humpfit                package:vegan                R Documentation

_N_o-_i_n_t_e_r_a_c_t_i_o_n _M_o_d_e_l _f_o_r _H_u_m_p-_b_a_c_k_e_d _S_p_e_c_i_e_s _R_i_c_h_n_e_s_s _v_s. _B_i_o_m_a_s_s

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

     Function 'humpfit' fits a no-interaction model for species
     richness vs. biomass data (Oksanen 1996). This is a null model
     that produces a hump-backed response as an artifact of plant size
     and density.

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

     humpfit(mass, spno, family = poisson, start)
     ## S3 method for class 'humpfit':
     summary(object, ...)
     ## S3 method for class 'humpfit':
     predict(object, newdata = NULL, ...) 
     ## S3 method for class 'humpfit':
     plot(x, xlab = "Biomass", ylab = "Species Richness", lwd = 2, 
         l.col = "blue", p.col = 1, type = "b", ...)
     ## S3 method for class 'humpfit':
     points(x, ...)
     ## S3 method for class 'humpfit':
     lines(x, segments=101,  ...)
     ## S3 method for class 'humpfit':
     profile(fitted, parm = 1:3, alpha = 0.01, maxsteps = 20, del = zmax/5, ...)

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

    mass: Biomass. 

    spno: Species richness.

   start: Vector of starting values for all three parameters.

  family: Family of error distribution. Any 'family' can be used, but
          the link function is always Fisher's diversity model, and
          other 'link' functions are silently ignored. 

x, object, fitted: Result object of 'humpfit'

 newdata: Values of 'mass' used in 'predict'. The original data values
          are used if missing.

xlab,ylab: Axis labels in 'plot'

     lwd: Line width

l.col, p.col: Line and point colour in 'plot'

    type: Type of 'plot': '"p"' for observed points, '"l"' for fitted
          lines, '"b"' for both, and '"n"' for only setting axes.

segments: Number of segments used for fitted lines.

    parm: Profiled parameters.

alpha, maxsteps, del: Parameters for profiling range and density.

     ...: Other parameters to functions.

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

     The no-interaction model assumes that the humped species richness
     pattern along biomass gradient is an artifact of plant size and
     density (Oksanen 1996). For low-biomass sites, it assumes that
     plants have a fixed size, and biomass increases with increasing
     number of plants. When the sites becomes crowded, the number of
     plants and species richness reaches the maximum. Higher biomass is
     reached by increasing the plant size, and then the number of
     plants and species richness will decrease. At biomasses below the
     hump, plant number and biomass are linearly related, and above the
     hump, plant number is proportional to inverse squared biomass. The
     number of plants is related to the number of species by the
     relationship ('link' function) from Fisher's log-series (Fisher et
     al. 1943).

     The parameters of the model are:

        1.  'hump': the location of the hump on the biomass gradient.

        2.  'scale': an arbitrary multiplier to translate the biomass
           into virtual number of plants.

        3.  'alpha': Fisher's alpha to translate the virtual number of
           plants into number of species.

     The parameters 'scale' and 'alpha' are intermingled and this
     function should not be used for estimating Fisher's alpha. 
     Probably the only meaningful and interesting parameter is the
     location of the 'hump'.

     Function may be very difficult to fit and easily gets trapped into
     local solutions, or fails with non-Poisson families, and function
     'profile' should be used to inspect the fitted models. If you have
     loaded 'package' 'MASS', you can use functions 'plot.profile.glm',
     'pairs.profile.glm' for graphical inspection of the profiles, and
     'confint.profile.glm' for the profile based confidence intervals. 

     The original model intended to show that there is no need to
     speculate about `competition' and `stress' (Al-Mufti et al. 1977),
     but humped response can be produced as an artifact of using fixed
     plot size for varying plant sizes and densities.

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

     The function returns an object of class '"humpfit"' inheriting
     from class '"glm"'. The result object has specific 'summary',
     'predict', 'plot', 'points' and 'lines' methods. In addition, it
     can be accessed by the following methods for 'glm' objects: 'AIC',
     'extractAIC', 'deviance', 'coef', 'residuals.glm' (except 'type =
     "partial"'), 'fitted', and perhaps some others. In addition,
     function 'ellipse.glm' (package 'ellipse') can be used to draw
     approximate confidence ellipses for pairs of parameters, if the
     normal assumptions look appropriate.

_N_o_t_e:

     The function is a replacement for the original 'GLIM4' function at
     the archive of Journal of Ecology.  There the function was
     represented as a mixed 'glm' with one non-linear parameter
     ('hump') and a special one-parameter link function from Fisher's
     log-series.  The current function directly applies non-linear
     maximum likelihood fitting using function 'nlm'.  Some expected
     problems with the current approach are:

        *  The function is discontinuous at 'hump' and may be difficult
           to optimize in some cases (the lines will always join, but
           the derivative jumps).

        *  The function does not try very hard to find sensible
           starting values and can fail. The user may supply starting
           values in argument 'start' if fitting fails.

        *  The estimation is unconstrained, but both 'scale' and
           'alpha' should always be positive.  Perhaps they should be
           fitted as logarithmic. Fitting 'Gamma' 'family' models might
           become easier, too.

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

     Jari Oksanen

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

     Al-Mufti, M.M., Sykes, C.L, Furness, S.B., Grime, J.P & Band, S.R.
     (1977) A quantitative analysis of shoot phenology and dominance in
     herbaceous vegetation. _Journal of Ecology_ 65,759-791.

     Fisher, R.A., Corbet, A.S. & Williams, C.B. (1943) The relation
     between the number of species and the number of individuals in a
     random sample of of an animal population. _Journal of Animal
     Ecology_ 12, 42-58.

     Oksanen, J. (1996) Is the humped relationship between species
     richness and biomass an artefact due to plot size? _Journal of
     Ecology_ 84, 293-295.

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

     'fisherfit', 'profile.glm', 'confint.glm'.

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

     ##
     ## Data approximated from Al-Mufti et al. (1977)
     ##
     mass <- c(140,230,310,310,400,510,610,670,860,900,1050,1160,1900,2480)
     spno <- c(1,  4,  3,  9, 18, 30, 20, 14,  3,  2,  3,  2,  5,  2)
     sol <- humpfit(mass, spno)
     summary(sol) # Almost infinite alpha...
     plot(sol)
     # confint is in MASS, and impicitly calls profile.humpfit.
     # Parameter 3 (alpha) is too extreme for profile and confint, and we
     # must use only "hump" and "scale".
     library(MASS)
     plot(profile(sol, parm=1:2))
     confint(sol, parm=c(1,2))

