betaff {VGAM} | R Documentation |
Estimation of the mean and precision parameters of the beta distribution.
betaff(A=0, B=1, lmu=if(A==0 & B==1) "logit" else "elogit", lphi="loge", emu=if(lmu=="elogit") list(min=A,max=B) else list(), ephi=list(), imu=NULL, iphi=NULL, method.init=1, zero=NULL)
A, B |
Lower and upper limits of the distribution.
The defaults correspond to the standard beta distribution
where the response lies between 0 and 1.
|
lmu, lphi |
Link function for the mean and precision parameters.
See below for more details.
See Links for more choices.
|
emu, ephi |
List. Extra argument for the respective links.
See earg in Links for general information.
|
imu, iphi |
Optional initial value for the mean and precision parameters
respectively. A NULL value means a value is obtained in the
initialize slot.
|
method.init, zero |
See CommonVGAMffArguments for more information.
|
The two-parameter beta distribution can be written f(y) =
(y-A)^(mu1*phi-1) * (B-y)^((1-mu1)*phi-1) / [beta(mu1*phi,(1-mu1)*phi) * (B-A)^(phi-1)]
for A < y < B, and beta(.,.) is the beta function
(see beta
).
The parameter mu1 satisfies
mu1 = (mu - A) / (B-A)
where mu is the mean of Y.
That is, mu1 is the mean of of a standard beta distribution:
E(Y) = A + (B-A)*mu1,
and these are the fitted values of the object.
Also, phi is positive and A < mu < B.
Here, the limits A and B are known.
Another parameterization of the beta distribution involving the raw
shape parameters is implemented in beta.ab
.
For general A and B, the variance of Y is (B-A)^2 * mu1 * (1-mu1) / (1+phi). Then phi can be interpreted as a precision parameter in the sense that, for fixed mu, the larger the value of phi, the smaller the variance of Y. Also, mu1=shape1/(shape1+shape2) and phi = shape1+shape2.
Fisher scoring is implemented.
If A and B are unknown then the VGAM family function
beta4()
can be used to estimate these too.
An object of class "vglmff"
(see vglmff-class
).
The object is used by modelling functions such as vglm
,
and vgam
.
The response must have values in the interval (A, B).
Thomas W. Yee
Ferrari, S. L. P. and Francisco C.-N. (2004) Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31, 799–815.
Documentation accompanying the VGAM package at http://www.stat.auckland.ac.nz/~yee contains further information and examples.
beta.ab
,
Beta
,
genbetaII
,
betaII
,
betabin.ab
,
betageometric
,
betaprime
,
rbetageom
,
rbetanorm
,
kumar
,
beta4
,
elogit
.
y = rbeta(n <- 1000, shape1=exp(0), shape2=exp(1)) fit = vglm(y ~ 1, betaff, trace = TRUE) coef(fit, matrix=TRUE) Coef(fit) # Useful for intercept-only models # General A and B, and with a covariate x = runif(n <- 1000) mu = logit(0.5-x, inverse=TRUE) prec = exp(3+x) # phi shape2 = prec * (1-mu) shape1 = mu * prec y = rbeta(n, shape1=shape1, shape2=shape2) Y = 5 + 8 * y # From 5 to 13, not 0 to 1 fit = vglm(Y ~ x, betaff(A=5,B=13), trace=TRUE) coef(fit, mat=TRUE)