Finite Mixtures of Beta Regression for Rates and Proportions

Fit finite mixtures of beta regression models for rates and proportions via maximum likelihood with the EM algorithm using a parametrization with mean (depending through a link function on the covariates) and precision parameter (called phi).

Usage

betamix(formula, data, k, subset, na.action, weights, offset,
  link = c("logit", "probit", "cloglog", "cauchit", "log",
    "loglog"), link.phi = "log",
  control = betareg.control(...), cluster = NULL,
  FLXconcomitant = NULL, FLXcontrol = list(), verbose = FALSE,
  nstart = if (is.null(cluster)) 3 else 1, which = "BIC", 
  ID, fixed, extra_components, ...)

extraComponent(type = c("uniform", "betareg"), coef, delta,
  link = "logit", link.phi = "log")

Arguments

formula: symbolic description of the model (of type y ~ x or y ~ x | z; for details see betareg).
data, subset, na.action: arguments controlling formula processing via model.frame.
weights: optional numeric vector of integer case weights.
offset: optional numeric vector with an a priori known component to be included in the linear predictor for the mean.
k: a vector of integers indicating the number of components of the finite mixture; passed in turn to the k argument of stepFlexmix.
link: character specification of the link function in the mean model (mu). Currently, "logit", "probit", "cloglog", "cauchit", "log", "loglog" are supported. Alternatively, an object of class "link-glm" can be supplied.
link.phi: character specification of the link function in the precision model (phi). Currently, "identity", "log", "sqrt" are supported. The default is "log" unless formula is of type y ~ x where the default is "identity" (for backward compatibility). Alternatively, an object of class "link-glm" can be supplied.
control: a list of control arguments specified via betareg.control.
cluster: Either a matrix with k columns of initial cluster membership probabilities for each observation; or a factor or integer vector with the initial cluster assignments of observations at the start of the EM algorithm. Default is random assignment into k clusters.
FLXconcomitant: concomitant variable model; object of class FLXP. Default is the object returned by calling FLXPconstant. The argument FLXconcomitant can be omitted if formula is a three-part formula of type y ~ x | z | w, where w specificies the concomitant variables.
FLXcontrol: object of class "FLXcontrol" or a named list; controls the EM algorithm and passed in turn to the control argument of flexmix.
verbose: a logical; if TRUE progress information is shown for different starts of the EM algorithm.
nstart: for each value of k run stepFlexmix nstart times and keep only the solution with maximum likelihood.
which: number of model to get if k is a vector of integers longer than one. If character, interpreted as number of components or name of an information criterion.
ID: grouping variable indicating if observations are from the same individual, i.e. the component membership is restricted to be the same for these observations.
fixed: symbolic description of the model for the parameters fixed over components (of type ~ x | z).
extra_components: a list containing objects returned by extraComponent().
...: arguments passed to betareg.control.
type: specifies if the component follows a uniform distribution or a beta regression model.
coef: a vector with the coefficients to determine the midpoint of the uniform distribution or names list with the coefficients for the mean and precision of the beta regression model.
delta: numeric; half-length of the interval of the uniform distribution.

Details

The arguments and the model specification are similar to betareg. Internally stepFlexmix is called with suitable arguments to fit the finite mixture model with the EM algorithm. See Grün et al. (2012) for more details.

extra_components is a list where each element corresponds to a component where the parameters are fixed a-priori.

Value

An object of class "flexmix" containing the best model with respect to the log likelihood or the one selected according to which if k is a vector of integers longer than 1.

References

Cribari-Neto F, Zeileis A (2010). Beta Regression in R. Journal of Statistical Software, 34(2), 1–24. doi:10.18637/jss.v034.i02

Grün B, Kosmidis I, Zeileis A (2012). Extended Beta Regression in R: Shaken, Stirred, Mixed, and Partitioned. Journal of Statistical Software, 48(11), 1–25. doi:10.18637/jss.v048.i11

Grün B, Leisch F (2008). FlexMix Version 2: Finite Mixtures with Concomitant Variables and Varying and Constant Parameters. Journal of Statistical Software, 28(4), 1–35. doi:10.18637/jss.v028.i04

Leisch F (2004). FlexMix: A General Framework for Finite Mixture Models and Latent Class Regression in R. Journal of Statistical Software, 11(8), 1–18. doi:10.18637/jss.v011.i08

Author

Bettina Grün and Achim Zeileis

Examples

options(digits = 4)

## data with two groups of dyslexic and non-dyslexic children
data("ReadingSkills", package = "betareg")

suppressWarnings(RNGversion("3.5.0"))
set.seed(4040)
## try to capture accuracy ~ iq relationship (without using dyslexia
## information) using two beta regression components and one additional
## extra component for a perfect reading score
rs_mix <- betamix(accuracy ~ iq, data = ReadingSkills, k = 3,
  nstart = 10, extra_components = extraComponent(type = "uniform",
  coef = 0.99, delta = 0.01))

## visualize result
## intensities based on posterior probabilities
prob <- 2 * (posterior(rs_mix)[cbind(1:nrow(ReadingSkills),
   clusters(rs_mix))] - 0.5)
## associated HCL colors
col0 <- hcl(c(260, 0, 130), 65, 45, fixup = FALSE)
col1 <- col0[clusters(rs_mix)]
col2 <- hcl(c(260, 0, 130)[clusters(rs_mix)], 65 * abs(prob)^1.5,
   95 - 50 * abs(prob)^1.5, fixup = FALSE)
## scatter plot
plot(accuracy ~ iq, data = ReadingSkills, col = col2, pch = 19,
   cex = 1.5, xlim = c(-2, 2))
points(accuracy ~ iq, data = ReadingSkills, cex = 1.5, pch = 1,
   col = col1)
## fitted lines
iq <- -30:30/10
cf <- rbind(coef(rs_mix, model = "mean", component = 1:2),
   c(qlogis(0.99), 0))
for(i in 1:3)
   lines(iq, plogis(cf[i, 1] + cf[i, 2] * iq), lwd = 2,
         col = col0[i])


## refit the model including a concomitant variable model using the
## dyslexia information with some noise to avoid complete separation
## between concomitant variable and component memberships
set.seed(4040)
w <- rnorm(nrow(ReadingSkills), 
           c(-1, 1)[as.integer(ReadingSkills$dyslexia)])

## The argument FLXconcomitant can be omitted when specifying
## the model via a three part formula given by
## accuracy ~ iq | 1 | w
## The posteriors from the previously fitted model are used
## for initialization.
library("flexmix")
rs_mix2 <- betamix(accuracy ~ iq, data = ReadingSkills,
  extra_components = extraComponent(type = "uniform",
  coef = 0.99, delta = 0.01), cluster = posterior(rs_mix),
  FLXconcomitant = FLXPmultinom(~w))
coef(rs_mix2, which = "concomitant")
#>   (Intercept)       w
#> 1      0.0000  0.0000
#> 2      0.8114  1.0778
#> 3     -0.1195 -0.3488
summary(rs_mix2, which = "concomitant")
#> $Comp.2
#>             Estimate Std. Error z value Pr(>|z|)  
#> (Intercept)    0.821      0.882    0.93     0.35  
#> w              1.078      0.497    2.17     0.03 *
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> $Comp.3
#>             Estimate Std. Error z value Pr(>|z|)
#> (Intercept)   -0.113      0.812   -0.14     0.89
#> w             -0.346      0.452   -0.77     0.44
#>