# GAMLSS model for a proportion response variable with point(s) mass at 0 and or 1. Function `gamlssInf0to1()` allows to fit inflated gamlss models when the response variable distribution is defined in the intervals [0,1), (0,1] and [0,1]. The `gamlssInf0to1` model for inflated proportion variables is a `gamlss` model provided of up to two extra parameters for the mass point(s). In the case of inflation point at zero (one), this is equivalent to fit two separate models, a gamlss model for the (0,1) part, and a logit model for zero (one) vs non-zero (non-one) part. When both zero and one are present, a multinomial model is involved to fit the non-(0,1) part. ```r gamlssInf0to1(y = NULL, mu.formula = ~1, sigma.formula = ~1, nu.formula = ~1,tau.formula = ~1, xi0.formula = ~1,xi1.formula = ~1, data = NULL, family = BE, weights = rep(1, length(Y_)), trace = FALSE, ...) ``` ## Arguments - `y`: the proportion response variable with inflation at zero and/or one - `mu.formula`: a model formula for `mu` - `sigma.formula`: a model formula for `sigma` - `nu.formula`: a model formula for `nu` - `tau.formula`: a model formula for `tau` - `xi0.formula`: a model formula for the probability at zero - `xi1.formula`: a model formula for the probability at one - `data`: a data frame containing the variables occurring in the formula. - `family`: any `gamlss` distribution family defined in (0,1) - `weights`: a vector of weights as in gamlss - `trace`: logical, if TRUE information on model estimation will be printed during the fitting - ``...``: for extra parameters ## Details The default family is a Beta distribution (BE), but other (0,1) distributions can be used, e.g. those generated from existing continuous gamlss family distributions by using `gen.Family` with link "logit". ## Returns returns a `gamlssInf0to1` object which has its own methods ## References Hossain, A., Stasinopoulos, M., Rigby, R. and Enea, M. (2015). Centile estimation for a proportion response variable. **Statistics in Medicine**, doi: 10.1002/sim.6748. Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), **Appl. Statist.**, 54 , part 3, pp 507-554. Stasinopoulos D. M., Rigby R.A. and Akantziliotou C. (2006) Instructions on how to use the GAMLSS package in R. Accompanying documentation in the current GAMLSS help files, (see also [http://www.gamlss.org/](http://www.gamlss.org/)). Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. **Journal of Statistical Software**, Vol. 23 , Issue 7, Dec 2007, [http://www.jstatsoft.org/v23/i07](http://www.jstatsoft.org/v23/i07). ## Author(s) Mikis Stasinopoulos, Robert Rigby, Abu Hossain and Marco Enea ## See Also `gamlss.family`, `BEINF`, `BE`, `BEo`, `BEZI`, `BEOI` ## Examples ```r # 1. An artificial example using simulated data # Firstly, we use function gen.Family() to create the logit skew # student t (logitSST) distribution defined in the (0,1) interval, # and function gen.Inf0to1() to create the 0-inflated logitSST # distribution defined in [0,1). gen.Family("SST", "logit") gen.Inf0to1("logitSST","Zero") #now we can generate the data and run the model set.seed(10) Y <- rlogitSSTInf0(500,mu=0.5,sigma=0.7,nu=0.5,tau=5,xi0=0.5,log=FALSE) dat <- data.frame(Y) dat$x <- rnorm(500) m1 <- gamlssInf0to1(y=Y,mu.formula=~x, sigma.formula=~x, nu.formula=~x, tau.formula=~x, xi0.formula=~x,data=dat, family=logitSST) summary(m1) # 2. Example of equivalent gamlss models for an inflated-at-1 Beta distribution Y <- rBEINF1(500,mu=0.5,sigma=0.7,nu=0.5) m2 <- gamlss(Y~1,sigma.formula=~1,nu.formula=~1,family=BEINF1) m3.1 <- gamlss(Y[Y<1]~1,sigma.formula=~1,family=BE) m3.2 <- gamlss(I(Y==1)~1,family=BI) m4 <- gamlssInf0to1(Y,mu.formula=~1,sigma.formula=~1,xi1=~1,family=BE) stopifnot(all.equal(deviance(m2),(deviance(m3.1)+deviance(m3.2))), all.equal(deviance(m2),deviance(m4))) ```

gamlssInf0to1 function