clm2

Cumulative link models

A new improved implementation of CLMs is available in clm.

Fits cumulative link models with an additive model for the location and a multiplicative model for the scale. The function allows for structured thresholds. A popular special case of a CLM is the proportional odds model. In addition to the standard link functions, two flexible link functions, "Arandar-Ordaz" and "log-gamma" are available, where an extra link function parameter provides additional flexibility. A subset of the predictors can be allowed to have nominal rather than ordinal effects. This has been termed "partial proportional odds" when the link is the logistic.

Usage

clm2(location, scale, nominal, data, weights, start, subset,
    na.action, contrasts, Hess = TRUE, model,
    link = c("logistic", "probit", "cloglog", "loglog",
    "cauchit", "Aranda-Ordaz", "log-gamma"), lambda,
    doFit = TRUE, control,
    threshold = c("flexible", "symmetric", "equidistant"), ...)

Details

There are methods for the standard model-fitting functions, including summary, vcov, predict, anova, logLik, profile, plot.profile, confint, update, dropterm, addterm, and an extractAIC method.

The design of the implementation is inspired by an idea proposed by Douglas Bates in the talk "Exploiting sparsity in model matrices" presented at the DSC conference in Copenhagen, July 14 2009. Basically an environment is set up with all the information needed to optimize the likelihood function. Extractor functions are then used to get the value of likelihood at current or given parameter values and to extract current values of the parameters. All computations are performed inside the environment and relevant variables are updated during the fitting process. After optimizer termination relevant variables are extracted from the environment and the remaining are discarded.

Some aspects of clm2, for instance, how starting values are obtained, and of the associated methods are inspired by polr from package MASS.

References

Agresti, A. (2002) Categorical Data Analysis. Second edition. Wiley.

Aranda-Ordaz, F. J. (1983) An Extension of the Proportional-Hazards Model for Grouped Data. Biometrics, 39, 109-117.

Genter, F. C. and Farewell, V. T. (1985) Goodness-of-link testing in ordinal regression models. The Canadian Journal of Statistics, 13(1), 37-44.

Christensen, R. H. B., Cleaver, G. and Brockhoff, P. B. (2011) Statistical and Thurstonian models for the A-not A protocol with and without sureness. Food Quality and Preference, 22, pp. 542-549.

Aliases

• clm2

Examples

# NOT RUN {
options(contrasts = c("contr.treatment", "contr.poly"))

## A tabular data set:
(tab26 <- with(soup, table("Product" = PROD, "Response" = SURENESS)))
dimnames(tab26)[[2]] <- c("Sure", "Not Sure", "Guess", "Guess", "Not Sure", "Sure")
dat26 <- expand.grid(sureness = as.factor(1:6), prod = c("Ref", "Test"))
dat26$wghts <- c(t(tab26))

m1 <- clm2(sureness ~ prod, scale = ~prod, data = dat26,
          weights = wghts, link = "logistic")

## print, summary, vcov, logLik, AIC:
m1
summary(m1)
vcov(m1)
logLik(m1)
AIC(m1)
coef(m1)
coef(summary(m1))

## link functions:
m2 <- update(m1, link = "probit")
m3 <- update(m1, link = "cloglog")
m4 <- update(m1, link = "loglog")
m5 <- update(m1, link = "cauchit", start = coef(m1))
m6 <- update(m1, link = "Aranda-Ordaz", lambda = 1)
m7 <- update(m1, link = "Aranda-Ordaz")
m8 <- update(m1, link = "log-gamma", lambda = 1)
m9 <- update(m1, link = "log-gamma")

## nominal effects:
mN1 <-  clm2(sureness ~ 1, nominal = ~ prod, data = dat26,
            weights = wghts, link = "logistic")
anova(m1, mN1)

## optimizer / method:
update(m1, scale = ~ 1, method = "Newton")
update(m1, scale = ~ 1, method = "nlminb")
update(m1, scale = ~ 1, method = "optim")
# }
# NOT RUN {
## threshold functions
mT1 <- update(m1, threshold = "symmetric")
mT2 <- update(m1, threshold = "equidistant")
anova(m1, mT1, mT2)

## Extend example from polr in package MASS:
## Fit model from polr example:
if(require(MASS)) {
    fm1 <- clm2(Sat ~ Infl + Type + Cont, weights = Freq, data = housing)
    fm1
    summary(fm1)
    ## With probit link:
    summary(update(fm1, link = "probit"))
    ## Allow scale to depend on Cont-variable
    summary(fm2 <- update(fm1, scale =~ Cont))
    anova(fm1, fm2)
    ## which seems to improve the fit
}

#################################
## It is possible to fit multinomial models (i.e. with nominal
## effects) as the following example shows:
if(require(nnet)) {
    (hous1.mu <- multinom(Sat ~ 1, weights = Freq, data = housing))
    (hous1.clm <- clm2(Sat ~ 1, weights = Freq, data = housing))

    ## It is the same likelihood:
    all.equal(logLik(hous1.mu), logLik(hous1.clm))

    ## and the same fitted values:
    fitHous.mu <-
        t(fitted(hous1.mu))[t(col(fitted(hous1.mu)) == unclass(housing$Sat))]
    all.equal(fitted(hous1.clm), fitHous.mu)

    ## The coefficients of multinom can be retrieved from the clm2-object
    ## by:
    Pi <- diff(c(0, plogis(hous1.clm$xi), 1))
    log(Pi[2:3]/Pi[1])

    ## A larger model with explanatory variables:
    (hous.mu <- multinom(Sat ~ Infl + Type + Cont, weights = Freq, data = housing))
    (hous.clm <- clm2(Sat ~ 1, nominal = ~ Infl + Type + Cont, weights = Freq,
                      data = housing))

    ## Almost the same likelihood:
    all.equal(logLik(hous.mu), logLik(hous.clm))

    ## And almost the same fitted values:
    fitHous.mu <-
        t(fitted(hous.mu))[t(col(fitted(hous.mu)) == unclass(housing$Sat))]
    all.equal(fitted(hous.clm), fitHous.mu)
    all.equal(round(fitted(hous.clm), 5), round(fitHous.mu), 5)
}

# }