rasch                  package:ltm                  R Documentation

_R_a_s_c_h _M_o_d_e_l

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

     Fit the Rasch model under the Item Response Theory approach.

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

     rasch(data, constraint = NULL, start.val, na.action = NULL, control = list())

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

    data: a data.frame (that will be converted to a numeric matrix
          using  'data.matrix()') or a numeric matrix of manifest
          variables. The binary responses must be in 0/1 format.

constraint: a two-column numeric matrix with at most p rows (where p is
          the number of items),  specifying fixed-value constraints.
          The first column represents the item (i.e., 1 denotes the
          first item, 2 the second, etc., and p+1 the discrimination
          parameter) and the second column the  value at which the
          corresponding parameter should be fixed. See *Examples* for
          more info.

start.val: a numeric vector of p+1 starting values for the algorithm.
          The first p values correspond to the easiness parameters
          while the last value corresponds to the discrimination 
          parameter. If it is not supplied randomly chosen starting
          values are used instead.

na.action: the 'na.action' to be used on 'dat'. In case of missing
          data, if  'na.action=NULL' the model uses the available
          cases, i.e., it takes into account the observed  part of
          sample units with missing values (valid under MAR mechanisms
          if the model is correctly specified).  If you want to apply a
          complete case analysis then use 'na.action=na.exclude'.

 control: a list of control values,

          _i_t_e_r._q_N the number of quasi-Newton iterations. Default 150.

          _G_H_k the number of Gauss-Hermite quadrature points. Default
               21.

          _m_e_t_h_o_d the optimization method to be used in 'optim'. Default
               "BFGS".

          _v_e_r_b_o_s_e logical; if 'TRUE' info about the optimization
               procedure are printed.

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

     The Rasch model is special case of the unidimensional latent trait
     model when all the discrimination  parameters are equal. This
     model was first discussed by Rasch (1960) and it is used mainly in
     educational  testing where the aim is to study the abilities of a
     particular set of individuals.

     The model is defined as follows 

                  logit (pi_i) = beta_{i} + beta z,

     where pi_i denotes the probability of responding correctly to the
     ith  item, beta_{i} is the easiness parameter for the ith item,
     beta is the  discrimination parameter (the same for all the items)
     and z denotes the latent ability.

     The fit of the model is based on approximate marginal Maximum
     Likelihood, using the Gauss-Hermite quadrature rule  for the
     approximation of the required integrals. The optimization
     algorithm works under the constraint that  the discrimination
     parameter is always positive.

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

     An object of class 'rasch' with components, 

coefficients: the parameter values at convergence.

 log.Lik: the log-likelihood value at convergence.

convergence: the convergence identifier returned by 'optim'.

 hessian: the approximate Hessian matrix at convergence returned by
          'optim'.

patterns: a list with two components: (i) 'mat' a numeric matrix  that
          contains the observed response patterns. (ii) 'dat' a
          data.frame that contains the observed and expected 
          frequencies for each observed response pattern.

      GH: a list with two components used in the Gauss-Hermite rule:
          (i) 'Z' a numeric matrix that contains  the quadrature
          points. (ii) 'GHw' a numeric vector that contains the
          corresponding  weights.

  max.sc: the maximum absolute value of the score vector at
          convergence.

       X: the responses data matrix.

 control: the values used in the 'control' argument.

    call: the matched call.

_W_a_r_n_i_n_g:

     In case the Hessian matrix at convergence is not positive
     definite, try to re-fit the model. 'rasch' will use new random
     starting values.

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

     Dimitris Rizopoulos dimitris.rizopoulos@med.kuleuven.be

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

     Baker, F. and Kim, S-H. (2004) _Item Response Theory_, 2nd ed. 
     New York: Marcel Dekker.

     Rasch, G. (1960) _Probabilistic Models for Some  Intelligence and 
     Attainment Tests_. Copenhagen: Paedagogiske  Institute.

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

     'coef.rasch', 'summary.rasch', 'anova.rasch', 'plot.rasch',
     'vcov.rasch', 'margins', 'factor.scores'

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

     ## The Rasch model for the Wirs data:
     rasch(Wirs)

     ## The Rasch model for the Wirs data, under the 
     ## constraint that: (i) the easiness parameter 
     ## for the first item equals 0.5, (ii) the easiness 
     ## parameter for the second item equals 1.2, and 
     ## (iii) the discrimination parameter equals 1
     (constr <- rbind(c(1, 0.5), c(2, 1.2), c(7, 1)))
     rasch(Wirs, constraint = constr)

     ## The Rasch model for the Lsat data:
     rasch(Lsat)

     ## The Rasch model for the Lsat data, under the 
     ## constraint that the discrimination parameter
     ## equals 1
     rasch(Lsat, constraint = rbind(c(6,1)))

     ## The Rasch model for the Abortion data:
     rasch(Abortion)

