Views of New Testament Textual Space (DRAFT)

Multivariate analysis of textual variation among witnesses of the New Testament reveals features of the textual space constituted by those witnesses. Originally written in Greek, the New Testament was copied by hand for almost fifteen centuries until the advent of mechanized printing provided an alternative means of propagation. Translations into other languages were produced as well. Some of these, such as the Latin, Coptic, Syriac, and Armenian versions, appeared early and so provide insights into ancient states of the text. Patristic citations form another class of evidence which allows varieties of the text to be associated with particular localities and epochs.

As with every widely read work from antiquity, the New Testament exhibits textual variation which has been introduced by scribes and correctors. Sites where textual variation occurs are identified by comparing extant witnesses of the text.[1] Alternative readings at a variation site may be classified as orthographic or substantive. Orthographic variations are often ignored as they only affect the surface forms of words (e.g. spelling) and not their meaning. Substantive variations do affect meaning: they are called variants. The list of witnesses which support a particular reading of a particular variation site is known as the reading's attestation. A list of all readings at a variation site along with the attestation of each reading is called a variation unit.

There is an ongoing effort to establish the initial text which stands behind the range of texts found among surviving witnesses of the New Testament.[2] The most important witnesses for establishing the initial text fall into these categories:

Greek manuscripts are the primary witnesses to the text of the New Testament. Ancient versions are early translations of the Greek text into languages such as Latin, Coptic, Syriac, and Armenian. It is often possible to establish which Greek variant a version supports by translating its text at a variation site back into Greek. Patristic citations are quotations of the scripture by Church Fathers. Which variant was in a Church Father's copy of the text at a particular variation site can often be discerned if that part of the text is covered by one of his quotations.

A large proportion of the textual evidence disappeared long ago. In general, the older the copy, the more likely it is to have been lost. Even the most comprehensive data sets are mere samples of what once existed. Results obtained by analysis of these data sets are therefore provisional because it is always possible that including further data would produce different results. Nevertheless, it is reasonable to expect that analysis results derived from a sufficiently large sample are approximately the same as results that would be obtained if a more comprehensive data set were analysed.

Even though much is lost, a stupendous amount of evidence remains. There are many thousands of manuscripts in Greek, Latin, Syriac, Armenian, and other languages. Patristic citations are also very numerous. Given such a great cloud of witnesses, it can be difficult to see where each one stands in relation to the others. Fortunately, various methods of statistical analysis can be applied to data sets which relate to textual variation in order to explore relationships among the witnesses.

Analysis might begin from a number of starting points. One is a critical apparatus which gives attestations (i.e. lists of witnesses) in support of variants found at variation sites. The information contained in the apparatus must first be encoded as illustrated by reference to this entry from the fourth edition of the United Bible Societies' Greek New Testament (UBS4):

Figure 1. Apparatus entry (Mark 1.1, UBS4)

The textual states of witnesses included in an apparatus entry may be encoded with numerals, letters, or other symbols. In this example, witnesses in the attestation list for the first variant are assigned the code 1, those in the second variant's list are given the code 2, and so on. The state of a witness is classified as undefined and given a code of NA (for not available) when it is not clear which variant the witness supports. For manuscripts this may be due to physical damage or because the manuscript does not include the section of text being examined; for versions, it may not be clear which state of the Greek text is supported by a back-translation of the version; for patristic citations, the reading of a Church Father's text may be unclear if the quotations are not exact (e.g. adaptations, allusions, or quotations from memory) or if different witnesses of the Church Father's text have different readings. In the present example, a number of versions (Latin, Syriac, Coptic) and patristic citations (e.g. those of Irenaeus, Ambrose, Chromatius, Jerome, and Augustine) are treated as undefined because it is not clear which variant each supports at this variation site.[3]

Figure 2. Part of a data matrix (Mark, UBS4)

The next step is to construct a distance matrix which tabulates the simple matching distance between every possible pair for the set of witnesses under examination. The simple matching distance between two witnesses is the proportion of disagreements between them in those variation units where the textual states of both are defined. Being a ratio of two pure numbers, this quantity is dimensionless (i.e. has no unit). It varies from a value of zero for complete agreement to a value of one for no agreement between the two witnesses.[4] A pair of witnesses only qualifies for inclusion in a distance matrix if both members of the pair share a minimum number of variation units at which the states of both are defined. This minimum requirement is intended to reduce sampling errors to a tolerable level. In the analysis performed here, the minimum required number is usually set at fifteen.[5]

Figure 3. Part of a distance matrix (Mark, UBS4)

A wide variety of multivariate analysis techniques are available to explore relationships between the objects whose distances from each other are tabulated in a distance matrix. Other generic terms such as observation, case, or item may also be used when refering to the objects being compared, which in this article are New Testament textual witnesses. Only three analysis techniques are presented here, namely classical multidimensional scaling (CMDS), divisive clustering (DC), and partitioning around medoids (PAM). My article on “How To Discover Textual Groups” provides an introduction to these techniques. While each method has its own merits and potential weaknesses, the ones used here serve to introduce results obtained when exploratory multivariate analysis is applied to distance matrices based on textual variation among New Testament witnesses. All of the statistical analysis in this study is performed using the R Language and Environment for Statistical Computing. The relevant programs are available here.

Classical multidimensional scaling finds the set of object coordinates which best reproduces the actual distances between objects in the distance matrix. A plot of these coordinates shows how the objects are disposed with respect to one another when all distances are considered. This study refers to such a plot as a map and uses the term textual space for the space obtained when the objects are textual witnesses. Achieving a perfect spatial representation of a distance matrix may require any number of dimensions up to one less than the number of objects. This presents a problem when a large number of objects is being examined because our spatial perception is three-dimensional. Fortunately, three dimensions is often sufficient to achieve a reasonable approximation to the actual situation. The CMDS analysis produces a coefficient called the proportion of variance which is the proportion of distance matrix information explained (i.e. accounted for) by a map. This coefficient ranges from a value of zero to one, with a value of one indicating that the map is a perfect representation of the actual distances.

Divisive clustering begins with a single cluster and ends with individual objects. The program documentation describes the clustering algorithm as follows:[6]

This type of analysis produces a dendrogram which shows the “heights” at which clusters divide into sub-clusters. A divisive coefficient which measures the amount of clustering structure is presented as well. The value of this coefficient ranges from zero to one with larger values indicating a greater degree of clustering. It should be emphasized that a DC dendrogram is not a genealogical tree of the type produced by phylogenetic analysis. Instead, it merely shows a reasonable way to progressively subdivide an all-encompassing cluster until every sub-cluster is comprised of a single object.[7]

Partitioning around medoids (PAM) builds clusters around representative objects called medoids. The program documentation provides this description:[8]

PAM analysis produces a statistic called the mean silhouette width (MSW) which indicates how many groups are in a data set.

The analysis results presented in this article are based on data extracted from a number of sources. Each source is assigned an identifier based on the responsible author or party. Results are keyed to these identifiers so that the basis of the relevant data set can be determined. Information on each source is given below under the heading of its identifier.

The sources use various data formats, including tables of percentage agreement and lists of pair-wise proportional agreements. These are converted to distance matrices before analysis. A number of the sources have tables of percentage agreement with blank entries where there is insufficient data to complete a calculation. This can happen if two witnesses do not cover a sufficient quantity of overlapping text (e.g. Old Latins e and k) or if the two are mutually exclusive (e.g. a scribe and corrector of the same manuscript). Analysis cannot proceed if a distance matrix has missing data. However, it is possible to produce multiple distance matrices from a table of percentage agreement which has missing entries so that each distance matrix has no missing data. For example, Fee's tables of percentage agreement include blank entries for cells relating to the scribe and corrector of a manuscript. Two distance matrices are produced from each table of this kind, one including entries for the scribes and the other containing entries for the correctors.

The following table presents data sets and analysis results based on the sources. The source column identifies the source of a data set, the matrices column presents data and dissimilarity matrices extracted from the data set, and the CMDS, DC, and PAM columns present analysis results. Data sets and some analysis results are accessed via links in the table. A distance matrix is always provided but a data matrix is only included if one has been constructed. If there is no data matrix then NA for not available is entered in the relevant column. Data and distance matrices are formatted as comma-separated vector (CSV) files so that they can be downloaded and imported into a spreadsheet program for inspection.

Some data sets represent manuscripts by Gregory-Aland numbers (e.g. 01, 02, 03, 044) while others use letters or latinized forms (e.g. Aleph, A, B, Psi). These symbols carry through to the analysis results. In INTF data, ECM or A (for Ausgangstext or initial text) represents the text of the Editio Critica Maior. The A for Ausgangstext in INTF data sets should not be confused with the A for Codex Alexandrinus in other data sets. Abbreviations UBS, WH, and TR stand for the texts of the United Bible Societies' Greek New Testament, Westcott and Hort's New Testament in the Original Greek, and the Textus Receptus, respectively. Maj, Byz, and Lect stand for majority, Byzantine, and lectionary texts, respectively. The relevant printed editions should be consulted for explanations of what these group symbols represent.

The analysis techniques employed in this study operate on a distance matrix, not a data matrix. If there is a data matrix for the source data then the corresponding distance matrix is obtained by calculating the simple matching distance between each pair of witnesses with enough defined variation sites. In other cases, the distance matrix is constructed from a table of percentage agreement or from a table which records the number of agreements and number of variation units where both witnesses of a pair are defined. Analysis cannot proceed if entries are missing from a distance matrix. Missing entries in tables of percentage agreement are therefore eliminated by an iterative procedure which at every step drops those witnesses with the most missing entries.[10] Three decimal places are used for distances regardless of whether this level of precision is warranted.

Results are presented for three modes of analysis: classical multidimensional scaling (CMDS), divisive clustering (DC), and partitioning around medoids (PAM). The proportion of variance coefficient associated with a CMDS map indicates the proportion of information contained in the distance matrix which is represented by the map. The divisive coefficient associated with a DC dendrogram indicates the degree of clustering of witnesses in the data set.[11]

The PAM column does not present partitions but instead gives indications of preferred numbers of groups for the data set. PAM analysis partitions cases (here, New Testament witnesses) into a number of groups which must be specified before the analysis begins. A statistic called the mean silhouette width (MSW) that is calculated during PAM analysis indicates preferable numbers of groups. Repeating the analysis for each possible number of groups shows which numbers are better suited to the data set because larger MSW values indicate more “natural” partitions. That is, local maxima (i.e. peaks) in a graph of MSW versus the number of groups correspond to preferable numbers of groups. The MSW column has a link to the MSW versus number of groups plot while the groups column gives a selection of preferred numbers of groups suggested by the graph.

A typical MSW plot for New Testament data contains numerous peaks. In this article, only those peaks which exceed the average of all MSW values for the plot are considered as eligible indicators of preferred numbers of groups.[12] Even with a reduced number of peaks, it can be difficult to choose how many groups to use for a partition. The highest peak often corresponds to only two or three groups but using such a small number produces some nebulous groups. By contrast, using a larger number tends to produce more coherent groups. Due to the tendency for MSW values to decrease as the number of groups increases, a peak to the right of another one has a claim to being preferable even if it has a smaller magnitude. Sometimes there is only one eligible peak, in which case only one number of groups can be given. While it would be reasonable to select the number of groups corresponding to any eligible peak of an MSW plot, the groups column usually gives only two suggestions:

A typical multivariate analysis result such as the CMDS map obtained from the UBS2 data set for Matthew's Gospel shows that groups exist among New Testament witnesses.

Figure 4. CMDS map (Matt, UBS2)

A term such as group, cluster, or nucleus might be used to describe a local maximum in the density of objects within a CMDS map. A line which joins two items might be called a trajectory, and a region between groups where there is a higher than usual concentration of witnesses might be called a stream or corridor.[14]

The vocabulary of tree structures is useful when discussing DC dendrograms.

Figure 5. DC dendrogram (Matt, UBS2)

Figure 6. MSW plot (Matt, UBS2)

The PAM algorithm identifies a medoid for each group. The medoid is the member for which the sum of distances from all other members of the group is a minimum. In groups with three or more members, it is fair to describe the medoid as the most central member of its group.[15] This article uses the medoid identifier as a label for the associated group.

As a data set is progressively divided into larger numbers of groups, it often happens that a parent group gives rise to narrower child groups. Quite often there is a single group which acts as a catch-all for witnesses that do not fall into narrower, more coherent groups. Group K of the 13-way partition is an example: if the same data set is partitioned into more groups then this group is likely to produce subgroups while retaining a core membership.[16] Dividing a data set into a large number of parts reveals group cores. Whichever groups remain are highly coherent, entirely comprised of closely related members. Dividing a data set into numerous groups often produces singletons, each comprised of a single member. A singleton is isolated, not having any close relatives among the items in the data set.

Adding the partition's number of groups to the group label produces a more specific identifier. For example, in partitions of the INTF data set for Matthew, 033 (2) refers to the group with medoid 033 in a two-way partition while 033 (24) refers to the group with medoid 033 in a 24-way partition. Corresponding groups such as 033 (2) and 033 (24) are often produced when the same data set is divided into different numbers of parts. However, the medoids of such groups are not necessarily the same. Adding or subtracting even a single member can cause the medoid of a group to change. Consequently, correspondence must be established on the basis of common membership, not shared medoids. If groups from different partitions have the same core membership but differing medoids then descendant groups can be labelled by chaining the respective medoids together. To give an example, the relevant MSW plot suggests that a 93-way partition is another reasonable choice for the INTF data set for Matthew's Gospel. The members of group 031 (93) are also found in group 045 (24) yet the respective medoids differ. Labelling the subgroup as 045/031 signifies the connection with the parent group from which its members are drawn.

Even if an item is placed in a group with others, it may not be a good fit. Poorly classified items are identified using the silhouette width statistic which is calculated for every item during PAM analysis. Its value ranges from +1 to -1: the closer it is to +1, the better the associated object fits into its assigned group; by contrast, the closer the statistic is to -1, the worse the fit. Items with a negative silhouette width are listed at the end of each PAM partition presented in this article to show that they are poorly classified. They are listed in order of increasingly negative silhouette width so the worst classified appears last.

A poor fit may indicate that a witness has a mixed text. It may also indicate that the chosen number of groups is too small for a text to be grouped with like texts alone. Another indication of a questionable classification is when the results of different analysis modes point to a range of possible affiliations. When the classification of a witness is uncertain, further information may produce a more definite result. However, if a witness actually does have a mixed text then it will remain difficult to classify as anything but a mixture. In the case of a partition, a mixed text will not fit well into any group unless other witnesses happen to possess the same kind of mixture of texts.

The respective views produced by CMDS, DC, and PAM analysis are often but not always consistent. If results obtained by all modes point to the same conclusion with respect to implied clustering then that can be taken as a reasonably sure result; however, if they differ then each result needs to be handled with due caution. The distance matrix is the final arbiter when the affiliation of a witness is not clearly indicated by concurrence of analysis results. A statistical analysis reveals the range of distances which normally occur between two “texts” comprised of states (i.e. readings) which are randomly selected from those available at each variation site. Distances outside the normal range indicate an adjacent or opposite relationship between two texts: adjacent if the distance is less than normal and opposite if greater.[17]

To explore the affiliation of a reference witness, the relevant row of the distance matrix is extracted and its entries are ranked in order of increasing distance from the reference. Any distance in the normal range is marked by an asterisk to show that it is not statistically significant. Proximity is still a useful indicator of affiliation even when the distances are not statistically significant. Lack of significance does not imply lack of relationship: all New Testament witnesses are ultimately related even though not all have close relationships. The relative size of the normal range of distances contracts as the number of places compared increases so some of the closest witnesses might become significantly close if a larger data set is examined.

In DC and PAM analysis, a slight change in the distance matrix can cause a witness which lies near the midpoint between two groups to jump from one to the other. If a text is largely comprised of a mixture of variants characteristic of particular groups then a CMDS map will locate it between the relevant groups, proportionally closer to those whose variants it most often contains. Singular readings and variants shared with only a few other witnesses tend to isolate a text, increasing its distance from all others. The more eccentric a text relative to others in the data set, the more remote its location in a CMDS map.

The analysis techniques used in this study do not provide direct guidance concerning the relative priority of groups or the witnesses they contain. Another technique such as the Coherence-based Genealogical Method (CBGM) developed by the INTF can be used to investigate whether the witnesses in one group are closer to the initial text than those in another.[18] While the analysis results presented here are not genealogical, they do reveal textual affiliation and the main streams of New Testament textual development. To use another analogy, a CMDS map is like a time-exposure where texts of differing date occupy the same picture. Even though the techniques used here do not indicate temporal order among the texts, they are valuable for exploring relative dispositions, with similar texts being close to each other in a CMDS map, in the same branch of a DC dendrogram, and the same group of a PAM result. That is, similar texts tend to collocate in results produced by all three analysis modes.

Clustering may be identified by inspecting a CMDS map, cutting a DC dendrogram, or choosing a local maximum in a plot of mean silhouette widths and producing a corresponding partition using PAM analysis. We have a natural facility for recognising group structure in a point cloud such as constitutes a CMDS map. However, we are prone to misinterpret a purely random arrangement of points as constituting a cluster. One way to avoid this kind of error is to know what analysis results look like for randomly generated analogues of the data sets being examined, an approach demonstrated in my “Groups” article.

The following discussion of analysis results typically begins with partitions obtained by PAM analysis and may consider CMDS and DC results as well. More comprehensive data sets for a book or section take priority although others may be examined if they contain versional or patristic evidence. Many of the groups found by multivariate analysis have been discovered before. Unless otherwise indicated, associations between known groups and those revealed by PAM analysis are established by reference to the table of “Profile Classification” contained in Frederik Wisse's Profile Method, 52-90. Wisse's test passages are from three chapters of Luke's Gospel so his classifications are not necessarily valid in other parts of the New Testament.

The analysis results presented here highlight variations between witnesses of the New Testament. This naturally raises the question of what difference the variations make to the meaning of the text. Many variations are of little consequence — whether an added or dropped article, a change of word order, or substitution of a synonymous phrase. Some variations have a larger effect, the two most extreme examples being Mark 16.9-20 and John 7.53-8.11 which are absent from a number of witnesses.

One way to convey how much difference the variations make is to provide translations of a number of textual varieties for the same section of text. The following table gives a parallel translation of four varieties of the first chapter of Mark, highlighting the variation sites identified in the fourth edition of the United Bible Societies Greek New Testament. This edition only presents a selection of textual variations:

The variation units presented in the UBS apparatus constitute a small proportion of the total number of variation units that exist. However, the ones given below should provide a reasonably good impression of how much the respective varieties of text differ in meaning. This is because the great majority of variations which are not presented in the UBS apparatus have only a slight semantic effect.

The textual varieties shown in the table consist of four clusters identified by reference to the DC dendrogram of UBS4 combined data for Mark:[39]

For each variation unit, the variant supported by a textual variety is taken to be the one that occurs most frequently among its members. To illustrate, suppose that a variation unit has three variants and that two witnesses in cluster C have the first, three have the second, and four have the third. The variant supported by cluster C would then be taken to be the third. For the purpose of this exercise, if a tie occurs then the supported variant is taken to be the one with the greatest tendency to isolate the variety.

Isaac Newton said, “If I have seen further it is only by standing on the shoulders of giants.” This sentiment truly applies to the results presented here. Our field owes a great debt to those who have compiled the information, both printed and electronic, upon which the data and distance matrices are based.

Compiling the basic data from which analysis proceeds is an arduous and painstaking task. Richard Mallett deserves special thanks in this respect, having encoded data matrices and transcribed tables of percentage agreement from numerous sources. Mark Spitsbergen helped to encode the UBS4 apparatus data for the first fourteen chapters of Matthew.

Maurice A. Robinson kindly provided tables of percentage agreement for the Gospels and Acts. These are derived from the apparatus of the second edition of the United Bible Societies' Greek New Testament. The exacting task of transforming the data into electronic format was performed by Claire Hilliard and Kay Smith.

A number of the results are produced from comprehensive data generously provided by the Institut für neutestamentliche Textforschung in Münster, Germany. Researchers at the INTF have spent many years on the gargantuan task of compiling this data. Holger Strutwolf, Klaus Wachtel, and Volker Krüger were instrumental in providing access to the data.

The analysis would scarcely have been possible without the marvellous R Language and Environment for Statistical Computing. Finally, thanks go to Gerald Donker for suggesting that the RGL plotting library be used to produce three-dimensional CMDS maps. He also encouraged me to take a less procrustean approach to missing data. As a consequence, the analysis results presented here include many more witnesses than they otherwise would.