Analyzing MaxDiff Using the Rank-Ordered Logit Model With Ties Using R

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R Setup

library(BiasedUrn)
setup.flat.data = function(x, number.alternatives){
   n = nrow(x)
   number.sets = ncol(x) / number.alternatives
   data = vector("list",n)
   for (i in 1:n)
   {
       temp.respondent.data = matrix(x[i,],byrow=TRUE,ncol = number.alternatives)
       respondent.data = vector("list",number.sets)
       for (s in 1:number.sets)
          respondent.data[[s]] = as.numeric(temp.respondent.data[s,])
       data[[i]] = respondent.data
		}
compress.data(data)}

compress.data = function(x){#Creates a vector for each set where the first entry was best and the last worst
  compress = function(x){
     x.valid = !is.na(x)
     x.position = (1:length(x))[x.valid]
     x.position[order(x[x.valid], decreasing = TRUE)]
  }
  n = length(x)
  number.sets = length(x[[1]])
  number.alternatives = length(x[[1]][[1]])
  data = vector("list",n)
  for (i in 1:n)
   {
       respondent.data = vector("list",number.sets)
       for (s in 1:number.sets)
          respondent.data[[s]] = compress(x[[i]][[s]])
       data[[i]] = respondent.data
		}
class(data) = "maxdiffData"
data}

d.marley = function(b,x){
  b.vector = b[x]
  k = length(b.vector)
  ediffs = exp(matrix(b.vector,k,k,byrow=FALSE) - matrix(b.vector,k,k,byrow=TRUE))
ediffs[1,k] / (sum(ediffs) - sum(diag(ediffs)))}

d.rlogit = function(b,x){
  eb = exp(b[x])
  k = length(eb)
  d.best = eb[1]/sum(eb)
  d.not.worst = dMWNCHypergeo(c(rep(1,k-2),0), rep(1,k-1),k-2,eb[-1], precision = 1E-7)
d.best * d.not.worst}

d.repeated.maxdiff = function(b,x, method){
  prod(as.numeric(lapply(x,b = b,switch(method,marley=d.marley, rlogit = d.rlogit))))}

ll.max.diff = function(b,x, maxdiff.method = c("marley","rlogit")[1]){
   b[b > 100] = 100
   b[b < -100] = -100
   sum(log(as.numeric(lapply(x,b = c(0,b),method=maxdiff.method, d.repeated.maxdiff))))
}

max.diff.rank.ordered.logit.with.ties = function(stacked.data){
  flat.data = setup.flat.data(stacked.data, ncol(stacked.data))
  solution = optim(seq(.01,.02, length.out = ncol(stacked.data)-1), ll.max.diff,  maxdiff.method  = "rlogit", gr = NULL, x = flat.data, method =  "BFGS", control = list(fnscale  = -1, maxit = 1000, trace = FALSE), hessian = FALSE)
  pars = c(0, solution$par)
  names(pars) = dimnames(stacked.data)[[2]]
  list(log.likelihood = solution$value, coef = pars)}


Reading the data into R

The code above assumes the data is in the stacked layout. Using the example used in Analyzing MaxDiff Using Standard Logit Models Using R:

mdData = matrix(c(NA,NA,1,0,-1,NA,1,NA,NA,0,-1,NA,NA,1,NA,0,NA,-1,0,NA,1,NA,NA,-1,0,-1,1,NA,NA,NA,NA,1,NA,NA,0,-1),6,byrow=TRUE, dimnames=list(Block=1:6,Alternatives = LETTERS[1:6]))
mdData
     Alternatives
Block  A  B  C  D  E  F
    1 NA NA  1  0 -1 NA
    2  1 NA NA  0 -1 NA
    3 NA  1 NA  0 NA -1
    4  0 NA  1 NA NA -1
    5  0 -1  1 NA NA NA
    6 NA  1 NA NA  0 -1


Estimating the model

The model is estimated using the following code:

max.diff.rank.ordered.logit.with.ties(mdData)

Interpreting the results

The results of the model are (this will differ from computer-to-computer, but the relativities will remain the same):

$log.likelihood
[1] -3.34057e-08

$coef
        A         B         C         D         E         F 
  0.00000 -18.32953  21.20740 -36.45326 -56.78844 -75.94036 

A few points about the interpretation:

  • The interpretation is identical to that from the logit model (click here for more information). In particular, the correct interpretation is to either look at ranks, or, to compute probabilities.
  • As was the case with the standard logit model, and for the same reasons, it is not appropriate to directly compare the parameters that are estimated between different respondents.
  • The actual substantive interpretation of these outputs, which is: C > A > B > D > E > F is identical to that obtained from the tricked multinomial logit model.
  • The magnitudes of these numbers differs from those of the tricked multinomial logit model. This is unimportant and is due to different levels of precision used in the different algorithms.
  • The log-likelihood computed using this model cannot be meaningfully compared with those from the other methods (they make different assumptions).

See also

See Category:MaxDiff for an overview of MaxDiff and links to key resources.