harsm: Friendly interface to softmax regression under Harville...

In ohenery: Modeling of Ordinal Random Variables via Softmax Regression

Friendly interface to softmax regression under Harville model.

Description

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

Usage

harsm(
  formula,
  data,
  group = NULL,
  weights = NULL,
  fit0 = NULL,
  na.action = na.omit
)

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

## S3 method for class 'harsm'
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.
fit0	An optional object of class harsm or of hensm 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.
na.action	How to deal with missing values in the outcomes, groups, weights, etc.
object	an object of class harsm.
...	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, under the Harville model. We fit \beta where odds are \eta = x^{\top}\beta for independent variables x. The probability of taking first place is then \mu=c\exp{\eta}, where the c is chosen so the \mu sum to one. Under the Harville model, conditional on the first place finisher having been observed, the probability model for second (and successive) places with the probabilities of the remaining participants renormalized.

The print method of the harsm object includes a display of the R-squared. This measures the improvement in squared errors of the expected rank from the model over the null model which posits that all odds are equal. When the formula includes an offset, a ‘delta R-squared’ is also output. This is the improvement in predicted ranks over the model based on the offset term. Note that the expected ranks are only easy to produce under the Harville model; under the Henery model, the summary R-squared statistics are not produced. Note that this computation uses weighted sums of squares, as the weights contribute to the likelihood term. However, the square sum computation does not take into account the standard error of the rank, and so unlike in linear regression, the softmax regression does not always give positive R-squareds, and the statistic is otherwise hard to interpret.

Value

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

Note

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.

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.

Author(s)

Steven E. Pav [e-mail]

See Also

harsmfit, harsmlik.

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
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 <- harsm(fmla,data,group=race)

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 <- harsm(fmla,data,group=race)

# with weights
data <- cbind(data.frame(outcome=y,race=g,eta0=eta0),as.data.frame(X))
data$wts <- runif(nrow(data),min=1,max=2)
fmla <- outcome ~ offset(eta0) + V1 + V2 + V3 + V4 + V5
fitm <- harsm(fmla,data,group=race,weights=wts)

# softmax on the Best Picture data
data(best_picture)
df <- best_picture
df$place <- ifelse(df$winner,1,2)
df$weight <- ifelse(df$winner,1,0)

fmla <- place ~ nominated_for_BestDirector + nominated_for_BestActor + Drama 
fit0 <- harsm(fmla,data=df,group=year,weights=weight) 

# warm start is a thing:
sub_fmla <- place ~ nominated_for_BestDirector + nominated_for_BestActor 
fit1 <- harsm(sub_fmla,data=df,group=year,weights=weight,fit0=fit0) 


# test against logistic regression
if (require(dplyr)) {
nevent <- 10000
set.seed(1234)
adf <- data_frame(eventnum=floor(seq(1,nevent + 0.7,by=0.5))) %>%
  mutate(x=rnorm(n()),
         program_num=rep(c(1,2),nevent),
         intercept=as.numeric(program_num==1),
         eta=1.5 * x + 0.3 * intercept,
         place=ohenery::rsm(eta,g=eventnum))

# Harville model
modh <- harsm(place ~ intercept + x,data=adf,group=eventnum)

# the collapsed data.frame for glm
ddf <- adf %>%
  arrange(eventnum,program_num) %>%
  group_by(eventnum) %>%
    summarize(resu=as.numeric(first(place)==1),
              delx=first(x) - last(x),
              deli=first(intercept) - last(intercept)) %>%
  ungroup()

# glm logistic fit
modg <- glm(resu ~ delx + 1,data=ddf,family=binomial(link='logit'))

all.equal(as.numeric(coef(modh)),as.numeric(coef(modg)),tolerance=1e-4)
all.equal(as.numeric(vcov(modh)),as.numeric(vcov(modg)),tolerance=1e-4)
}

ohenery documentation built on Oct. 25, 2024, 9:07 a.m.