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Statistical Validation of Lukan Priority

Statistical Validation of Lukan Priority

A statistical analysis of the Synoptic Gospels validates Lukan Priority

On the website JerusalemPerspective.com, Halvor Ronning, founding member and past director of Jerusalem School of Synoptic Research (Jerusalem School) published a four-part series entitled “A Statistical Approach to the Synoptic Problem” in which he conducts an analysis of all the potential options regarding the sequence of the Synoptic Gospels. The results of his analysis indicate the correct sequence and dependency of the gospels is Luke-Mark-Matthew with Luke having priority as the first gospel, serving as a direct reference to Mark and an indirect reference to Matthew.   

A summary is provided below. The four-part series of articles by Halvor Ronnning are at the following Links:

The six theoretically possible relationships he examines are as follows:
 
  • Matthew→Luke→Mark
  • Mark→Luke→Matthew
  • Luke→Matthew→Mark
  • Mark→Matthew→Luke
  • Luke→Mark→Matthew
  • Matthew→Mark→Luke

Although not all of these theoretical options of linear dependence have been proposed by scholars historically. The point of the analysis is to see if any of the options can survive the empirically observable statistical facts. The body of empirical data is the thousands of words exhibited in the gospels of Mark, Matthew, and Luke in calculations pertaining to their similarities (material of Identical Form and Sequence (IFS) and calculating their differences (Contrary Agreements (CA)). With the aim of providing a quantifiable means of evaluating the six options, Ronning adopted several new methods of evaluating and testing synoptic theories. A seven-step method has been devised as a basis for the comparison of the six options of linear dependence. See the referenced articles for a complete description of the methodology. 

Abbreviations Used in the analysis are as follows:
 
  • IFS: Identical Form and Sequence
  • TT: Triple Tradition, text paralleled in all three Synoptic Gospels
  • DT: Double Tradition, text paralleled in two of the three Synoptic Gospels
  • ST: Single Tradition, text found in only one Gospel
  • CA: Contrary agreements, all instances of two Gospels agreeing with each other contrary to the third. The term “Contrary Agreement” includes the “minor agreements” of Luke and Matthew against Mark, but this term also includes the other two possibilities: a) Luke and Mark against Matthew; and b) Mark and Matthew against Luke.
Below is the summary of the results for the Sequence of Luke – Mark – Matthew. The referenced articles include the complete results for all six options.

Part 1—Triple Tradition, Results of Luke→Mark→Matthew

  1. 32.4% of Mark is Identical Form and Sequence (IFS) with Luke (2555 of 7889 words).
  2. 48.1% of Matthew is Identical Form and Sequence (IFS) with Mark (3215 of 6882 words).
  3. 19.5% of Matthew is Identical Form and Sequence (IFS) with Luke (1307 of 6682 words).
  4. 3.8% of Matthew is Identical Form and Sequence (IFS) with Luke in Contrary Agreements (CAs) (256 of 6682 words), not Mark.

The prediction:

  • 15.6% of Matthew should be Identical Form and Sequence (IFS) with Luke and Mark.

The actual results:

  • 15.7% of Matthew is Identical Form and Sequence (IFS) with Luke and Mark (1051 of 6682 words). 

Ronning’s summary of the conclusion from these results from Triple Tradition analysis is as follows:

We are forced to conclude that the Luke→Mark→Matthew scenario is nearly perfectly in line with the empirical facts. This empirically observed result is almost exactly the theoretically anticipated result. The deviation is only 1/10th of 1%!

 

But this is not the full picture even yet. There are over 2,000 words of Mark which are not found in the Lukan and Matthean parallels. These additional words of Mark could have been included in the texts of Matthew and Luke, according to the theory of Markan Priority, because these words were supposedly seen in Mark by the writers of Matthew and Luke, but the writers of Matthew and Luke decided to omit these words.

By agreeing, according to Markan Priority, to drop words out of the text of Mark at exactly the same points in parallel contexts, the writers of Matthew and Luke not only demonstrated some peculiar facility for knowing to agree at these places, they also deprived themselves of a considerable amount of text. Another two thousand words which Matthew could have included, but dropped, would have increased the length of his TT text by 29.9% (2000/6682) and these potential additional two thousand words would have increased the length of Luke’s TT text by 30.9% (2000/6482).

 

 

The “minor agreement” evidence has now piled up to a total picture of 17.4% of Matthew’s existing TT text and 17.9% of Luke’s existing TT text, plus a potential additional 30% of text that Matthew and Luke could have included, but according to Markan Priority they supposedly decided to drop at the same points. The positive and negative agreements together affect over 45% of the potential text of Matthew and Luke (17.4% + 29.9% = 47.3% of Matthew’s potential and 17.9% + 30.9% = 48.8% of Luke’s potential). “Minor agreements” is surely a misnomer for these powerfully significant Matthean-Lukan agreements against Mark. They are rather precisely the Key Agreements for unlocking the Synoptic Problem! They establish that Mark is in the middle between the other two Synoptic Gospels.

 

(Halvor Ronning, “A Statistical Approach to the Synoptic Problem: Part 1—Triple Tradition,“ Jerusalem Perspective (2015), https://www.jerusalemperspective.com/15295/)

Part 2 – Double Tradition, Scenario of  Luke→Mark→Matthew

 

In the second article, Ronning brings into consideration the statistics of “Double Tradition” (DT). His use of this general term is regarding all Synoptic Gospel material that is attested by two Gospels, but not found in the third Gospel. The amount of material shared exclusively by Matthew and Luke is much greater than the material shared by the other two pairs. For this reason, the double tradition (DT) material has received the most attention in critical scholarship.

Results

Mark: 

  • 27% is Identical Form and Sequence (IFS) with Luke only (no Matthean parallel).
  • 32% is Identical Form and Sequence (IFS) with Luke when Matthew is parallel.

Matthew: 

  • 50% is Identical Form and Sequence (IFS) with Mark only (no Lukan parallel).
  • 48% is Identical Form and Sequence (IFS) with Mark when Luke is parallel.
  • 47% is Identical Form and Sequence (IFS) with Luke only (no Markan parallel).
  • 20% is Identical Form and Sequence (IFS) with Luke when Mark is parallel.
Ronning’s summary of the conclusion from these results from the Double Tradition analysis is as follows:

This scenario gives a totally consistent picture of Matthew in three of the four contexts. The exception is when Matthew writes a story that has a parallel not only in Luke, but also in Mark. Then Matthew’s average of verbal identity with Mark remains high (48% as just indicated), but Matthew’s average verbal identity with Luke drops radically from 47% to 20%, just because Mark happens to have the same stories! This is a radical change because it is less than half of Matthew’s average verbal identity with Luke in those parallels shared with Luke alone without the presence of Mark.

 

 

This radical change is elegantly explained by the Luke→Mark→Matthew scenario. The dramatic shift between Matthew’s ability to replicate approximately 50% of Luke’s wording when there is no parallel in Mark to a 20% rate when Mark is parallel to Luke is explained as Matthew’s continuing in precisely the same overall steady faithfulness to his sources as always. It becomes clear that it is precisely Matthew’s faithfulness to Mark, the immediate source before him, which was the very cause of the distance from Luke.

 

 

Matthew is consistent in his average use of Mark’s wording whether Luke has a parallel or not. It is Mark’s relationship to Luke that causes the distance, not Matthew’s relationship to Luke. It has nothing to do with any change in Matthew’s attitude toward Lukan material. Matthew treats Lukan material the same way he treats Markan material: with the same consistent 50% average—as long as Mark is not present. But since Matthew has chosen (for whatever reason) to follow the Markan material as primary, this decision causes Matthew himself automatically to reflect Mark’s distance from Luke. Indeed it is precisely the amount of distance from Luke that demonstrates the fact that Matthew got nearly all of his Lukan words in Triple Tradition though Mark, because the amount of distance fits so perfectly with what could be predicted from Matthew’s use of Markan wording.

 

 

It is most impressive to note the near perfect predictability of the distance between Matthew and Luke precisely because Matthew kept relating exactly the same way to Markan material whether Luke was present or not. Since Mark’s wording has only a 32% relationship to Luke’s TT material, and Matthew has a 50% relationship to Mark’s material, this allowed us to predict that Matthew’s relationship to Lukan material would drop to 16%, because the Lukan material is not coming directly to Matthew, but only through Mark as the intermediary.

(Halvor Ronning, “A Statistical Approach to the Synoptic Problem: Part 2—Double Tradition,” Jerusalem Perspective (2016), https://www.jerusalemperspective.com/15304/)

Part 3 – Single Tradition and Semitic Influence Analysis

In this part of Ronning’s analysis, his approach was to attempt to quantify the degree of Semitic influence in the Synoptic Gospels and to compare the level of Semitic influence in the three Single Traditions ((Matthean ST, Markan ST, Lukan ST) with the level of Semitic influence in the one Triple Tradition and the three Double Traditions. He then considers whether the information derived from the analysis is consistent with or contrary to the Luke→Mark→Matthew scenario of literary dependence.

Ronning noted that when evaluating the direction of dependence among the Gospels of Matthew, Mark, and Luke, it is theoretically possible that a chronologically later author could preserve more original wording than a chronologically earlier author. This can be the case if the chronologically later author quoted a source older than the chronologically earlier Gospel.

Ronning also noted that the level of Semitic influence is not necessarily a straightforward clue as to the relative age of a document. He observed that even when a high level of Semitic influence is accepted as evidence of an older wording of a Gospel passage, this still teaches us nothing about whether the document in which it was found is either earlier or later than a document which Greek version of the same material.

It is our natural tendency to assume that a high level of Semitic influence is a sign of chronological priority in terms of the wording of a passage, this it is not necessarily valid to assume the chronological priority of the document in which the Semitic wording is embedded. This specifically pertains to Matthew which exhibits a higher level of Semitic influence but can clearly be seen to depend on Mark which has a lesser degree of Semitic influence.

However, a Gospel with high levels of Semitic influence is likely to be based on ancient sources that originated among Jesus’ earliest followers. Moreover, if one Gospel, in particular, proves to be more Semitic in comparison to the other two Synoptics across all of its different parts including triple, double, and single traditions, then it is likely to be influenced by a source or sources outside the other two Synoptic Gospels. 

Martin’s Criteria as a basis for quantifying Semitic Influence

Ronning noted that Raymond Martin had achieved a breakthrough by developing seventeen criteria from Greek syntax for identifying and quantifying Semitic influence in Greek texts. Employing Martin’s criteria makes it practical and objective to quantify Semitic and Greek syntactical features in the Synoptic Gospels. (Raymond Martin, Syntax Criticism of the Synoptic Gospels (Lewiston, N.Y.: Edwin Mellen, 1987))

 Using Martin’s criteria provides differentiation between three kinds of Greek Texts as follows:

  • Original Greek Composition: When certain syntactical features that are typical of texts translated from Semitic originals are either completely lacking in a Greek text or fall below a certain percentage, then one can be certain that the source was originally composed in Greek.
  • Greek Translation of a Semitic Source: When the usage of these syntactical indicators are present in a text at a high percentage, this constitutes evidence that the Greek text was translated from a Semitic original. In fact, these “translation” Greek features are so typical that if a certain percentage of these features is present, then one can be certain that a particular Greek document is not originally Greek but was translated from Hebrew or Aramaic or directly based on such a translation.
  • Uncertain Texts Which Are Not Clearly Either of the Above: Martin also allows a wide range of doubt when considering texts which are neither clearly pure Greek composition nor translation from a Semitic original.

Results

  • Luke Single Tradition (ST) exhibited the most Semitic syntactical features in excess of Greek syntactical features of a value of -7 (the Lower the number the greater the amount of Semitic syntactical features in excess of Greek  syntactical features)
  • Matthew Single Tradition (ST) exhibited More Greek syntactical features than Semitic syntactical features with a value of 2
  • Luke Double Tradition with respect to Matthew (DT-Mt = -4) exhibited more Semitic syntactical features than Matthew Double Tradition with respect to Luke (DT-Lk = -2)
  • Luke Triple Tradition (TT) texts are more Semitic than Matthew Triple Tradition (TT) texts (-7 vs -4)
  • Mark exhibited the highest number of Greek syntactical features in excess of Semitic syntactical features (3)
Martin’s predisposition prior to completing the analysis was Markan Priority. However, on account of a full analysis of Mark, Martin ceased to speak confidently about Markan Priority stating “…somewhat surprising is the fact that the net frequencies of both Matthew’s and Luke’s accounts are much more Semitic, falling into the clearly translation Greek area!” (Martin, Syntax Criticism of the Johannine Literature, the Catholic Epistles, and the Gospel Passion Accounts, 45)

Conclusion

Ronning’s summary of the conclusion from the results of using Martin’s criteria in conducting a Semitic influence analysis are as follows:

From the data culled from Martin’s statistical analyses it is evident that the Gospel of Luke emerges as the most Semitic Gospel, whether in Lukan ST, Matthean-Lukan DT, or TT parallel to Mark 10-16. Although we are lacking statistics for Markan-Lukan DT and for TT parallel to Mark 1-10, the evidence we can glean from Martin’s work is in no way inconsistent with the Luke→Mark→Matthew scenario of linear dependence.
 
Triple Tradition: The Luke→Mark→Matthew scenario can account for Luke’s higher level of Semitic influence in TT parallel to Mark 10-16 by supposing that Luke was independent of Mark. Mark, who copied Luke, eliminated some of the Semitisms in Luke’s text. Matthew’s higher level of Semitic influence in TT parallel to Mark 10-16 can be explained on the presumption that, in addition to Mark, the author of Matthew used a source parallel to Mark 10-16 that was more Semitic and that had also been used by the author of Luke (which would account for the “minor agreements”).
 
Matthean-Lukan DT: The Luke→Mark→Matthew scenario can account for Luke’s higher level of Semitic influence in Matthean-Lukan DT by supposing that Luke was generally more faithful to the highly Semitic Greek source underlying these shared non-Markan pericopae.
 
Lukan ST: The Luke→Mark→Matthew scenario can account for Luke’s highly Semitic unique material by supposing that he copied it from the Semitic source (or sources) upon which the author of Luke based his Gospel.
 
(Halvor Ronning, “A Statistical Approach to the Synoptic Problem: Part 3—Single Tradition,” Jerusalem Perspective (2016), https://www.jerusalemperspective.com/15475/)

Part 4—Non-Linear Hypotheses

In the opening part of the final article, Ronning summarizes the previous findings as to three strong indications of the Luke→Mark→Matthew model:

  1. The elegance with which Lukan Priority explains the verbal identity relationships between the Synoptic Gospels, including its promotion of the minor agreements from a problem to a key part of the solution because of seeing Mark in the middle position (Part 1).
  2. The ability of Lukan Priority to give a consistent picture of each writer’s relationship to his parallel texts (Part 2).
  3. The ability of Lukan Priority to account for the level of Semitic influence in the various parts of each Gospel (Part 3)

In the fourth article, Ronning turns to non-linear scenarios and the question of whether these scenarios can explain the statistics of verbal identity and the levels of Semitic influence in each Gospel, as well as the Lukan Priority model. One of the most prominent non-linear hypothesis is Markan Priority (also known as the Two-Source Hypothesis). A less common theory is Markan Conflation (also known as the Two Gospel Hypothesis, or Matthean Priority). In this article, Ronning addresses both these hypotheses by subjecting them to statistical analysis. He does this by examining the ability of the non-linear hypothesis to satisfactorily explain the statistics of verbal identity (based on ISF) and the levels of Semitic Influence in the three Synoptic Gospels. Regarding Semitic influence, his assessment is based on the relative ease or difficulty of translating the Greek texts of the Gospels into Hebrew that Lindsey observed. That is, the ease of translating the Synoptic Gospels back into Hebrew. 

Markan Priority (Two-Source) Scenario

The theory of Markan Priority, or the Two-source Hypothesis, supposes that the authors of Matthew and Luke both relied on the Gospel of Mark as their primary source. Mark’s Gospel not only supplied the wording of the stories Matthew and Luke copied from Mark. The Two-source Hypothesis further supposes that in addition to Mark, the authors of Matthew and Luke copied from a source called “Q” document. “Q” is the source of the Matthean-Lukan Double Tradition (DT) pericopae, but since neither Matthew nor Luke were aware of one another’s work, they inserted the Q sayings into different slots within Mark’s narrative framework.

1. Identical Form and Sequence (ISF) Analysis

50% of Matthew’s text in parallels with Luke alone (Matthean-Lukan DT) is composed of words that are Identical Form and Sequence (IFS) with Luke. But in great contrast to Matthew’s text in parallels with Luke which also have parallels in Mark (DT within TT), the percentage of Matthew’s text that retains Identical Form and Sequence (IFS) with Luke, according to the Markan Prioity theory, drops to 20%.

Ronning Notes

This shift from 50% to 20% in Matthew’s relationship to Luke just because Mark happens to have the same material demands an explanation. One is forced to conclude that although Matthew is able to maintain a high verbal identity with Lukan parallels, Matthew reflects the wording of Mark whenever Mark is present (at the same consistent 50% average), regardless of what happens in relationship to Luke. Additionally one must explain the origin of the minor agreements, i.e., how it is that 4% of Matthew’s text is composed of verbal identities with Luke that are not shared with Mark.

(Halvor Ronning, “A Statistical Approach to the Synoptic Problem: Part 4—Non-Linear Hypotheses,” Jerusalem Perspective (2016), https://www.jerusalemperspective.com/15729/)

2. Ease of Retroversion Assessment

Ronning notes that Mark is typically erratic throughout his Gospel in terms of being easily back-translated from Greek to Hebrew. In areas where there is a Triple Tradition (TT), Matthew is almost as erratic as Mark, although Luke is also parallel. However, in areas where Matthew only parallels with Luke and not with Mark, Matthew is often as Hebraic as Luke. Matthew’s Single Tradition (ST) is also more Hebraic than Mark. 

Luke is more consistently Hebraic throughout than any of the other Synoptic Gospels, but this is a problem for a theory of Markan Priority that supposes Luke is dependent on Mark. The implication is that Luke would have to be rewriting the erratic Markan material to force it to become more consistently Hebraic. Too much correcting of Greek syntax back into Semitic is necessarily exhibited by Luke to sustain the theory that Luke sought to impose a Semitic style artificially on his materials. The theory that Luke used snippets from various Semitic sources and interspersed with his Greek redactional comments is a better fit for the data supported by Martin and Lindsey. 

Matthean Priority Scenario

This view postulates that Matthew is the oldest Gospel and that the author of Luke used Matthew in addition to other traditions he had received. This view regarding Mark is that he drew from both Matthew and Luke, often harmonizing them in the process. 

1. Identical Form and Sequence (ISF) Analysis

  • According to this theory, when Luke is leaning on Matthew alone (Matthean-Lukan DT), 50% of Luke’s text is composed of words that are Identical Form and Sequence (IFS) with the Matthean parallels,
  • However, when Luke is leaning on Matthew when Mark also is parallel (DT within TT), only 20% of Luke’s text is composed of Identical Form and Sequence (IFS with Matthean wording).
  • According to this theory, there is a substantial difference between these two data points.  To justify this scenario, one must explain why Luke treated his Matthean material inconsistently. The theory fails to provide support for the evidence that Mark is the cause of this inconsistency since the theory postulates that Mark came last. 

2. Ease of Retroversion Assessment

Some parts of Matthew are relatively easy to revert to Hebrew, but when parallel to Mark, retroversion of the Matthean text becomes much more difficult. Matthean Priority fails to account for this fluctuation in Matthew, since the Markan Conflation theory holds that since Matthew was prior to Mark, there was no way for Matthew to know when Mark would copy his material and when he would not. One is forced to explain why Mark accepted only the parts of Matthew that are difficult to revert to Hebrew (where they remain equally reversion resistant in Mark).

According to this scenario, when Luke accepts Matthean material that is difficult to revert to Hebrew (as is almost all 36.52% of Matthew’s text which is Matthean-Lukan with Markan parallels [DT within TT], some 6,682 words of TT), Luke rewrites this material in such a way that it becomes easier to revert to Hebrew.

Ronning notes the following concerning ease of retroversion.  

Markan Conflation faces the same serious problem we noted above under Markan Priority: the need to explain how Luke managed to convert a source text that is difficult to revert to Hebrew into a text that is easy to revert to Hebrew. The difference here is that according to Markan Conflation Luke must have been re-Hebraizing parts of Matthew that resist retroversion to Hebrew, whereas according to Markan Priority, Luke must have been re-Hebraizing parts of Mark that resist Hebrew retroversion. (Halvor Ronning, “A Statistical Approach to the Synoptic Problem: Part 4—Non-Linear Hypotheses,” Jerusalem Perspective (2016), https://www.jerusalemperspective.com/15729/)

Lukan Priority Scenario

This is the view that Luke is the oldest of the canonical Gospels that Mark was in view of Luke and that Matthew was made in view of Mark (and likely was only indirectly influenced by Luke in terms of what Mark incorporated from Luke). The implication, with respect to Mark, is if the author was highly selective in what material of Luke to incorporate, and he was highly creative by recasting the stories he selected in his words. The implication with respect to Matthew is that the author was able to restore some of the same stories in Luke that Mark had omitted, and also that some Mark had rewritten by utilizing one of the sources the author of Luke had when writing his Gospel. 

 

1. Identical Form and Sequence (ISF) Analysis

  • 50% of Matthew’s text is made up of words that are Identical Form and Sequence (IFS) with the Lukan parallels in Matthean-Lukan DT.
  • The high frequency of IFS with Luke would seem to point to the direct dependence of Matthew on Luke were it not for the fact that Matthew’s relationship to Luke when Mark is parallel (DT within TT) is entirely different. When Mark is also parallel only 20% of Matthew’s text is made up of words that are IFS with Luke.

Ronning states the advantage of the Lukan Priority scenario as follows.

The advantage of the Lukan Priority scenario is that Matthew’s changing relationship to Lukan material depending on whether or not Mark is also present is not a challenge to the Luke→Mark→Matthew scenario, but a proof of its viability. When one takes a closer look at the 20% it turns out that only 16% is left after one subtracts the “minor agreements” of Matthew and Luke against Mark, that is, the words that have not come from Luke via Mark into Matthew. This 16% is exactly what the Luke→Mark→Matthew scenario predicts as Lukan IFS arriving into Matthew via Mark. This prediction is exactly what the statistical facts later confirmed from our research.(Halvor Ronning, “A Statistical Approach to the Synoptic Problem: Part 4—Non-Linear Hypotheses,” Jerusalem Perspective (2016), https://www.jerusalemperspective.com/15729/)

2. Ease of Retroversion Assessment

The observation of Martin and Lindsey as that throughout the Gospel of Luke, it is easier to revert to Hebrew than either Mark or Matthew. Their observations confirm that Mark is consistently more difficult to revert to Hebrew than Luke. 

According to the Lukan Priority Scenario, the indications are:

  •  Lukan Priority is better able to explain the greater ease with which Luke reverts to Hebrew than Mark by supposing that the author of Mark consistently paraphrased Luke’s wording, with the result that his test is further removed from Hebrew idiom and word order than the text of Luke.
  • With respect to the author of Matthew, he follows his sources in such a way that he exhibits the same resistance to Hebrew retroversion as Mark in all parallels with Mark. In some cases, by relying an earlier, more Hebraic source (consistent with Luke), Matthew sometimes achieved a more Hebraic text than Mark’s. In the Matthean-Lukan Double Tradition (DT), when Matthew is not influenced by Mark, Matthew reverts to Hebrew with about the same ease as Luke. 

Summary

After measuring the Markan, Matthean, and Lukan Priority scenarios Ronning concludes, “The consistence and elegance of the Lukan Priority scenario contrast sharply with the difficulties faced by the other two scenarios.” Regarding Markan Priorists, he adds that they, “have been extremely feeble in their efforts to explain away the “minor agreements.”

Ronning further acknowledges some aspects of other theories being correct, including Mark having priority over Matthew, the likely existence of other source documents, and Mark exhibiting a conflationary style of redaction that existed in the first century. 

Ronning’s concluding remarks are as follows:

Neither Markan Priority nor Markan Conflation can explain the publicly observable facts and verifiable statistics as well as Lindsey’s hypothesis of Lukan Priority. There is a great advantage for synoptic research when it is understood that Mark is in the middle. The immeasurable advantage to be gained by seeing Mark in the middle is the way it can account for the so-called “minor agreements.” For Lukan Priority the Matthean-Lukan agreements against Mark in TT pericopae are no longer a problem, but part of the elegant solution made possible by these “key agreements.”

 

When Mark is placed in the middle the Matthean-Lukan agreements against Mark can now be easily understood as the “corrections” of the final Gospel writer. The final writer—the author of the canonical Greek Matthew—is to be imagined as working with Mark, his immediate source, in front of him. But out of his awareness of Luke’s Hebraic source, Matthew reinserted many of the exact words that Luke had copied from that first Gospel, words that Mark had omitted. Matthew reinserted them right back into the exact same positions as in Luke. Not only did Matthew make many reinsertions, he also frequently omitted words and phrases that Mark had inserted, words and phases that were not present in that earlier Gospel that stands behind the Gospel of Luke. This all falls into place as we see the statistical support for the mediating position of Mark between Luke and Matthew. 

 

(Halvor Ronning, “A Statistical Approach to the Synoptic Problem: Part 4—Non-Linear Hypotheses,” Jerusalem Perspective (2016), https://www.jerusalemperspective.com/15729/)