As you can see, there is a pattern in the data for each group. After repeated testing, while the first group learns and their data get better over time, the second gets fatigued and so their data get worse over time. If you only looked at the overall average performance plot on the right, you would conclude that the performance is flat, but this would be wrong, as the data hide two separate groups.

If we fit a repeated-measures regression aggregating across students, each of them is classified into one or the other cluster. From the ratio below, we can see two distinct classifications (about 91-100%).

Call:
flexmix(formula = value ~ time | as.factor(sub), data = long, 
    k = 2, control = list(iter.max = 10))

       prior size post>0 ratio
Comp.1   0.5  100    100 1.000
Comp.2   0.5  100    110 0.909

'log Lik.' -382.1782 (df=7)
AIC: 778.3564   BIC: 801.4446 

Model parameters

                     Comp.1    Comp.2
coef.(Intercept)  9.9607590 5.1754053
coef.time        -0.4886075 0.4633109
sigma             1.5150882 1.5525637

Clusters

   
     1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20
  1  0  0  0  0  0  0  0  0  0  0 10 10 10 10 10 10 10 10 10 10
  2 10 10 10 10 10 10 10 10 10 10  0  0  0  0  0  0  0  0  0  0

The Bayesian Information Criterion (BIC) is a statistic that is based on both the likelihood of the data and a factor related to the number of parameters used. Using the BIC statistic, you can compare across models and determine which one provides the best trade-off between accuracy and complexity by selecting the smallest BIC.

1 : * * * * * * * * * *
2 : * * * * * * * * * *
3 : * * * * * * * * * *
4 : * * * * * * * * * *

Call:
stepFlexmix(value ~ time | as.factor(sub), data = long, k = 1:4, 
    nrep = 10)

  iter converged k k0    logLik      AIC      BIC      ICL
1    2      TRUE 1  1 -428.4623 862.9246 872.8195 872.8195
2    6      TRUE 2  2 -382.1782 778.3564 801.4446 801.4472
3    7      TRUE 3  3 -382.1770 786.3539 822.6354 829.2521
4    8      TRUE 3  4 -382.1759 786.3517 822.6332 832.0410

The best trade-off between accuracy and complexity can be visualized by plotting performance measures such as the Bayesian Information Criterion (BIC), the Akaike Information Criterion (AIC), and the Integrated Complete-data Likelihood (ICL) for different models. As you can see, BIC, AIC and ICL are all optimal (minimum value) for two groups or clusters.

In a rootogram, the height of the bars correspond to square roots of counts rather than the counts themselves, so that low counts are more visible and peaks less so. A peak near probability 1 indicates that many of the points are overwhelmingly well-represented by that cluster. A peak near 0 would indicate that many points clearly don’t fit the category (which is also usually good). Points in the middle indicate a lack of separation–there are points that are only moderately well-described by that group, which should be considered with respect to the number of groups you think exist.