A more detailed discussion of the results in the sayings categories.
The sayings categories 102, 202, and 201, and sondergut Matthew, 200, are particularly important for distinguishing between the similar alternatives of the FH, the 2SH and the 3SH.
(see category definitions
here.)
(see hypotheses
here.)
(see all results
here.)
(see a good site about the gospel of Thomas, the sayings source we
do know existed, here.)

None of these categories show any strong relationship with the narrative triple tradition categories. The closest is 200-211 with a probability of  .2288 . This may be because we are looking at a separate author, distinct from Matthew, Mark and Luke, or because the genra, and subject of the sayings categories is sufficiently distinct from the narrative categories to cause this difference. For example different pronouns are used when Jesus is speaking than when his actions are being described in narrative form.

We should note here that 002 does show a strong relation to a narrative category. 002 and 112 have an associated probability of 2E-10. This may be because 002 is more narrative in character that the sayings categories or sondergut Matthew. But it also suggests, that at least in this case, the subject matter does not cause strong differences, since 002 and 112 cover different topics.

Category 102 does not show a strong relationship to any other category. The closest is 202 with a probability of .0031, next is 002 with a probability of  .213. Again the fact that 102 does not group with other categories related to Luke, may be because of the difference in genre.

It is interesting to note that 102 is about as related to 202 and 201 is.
202-201 .002
202-102 .0058

This symmetry suggests the saying source Q. However, the lack of a strong 201-202 relationship may suggest a source other than Matthew, but it is difficult to interpret a lack of a finding, and I dont believe the FH can be eliminated based on this. All we can say is that a solution with a sayings source gives a more natural fit here, and may be more probable than the FH, based on this.

The results for category 200, are also important.

202-200 2E-06
201-200 2E-10

The 202-200 relationship should not be there on the 2SH, at least at first glance. On the 2SH, 200 is Matthew, and 202 is Q. But the 200-202 relationship is strong in spite of the fact that they cover different topics, and at least a large part of 200 is narrative. i.e. Matthew chapters 1 and 2. The only recourse I see for the 2SH is to say that 200 also contains significant amounts of Q material that Luke did not use. For example, there are cases where a Q phase is used multiple times in Matthew. This re-use by Matthew tends to draw 200 and 202 together.
Mark Goodacre has pointed out that this phenomenon makes the style attributed to Q and the style attributed to Matthew similar. However, I think we would also have to speculate that other parts of 200 were also in Q, and that Luke did not use them even once, in order to account for the strong observed relationship.

Now Id like to present the way the 3SH would view these relationships. On the 3SH 202 is mostly the saying source S, with a little Matthew mixed in. 102 is more than half Luke, and less than half S. 201 is less than half S and more than half Matthew. 200 is also some S and mostly Matthew. 

Based on this we would not expect 102 to look much like the narrative categories of  Luke, since there is a difference in genre, a difference in topic, and a difference in authorship for something like half of the material. 102 and 201 would show some symmetry around 202, since both are about half S, and share the same genre and topics. 200 would show a strong relationship to 201, since both are mostly Matthew, with some S. And finally, 200 and 202 would be related, but less strongly that 201-200, since 202 would be mostly S with some Matthew, and 200 would be mostly Matthew, with some S.

Thus, the 3SH most naturally explains the observed results, while the FH and 2SH can not be eliminated.
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