as opposed to the transpose of this matrix).
Logical. If TRUE, then parallel coefficient terms
will be included in the model. parallelTerms and nonparallelTerms
cannot both be FALSE.
Logical. if TRUE, then nonparallel coefficient terms
will be included in the model. parallelTerms and nonparallelTerms
cannot both be FALSE.
Nonnegative numeric value equal to one by
default. The penalty on all parallel terms is scaled by this factor (as well
as variable-specific penaltyFactors). Only used if
parallelTerms=TRUE.
An optional user-specified lambda sequence (vector). If NULL,
a sequence will be generated based on nLambda and lambdaMinRatio.
In this case, the maximum lambda is the smallest value that sets all penalized
coefficients to zero, and the minimum lambda is the maximum value multiplied
by the factor lambdaMinRatio.
Positive integer. The number of lambda values in the solution path.
Only used if lambdaVals=NULL.
A factor greater than zero and less than one. Only used
if lambdaVals=NULL.
Logical. If TRUE, then zero is added to the end
of the sequence of lambdaVals. This is not done by default because
it can significantly increase computational time. An unpenalized saturated model
may have infinite coefficient solutions, in which case the fitting algorithm
will still terminate when the relative change in log-likelihood becomes small.
Only used if lambdaVals=NULL.
max(alpha, alphaMin) is used to calculate the starting
lambda value when lambdaVals=NULL. In this case, the default lambda
sequence begins with the smallest lambda value such that all penalized
coefficients are set to zero (i.e. the value where the first penalized
coefficient enters the solution path). The purpose of this argument is to
help select a starting value for the lambda sequence when alpha = 0,
because otherwise it would be infinite. Note that alphaMin is only
used to determine the default lamba sequence and that the model is always fit
using alpha to calculate the penalty.
Value greater than zero, but much less than one. During the optimization routine, the Fisher information is calculated using fitted probabilities. For this calculation, fitted probabilities are capped below by this value to prevent numerical instability.
In the relative log-likelihood change between successive lambda values falls below this threshold, then the last model fit is used for all remaining lambda.
Convergence threshold for the coordinate descent outer loop.
The optimization routine terminates when the relative change in the
penalized log-likelihood between successive iterations falls below this threshold.
It is recommended to set theshOut equal to threshIn.
Convergence threshold for the coordinate descent inner loop. Each
iteration consists of a single loop through each coefficient. The inner
loop terminates when the relative change in the penalized approximate
log-likelihood between successive iterations falls below this threshold.
It is recommended to set theshOut equal to threshIn.
Maximum number of outer loop iterations.
Maximum number of inner loop iterations.
Logical. If TRUE, the optimization routine progress is
printed to the terminal.
Logical. If TRUE, coefficient estimates are printed
after each coordinate descent outer loop iteration.
Logical. If TRUE, the following warning message is displayed
when fitting a cumulative probability model with nonparallelTerms=TRUE
(i.e. nonparallel or semi-parallel model).
"Warning message: For out-of-sample data, the cumulative probability model
with nonparallelTerms=TRUE may predict cumulative probabilities that are not
monotone increasing."
The warning is displayed by default, but the user may wish to disable it.
Logical. If TRUE, then x and y
are saved with the returned "ordinalNet" object. This allows
predict.ordinalNet to return fitted values for the training data
without passing a newx argument.
An object with S3 class "ordinalNet". Model fit information can be accessed
through the coef, predict, and summary methods.
Matrix of coefficient estimates, with each row corresponding to a lambda value.
(If covariates were scaled with standardize=TRUE, the coefficients are
returned on the original scale).
Sequence of lambda values. If user passed a sequence to the
lambdaVals, then it is this sequence. If lambdaVals argument
was NULL, then it is the sequence generated.
Log-likelihood of each model fit.
Number of nonzero coefficients of each model fit, including intercepts.
AIC, defined as -2*loglik + 2*nNonzero.
BIC, defined as -2*loglik + log(N)*nNonzero.
Percentage deviance explained, defined as \(1 - loglik/loglik_0\), where \(loglik_0\) is the log-likelihood of the null model.
Number of coordinate descent outer loop iterations until convergence for each lambda value.
Number of coordinate descent inner loop iterations on last outer loop for each lambda value.
Relative improvement in objective function on last outer loop
for each lambda value. Can be used to diagnose convergence issues. If iterOut
reached maxiterOut and dif is large, then maxiterOut should
be increased. If dif is negative, this means the objective did not improve
between successive iterations. This usually only occurs when the model is
saturated and/or close to convergence, so a small negative value is not of concern.
(When this happens, the algorithm is terminated for the current lambda value,
and the coefficient estimates from the previous outer loop iteration are returned.)
Number of response categories.
Number of covariates in x.
Covariate names.
List of arguments passed to the ordinalNet function.
The ordinalNet function fits regression models for a categorical response
variable with \(K+1\) levels. Conditional on the covariate vector \(x_i\)
(the \(i^{th}\) row of x), each observation has a vector of \(K+1\)
class probabilities \((p_{i1}, \ldots, p_{i(K+1)})\). These probabilities
sum to one, and can therefore be parametrized by \(p_i = (p_{i1}, \ldots, p_{iK})\).
The probabilities are mapped to a set of \(K\) quantities
\(\delta_i = (\delta_{i1}, \ldots, \delta_{iK}) \in (0, 1)^K\), which depends on the choice
of model family. The elementwise link function maps
\(\delta_i\) to a set of \(K\) linear predictors. Together, the family
and link specifiy a link function between \(p_i\) and \(\eta_i\).
Model families:
Let \(Y\) denote the random response variable for a single observation, conditional on the covariates values of the observation. The random variable \(Y\) is discrete with support {\(1, \ldots, K+1\)}. The following model families are defined according to these mappings between the class probabilities and the values \(\delta_1, \ldots, \delta_K\):
\(\delta_j = P(Y \le j)\)
\(\delta_j = P(Y \ge j + 1)\)
\(\delta_j = P(Y = j | Y \ge j)\)
\(\delta_j = P(Y=j + 1 | Y \le j + 1)\)
\(\delta_j = P(Y > j | Y \ge j)\)
\(\delta_j = P(Y < j | Y \le j)\)
\(\delta_j = P(Y = j + 1 | j \le Y \le j+1)\)
\(\delta_j = P(Y = j | j \le Y \le j+1)\)
Parallel, nonparallel, and semi-parallel model forms:
Models within each of these families can take one of three forms, which have different definitions for the linear predictor \(\eta_i\). Suppose each \(x_i\) has length \(P\). Let \(b\) be a length \(P\) vector of regression coefficients. Let \(B\) be a \(P \times K\) matrix of regression coefficient. Let \(b_0\) be a vector of \(K\) intercept terms. The three model forms are the following:
\(\eta_i = b_0 + b^T x_i\) (parallelTerms=TRUE, nonparallelTerms=FALSE)
\(\eta_i = b_0 + B^T x_i\) (parallelTerms=FALSE, nonparallelTerms=TRUE)
\(\eta_i = b_0 + b^T x_i + B^T x_i\) (parallelTerms=TRUE, nonparallelTerms=TRUE)
The parallel form has the defining property of ordinal models, which is that a single linear combination \(b^T x_i\) shifts the cumulative class probabilities \(P(Y \le j)\) in favor of either higher or lower categories. The linear predictors are parallel because they only differ by their intercepts (\(b_0\)). The nonparallel form is a more flexible model, and it does not shift the cumulative probabilities together. The semi-parallel model is equivalent to the nonparallel model, but the elastic net penalty shrinks the semi-parallel coefficients toward a common value (i.e. the parallel model), as well as shrinking all coefficients toward zero. The nonparallel model, on the other hand, simply shrinks all coefficients toward zero. When the response categories are ordinal, any of the three model forms could be applied. When the response categories are unordered, only the nonparallel model is appropriate.
Elastic net penalty:
The elastic net penalty is defined for each model form as follows. \(\lambda\)
and \(\alpha\) are the usual elastic net tuning parameters, where \(\lambda\)
determines the degree to which coefficients are shrunk toward zero, and \(\alpha\)
specifies the amound of weight given to the L1 norm and squared L2 norm penalties.
Each covariate is allowed a unique penalty factor \(c_j\), which is specified with the
penaltyFactors argument. By default \(c_j = 1\) for all \(j\).
The semi-parallel model has a tuning parameter \(\rho\) which determines the degree to
which the parallel coefficients are penalized. Small values of \(\rho\) will
result in a fit closer to the parallel model, and large values of \(\rho\)
will result in a fit closer to the nonparallel model.
\(\lambda \sum_{j=1}^P c_j \{ \alpha |b_j| + \frac{1}{2} (1-\alpha) b_j^2 \}\)
\(\lambda \sum_{j=1}^P c_j \{ \sum_{k=1}^K \alpha |B_{jk}| + \frac{1}{2} (1-\alpha) B_{jk}^2 \}\)
\(\lambda [ \rho \sum_{j=1}^P c_j \{ \alpha |b_j| + \frac{1}{2} (1-\alpha) b_j^2 \} + \sum_{j=1}^P c_j \{ \sum_{k=1}^K \alpha |B_{jk}| + \frac{1}{2} (1-\alpha) B_{jk}^2 \}]\)
ordinalNet minimizes the following objective function. Let \(N\) be
the number of observations, which is defined as the sum of the y elements
when y is a matrix.
$$objective = -1/N*loglik + penalty$$
# NOT RUN {
# Simulate x as independent standard normal
# Simulate y|x from a parallel cumulative logit (proportional odds) model
set.seed(1)
n <- 50
intercepts <- c(-1, 1)
beta <- c(1, 1, 0, 0, 0)
ncat <- length(intercepts) + 1 # number of response categories
p <- length(beta) # number of covariates
x <- matrix(rnorm(n*p), ncol=p) # n x p covariate matrix
eta <- c(x %*% beta) + matrix(intercepts, nrow=n, ncol=ncat-1, byrow=TRUE)
invlogit <- function(x) 1 / (1+exp(-x))
cumprob <- t(apply(eta, 1, invlogit))
prob <- cbind(cumprob, 1) - cbind(0, cumprob)
yint <- apply(prob, 1, function(p) sample(1:ncat, size=1, prob=p))
y <- as.factor(yint)
# Fit parallel cumulative logit model
fit1 <- ordinalNet(x, y, family="cumulative", link="logit",
parallelTerms=TRUE, nonparallelTerms=FALSE)
summary(fit1)
coef(fit1)
coef(fit1, matrix=TRUE)
predict(fit1, type="response")
predict(fit1, type="class")
# Fit nonparallel cumulative logit model
fit2 <- ordinalNet(x, y, family="cumulative", link="logit",
parallelTerms=FALSE, nonparallelTerms=TRUE)
fit2
coef(fit2)
coef(fit2, matrix=TRUE)
predict(fit2, type="response")
predict(fit2, type="class")
# Fit semi-parallel cumulative logit model (with both parallel and nonparallel terms)
fit3 <- ordinalNet(x, y, family="cumulative", link="logit",
parallelTerms=TRUE, nonparallelTerms=TRUE)
fit3
coef(fit3)
coef(fit3, matrix=TRUE)
predict(fit3, type="response")
predict(fit3, type="class")
# }
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