cauchit                 package:VGAM                 R Documentation

_C_a_u_c_h_i_t _L_i_n_k _F_u_n_c_t_i_o_n

_D_e_s_c_r_i_p_t_i_o_n:

     Computes the cauchit (tangent) link transformation, including its
     inverse and the first two derivatives.

_U_s_a_g_e:

     cauchit(theta, earg = list(bvalue= .Machine$double.eps),
             inverse = FALSE, deriv = 0, short = TRUE, tag = FALSE)

_A_r_g_u_m_e_n_t_s:

   theta: Numeric or character. See below for further details. 

    earg: List. Extra argument for passing in additional information.
          Values of 'theta' which are less than or equal to 0 can be
          replaced by the 'bvalue' component of the list 'earg' before
          computing the link function value. Values of 'theta' which
          are greater than or equal to 1 can be replaced by 1 minus the
          'bvalue' component of the list 'earg' before computing the
          link function value. The component name 'bvalue' stands for
          ``boundary value''. See 'Links' for general information about
          'earg'.

 inverse: Logical. If 'TRUE' the inverse function is computed.

   deriv: Order of the derivative. Integer with value 0, 1 or 2.

   short: Used for labelling the 'blurb' slot of a 'vglmff-class'
          object.

     tag: Used for labelling the linear/additive predictor in the
          'initialize' slot of a 'vglmff-class' object. Contains a
          little more information if 'TRUE'.

_D_e_t_a_i_l_s:

     This link function is an alternative link function for parameters
     that lie in the unit interval. This type of link bears the same
     relation to the Cauchy distribution as the probit link bears to
     the Gaussian. One characteristic of this link function is that the
     tail is heavier relative to the other links (see examples below).

     Numerical values of 'theta' close to 0 or 1 or out of range result
     in 'Inf', '-Inf', 'NA' or 'NaN'.  The arguments 'short' and 'tag'
     are used only if 'theta' is character.

_V_a_l_u_e:

     For 'deriv = 0', the tangent of 'theta', i.e., 'tan(pi *
     (theta-0.5))' when 'inverse = FALSE', and if 'inverse = TRUE' then
     '0.5 + atan(theta)/pi'.

     For 'deriv = 1', then the function returns _d_ 'theta' / _d_ 'eta'
     as a function of 'theta' if 'inverse = FALSE', else if 'inverse =
     TRUE' then it returns the reciprocal.

_N_o_t_e:

     Numerical instability may occur when 'theta' is close to 1 or 0.
     One way of overcoming this is to use 'earg'.

     As mentioned above, in terms of the threshold approach with
     cumulative probabilities for an ordinal response this link
     function corresponds to the Cauchy distribution (see 'cauchy1').

_A_u_t_h_o_r(_s):

     Thomas W. Yee

_R_e_f_e_r_e_n_c_e_s:

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

_S_e_e _A_l_s_o:

     'logit', 'probit', 'cloglog', 'loge', 'cauchy1'.

_E_x_a_m_p_l_e_s:

     p = seq(0.01, 0.99, by=0.01)
     cauchit(p)
     max(abs(cauchit(cauchit(p), inverse=TRUE) - p)) # Should be 0

     p = c(seq(-0.02, 0.02, by=0.01), seq(0.97, 1.02, by=0.01))
     cauchit(p)  # Has no NAs

     ## Not run: 
     par(mfrow=c(2,2))
     y = seq(-4, 4, length=100)

     for(d in 0:1) {
         matplot(p, cbind(logit(p, deriv=d), probit(p, deriv=d)),
                 type="n", col="purple", ylab="transformation",
                 lwd=2, las=1, main=if(d==0) "Some probability link functions"
                 else "First derivative")
         lines(p, logit(p, deriv=d), col="limegreen", lwd=2)
         lines(p, probit(p, deriv=d), col="purple", lwd=2)
         lines(p, cloglog(p, deriv=d), col="chocolate", lwd=2)
         lines(p, cauchit(p, deriv=d), col="tan", lwd=2)
         if(d==0) {
             abline(v=0.5, h=0, lty="dashed")
             legend(0, 4.5, c("logit", "probit", "cloglog", "cauchit"),
                    col=c("limegreen","purple","chocolate", "tan"), lwd=2)
         } else
             abline(v=0.5, lty="dashed")
     }

     for(d in 0) {
         matplot(y, cbind(logit(y, deriv=d, inverse=TRUE),
                          probit(y, deriv=d, inverse=TRUE)),
                 type="n", col="purple", xlab="transformation", ylab="p",
                 main=if(d==0) "Some inverse probability link functions"
                 else "First derivative", lwd=2, las=1)
         lines(y, logit(y, deriv=d, inverse=TRUE), col="limegreen", lwd=2)
         lines(y, probit(y, deriv=d, inverse=TRUE), col="purple", lwd=2)
         lines(y, cloglog(y, deriv=d, inverse=TRUE), col="chocolate", lwd=2)
         lines(y, cauchit(y, deriv=d, inverse=TRUE), col="tan", lwd=2)
         if(d==0) {
             abline(h=0.5, v=0, lty="dashed")
             legend(-4, 1, c("logit", "probit", "cloglog", "cauchit"),
                    col=c("limegreen","purple","chocolate", "tan"), lwd=2)
         }
     }
     ## End(Not run)

