Views of New Testament Textual Space

The Greek 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, and Syriac versions — appeared early and thus preserve ancient states of the text. Patristic citations form another class of evidence that allows varieties of the text to be associated with particular localities and epochs. Analysis of textual variation allows relationships between these three classes of witnesses — manuscripts, versions, and patristic citations — to be explored. This article applies various kinds of analysis to textual variation data collected from a variety of sources. The analysis results offer complementary views of the textual space occupied by these venerable witnesses to the New Testament text.

As with every widely read work from antiquity, the New Testament exhibits textual variation introduced by scribes and correctors. Sites where textual variation occurs are identified by comparing extant witnesses of the text.[1] Differences may be classified as orthographic or substantive. Orthographic variations are often ignored as they only affect the surface form of a text and not its meaning. Substantive variations do affect meaning: they are often called readings or variants The list of witnesses that 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. Critical editions often present variation units in an apparatus. The present study is based on analysis of data sets extracted from critical editions, monographs, journal articles, dissertations, or online databases.

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 reading 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 reading 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. Even a comprehensive data set that includes all readings of all extant witnesses is still a mere sample of what once existed. In general, the older the copy, the more likely it is to have been lost. This loss of data presents a fundamental problem: if extant texts do not represent the oldest copies then the survivors will give a skewed impression of the initial text. Happily, there is a way forward: if like texts are grouped and more or less accurate representations of the archetypal texts that gave rise to the groups can be reconstructed then these archetypes have a claim to being more representative of the ancient text. In addition, the extant copies provide data for estimating how accurately the copyists practiced their art. Armed with a knowledge of the number of generations of copies and typical rates of substituting one reading for another per generation, it is possible to say how much the original text is likely to differ from the initial text recoverable from extant copies. While raw data that could be used for the purpose is presented below, I do not propose to reconstruct hypothetical archetypes or estimate rates of change here. Instead, this article will focus on presentation and analysis of available data to explore grouping among the extant texts.[3]

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 suitable place to begin is a data set derived from a critical apparatus which gives attestations (i.e. lists of witnesses) in support of readings found at variation sites. In nearly all cases, practical considerations restrict an apparatus to presenting a sample of extant texts. Results obtained by analysis of these data sets are therefore provisional because it is always possible that including further data would produce different results. However, it is reasonable to expect that analysis results will approximate those that would be obtained if a more comprehensive data set were analysed provided that the sample is sufficiently large and has been selected without systematic bias.

The information contained in an 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)

When it comes to encoding apparatus entries, the textual states found among the witnesses can be represented by numerals, letters, or other symbols. In the present example, the first reading is encoded as 1, the second as 2, and so on. The state of a witness is classified as undefined and encoded as NA (for not available) when it is not clear which reading 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 readings they support at this variation site.[4]

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 each pair of witnesses sufficiently represented in the data set. The simple matching distance between two witnesses is the proportion of disagreements between them at those variation sites 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.[5] A witness only qualifies for inclusion in a distance matrix if all distances for that witness are calculated from at least a minimum number of variation sites. This constraint is intended to reduce sampling error to a tolerable level. It is enforced by a vetting algorithm that progressively drops witnesses with the least numbers of defined variation sites until all distances in the distance matrix are guaranteed to have been calculated from a minimum acceptable number of sites. The minimum acceptable number for the distance matrices of this study is nearly always set at fifteen.[6]

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

Various analytical methods can applied to a data set derived from a critical apparatus to explore relationships between witnesses. All of the results presented in this article are obtained using a statistical computing language called R. The analysis is performed by means of R scripts written by the author which are available here. The R program and additional packages (e.g. cluster, rgl, ape) required to run the scripts can be installed using instructions provided at the R web site.

Readers are encouraged to use the scripts. There are various ways to run a script once the R environment is installed. For users who prefer a command line interface, typing R into a terminal window provides an R prompt. (It helps to change to the directory which holds the scripts before launching R.) A command can then be entered in order to run a script. As an example, the command source("dist.r") typed at the R prompt causes the dist.r script to construct a distance matrix from the specified data matrix. Parameters such as paths to input and output files are specified in the scripts, which users are free to edit.

The data sets analysed in this article derive from various sources. Each source is assigned an identifier based on the author or party who produced it. A source is often used to produce data sets for a number of New Testament sections such as individual gospels and letters. Each analysis result is keyed to the relevant section and source identifier so that its underlying data set can be identified.

The data sets generally retain the symbols used by their associated sources to represent New Testament witnesses. Some 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. Also beware of confusing texts when the same letter (e.g. D, E, F, G, H, K, L, P) refers to different manuscripts in different parts of the New Testament. 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.

A source may be in the form of apparatus entries, tables of percentage agreement, or lists of pairwise proportional agreement. If the source is an apparatus then it is used to construct one data matrix per desired section. Each data matrix includes those witnesses and variation sites covered by the apparatus, using symbols such as numerals or letters to encode reported textual states (i.e. readings). A distance matrix is then constructed from the data matrix. If the source only reports percentage or proportional agreement between witnesses then a distance matrix is constructed directly from the agreement data and no data matrix is produced. Distances are usually specified to three decimal places regardless of whether this level of precision is warranted.

Analysis cannot proceed if a distance matrix has missing entries. This problem can be avoided by manually producing multiple distance matrices from the same source data, each omitting a particular witness whose inclusion would create an empty cell. This is done for a number of the distance matrices presented below, including Brooks' table for John (where there is a missing cell for C and Old Latin j) and Fee's table for John 1-8 (which lacks cells where the first hand and corrector intersect for P66 and Aleph).

Distance matrices are normally obtained by applying the default vetting algorithm, which drops the least defined witness of each pair used to calculate a distance until all distances are calculated from the minimum acceptable number of variation sites where both are defined, which is normally fifteen. In some cases, an alternative approach is used which forces a particular witness to be retained provided it has enough defined variation sites at the outset. Examples include UBS2 distance matrices for Matthew, Mark, John, and Acts where Alexandrinus (A), Ephraemi Rescriptus (C), Sinaiticus (Aleph), and P45 have been retained due to their importance.

It is helpful to know what analysis results look like when there is no clustering among the objects being analysed. (Generic terms such as object, observation, case, or item may be used for the things being compared when they are not necessarily New Testament witnesses.) We have a natural facility for recognising group structure but are also prone to mistake a purely random distribution of items for a cluster. One way to avoid this kind of error is to be familiar with analysis results produced from a data set that has no group structure. With this purpose in mind, a control data set may be generated which is analogous to its model data set in various respects (e.g. number of objects, number of variables, mean distance between objects) but has no actual clustering among its objects.

A control data set is generated by performing c trials to randomly select one of two possible states (1 and 2) then repeating this r times to produce a data matrix with r rows of objects and c columns of variables. The generator aims to produce objects which have a mean distance of d between them. Values for r, c, and d are derived from the model: r is the number of objects in the model distance matrix; c is the rounded mean number of variables in the objects from which the model distance matrix was calculated; and d is the mean of distances in the model distance matrix. The control data matrix is then used to calculate a control distance matrix which has the same number of objects as the model and approximately the same mean distance between objects.[7]

The binomial distribution predicts the range of distances expected to occur between pairs of objects generated in this way. A 95% confidence interval is the range of distances expected to occur for 95% of randomly generated cases. Only 5% of distances between two randomly generated objects fall outside the upper and lower limits defined by this interval. A distance outside this range, either less or more, is statistically significant in the sense that it is unlikely to happen by chance (though there is a 5% chance it will). A distance outside the normal range defined by the 95% confidence interval indicates an adjacent or opposite relationship between two objects: adjacent if the distance is less than normal and opposite if greater.[8]

While distances outside the normal range are unlikely to occur by chance, a distance inside that range does not necessarily imply lack of relationship between two objects: a relationship between the two may exist but it is not possible to say so with confidence. The relative size of the normal range contracts as the number of places compared increases so a distance which is not statistically significant in one data set may be statistically significant in another which includes more variation sites.

The following table presents the data sets and their sources. Links in the table provide access to data and distance matrices which are formatted as comma-separated vector (CSV) files so that they can be downloaded and imported into a spreadsheet program. 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.

This study presents results obtained by applying the following analysis methods to the data sets:

These analysis modes will now be introduced by reference to two data sets:

Clusters may be isolated by inspecting a CMDS or NJ plot, cutting a DC dendrogram, or producing a partition using PAM analysis. Similar objects tend to be similar distances from a reference object, near each other in a CMDS plot, in the same branch of DC and NJ plots, and in the same group of a PAM partition. The more eccentric an object when compared to others in the data set, the more isolated it will appear in analysis results. If an object is mixed, being comprised of a mixture of states characteristic of differing groups, then a CMDS result will locate it between the relevant groups, proportionally closer to those whose characteristics it most often contains. In DC, NJ, and PAM analysis, a slight change in the distance matrix can cause a mixed object to leap from one branch, cluster, or group to another.

The respective analysis results are often but not always consistent. If all of the analysis results point to the same conclusion with respect to implied clustering then that can be taken as a firm result; if they differ then each result needs to be handled with due caution. The distance matrix remains the final arbiter when the affiliation of an object is not clearly indicated by concurrence of analysis results. When the classification of an object is uncertain, further information may produce a more definite result. However, if an object has a mixed nature then it may remain difficult to classify as anything but a mixture. A mixed object will tend to be isolated unless other objects happen to have similar mixtures of states.

One aim of New Testament textual research is to recover the initial text, namely the common ancestor of extant New Testament texts. Some aspects of the results produced by the analysis modes used in this study can be interpreted in terms of temporal development. In particular, there may be points of contact between the the family tree of New Testament texts and the tree-like structures produced by divisive clustering and neighbour joining. However, these tree-like analysis results do not provide unequivocal guidance on the location of the initial text. Any node (i.e. junction) or leaf (i.e. terminal) of a DC or NJ tree could be closest to the initial text. If one were to make a string model of such a tree, with knots at every node tying together string segments of the appropriate lengths, the model could be picked up at any node or leaf. The point being held would become a new tree root so there would be as many possible trees as the number of nodes and leaves. The trick is to decide where the root of the tree is located, a topic which will occupy the field of New Testament textual research for some time to come. The Coherence-based Genealogical Method (CBGM) developed by the INTF can be used to investigate whether the witnesses in one branch are closer to the initial text than those in another.[9] Phylogenetic techniques such as described in Spencer, Wachtel and Howe's “Greek Vorlage of the Syra Harclensis” can also be used to investigate the priority of texts. Yet another possibility is to see where texts reconstructed from early patristic citations are located in trees produced by DC and NJ analysis.

Classical Multidimensional Scaling (CMDS)

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 reasonably good approximation to the actual situation. CMDS analysis produces a coefficient called the proportion of variance which indicates how much of the information contained in a distance matrix is explained by the associated 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 entire set of actual distances.

The CMDS map obtained from the UBS4 data set for Mark's Gospel shows that the textual space formed by New Testament witnesses has structure. The associated proportion of variance figure is 0.51, meaning that about half of the entire distance information is captured in the plot.[12]

Figure 4. CMDS (Mark, UBS4)

The galactic imagery Eldon J. Epp uses to describe text-types seems apt for the clusters evident in this analysis result:

A text-type is not a closely concentrated entity with rigid boundaries, but it is more like a galaxy — with a compact nucleus and additional but less closely related members which range out from the nucleus toward the perimeter. An obvious problem is how to determine when the outer limits of those more remote, accompanying members have been reached for one text-type and where the next begins.[13]

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]

CMDS analysis of the control distance matrix produces the following result:

Figure 5. CMDS (Mark, UBS4, control)

Any appearance of clustering in the control map is illusory: its objects are by definition unrelated, having been randomly generated. There are various differences between the model and control maps. The model map has an irregular shape while the control map is globular. Another difference relates to the respective map diameters: the volume enclosed by map axes is greater for the model than the control. This indicates that dispersion among New Testament texts is greater than would be expected if those texts resulted from random selection among alternative readings. Yet another difference is the proportion of variance figures for the model and control maps, which are respectively 0.51 and 0.16. The dimensionality of the New Testament distance data is lower than for the control data, making it easier to squeeze into only three dimensions.

Divisive Clustering (DC)

Divisive clustering begins with a single cluster and ends with individual objects. The R program documentation describes the clustering algorithm as follows:[15]

At each stage, the cluster with the largest diameter is selected. (The diameter of a cluster is the largest dissimilarity between any two of its observations.) To divide the selected cluster, the algorithm first looks for its most disparate observation (i.e., which has the largest average dissimilarity to the other observations of the selected cluster). This observation initiates the "splinter group". In subsequent steps, the algorithm reassigns observations that are closer to the "splinter group" than to the "old party". The result is a division of the selected cluster into two new clusters.

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 is presented as well. The value of this coefficient ranges from zero to one with larger values indicating a greater degree of clustering. A DC dendrogram does not necessarily reflect the family tree of objects in the underlying data set. Instead, it merely shows a reasonable way to progressively subdivide an all-encompassing cluster until every sub-cluster is comprised of a single object.[16]

Figure 6. DC (Mark, UBS4)

The vocabulary of tree structures is useful when discussing DC dendrograms. A branching point is called a node, each structure which descends from a node is called a branch, and terminals are called leaves. The dendrograms produced by analysing New Testament data have a self-similar character where, apart from scale, smaller parts have the same appearance as larger parts. Each branch contains its own sub-branches, unless terminated by leaves (i.e. individual witnesses).

A partition based on a DC dendrogram is obtained by means of a horizontal line which cuts across the dendrogram at some height to produce a set of separate branches. One possible height to cut a DC dendrogram is the upper critical limit of distances. Such a large distance is seldom encountered among unrelated objects. Cutting at the upper critical limit produces the following partition of the model data set.[17]

Table 5. DC partition (Mark, UBS4, upper critical limit)

Group no.	Members
1	UBS Aleph B C L Delta Psi 892 2427 cop-sa cop-bo
2	A 33 157 180 579 597 700 1006 1010 1071 1241 1243 1292 1342 1424 1505 Byz E F G H N Sigma Lect syr-p syr-h eth slav
3	D Theta 565 it-a it-aur it-b it-c it-d it-f it-ff-2 it-i it-l it-q it-r-1 vg Augustine
4	W
5	f-1 f-13 28 205
6	it-k
7	syr-pal
8	syr-s arm geo

Performing DC analysis on the control distance matrix produces this dendrogram:

Figure 7. DC (Mark, UBS4, control)

The model and control dendrograms seem quite similar at first glance although there are important differences: nearly all of the branching heights in the control dendrogram are in the normal range [0.374, 0.553], and the divisive coefficient for the model (0.74) is much larger than for the control (0.33).

The objects in the control can be grouped even though it is pointless to do so: if nearly all distances between objects fall within the normal range then partitioning may well be futile. In the present case, group sizes are more uniform for the control than model although there is no reason why a data set with actual groups cannot have uniform group sizes.

Table 6. DC partition (Mark, UBS4, control, upper critical limit)

Group no.	Members
1	R1 R5 R11 R15 R20 R22 R23 R25 R27 R32 R38 R50 R52 R54 R57 R64
2	R2 R3 R4 R10 R18 R30 R34 R41 R44 R47 R51 R62
3	R6 R7 R12 R17 R19 R29 R33 R39 R40 R48 R49 R58 R65
4	R8 R13 R14 R28 R31 R36 R37 R43 R45 R53 R55 R59
5	R9 R16 R21 R24 R26 R42 R46 R56 R60 R61
6	R35 R63

Neighbour Joining (NJ)

Neighbour joining (NJ) is an iterative process that begins with a starlike tree. A pair of neighbours is chosen at every step, being that pair of objects which gives the smallest sum of branch lengths. A node is then inserted between this pair, which node is regarded as a single object for subsequent steps. The procedure seeks to find the minimum-evolution tree, being that tree which most economically accounts for the observed set of distances between objects. While the method “produces a unique final tree under the principle of minimum evolution,” it does not always produce the minimum-evolution tree. However, computer simulations show that it “is quite efficient in obtaining the correct tree topology.”[18]

As with DC dendrograms, the vocabulary of tree structures is useful for discussing NJ analysis results. The NJ procedure produces an unrooted tree, meaning that any node or terminal in the result could be closest to the common ancestor of the entire tree.

Applying the NJ procedure to the model distance matrix produces a tree whose branches correspond to clusters seen in the CMDS and DC results obtained from the same distance matrix:[19]

Figure 8. NJ (Mark, UBS4)

The tree obtained from the control distance matrix retains the NJ algorithm's initial starlike structure. This shows what kind of topology (i.e. shape) to expect for an NJ result derived from a data set comprised of unrelated objects. The marked difference from the model result is another indication that clustering exists among texts of Mark's Gospel.

Figure 9. NJ (Mark, UBS4, control)

Partitioning Around Medoids (PAM)

Partitioning around medoids (PAM) builds clusters around representative objects called medoids. The program documentation provides this description:[20]

The ‘pam’-algorithm is based on the search for ‘k’ representative objects or medoids among the observations of the dataset. These observations should represent the structure of the data. After finding a set of ‘k’ medoids, ‘k’ clusters are constructed by assigning each observation to the nearest medoid. The goal is to find ‘k’ representative objects which minimize the sum of the dissimilarities of the observations to their closest representative object.

PAM analysis can be used to divide a data set into any number of clusters between two and the number of cases in the data set. The standard procedure in this study will be to partition data sets into two, three, four, five, and twelve clusters. The progression from two to five shows which groups separate first, while the twelve-way partition is useful for revealing core group members.[21]

Table 7. PAM (Mark, UBS4)

No.	Groups with their medoids in [brackets]	Singletons	Poorly classified (worst last)
2	UBS Aleph B D L W Theta 565 2427 it-a it-aur it-b it-c it-d it-f it-ff-2 it-i it-k it-l it-q it-r-1 [vg] syr-pal syr-s cop-sa cop-bo arm eth geo Augustine A C Delta Psi f-1 f-13 28 33 157 180 205 579 597 700 892 1006 1010 1071 1241 1243 1292 1342 1424 1505 [Byz] E F G H N Sigma Lect syr-p syr-h slav		Theta 2427 arm 565 it-aur L Augustine geo it-l it-q vg it-f syr-pal eth
3	UBS Aleph B C L W Delta [Psi] 892 2427 it-k syr-s cop-sa cop-bo A f-1 f-13 28 33 157 180 205 579 597 700 1006 1010 1071 1241 1243 1292 1342 1424 1505 [Byz] E F G H N Sigma Lect it-aur it-f it-l syr-p syr-h syr-pal eth geo slav Augustine D Theta 565 it-a it-b it-c it-d it-ff-2 [it-i] it-q it-r-1 vg arm		C syr-s arm vg
4	UBS Aleph B C L W Delta [Psi] 892 2427 it-k cop-sa cop-bo A f-1 f-13 28 33 157 180 205 579 597 700 1006 1010 1071 1241 1243 1292 1342 1424 1505 [Byz] E F G H N Sigma Lect syr-p syr-h slav D Theta it-a it-b it-d it-ff-2 [it-i] it-r-1 565 it-aur it-c it-f it-l it-q [vg] syr-pal syr-s arm eth geo Augustine		syr-pal C W it-q Theta it-c eth
5	UBS Aleph [B] L Delta Psi 2427 cop-sa cop-bo A C f-13 33 157 180 579 597 700 892 1006 1010 1071 1241 1243 1292 1342 1424 1505 [Byz] E F G H N Sigma Lect syr-p syr-h slav D it-a it-b it-c it-d [it-ff-2] it-i it-k it-r-1 W [f-1] 28 205 565 syr-s arm geo Theta it-aur it-f it-l it-q [vg] syr-pal eth Augustine		eth 892
12	UBS Aleph [B] Psi 2427 A f-13 33 157 180 579 597 700 1006 1010 1071 1241 1243 1292 1342 1424 1505 [Byz] E F G H N Sigma Lect syr-p syr-h slav C L [Delta] D it-a [it-d] [Theta] 565 syr-pal f-1 28 [205] 892 cop-sa [cop-bo] it-aur it-c it-f it-l it-q [vg] eth Augustine it-b it-ff-2 [it-i] it-r-1 syr-s [arm] geo	W it-k	892 eth it-c L

Brackets mark the medoid of each group. A medoid has the minimum mean distance to other group members and is the most central one for groups of three or more items. For two member groups, the PAM algorithm chooses one as the medoid.

Note

This study uses the bracketed medoid identifier as a label for the associated group. For example, [vg] refers to the first group in the above two-way partition.

A singleton is a solitary item which forms its own group. It is isolated, not having any close relatives within the data set. Singletons are listed under a separate heading, and the medoid of a singleton group is the sole member itself. The total number of groups in a partition equals the sum of numbers of singletons and multiple member groups.

Not all members of a group need be a good fit. PAM analysis calculates a statistic called the silhouette width for each object in the data set being partitioned into a chosen number of groups. Its value ranges from +1 to -1: the closer it is to +1, the better the associated case fits into its assigned group; by contrast, the closer the statistic is to -1, the worse the fit. Like hammering square pegs into round holes (or vice versa), negative silhouette widths indicate that the affected cases are not well suited to their assigned places. The last column in the table lists witnesses with negative silhouette widths, putting those with the most negative values last. The worst classified witnesses lie farthest to the right in such a list. A poor fit may indicate that a witness has a mixed text or that the chosen number of groups is too small for a text to be grouped with like texts alone.

As a data set is partitioned into larger numbers of groups, parent groups tend to spawn child groups while themselves contracting into narrower, more coherent groups. Group [Byz] is an example: as the same data set is partitioned into more and more groups, this group contributes items to various other groups while retaining a core membership. Partitioning a data set into a large number of groups reveals coherent cores comprised of close confederates.

Adding the partition's number of groups to the group label produces a more specific identifier. For example, [Byz] (3) refers to the group with medoid Byz in a three-way partition while [Byz] (12) refers to the group with medoid Byz in a twelve-way partition. Corresponding groups such as [Byz] (3) and [Byz] (12) 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 shared membership, not common 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 from the table above, [Psi] (3) and [B] (5) share members but their medoids differ. One might label the subgroup as [Psi-B] (5) to indicate the connection with the supergroup from which its members are drawn.

Some numbers of groups are more suitable than others. Plotting a statistic called the mean silhouette width against each possible number of groups indicates which numbers of groups are more natural for the data set. The plot for the model data set indicates that three, six, eleven, and twenty-four are among the more preferable numbers of groups.[22]

Figure 10. Mean silhouette width versus number of groups (Mark, UBS4)

The MSW plot for the control data set also has a number of peaks even though that data set has no actual groups.

Figure 11. Mean silhouette width versus number of groups (Mark, UBS4, control)

Comparing the model and control MSW plots reveals a great difference in the respective magnitudes of the MSW statistic. While the control data set is randomly generated and consequently contains no actual groups, there is nevertheless random clustering which accounts for the peaks seen in the associated MSW plot. The MSW plot for the control data set establishes a noise level: peaks with such small magnitudes are worthless as indicators of grouping.

This section presents results obtained by analysing the data sets referenced above using the methods described in the preceding section. The results are given in three parts:

PAM results are presented for a series of data sets selected for their broad coverage of witnesses and variation sites in respective sections of the New Testament. For patristic data sets, ranked distance is the preferred analysis mode.

This discussion focusses on two categories of analysis results given above:

An attempt will be made to cover what seem to be the most important features of the textual landscape revealed by analysis of the more comprehensive data sets. Analysis results for patristic texts will be discussed as well even though the associated data sets tend not to have a broad coverage of witnesses or variation sites.

Discussion typically begins with PAM results for a data set, though CMDS, DC, NJ, and ranked distance results are often covered as well. Patristic data sets are an exception to this general approach. Due to the underlying vagaries of this hard won class of information, the pictures produced by most of the analysis techniques tend to be uninformative. To give an example of the difficulties encountered, PAM analysis often isolates a patristic witness as a singleton as soon as the number of groups is increased beyond a few. Ranked distances provide a viable alternative in these circumstances, allowing some sense of the textual nature of a patristic witness to be gained by identifying its near neighbours. However, the accuracy and precision of the impressions thus obtained may be open to question due to the inherent uncertainties associated with establishing a Church Father's text from quotations.

A few warnings need to be issued before launching into the disquisition. The results are provisional in the sense that any change to the inputs will produce changes in the outputs. If more comprehensive data sets are analysed then their results will vary from those presented here. The important question is, how much will they vary? If the data set analysed here is already quite comprehensive and is not marred by some form of systematic bias then the corresponding results can be expected to be consistent with any obtained from more comprehensive data sets.[23]

Each analysis method presents its own view of the textual landscape occupied by New Testament witnesses. While these views are generally consistent, there are cases where the analysis modes differ with respect to the alignments of particular witnesses. If this happens then due caution should be exercised when drawing conclusions about the textual complexions of those witnesses. For analysis methods which divide texts into branches or groups (i.e. DC, NJ, and PAM), any text which shares characteristics of multiple branches or groups (sometimes called a "mixed" text) is prone to jump from one place to another if even slight changes are made to the data upon which the analysis is based. The respective analysis techniques will often agree on the core members of a group or branch but disagree on the placement of peripheral ones.

All this raises the question of how best to describe a witness when the different analysis modes present conflicting views of its affiliations. The ranked distance result acts as an arbiter in such cases, providing a standard against which to assess aspects of textual relationships indicated by the respective modes of analysis.[24]

The analysis results presented above 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. Most variations are of little consequence — whether an added or dropped article, change of word order, or substitution of a synonymous phrase. Other variations have larger semantic effect, the two most extreme examples being Mark 16.9-20 and John 7.53-8.11 that 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 parallel translation of four varieties of the first chapter of Mark highlights the variation sites given in the United Bible Societies Greek New Testament. This edition only presents a selection of textual variations:

The variation sites presented in the UBS apparatus constitute a small proportion of the total number that exist. Nevertheless, those presented are the most important from a semantic point of view, and the great majority of variations not presented in the UBS apparatus have negligible semantic effect. Consequently, looking at these variation sites should give a reliable impression of the magnitude of difference in meaning between textual varieties.

The textual varieties shown in the parallel translation consist of four clusters identified by reference to the DC dendrogram of the UBS4 data set for Mark:[28]

For each variation unit, the reading 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 readings and that two witnesses in cluster C have the first, three have the second, and four have the third. The reading 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 reading is taken to be the one with the greatest tendency to isolate the variety.

The highlighted passages show how much the more important differences encountered in the first chapter of Mark's Gospel affect the meaning of the text. Most of the differences are hardly worth a second thought, though a few do convey a different shade of meaning. If this one chapter is representative (and there is no reason to think it is not) then it is fair to say that most of the textual variation in the New Testament has little semantic effect. That said, there are a few places (such as Mark 16.9-20 and John 7.53-8.11) where the differences are significant.

Finally, a plea. Please share information in a format that is useful to others. Given the volume of data and human cost of manual transcription, sharing data as electronic files is a Good Thing. (CSV is a good choice; so is XML.) Please present data sets as data matrices or something like the following XML that can be processed to produce a data matrix using a language such as XQuery:

Having information that can be electronically processed to produce data matrices allows a broad range of analysis techniques to be applied. If unable to release data matrices then distance matrices are useful, though the range of applicable analysis techniques is narrower. (Tables of percentage agreement or proportional agreement are readily converted to distance matrices.) An important adjunct to distance matrices (or tables of agreements) is a table of counts saying how many data points were used to calculate each distance. Not supplying this information leaves others in the dark concerning the statistical significance of the distances.

Another important aspect is to present apparatus data in a manner that allows presence or absence of all witnesses chosen for citation to be readily established. (See, for instance, the category with text "undefined" in the above XML example.) The UBS Greek New Testament apparatus is most useful in this respect: one knows that if a sometimes cited witness does not appear in an apparatus entry then its reading cannot be established at that place. The Nestle-Aland Novum Testamentum Graece apparatus is less easy to use for constructing data matrices because there are a number of reasons why a witness might not be cited at a variation site:[29]

In the absence of an algorithm to establish why a witness is not cited at a variation site, this kind of apparatus is not amenable to producing data matrices.

Richard Mallett deserves special thanks for encoding data matrices and transcribing tables of percentage agreement from numerous sources. Compiling the basic data from which analysis proceeds is an arduous and painstaking task, and he has done great service in this respect. Mark Spitsbergen also deserves thanks for helping to encode 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.

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.

The analysis would scarcely have been possible without the marvellous R Language and Environment for Statistical Computing.

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.