gammaRMean {VGAMextra}R Documentation

2–parameter Gamma Distribution

Description

Estimates the 2–parameter gamma distribution by maximum likelihood estimation by modelling its mean.

Usage

        gammaRMean(zero = "rate", lmu = "gammaRMeanlink",
                   lrate = "loge", lshape = NULL,
                   irate = NULL,   ishape = NULL,
                   lss = TRUE)
                       

Arguments

zero

Specifies the parameters to be modelled as intercept–only.

See CommonVGAMffArguments for further information.

lmu

The link function applied to model the mean of this distribution: gammaRMeanlink.

lrate, lshape, irate, ishape, lss

Same as gammaR.

Details

This family function slightly enlarges the functionalities of gammaR allowing the extent of the linear predictor to model the mean of the 2–parameter gamma distribution, with density

f(y; alpha, beta) = rate^(shape) exp(-rate * y) y^(shape - 1) / Γ(shape),

Here, shape and rate are positive shape and rate parameters as in gammaR. The mean is given by μ = shape / rate.

Unlike gammaR where the default linear predictors are η1 = log (shape) and η2 = log (rate), this family function re–defines this structure by setting up η1 = gammaRMeanlink(shape; rate) and η2 = log (rate) by default, where gammaRMeanlink is the link function for the mean of Y.

To mimic the work of gammaR, set lmu = NULL and lshape = "loge". Particularly, lss works exactly as with gammaR

For further choices on link functions for η2, see Links.

This family function also differs from gamma2. The latter is re-parametrization of the gamma distribution to estimate μ and shape. That is, the density is re–expressed in terms of μ and shape specifically.

Notice, to model the mean of the gamma distribution with this family function, the link gammaRMeanlink must be necessarily used via lmu. Here, lmu overrides the work of lshape. Then, the transformed mean

gammaRMeanlink(shape; rate)

are returned as the fitted values, for estimated shape and rate.

Value

An object of class "vglm". See vglm-class for full details.

Note

The fitted values returned are gammaRMeanlink–transformed, provided the mean is modelled via lmu

The parameters shape and rate match with the arguments shape and rate of rgamma.

Multiple responses are handled.

Author(s)

V. Miranda and Thomas W. Yee.

References

Yee, T. W. (2015) Vector Generalized Linear and Additive Models: With an Implementation in R. Springer, New York, USA.

See Also

gammaRMeanlink, CommonVGAMffArguments, gammaR, gamma2, Links.

Examples

 
  ### Modelling the mean in terms of x2, two responses.
  
    set.seed(2017022101)
    nn <- 80
    x2 <- runif(nn)
    mu <- exp(1 + 0.5 * x2)
  
  # Shape and rate parameters in terms of 'mu'
    rate <- rep(exp(1), nn)
    shape <- gammaRMeanlink(theta = mu, rate = rate,
                            inverse = TRUE, deriv = 0)
  
  # Generate some random data
    y1 <- rgamma(n = nn, shape = shape, rate =  rate)
    gdata <- data.frame(x2 = x2, y1 = y1)
    rm(y1)

  # lmu = "gammaRMeanlink" replaces lshape, whilst lrate = "loge"
    fit1 <- vglm(cbind(y1, y1) ~ x2,
                 gammaRMean(lmu = "gammaRMeanlink", lss = FALSE, zero = "rate"),
                 data = gdata, trace = TRUE, crit = "log")
     coef(fit1, matrix = TRUE)
     summary(fit1)
    
  # Compare fitted values with true values.
    compare1 <- cbind(fitted.values(fit1)[, 1, drop = FALSE], mu)
    colnames(compare1) <- c("Fitted.vM1", "mu")
    head(compare1)
 
  
  ### Mimicking gammaR. Notice lmu = NULL.
    fit2 <- vglm(y1 ~ x2, gammaRMean(lmu = NULL, lshape = "loge",
                                     lrate = "loge", lss = FALSE, zero = "rate"),
                 data = gdata, trace = TRUE, crit = "log")
 
  # Compare fitted values with true values.
    compare2 <- with(gdata, cbind(fitted.values(fit2), y1, mu))
    colnames(compare2) <- c("Fitted.vM2", "y", "mu")
    head(compare2)
 
    
  ### Fitted values -- Model1 vs Fitted values -- Model2
    fit1vsfit2 <- cbind(fitted.values(fit1)[, 1, drop = FALSE], 
                        fitted.values(fit2))
    colnames(fit1vsfit2) <- c("Fitted.vM1", "Fitted.vM2")
    head(fit1vsfit2)


[Package VGAMextra version 0.0-1 Index]