hensm: Friendly interface to softmax regression under Henery model.

In ohenery: Modeling of Ordinal Random Variables via Softmax Regression

Friendly interface to softmax regression under Henery model.

Description

A user friendly interface to the softmax regression under the Henery model.

Usage

hensm(
  formula,
  data,
  group = NULL,
  weights = NULL,
  ngamma = 4,
  fit0 = NULL,
  reg_wt = NULL,
  reg_zero = NULL,
  reg_power = NULL,
  reg_coef_idx = NULL,
  reg_standardize = FALSE,
  na.action = na.omit
)

## S3 method for class 'hensm'
vcov(object, ...)

## S3 method for class 'hensm'
print(x, ...)

Arguments

formula	an object of class "formula" (or one that can be coerced to that class): a symbolic description of the model to be fitted. The details of model specification are given under ‘Details’.
data	an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which lm is called.
group	the string name of the group variable in the data, or a bare character with the group name. The group indices need not be integers, but that is more efficient. They need not be sorted.
weights	an optional vector of weights, or the string or bare name of the weights in the data for use in the fitting process. The weights are attached to the outcomes, not the participant. Set to NULL for none.
ngamma	The number of gammas to fit. Should be at least 2.
fit0	An optional object of class hensm or of harsm with the initial fit estimates. These will be used for ‘warm start’ of the estimation procedure. A warm start should only speed up estimation, not change the ultimate results. When there is mismatch between the coefficients in fit0 and the model being fit here, the missing coefficients are initialized as zero. If ngamma is NULL and fit0 is given, we default to the number of gammas in the initial fit, otherwise we fill any missing gammas with 1. If a harsm object is given, then ngamma must be non-null.
reg_wt	the multiplicative weight(s) of the regularization terms. must be non-negative. May be a scalar or vector, but will be recycled to the length of the reg_coef_idx.
reg_zero	the ‘zero’ of the regularization terms. This would usually be zero if you want to shrink to zero, but in some cases you may with to shrink to 1 for example. May be a scalar or vector, but will be recycled to the length of the reg_coef_idx. If NULL is given, defaults to zeroes for beta terms, and ones for gamma terms in a Henery model fit, all zeroes for Harville model fits.
reg_power	the power of the regularization terms, 2 for ridge regression, 1 for lasso.
reg_coef_idx	the index of the coefficient which the corresponding regularization term is applied to. For the Harville model, the indices only refer to the \beta coefficient vector. For the Henery model, the indices refer to the beta coefficient vector and \gamma coefficient vector concatenated together. The other regularization parameters may be recycled as needed, but not the reg_coef_idx values.
reg_standardize	if true, the reg_wt are normalized, or ‘standardized’ with respect to the standard deviation of the corresponding columns of the design matrix. That is, the weight used is the given weight times the standard deviation of the corresponding independent variable to the corresponding reg_power. Only terms associated with the betas are so normalized.
na.action	How to deal with missing values in y, g, X, wt, eta0.
object	an object of class hensm.
...	For lm(): additional arguments to be passed to the low level regression fitting functions (see below).
x	an object used to select a method.

Details

Performs a softmax regression by groups, via Maximum Likelihood Estimation. It is assumed that successive sub-races maintain the proportional probability of the softmax, up to some gamma coefficients, \gamma_2, \gamma_3, ..., \gamma_n, which we fit. This model nests the Harville model fit by harsm, by fixing all the gammas equal to 1.

Value

An object of class hensm, but also of maxLik with the fit.

Regularization

The regularization term is of the form

\sum_i w_i |\nu_{c_i} - z_i|^{p_i},

where w_i are the reg_wt weights, z_i are the reg_zero zeroes, p_i are the reg_power powers, and c_i are the reg_coef_idx coefficient indices. Note that the coefficient indices can be repeated so that the regularization term can include multiple contributions from one coefficient element. This allows ‘elasticnet’ regularization. The \nu here refer to the regression coefficients \beta concatenated with the \gamma coefficients in the Henery model.

Note

The fit functions return an object of type maxLik even when regularization penalties are applied; the statistical inference functions are not valid when regularization is used. The user is warned.

This regression may give odd results when the outcomes are tied, imposing an arbitrary order on the tied outcomes. Moreover, no warning may be issued in this case. In future releases, ties may be dealt with differently, perhaps in analogy to how ties are treated in the Cox Proportional Hazards regression, using the methods of Breslow or Efron.

To avoid incorrect inference when only the top performers are recorded, and all others are effectively tied, one should use weighting. Set the weights to zero for participants who are tied non-winners, and one for the rest So for example, if you observe the Gold, Silver, and Bronze medal winners of an Olympic event that had a starting field of 12 participants, set weights to 1 for the medal winners, and 0 for the others. Note that the weights do not attach to the participants, they attach to the place they took.

Since version 0.1.0 of this package, the normalization of weights used in this function have changed under the hood. This is to give correct inference in the case where zero weights are used to signify finishing places were not observed. If in doubt, please confirm inference by simulations, taking as example the simulations in the README.

When regularization is used, the first gamma coefficient is not shrunk, as it always equals one in the Henery model, and is not estimated from the data.

Author(s)

Steven E. Pav [e-mail]

See Also

harsm, smlik.

Examples

nfeat <- 5
set.seed(1234)
g <- ceiling(seq(0.1,1000,by=0.1))
X <- matrix(rnorm(length(g) * nfeat),ncol=nfeat)
beta <- rnorm(nfeat)
eta <- X %*% beta
# 2FIX: do rhenery
y <- rsm(eta,g)
# now the pretty frontend
data <- cbind(data.frame(outcome=y,race=g),as.data.frame(X))

fmla <- outcome ~ V1 + V2 + V3 + V4 + V5
fitm <- hensm(fmla,data,group=race)

# with offset
eta0 <- rowMeans(X)
data <- cbind(data.frame(outcome=y,race=g,eta0=eta0),as.data.frame(X))
fmla <- outcome ~ offset(eta0) + V1 + V2 + V3 + V4 + V5
fitm <- hensm(fmla,data,group=race)

# on horse race data
library(dplyr)
data(race_data)
df <- race_data %>%
	group_by(EventId) %>%
		mutate(eta0=log(WN_pool / sum(WN_pool))) %>%
	ungroup() %>%
	mutate(weights=ifelse(!is.na(Finish),1,0)) %>%
	mutate(fac_age=cut(Age,c(0,3,5,7,Inf),include.lowest=TRUE))

# Henery Model with market efficiency
hensm(Finish ~ eta0,data=df,group=EventId,weights=weights,ngamma=3)

# look for age effect not captured by consensus odds.
fmla <- Finish ~ offset(eta0) + fac_age
fit0 <- hensm(fmla,data=df,group=EventId,weights=weights,ngamma=2)
# allow warm start.
fit1 <- hensm(fmla,data=df,group=EventId,weights=weights,fit0=fit0,ngamma=2)
# allow warm start with more gammas.
fit2 <- hensm(fmla,data=df,group=EventId,weights=weights,fit0=fit0,ngamma=3)
# or a different formula
fit3 <- hensm(update(fmla,~ . + PostPosition),data=df,group=EventId,weights=weights,fit0=fit0)

# warm start from harsm object
fit0_har <- harsm(fmla,data=df,group=EventId,weights=weights)
fit4 <- hensm(fmla,data=df,group=EventId,fit0=fit0_har,weights=weights)

# regularization examples
fmla <- Finish ~ offset(eta0) + fac_age + PostPosition
# no regularization
fitr0 <- hensm(fmla,data=df,group=EventId,weights=weights,ngamma=3)
# ridge regression on the betas (there are two)
fitr2 <- hensm(fmla,data=df,group=EventId,weights=weights,ngamma=3,
  reg_wt=rep(1,2), reg_power=2, reg_zero=0, reg_coef_idx=c(1,2))
# l1 regression on the betas (there are two)
fitr1 <- hensm(fmla,data=df,group=EventId,weights=weights,ngamma=3,
  reg_wt=rep(1,2), reg_power=1, reg_zero=0, reg_coef_idx=c(1,2))
# elasticnet regression on the betas (there are two)
fitr12 <- hensm(fmla,data=df,group=EventId,weights=weights,ngamma=3,
  reg_wt=rep(1,4), reg_power=c(1,1,2,2), reg_zero=0, reg_coef_idx=c(1,2,1,2))
# l1 regression on the gammas, shrinking to one (there are two estimated)
fitrg1 <- hensm(fmla,data=df,group=EventId,weights=weights,ngamma=3,
  reg_wt=rep(1,2), reg_power=1, reg_zero=1, reg_coef_idx=c(3,4))
# l2 regression on the betas and gammas, shrinking beta to 0, gamma to 1.
fitrg2 <- hensm(fmla,data=df,group=EventId,weights=weights,ngamma=3,
  reg_wt=rep(1,4), reg_power=2, reg_zero=c(0,0,1,1), reg_coef_idx=1:4)

ohenery documentation built on Sept. 9, 2025, 5:56 p.m.