BC.5                   package:drc                   R Documentation

_T_h_e _B_r_a_i_n-_C_o_u_s_e_n_s _h_o_r_m_e_s_i_s _m_o_d_e_l_s

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

     'BC.4' and 'BC.5' provide the Brain-Cousens modified log-logistic
     models for describing u-shaped hormesis.

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

       BC.5(fixed = c(NA, NA, NA, NA, NA), names = c("b", "c", "d", "e", "f"), ...)

       BC.4(fixed = c(NA, NA, NA, NA), names = c("b", "d", "e", "f"), ...)

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

   fixed: numeric vector specifying which parameters are fixed and at
          which values they are fixed.  NAs designate parameters that
          are not fixed.

   names: a vector of character strings giving the names of the
          parameters.

     ...: additional arguments to be passed from the convenience
          functions.

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

     The model function for the Brain-Cousens model (Brain and Cousens,
     1989) is


     f(x, b,c,d,e,f) = c + frac{d-c+fx}{1+exp(b(log(x)-log(e)))}

     ,

     and it is a five-parameter model, obtained by extending the
     four-parameter log-logistic model ('LL.4' to take into account
     inverse u-shaped hormesis effects.

     The parameters have the following interpretations

        *  b: Not direct interpretation

        *  c: Lower horizontal asymptote

        *  d: Upper horizontal asymptote

        *  e: Not direct interpretation

        *  f: Size of the hormesis effect: the larger the value the
           larger is the hormesis effect. f=0 corresponds to no
           hormesis effect and the resulting model is the
           four-parameter log-logistic model.  This parameter should be
           positive in order for the model to make sense.

     Fixing the lower limit at 0 yields the four-parameter model 


           f(x) = 0 + frac{d-0+fx}{1+exp(b(log(x)-log(e)))}


     used by van Ewijk and Hoekstra (1993).

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

     See 'braincousens'.

_N_o_t_e:

     This function is for use with the function 'drm'.

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

     Christian Ritz

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

     Brain, P. and Cousens, R. (1989) An equation to describe dose
     responses  where there is stimulation of growth at low doses, 
     _Weed Research_, *29*, 93-96.

     van Ewijk, P. H. and Hoekstra, J. A. (1993)  Calculation of the
     EC50 and its Confidence Interval When Subtoxic Stimulus Is
     Present, _Ecotoxicology and Environmental Safety_, *25*, 25-32.

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

     More details are found for the general model function
     'braincousens'.

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

     ## Fitting the data in van Ewijk and Hoekstra (1993)
     lettuce.bcm1 <- drm(weight ~ conc, data = lettuce, fct=BC.5())
     modelFit(lettuce.bcm1)
     plot(lettuce.bcm1)

     lettuce.bcm2 <- drm(weight ~conc, data = lettuce, fct=BC.4())
     summary(lettuce.bcm2)
     ED(lettuce.bcm2, c(50))  
     # compare the parameter estimate and 
     # its estimated standard error 
     # to the values in the paper by 
     # van Ewijk and Hoekstra (1993)

     ## Brain-Cousens model with the constraint b>3
     ryegrass.bcm1 <- drm(rootl ~conc, data = ryegrass, fct = BC.5(), 
     lower = c(3, -Inf, -Inf, -Inf, -Inf), control = drmc(constr=TRUE))

     summary(ryegrass.bcm1)

     ## Brain-Cousens model with the constraint f>0 
     ## (no effect as the estimate of f is positive anyway)
     ryegrass.bcm2 <- drm(rootl ~conc, data = ryegrass, fct = BC.5(), 
     lower = c(-Inf, -Inf, -Inf, -Inf, 0), control = drmc(constr=TRUE))

     summary(ryegrass.bcm2)

