capscale                package:vegan                R Documentation

[_P_a_r_t_i_a_l] _C_o_n_s_t_r_a_i_n_e_d _A_n_a_l_y_s_i_s _o_f _P_r_i_n_c_i_p_a_l _C_o_o_r_d_i_n_a_t_e_s

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

     Constrained Analysis of Principal Coordinates (CAP) is an
     ordination method similar to Redundancy Analysis ('rda'), but it
     allows non-Euclidean dissimilarity indices, such as Manhattan or
     Bray-Curtis distance. Despite this non-Euclidean feature, the
     analysis is strictly linear and metric. If called with Euclidean
     distance, the results are identical to 'rda', but 'capscale' will
     be much more inefficient. Function 'capscale' may be useful with
     other  dissimilarity measures, since Euclidean distances inherent
     in 'rda' are generally poor with community data

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

     capscale(formula, data, distance = "euclidean", comm = NULL, ...)

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

 formula: Model formula. The function can be called only with the
          formula interface. Most usual features of 'formula' hold,
          especially as defined in 'cca' and 'rda'. The LHS must be
          either a community data matrix or a dissimilarity matrix,
          e.g., from 'vegdist' or 'dist'. If the LHS is a data matrix,
          function 'vegdist' will be used to find the dissimilarities.
          RHS defines the constraints. The constraints can be
          continuous or factors, they can be transformed within the
          formula, and they can have interactions as in typical
          'formula'. The RHS can have a special term 'Condition' that
          defines variables ``partialled out'' before constraints, just
          like in 'rda' or 'cca'. This allows the use of partial CAP.

    data: Data frame containing the variables on the right hand side of
          the model formula. 

distance: Dissimilarity (or distance) index  in 'vegdist' used if the
          LHS of the 'formula' is a data frame instead of dissimilarity
          matrix. 

    comm: Community data frame which will be used for finding species
          scores when the LHS of the 'formula' was a dissimilarity
          matrix. This is not used if the LHS is a data frame. If this
          is not supplied, the ``species scores'' are the axes of
          initial metric scaling ('cmdscale') and may be confusing.

     ...: Other parameters passed to 'rda'. 

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

     The Canonical Analysis of Principal Coordinates (CAP) is simply a
     Redundancy Analysis of results of Metric (Classical)
     Multidimensional Scaling (Anderson & Willis 2003). Function
     capscale uses two steps: (1) it ordinates the dissimilarity matrix
     using 'cmdscale' and (2) analyses these results using 'rda'. If
     the user supplied a community data frame instead of
     dissimilarities, the function will find the needed dissimilarity
     matrix using 'vegdist' with specified 'distance'. However, the
     method will accept dissimilarity matrices from 'vegdist', 'dist',
     or any other method producing similar matrices. The constraining
     variables can be continuous or factors or both, they can have
     interaction terms, or they can be transformed in the call.
     Moreover, there can be a special term 'Condition' just like in
     'rda' and 'cca' so that ``partial'' CAP can be performed.

     The current implementation  differs from the method suggested by
     Anderson & Willis (2003) in three major points:

        1.  Anderson & Willis used orthonormal solution of 'cmdscale',
           whereas 'capscale' uses axes weighted by corresponding
           eigenvalues, so that the ordination distances are best
           approximations of original dissimilarities. In the original
           method, later ``noise'' axes are just as important as first
           major axes.

        2.  Anderson & Willis take only a subset of axes, whereas 
           'capscale' uses all axes with positive eigenvalues. The use
           of subset is necessary with orthonormal axes to chop off
           some ``noise'', but the use of all axes guarantees that the
           results are the best approximation of original
           dissimilarities.

        3.  Function 'capscale' adds species scores as weighted sums of
           (residual) community matrix (if the matrix is available),
           whereas Anderson & Willis have no fixed method for adding
           species scores.

     With these definitions, function 'capscale' with Euclidean
     distances will be identical to 'rda' in eigenvalues and in site,
     species and biplot scores (except for possible sign reversal). 
     However, it makes no sense to use 'capscale' with Euclidean
     distances, since direct use of 'rda' is much more efficient. Even
     with non-Euclidean dissimilarities, the rest of the analysis will
     be metric and linear.

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

     The function returns an object of class 'capscale' which is
     identical to the result of 'rda'. At the moment, 'capscale' does
     not have specific methods, but it uses 'cca' and 'rda' methods
     'plot.cca', 'summary.rda' etc. Moreover, you can use 'anova.cca'
     for permutation tests of ``significance'' of the results.

_N_o_t_e:

     Function 'rda' usually divides the ordination results by number of
     sites minus one. In this way, the inertia is variance instead of
     sum of squares, and the eigenvalues sum up to variance. Many
     dissimilarity measures are in the range 0 to 1, so they have
     already made a similar division. If the largest original
     dissimilarity is less or equal to 4 (allowing for 'stepacross'),
     this division is undone in 'capscale' and original dissimilarities
     are used. The inertia is called as 'squared dissimilarity' (as
     defined in the dissimilarity matrix), but keyword 'mean' is added
     to the inertia in cases where division was made, e.g. in Euclidean
     and Manhattan distances.

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

     Jari Oksanen

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

     Anderson, M.J. & Willis, T.J. (2003). Canonical analysis of
     principal coordinates: a useful method of constrained ordination
     for ecology. _Ecology_ 84, 511-525.

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

     'rda', 'cca', 'plot.cca', 'anova.cca', 'vegdist', 'dist',
     'cmdscale'.

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

     data(varespec)
     data(varechem)
     vare.cap <- capscale(varespec ~ N + P + K + Condition(Al), varechem, dist="bray")
     vare.cap
     plot(vare.cap)
     anova(vare.cap)

