2-parameter Gamma Regression Family Function

Estimates the 2-parameter gamma distribution by maximum likelihood estimation.

Usage

gamma2(lmu = "loglink", lshape = "loglink", imethod = 1,  ishape = NULL,
       parallel = FALSE, deviance.arg = FALSE, zero = "shape")

Arguments

lmu, lshape

Link functions applied to the (positive) mu and shape parameters (called $\mu$ and $a$ respectively). See Links for more choices.

ishape

Optional initial value for shape. A NULL means a value is computed internally. If a failure to converge occurs, try using this argument. This argument is ignored if used within cqo; see the iShape argument of qrrvglm.control instead.

imethod

An integer with value 1 or 2 which specifies the initialization method for the $\mu$ parameter. If failure to converge occurs try another value (and/or specify a value for ishape).

deviance.arg

Logical. If TRUE, the deviance function is attached to the object. Under ordinary circumstances, it should be left alone because it really assumes the shape parameter is at the maximum likelihood estimate. Consequently, one cannot use that criterion to minimize within the IRLS algorithm. It should be set TRUE only when used with cqo under the fast algorithm.

zero

See CommonVGAMffArguments for information.

parallel

Details at CommonVGAMffArguments. If parallel = TRUE then the constraint is not applied to the intercept.

Details

This distribution can model continuous skewed responses. The density function is given by $$f(y;\mu,a) = \frac{\exp(-a y / \mu) \times (a y / \mu)^{a-1} \times a}{ \mu \times \Gamma(a)}$$ for $\mu > 0$, $a > 0$ and $y > 0$. Here, $\Gamma(\cdot)$ is the gamma function, as in gamma. The mean of Y is $\mu=\mu$ (returned as the fitted values) with variance $\sigma^2 = \mu^2 / a$. If $0<a<1$ then the density has a pole at the origin and decreases monotonically as $y$ increases. If $a=1$ then this corresponds to the exponential distribution. If $a>1$ then the density is zero at the origin and is unimodal with mode at $y = \mu - \mu / a$; this can be achieved with lshape="logloglink".

By default, the two linear/additive predictors are $\eta_1=\log(\mu)$ and $\eta_2=\log(a)$. This family function implements Fisher scoring and the working weight matrices are diagonal.

This VGAM family function handles multivariate responses, so that a matrix can be used as the response. The number of columns is the number of species, say, and zero=-2 means that all species have a shape parameter equalling a (different) intercept only.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm and vgam.

References

The parameterization of this VGAM family function is the 2-parameter gamma distribution described in the monograph

McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. London: Chapman & Hall.

Author

T. W. Yee

Note

The response must be strictly positive. A moment estimator for the shape parameter may be implemented in the future.

If mu and shape are vectors, then rgamma(n = n, shape = shape, scale = mu/shape) will generate random gamma variates of this parameterization, etc.; see GammaDist.

Examples

# Essentially a 1-parameter gamma
gdata <- data.frame(y = rgamma(n = 100, shape = exp(1)))
fit1 <- vglm(y ~ 1, gamma1, data = gdata)
fit2 <- vglm(y ~ 1, gamma2, data = gdata, trace = TRUE, crit = "coef")
#> Iteration 1: coefficients = 1.3814077, 1.3970382
#> Iteration 2: coefficients = 1.05717456, 0.42129181
#> Iteration 3: coefficients = 0.99173966, 0.84887395
#> Iteration 4: coefficients = 0.98950307, 1.00245425
#> Iteration 5: coefficients = 0.98950057, 1.01713645
#> Iteration 6: coefficients = 0.98950057, 1.01725643
#> Iteration 7: coefficients = 0.98950057, 1.01725644
coef(fit2, matrix = TRUE)
#>             loglink(mu) loglink(shape)
#> (Intercept)   0.9895006       1.017256
c(Coef(fit2), colMeans(gdata))
#>       mu    shape        y 
#> 2.689891 2.765597 2.689891 

# Essentially a 2-parameter gamma
gdata <- data.frame(y = rgamma(n = 500, rate = exp(-1), shape = exp(2)))
fit2 <- vglm(y ~ 1, gamma2, data = gdata, trace = TRUE, crit = "coef")
#> Iteration 1: coefficients = 3.1629118, 2.7427322
#> Iteration 2: coefficients = 2.9880070, 1.0416195
#> Iteration 3: coefficients = 2.9708050, 1.6282963
#> Iteration 4: coefficients = 2.9706553, 1.9172174
#> Iteration 5: coefficients = 2.9706553, 1.9712896
#> Iteration 6: coefficients = 2.9706553, 1.9728766
#> Iteration 7: coefficients = 2.9706553, 1.9728779
#> Iteration 8: coefficients = 2.9706553, 1.9728779
coef(fit2, matrix = TRUE)
#>             loglink(mu) loglink(shape)
#> (Intercept)    2.970655       1.972878
c(Coef(fit2), colMeans(gdata))
#>        mu     shape         y 
#> 19.504697  7.191343 19.504697 
summary(fit2)
#> 
#> Call:
#> vglm(formula = y ~ 1, family = gamma2, data = gdata, trace = TRUE, 
#>     crit = "coef")
#> 
#> Coefficients: 
#>               Estimate Std. Error z value Pr(>|z|)    
#> (Intercept):1  2.97066    0.01668  178.13   <2e-16 ***
#> (Intercept):2  1.97288    0.06183   31.91   <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Names of linear predictors: loglink(mu), loglink(shape)
#> 
#> Log-likelihood: -1677.582 on 998 degrees of freedom
#> 
#> Number of Fisher scoring iterations: 8 
#>