The behavior of two topological and four character-based congruence measures was explored using different indel treatments in three empirical data sets, each with different alignment difficulties. The analyses were done using direct optimization within a sensitivity analysis framework in which the cost of indels was varied. Indels were treated either as a fifth character state, or strings of contiguous gaps were considered single events by using linear affine gap cost. Congruence consistently improved when indels were treated as single events, but no congruence measure appeared as the obviously preferable one. However, when combining enough data, all congruence measures clearly tended to select the same alignment cost set as the optimal one. Disagreement among congruence measures was mostly caused by a dominant fragment or a data partition that included all or most of the length variation in the data set. Dominance was easily detected, as the character-based congruence measures approached their optimal value when indel costs were incremented. Dominance of a fragment or data partition was overwhelmed when new sequence length-variable fragments or data partitions were added.