Convergence: Statistics or Substance?

The RMT articles describes a situation common to all estimation processes which use an iterative technique. The convergence criteria must be set tight enough for the problem at hand. Iterative processes are used widely in statistics and engineering. If in doubt, set the convergence criteria too tight. The only down-side may be that the analysis takes longer to run than necessary.

Estimation frequently requires iterative procedures: the more iterations, the more accurate estimates. But when are estimates accurate enough? When can iteration cease? My the rule has become "Convergence is reached when more iterations do not change my interpretation of the estimates".

There is a trade-off between accuracy and speed. Greater accuracy requires more iterations - more time and computer resources. The specification of estimation accuracy is a compromise. Frequently, squeezing that last bit of inaccuracy out of estimates only affects the least significant digits of printed output, has no noticeable effect on model-data fit, and does not alter interpretation. Three numerical convergence rules are often employed:

1) Estimates are pronounced "accurate enough" when a predetermined "maximum" number of iterations have been performed.

2) Estimates are deemed converged when no estimate changes more than a small pre-set "tolerance" value during an iteration.

3) Estimates have converged when there is less residual difference between the observed data and that expected than can actually be observed.

Be wary! In a recent analysis of responses to a set of math tests, linked in block diagonal matrix form, I set these three convergence criteria to reasonable values. The computer program BIGSTEPS ran smoothly. All appeared well. The outcome is shown in Figure 1. As most of us would expected, both the 2995 children and the 1031 math items appear close to normally distributed. The children were from 9 grades, so the spread of 7 logits across the examinees could be right.

A question arose, however, when I went back and inspected the linking design. Children in the lower and higher grades had been deliberately over-sampled in order to get good child measures and item calibrations at the extremes. Yet this bias towards the extremes does not appear in Figure 1!

After eliminating other theories for this unexpected result, suspicion focussed on the analysis itself. Perhaps the familiar values for the convergence criteria were not stringent enough in this case. Accordingly, the criteria were made more stringent, and estimates were again obtained. The initial run used 50 iterations. The revised run, 263 iterations. The second outcome is shown in Figure 2. Now both the child and item distributions are clearly bimodal. The range of child abilities is about 9 logits, an increase of 2 logits. This result makes much better sense.

Establishing convergence is more than a statistical nicety. It can have profound substantive implications.

Ong Kim Lee

                       CHILDREN     MATH ITEMS
 5                             . +
                               . +
                               . +
 4                             . + #.
                             .## + #
                            .### + ####
 3                       .###### + ######.
                     .########## + ##############.
                   .############ + ##############.
 2              .############### + ##################.
       .######################## + ###############.
      .######################### + #################
 1   .########################## + #######################.
    .########################### + ######################.
   ############################# + ##########################
 0     .######################## + #########################
      .######################### + ######################
     .########################## + ####################.
-1        .##################### + ###############.
                .############### + ##################.
                   .############ + ###################.
-2                    .######### + ##################.
                          .##### + ############.
                            .### + ############
-3                            .# + #####.
                               . + ####
                                 + ##.
-4                               + #.
                                 + #
                                 + .
-5                               +
                                 +
                                 +
-6                               + .
                       CHILDREN  +  MATH ITEMS

Figure 1. Statistically converged estimates.

                         PERSONS +   ITEMS
 5                           ### + ###
                            .### + ##
                          .##### + #########
 4                   .########## + ################.
                     .########## + #################.
                 .############## + #####################.
 3             ################# + ######################
          .##################### + ################
          .##################### + ##################.
 2         .#################### + #####################
         ####################### + ######################
              .################# + ###################
 1              .############### + ################.
                .############### + #############.
                    .########### + #####################.
 0          .################### + ######################.
          .##################### + ###################.
        .####################### + #####################.
-1      .####################### + #######################
         .###################### + ###################
      .######################### + ########################
-2       .###################### + #######################
               .################ + #######################.
                  .############# + ####################.
-3                   .########## + ##########################
                        .####### + ###############.
                          .##### + ##############
-4                            .# + ######
                               # + ########
                                 + ###
-5                               + #.
                                 + ###
                                 + #
-6                               +
                         PERSONS +   ITEMS

Figure 2. Substantively converged estimates.



Convergence: Statistics or Substance?, O K Lee … Rasch Measurement Transactions, 1991, 5:3 p. 172




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