An alternative to pilesorts for measuring similarity is the triad test. Here, the items in a domain are presented to the respondent in groups of three. For each triple, the respondent must pick out the one she judges to be the most different. For example, one triple drawn from the domain of animals might be:

 Dog Seal Shark

Picking any item is equivalent to voting for the similarity of the other two. Hence, choosing "dog" would indicate that "seal" and "shark" were similar, while choosing "shark" would indicate that "seal" and "dog" were similar. If all possible triples are presented, each pair of items will occur N-2 times(14), each time "against" a different item. If a pair of items is really similar it will "win" each of those contests and will be voted most similar a total of N-2 times. If the pair is extremely dissimilar, it will never win. For example, "oyster" and "elephant" might occur in the following triples:

 Oyster Elephant Dog Oyster Elephant Shrimp Oyster Elephant Ostrich

In the first one, the respondent might choose "oyster" as the most different. In the second, the respondent might choose "elephant". In the third, the respondent might choose "oyster" again, and so on. Hence, the triad test in which every possible triple is presented will yield a similarity score for each pair of items that ranges from zero to N-2.

The problem with presenting all possible triples is that there are N(N-1)(N-2)/6 of them, which is a quantity that grows with the cube of the number of items. If the domain has 30 items in it, the number of triples is 4,060, which is too many for an informant to respond to, even over a period of days. The solution is to take a manageable sample of triples. For example, out of the 4,060 triples, we might randomly select 200 for the respondent to work with. However, a random sample would allow some pairs of items to appear in several triples, and allow others not to occur it all. The latter would be a real problem because the purpose of the task is to measure the perceived similarity between every pair of items.

The solution is to use a balanced incomplete block design (BIB design). In a BIB design, every pair of items occurs a fixed number of times. The number of times the pair occurs is known as lambda (). In a complete design (where all possible triples occur), obviously equals N-2, since each pair occurs against every other item in the domain. When equals 1, we have the smallest possible BIB design, where each pair of items occurs only once. For a domain with 30 items, a =1 design would have only 435 triads -- still a lot, but a considerable savings over 4,060.

In general, however, =1 designs should be avoided, because the similarity of each pair of items will be completely determined by their relation to whichever item happens to turn up as the third item. For example, if Elephant and Mouse occur in this triple:

 Mouse Elephant Rat

it is likely that they will be measured as not similar, since "elephant" is likely to be chosen as most different. But if instead they happen to occur in this triple:

 Mouse Elephant Oyster

it is likely that they will be measured as similar. Thus, it is much better to have at least a =2 design, where each pair of items occurs against two different third items. The only exception to this rule of thumb is when you give each respondent in a culturally homogeneous sample a completely different triad test, based on the same domain but containing different triples. For example, respondent #1 might get Mouse and Elephant paired with Oyster, but respondent #2 might get Mouse and Elephant paired with Dog. In a way, this is like taking a complete design and spreading it out across multiple respondents. This can work well, but means that you cannot compare respondents' answers with each other to assess similarity of views, since each person was given a different questionnaire.

A nice feature of the triads task is that, unlike the simple pilesort, it yields degrees of similarity for pairs of items for each respondent. In the simple pilesort, each respondent essentially gives a "yes they are similar" or "no they are not" vote. In the triads, the range of values obtained for each pair of items goes from zero to . Hence, for a =3 design, each pair of items is assigned an ordinal similarity score of 0, 1, 2, or 3. This means that we can sensibly construct separate multidimensional scaling maps for each respondent.(15)

One problem with triad tasks is that respondents often find them tiring and repetitive. They will swear that a certain triad has already occurred, and will suspect that you are trying to see if they are responding consistently, which makes them nervous. Another problem is that respondents tend to become aware of their own thought processes as they proceed, and start feeling uncomfortable about using varying criteria (which is unavoidable) to pick the item most different in each triple. This makes them feel that they are not doing a good job. In general, triads are only useful for very small domains (12 items or less) or for testing hypotheses (where it is important that every respondent make an active judgement regarding the similarities among a certain set of items).

Perhaps the most interesting use of triads was by Romney and D'Andrade (1964) who used them to test two theories of cognition about American male kinship roles, such as grandfather, father, son, grandson, uncle, brother, nephew and cousin. One theory, by Wallace and Atkins (1960), held that Americans use two attributes --- generation and lineality --- to distinguish among the roles, as shown in Table 5.

Table 5. Wallace and Atkins Model of American Kinship

Generation Lineal Collineal Ablineal
2 gen. above ego grandfather
1 gen. above ego father uncle
same as ego   brother cousin
1 gen. below ego son nephew
2 gen. below ego grandson

If the theory is true, in a triads test that included the triple

 grandfather grandson father

Americans should choose "grandson" as the one most different because grandfather and grandson are the least different with respect to the two attributes in the model.

In contrast, Romney and D'Andrade propose a model with three attributes --- generational distance, lineality, and reciprocal roles --- as shown in Table 6.

Table 6. Romney and D'Andrade Model of American Kinship

 Direct Collateral - Reciprocal + Reciprocal - Reciprocal + Reciprocal Gen ± 2 grandson grandfather Gen ± 1 son father uncle nephew Gen. 0 brother cousin

According to the Romney and D'Andrade model, when faced with the same triad given above, Americans should choose, with equal probability, either "grandson" or "father" as the item most different, and should never choose "grandfather". Given these predictions, it was a simple matter to test the theories by giving the triads to a sample of Americans and seeing which theory best predicted the actual answers on the triads test. The best theory turned out to be the Romney and D'Andrade model.

So far, I have only described the formal use of the triads task, which results in the generation of similarities among items. Another way to use triads is as a device to spark discussion of the underlying attributes that people use to distinguish among items in the domain. To do this, we present informants with a small random sample of triples, one at a time. For each triple, the informant is asked to explain in what ways each item is different from the other two. This is an extraordinarily effective way to elicit the attributes that people use to think about the domain. For example, consider this triple:

 Cancer Syphilis Measles

This triple can elicit a number of perceived attributes of illnesses including seriousness ("cancer is fatal"), age of the afflicted ("measles is something that kids get"), morality ("you get syphilis from sleeping around too much"), contagiousness ("you can catch syphilis and measles from other people"), and so on. It is easy to see that it only requires a handful of triples to elicit dozens and dozens of attributes.

### Footnotes

14. Again, N is the number of items in the domain.

15. The same was true for the successive pilesort techniques described earlier.