Textual Variants in Mark (INTF)

The fourth example is based on textual variation data collected by the INTF for the Parallel Pericopes installment of the ECM (Strutwolf and Wachtel 2011). The data matrix in Table 11 was generated from an electronic file made available by the INTF at their website. It records the states of 333 texts at 503 variation sites. (While most texts of this data set relate to the first hand of a Greek manuscript, others record supplementary, corrected, alternative, additional, and commentary readings. A complete description is in the “Introduction,” 5*–7*. The electronic file is located at http://intf.uni-muenster.de/PPApparatus/.) The corresponding distance matrix in Table 12 retains only 151 of those texts, the other 182 having been eliminated to reduce sampling error. Its distances range from 0.002 to 0.413 and have a mean value of 0.159.

Analysing this distance matrix produces a CMDS map, found in Figure 11, with a similar appearance to the one obtained for the UBS4 data on Mark’s Gospel. Both maps have three lobes of relatively high witness density diverging away from, or converging towards, a dense concentration of Byzantine witnesses. Once again, regions between the three non-Byzantine lobes are practically vacant. The R-squared value for this map is 0.73, implying that it accounts for almost three quarters of the information contained in the distance matrix.

Figure 11

CMDS Map for Textual Variants in Mark (INTF). Animated version available here:

In the DC dendrogram extracted by analysis of the INTF distance matrix, manuscripts 05 and 032 split away first to form solitary branches. Seen in Figure 12, the remaining witnesses split three ways at a group-to-group distance of about 0.35. One group contains a number of manuscripts often styled “Alexandrian.” Another is comprised of 038 and 565, which B. H. Streeter listed as primary authorities of the “Caesarean” text. (See the chart on page 108 of Streeter’s Four Gospels (1924) for lists of witnesses that he regarded as primary, secondary, and tertiary authorities of the “Caesarean” text.) The third group contains a number of known textual complexes including the Byzantine text (011, 07, …, 1326), Family Π (017, 041, …, 021), Family 1 (1, 1582, …, 28), Family 13 (124, 13, …, 983), Family 1424 (1424, 1675, …, 954), and von Soden’s Iβ (1279, 1528, …, 752). A number of the dendrogram branches correspond to regions of higher witness density found in the associated CMDS map. The divisive coefficient for this dendrogram is 0.8.

Figure 12

DC Dendrogram for Textual Variants in Mark (INTF). Archival version available here:

The control data matrix, found in Table 13, was produced by configuring the generating script to make 151 cases with 503 variables per case while aiming for a mean distance between cases of 0.159. The distance matrix obtained from this data matrix, represented here in Table 14, has values ranging from 0.099 to 0.235, less than a third of the distance range found in the primary data set of this example. The mean distance between cases hits the mark of 0.159 that the generating script aimed to produce. Analysis of this distance matrix produces a CMDS map with a roughly spherical inner core surrounded by numerous outliers, shown here in Figure 13. The R-squared value for this map is 0.38, about half the value of 0.73 obtained for the map derived from INTF data.

Figure 13

Control CMDS Map for Textual Variants in Mark (INTF). Animated version available here:

The control DC dendrogram, seen in Figure 14, has a divisive coefficient of 0.37, considerably less than the corresponding value of 0.8 obtained for the primary example. The contrast between the INTF data set for Mark’s Gospel and the analogous data set comprised of randomly produced cases again points to the existence of grouping among New Testament witnesses.

Figure 14

Control DC Dendrogram for Textual Variants in Mark (INTF). Archival version available here:

Comparing CMDS and DC Results for UBS4 and INTF Data

The CMDS maps obtained for the Gospel of Mark using the UBS4 and INTF data sets have a number of similarities. However, there are conspicuous differences as well. Two of the four regions of higher witness density found in the two maps can be identified with each other. What to call each region presents a problem but conventional labels will do for now. In the UBS4 map, the regions of higher density may be labelled as follows:

1. “Byzantine”

2. “Alexandrian” (e.g. 01, 03, 04, 019)

3. “Western” (e.g. 05, it-a, it-b, it-d)

4. “Family 1” (e.g. f-1, 28, 205).

Regions of the INTF map may be labelled in a similar way:

1. “Byzantine”

2. “Alexandrian” (e.g. 01, 03, 04, 019)

3. “Family 1” (e.g. 1, 205, 209, 1582)

4. “Family 13” (e.g. 13, 69, 346, 543).

The correspondence between the two maps for regions labelled as “Byzantine” and “Alexandrian” is plain enough to require no further comment. As for differences, the INTF map does not have a counterpart for the “Western” group of the UBS4 map but, surprisingly, puts Codex Bezae (05) in the vicinity of Family 1. Also, the UBS4 map does not have a counterpart for the Family 13 group found in the INTF map. Instead, the entity that represents Family 13 in the UBS4 apparatus (f-13) is located near the Family 1 group.

There is an explanation for these differences. Each CMDS map reveals groups found in the corresponding data set. The UBS4 data set has only a single entity to represent Family 13 (i.e. f-13) and the INTF data set has only a single representative of the “Western” family of texts (i.e. 05). If the data sets incorporated more witnesses of the respective families, then CMDS analysis results would contain corresponding groups. It seems that in the absence of multiple representatives of a group, CMDS analysis can place a solitary case closer to other groups than would occur if more members of its tribe were included. Perhaps the difference in location, which would be expressed if more members of a group were included, is being pushed into higher dimensions than those presented in a three-dimensional analysis result.

The DC dendrograms derived from the UBS4 and INTF data sets both contain “Alexandrian” (e.g. 01, 03, 04, 019) and “Byzantine” (e.g. 07, 09, 011, 013) branches. There is also consensus concerning the membership of a number of other branches when witnesses present in both data sets are considered. For example, the following branches which appear in the UBS4 dendrogram have counterparts in the INTF dendrogram: 022 and 042; 1241, 1424, and slav; 038 and 565. Apart from inclusion of the Family 13 entity (f-13), the branch of the UBS4 dendrogram comprised of f-1, f-13, 28, and 205 is comparable to the Family 1 branch of the INTF dendrogram (1, 28, …, 2542).

One difference between the dendrograms relates to 05 and 032, which are solitary in the INTF dendrogram but occupy the same branches as other witnesses in the UBS4 dendrogram. The lack of companions for 05 is consistent with the absence of other “Western” representatives in the INTF data set. The INTF distance matrix confirms that 032 is solitary, being located a relatively large distance from all other witnesses. When witnesses are ranked by distance from 032, the closest (022) belongs to the “Byzantine” complex while the next closest six (2193, 205, 209, 28, 1, 1582) are all members of Family 1. These factors help to explain why 032 is solitary in the INTF dendrogram but shares the same branch as Family 1 in the UBS4 dendrogram. Rather than being contradictory, both dendrograms reveal actual characteristics of 032.

Another difference relates to Family 13, which constitutes a separate branch of the INTF dendrogram. By contrast, the entities that represent Families 1 and 13 in the UBS4 apparatus (i.e. f-1 and f-13) occupy the same branch in the UBS4 dendrogram. The distance between these entities in the UBS4 distance matrix is 0.360; in the INTF distance matrix, minuscules 1 and 13 are a distance of 0.209 apart. By comparison, the mean witness-to-witness distance is 0.471 for the UBS4 distance matrix and 0.159 for the INTF distance matrix. That is, the distance from f-1 to f-13 is less than the mean distance for the UBS4 data set while the distance from minuscule 1 to minuscule 13 is greater than the mean distance for the INTF data set. This suggests an inconsistency between the two data sets that affects Family 1 or 13 or both. Perhaps one of the entities that represent these families in the UBS4 apparatus does not adequately represent its family? The disparity might also occur if minuscules 1 and 13 were not central members of their respective families.

Yet another difference is that the UBS4 dendrogram puts minuscule 28 in the same branch as the entity which represents Family 13 (f-13) while the INTF dendrogram locates 28 in the Family 1 branch. According to the UBS4 distance matrix, the closest three items to 28 are f-13, 205, and f-1. For the INTF distance matrix, fifteen of the nineteen closest witnesses to 28 are members of Family 1 or 13, with Family 1 members tending to precede those of Family 13. These fifteen include all but one of the members of Families 1 and 13 identified by the relevant branches of the INTF dendrogram, minuscule 983 being the only one left out. Thus, both dendrograms accurately reflect the situation of minuscule 28 relative to Families 1 and 13 implied by the associated distance matrices. The INTF data set, which is more comprehensive with respect to Greek manuscripts, shows that minuscule 28 is more closely related to Family 1 than Family 13.

Comparing these analysis results has been instructive. The cases of Codex Bezae and Family 13 show how sensitive results can be to the mix of witnesses selected for inclusion in a data set. The case of Codex Bezae also shows that an apparent affiliation indicated by one analysis method should be regarded with suspicion if not confirmed by other methods. Recourse to the distance matrix often provides a better understanding of cases for which analysis results are puzzling.

Both the UBS4 and INTF data sets exhibit weaknesses with respect to representing the New Testament textual tradition of Mark’s Gospel. The UBS4 data set suffers from a relative lack of variation sites and Greek manuscripts, and there may be a problem with the entities it uses to represent Families 1 and 13. At the same time, the INTF data set lacks early versions and patristic citations, which offer a valuable context for understanding affiliations among the Greek manuscripts.

Jerome’s Early Manuscripts

Jerome says in his prologue to the Vulgate version of the Four Gospels:

For if we are to pin our faith to the Latin texts, it is for our opponents to tell us which; for there are almost as many forms of texts as there are copies. If, on the other hand, we are to glean the truth from a comparison of many, why not go back to the original Greek and correct the mistakes introduced by inaccurate translators, and the blundering alterations of confident but ignorant critics, and, further, all that has been inserted or changed by copyists more asleep than awake? … I therefore promise in this short Preface the four Gospels only, which are to be taken in the following order, Matthew, Mark, Luke, John, as they have been revised by a comparison of the Greek manuscripts. Only early ones have been used. (Jerome 1892, 488)

The CMDS map for textual variants in Mark (UBS4) shows that Jerome’s revision (vg) lies close to a trajectory which runs between a cluster of Old Latin texts such as Vercellensis (it-a), Veronensis (it-b), Colbertinus (it-c), and Bezae (it-d) at one end and “Byzantine” texts at the other. If these Old Latin texts represent the Latin exemplars used by Jerome, it seems that the “early” Greek manuscripts he used to revise the Latin text of Mark were of the Byzantine variety.

A Simple Model

Adopting the last approach and assuming the common case of only two possible readings per variation unit results in a particularly simple model where each copying event is represented by a sequence of trials, each trial comprised of selecting one alternative from two equally probable states. The model applies to a copyist selecting a series of readings from an exemplar whose variation sites each have only two readings with apparently equal merit. From a statistical perspective, the model is equivalent to a series of coin tosses using an unbiased coin. This equivalence allows a minimum standard to be established for what constitutes a statistically significant level of disagreement between two witnesses.

If there are two equally probable states (i.e. readings) for each trial (i.e. selection of a reading at a variation site), the chance of disagreement at each place where a choice has to be made is one half. This is because there are four possible combinations of two states chosen in two trials, half of which constitute disagreement. To illustrate, if the two states are represented by the numerals 1 and 2 then the four possible combinations are (1, 1), (1, 2), (2, 1), and (2, 2), the second and third of which disagree.

The binomial distribution applies to the outcomes of multiple independent trials when the outcome of each trial can have only two states and the respective probabilities of the two states are the same for each trial. By convention, the two states are labelled success and failure. Given a particular number of trials and a fixed probability of success, the binomial distribution describes how frequently each number of successes occurs. Using this distribution, it is possible to obtain critical limits, which are the upper and lower bounds of a confidence interval that specifies the range of numbers of successes that can be confidently attributed to chance. Before obtaining the limits, it is necessary to select an alpha value, which represents an acceptable level of error. While any number of successes between zero and the number of trials can occur, only a central range of numbers of successes is likely. Over many repeats of an experiment consisting of a set number of trials, numbers of successes outside this central range will occur with a relative frequency equal to the alpha value. If the alpha value is small enough, then it is reasonable to assert that a number of successes outside the range defined by the confidence interval is not due to chance. However, such an assertion is expected to be wrong in the proportion of cases corresponding to the alpha value. For this article, an alpha value of 5% is used, producing a 95% confidence interval. Given this alpha value, one expects to be wrong only 5% of the time when asserting that a value outside the 95% confidence interval is not due to chance. (An alpha value of 5% is common for work where the consequences of false positives are not too dire.)

Dividing a number of successes by the total number of trials produces a proportion of success. Table 16 presents 95% confidence intervals for proportions of success expected to occur by chance for various numbers of trials where each trial has a probability of success equal to one half. Each interval uses the notation [lower, upper], where lower and upper are the inclusive limits of the range. The intervals given in the table relate to the simple model where each trial consists of two random selections from two equally probable states. As a success corresponds to a disagreement between two randomly selected states, the proportion of successes corresponds to the proportion of disagreements, which is the simple matching distance.

These intervals only apply to the special case of each variation unit having two equally probable readings. Nevertheless, the table illustrates some important points:

1. Calculating a distance from too few variation units is a futile exercise because any distance thus obtained is reasonably attributable to chance. In this case, if five or less variation units are being compared, then no distance is outside the range expected to occur when states are randomly selected.

2. The relative size of the confidence interval bounded by the upper and lower critical limits decreases as the number of trials increases. Here, the relative size of the interval is 100% for five, just under 50% for fifteen, and 20% for one hundred trials.

3. Just as a distance less than the lower bound of a confidence interval is statistically significant, so is one greater than the upper bound. To use the example of one hundred trials given in this table, a distance between two witnesses which is larger than 0.6 is just as unexpected as one less than 0.4.

A More Realistic Approach

The simple model is based on a number of assumptions that do not apply to data derived from large-scale collations. For example, a typical scribe would only have known about a fraction of the variations that manifest when a large number of copies are compared. Also, a typical scribe did not have an equal preference for alternatives at every variation site. Furthermore, editorial definition of variation site boundaries affects the distribution of the number of alternative textual states per variation unit. A more realistic approach establishes critical limits by reference to the distribution of witness-to-witness distances found in the data set being studied.

The histograms, Figures 15 and 16, show distributions of simple matching distances for the UBS4 and INTF data sets for Mark. A blue binomial curve is superimposed on each histogram to show the range of distances expected to occur for unrelated pairs of witnesses. These curves were generated by an R expression that uses the following parameters to obtain relative frequencies from the binomial distribution:

Figure 15

Histogram (UBS4, Mark, SMD).

Figure 16

Histogram (INTF, Mark, SMD).

1. the number of trials as estimated by the rounded mean number of variation units from which entries in the distance matrix were calculated

2. the probability of success as estimated by the mean value of distances in the distance matrix.

The R script named groups-scripts-hist.R was used to produce the histogram and binomial curve in each case. The mean number of variation units is rounded because the number of trials used to define a binomial distribution must be an integer. Each binomial curve has been scaled up to the maximum height of the corresponding histogram; if not scaled, the vertical height is much less, making it more difficult to see the horizontal limits implied by the binomial distribution. Due to the scaling, the vertical scale does not give probability values for the binomial curves.

A binomial curve generated from mean values for the number of variation units and distance between witnesses shows the typical range of distances which is expected to occur by chance. However, there are reasons why such a range should not be expected to apply to all pairs of witnesses in the data set. First, the readings of a witness might not be well defined at every variation unit so the number of variation units used to calculate a distance could vary from one pair of witnesses to the next. Second, the expected distance between witnesses may vary. Fortunately, the curves are not particularly sensitive to changes in these parameters. Nevertheless, if the values for the entire data set are far different to those appropriate for a particular set of witnesses then it is better to use the particular values to obtain the corresponding expected range of distances. To illustrate, if one were interested in a fragmentary witness, then the number of trials would be constrained by the number of variation units at which its readings are defined. As another example, if one were studying a closely related subset of witnesses, then the probability of success would be the mean distance between members of that subset.

Table 17, gives the 95% confidence intervals obtained for the UBS4 and INTF data sets by using this approach. The critical limits correspond to an alpha value of 5%, a number of trials equal to the mean number of variation units, and a probability of success equal to the mean distance between pairs of witnesses. The critical limits were obtained with the same R script as used to produce the histograms and binomial curves. The relevant R expression is the same as was used to obtain critical limits for the simple model. There, the probability value was 0.5; here, it is the mean distance.

Any distance between the upper and lower limits of an interval is normal in the sense of not being unexpected if readings are randomly chosen. A distance outside this range is not expected to happen by chance: if less than the lower limit, then the relevant pair of witnesses are adjacent, being closer together than normal; if greater than the upper limit there is an opposite relationship, the pair being further apart than normal. An active process is implied for distances outside the normal range, one that has either driven texts closer together or further apart than would be expected if readings had been chosen at random. For the UBS4 data set, the lower limit is 0.382, corresponding to a percentage agreement of 61.8%. Thus, for this data set, a percentage agreement greater than 61.8% is significant in a statistical sense. For the INTF data set, a distance less than 0.127, corresponding to a percentage agreement greater than 87.3%, is statistically significant.

We are now in a position to provide another answer to the question posed earlier: “At what height should a dendrogram be cut to produce groups where members are actually related?” A reasonable value to use for this purpose is the lower bound of the range of distances that are likely to occur by chance for a given data set. To illustrate, the control for the well-defined groups example is comprised of twelve cases with fifteen variables per case and a mean distance between cases of 0.481. The corresponding lower critical limit is 0.267. (The limit is obtained using the R expression qbinom(alpha/2, n, p)/n, where the alpha value is 0.05, the number of trials n is 15, and the probability of success p is 0.481.) Cutting the relevant dendrogram at this height succeeds in assigning most cases to solitary groups, as is appropriate because the cases are unrelated. However, cutting at that height also produces two spurious groups as well, one containing R4 and R8, the other R5 and R11. This serves as a reminder that pairs of randomly generated cases can occasionally be closer together or further apart than expected. In the long run average of many trials, the frequency of such cases approaches the alpha value.

It is also reasonable to use the upper critical limit in the context of dendrograms. Branches obtained by cutting a dendrogram at the upper limit represent super-structures that are more dissimilar than would be expected if their members were comprised of randomly chosen states. Using the UBS4 dendrogram as an example, cutting at the upper critical limit (0.561) produces the partition seen in Table 18.

Sampling Error

The sampling error of a distance between two texts is the difference between the actual distance that would be obtained by examining the entire population of variation sites and the estimated distance obtained from a sample of the places where the two texts vary. The actual distance is a parameter obtained from the entire population of variations, while the estimated distance is a statistic obtained from a sample. In practice, almost every text-to-text distance is a mere estimate because of two common circumstances that prevent the entire population of variations being examined:

1. witnesses are compared by reference to a critical apparatus which does not list every variation

2. one or both of the witnesses being compared may be fragmentary.

In the first case, only a sample of variation units is used for the comparison. In the second, a smaller number of variation units is compared than would be the case if both witnesses were complete.

The actual distance can only be determined by examining the entire population of variations. Nevertheless, a distance estimated from a sample is still useful provided that one knows how well it is expected to approximate the actual value. This piece of information is supplied by a statistical analysis of the sample to produce a confidence interval that probably contains the actual distance. The interval’s limits are calculated from the binomial distribution using the estimated distance as the probability of success and the number of compared variation units as the number of trials. To illustrate, Table 19 gives a distance estimate and 95% confidence interval obtained when variables are randomly selected from two of the cases found in one of the example data sets.

Another way to express a confidence interval is by giving an estimated value and a margin of error. The lower margin of error is the difference between the lower critical limit and the estimate while the upper margin of error is the difference between the estimate and upper critical limit. If the interval is symmetrical with respect to the estimate then the upper and lower margins are the same and the expression estimate ± margin can be used to specify the estimate and its confidence interval.

The magnitude of the margin of error for simple matching distances and the binomial distribution is approximately t (p(1–p))^1/2/n^1/2, where t is the appropriate t-distribution value, p the probability, and n the number of trials. The value of t tends towards the corresponding z value for the normal distribution (e.g. 1.96 for an alpha value of 0.05) as the number of trials increases. For smaller values of n, the t value may be obtained using the R expression qt(1 – alpha/2, df = n–1). Taking the seventh row of the table as an example, when n is 50 and the alpha value is 0.05 then the t value is 2.01. Using the estimated distance (0.22) as the probability produces a margin of error estimate of 2.01 × (0.22 × 0.78)^1/2/50^1/2, which is 0.118. This agrees quite well with the margins of 0.10 (lower) and 0.12 (upper) obtained with the binomial distribution.

As can be seen from the table, estimates based on only a few places of comparison are unreliable because the range of values expected to occur for an estimate (i.e. the confidence interval) covers a large part of the range of possible values. Increasing the number of places compared makes the relative size of the confidence interval decrease. It is therefore desirable to use as many variation units as possible when estimating distances between witnesses. However, one is sometimes forced to use a lesser number, as when fragmentary witnesses are involved. What then is an acceptable lower limit for the number of variation units? There is no absolute guide. In this article, a distance is only used if calculated from a minimum of fifteen variation units where both witnesses are defined. In order to satisfy this standard, the script used to calculate distance matrices for this article (i.e groups-scripts-dist.R) goes through an iterative process, dropping the least well-defined member of the least well-defined pair at every step until all remaining distances are calculated from at least fifteen variation units.

Ranking Witnesses by Distance from a Reference

Given a distance matrix, it is straightforward to rank witnesses by distance from a reference witness. Furthermore, a confidence interval can be established for each distance estimate using the number of variation units from which it is calculated. It is then possible to identify adjacent and opposite witnesses with respect to a reference witness. Those that are adjacent are less distant than the lower limit of the relevant interval, while those that are opposite are more distant than the upper limit. Table 20 ranks witnesses by distance from the entity which represents Family 1 in the UBS4 data set (i.e. f-1). This table shows that f-1 is adjacent to 205, 28, Lect, f-13, 1424, 1241, geo, 1505, slav, and G (011). Many witnesses (1243, …, it-i) occupy the middle ground with distances from f-1 that are not statistically significant; there may be a relationship with f-1 in each case, but the sampling error associated with the available number of defined variation units in the data set is too large to allow a confident decision to be made on the matter. At the other end of the scale, f-1 is opposite to Delta (037), cop-bo, it-ff-2, it-b, it-c, it-r-1, 2427, it-a, Aleph, B, D, it-k, and it-d. This indicates that Family 1 is non-Western and non-Alexandrian in the Gospel of Mark. (One assumes that the entity used to represent Family 1 in the UBS4 apparatus (i.e. f-1) does represent that family of texts.)

The Random Walk

The random walk is a statistical problem that considers how far from the starting point a thing will end up if every movement is random. The classic example is a drunk staggering along a gutter. The man is so drunk that a forward or backward step is equally likely. How far from the beginning will the drunk end up? If this scenario is extended into two dimensions (e.g. a level field) then the drunk could end up anywhere on a flat surface within a maximum distance of his beginning point, that maximum being the number of steps times the average step length. For three dimensions (involving stairs or ladders), the final location would be anywhere within a sphere of the same maximum radius. While possible for the drunk to take a step in the same direction every time, it is unlikely. In fact, the drunk will probably end up somewhere within a smaller distance of the origin, which distance is the order of the square root of the number of steps times the average step length.

To the extent that the New Testament textual tradition can be modelled as random choices among readings, one might expect the diameter of the point cloud in a CMDS diagram to be about the same as obtained for a random data set of the kind generated for the controls presented above. As it happens, the diameters of a number of the major clusters in the CMDS maps of the UBS4 and INTF data sets are approximately the same as the diameters of the relevant control maps, which are about 0.3 and 0.1, respectively. (These diameters were estimated by inspection of the CMDS maps for the UBS4 and INTF control data sets.) However, the distances between the outermost witnesses in the CMDS maps of the UBS4 and INTF data sets are greater than the relevant control CMDS map diameters, implying that these witnesses are further apart than would be expected if random processes alone were to blame for the differences.

What might explain these larger than expected differences? One possibility is conscious selection among readings that resulted in distinctive texts, perhaps due to theological differences between the users (or promulgators) of those texts. Another possibility is suggested by the apparent association of certain clusters with early versions, seen in Table 21. Perhaps the early versions were players in the New Testament’s divergence into some of the major textual streams seen in the analysis results? It is not unreasonable to expect that a scribe copying a Greek manuscript in a region where a particular version prevailed would tend to make the Greek conform to a back-translation of that version.

Distances Between Cities

Plotting the MSW versus number of groups for the first example produces the results seen in Figure 17. The MSW value is given for each number of groups from two up to one less than the number of cases in the data set. The tendency for the MSW to decrease as the number of groups increases is apparent. Local maxima occur for three, seven, eleven, and seventeen groups.

Figure 17

MSW Plot (Cities).

Using PAM to partition the data set into three groups produces the divisions shown in Table 22. Comparing with the corresponding CMDS map shows that this partition makes sense with respect to the geographical distribution of the cities, having isolated North American, Asian, and European groups. A similar partition is obtained by cutting the corresponding DC dendrogram at a height of 6,000 km although DXB (Dubai) and SYD (Sydney) form solitary branches when that is done. The medoids identified by PAM analysis are DFW (Dallas and Fort Worth), HKG (Hong Kong), and FRA (Frankfurt), which stand near the geographical centres of the regions associated with the groups.

The next local maximum occurs for seven groups. The corresponding partition in Table 23 also makes sense when compared with the geographical data. The North American group is now split east-west, the Asian group is split north-south, and the two most isolated cases (DXB and SYD) form singletons. (A singleton is a set that contains only one element.)

While the exercise could be continued with the other numbers of groups identified by the MSW plot, these two partitions suffice to show the merit of the approach. The example of distances between cities shows that sensible groupings are obtained even though there is no “correct” number of groups for the data set.

Well-Defined Groups

The well-defined groups example does have a “correct” number of groups, which is four. The MSW plots for this example’s primary and control data sets can be seen in Figures 18 and 19. At first glance these two do not seem very different. Both exhibit a tendency for the mean silhouette width to decrease as the number of groups increases. However, there is a major difference. The highest peak for the primary example (0.728) is more than three times greater than the highest peak of the control (0.190). Given that the control is based on randomly generated cases, it is prudent to ignore any peak in the primary example if its magnitude is not much greater than the value obtained for the same number of groups in the control. The highest peak in the primary example’s MSW plot correctly identifies the number of groups in the data set, and the corresponding partition correctly assigns the cases to their respective groups, seen here in Table 24.

Figure 18

MSW Plot (Well-Defined Groups, Primary).

Figure 19

MSW Plot (Well-Defined Groups, Control).

Variants in Mark (UBS4)

Comparing MSW plots for the primary example and control of the UBS4 data set for the Gospel of Mark, shown in Figures 20 and 21 respectively, shows that peaks in the first plot are well above the noise level indicated by the second plot until the number of groups reaches about sixty.

Figure 20

MSW Plot (Mark, UBS4, Primary).

Figure 21

MSW Plot (Mark, UBS4, Control).

The first plot has quite prominent peaks that exceed the noise level for three, six, eleven, twenty-four, thirty-four, forty-two or forty-three, and forty-nine groups. As far as grouping is concerned, this data set is like the example for inter-city distances rather than the one for well-defined groups in that there is no clear winner, no “correct” number of groups. Instead, certain numbers of groups have greater claim than others to be “natural” when partitioning the data set. For such a data set, peaks in the MSW plot are suggestive rather than emphatic.

One is then left wondering what number of groups is best. In this case, the peak at twenty-four groups seems particularly conspicuous in view of the general tendency for the mean silhouette width to decrease as the number of groups increases. However, dividing the witnesses into so many groups tends to dissolve larger entities, which, though not as coherent as ones that remain together, are nevertheless important for comprehending the broad structure of the textual tradition. Partitions of the UBS4 data set based on three, six, eleven, and twenty-four groups will therefore be presented below.

Using PAM analysis to split the UBS4 data set into three groups produces the divisions in Table 25. Being the most central witness makes the medoid a useful representative of its group. In addition, the siglum of the medoid serves as a label. There are reasons why it is better to use a medoid siglum rather than a conventional name such as “Alexandrian,” “Byzantine,” “Caesarean,” “Western,” “Family 1,” or “Family 13” to label a group. For one thing, partitioning a data set into a large number of groups tends to split any structure for which a broad categorical label such as “Alexandrian” might be apt. For another, the most central witness of a textual family is often not the one that the family is named after. Sometimes, however, corresponding groups in different partitions do not have the same medoids. For example, the group with medoid 044 in the three-way partition of the UBS4 data set has the same core members as the one with medoid 03 in a six-way partition of the same data set. A group’s medoid can change if even a single case is added or removed because another case can then become the most central one. Consequently, while the medoid does serve as a convenient and appropriate label for a group, it is not a reliable guide to identifying corresponding groups in different partitions of the same data set. A better approach for this purpose is to look for common constituents. If the medoid of a textual complex does change from one partition to the next, then the sequence of medoids that complex has for different numbers of groups can be chained together to form a label.

From this point forwards, the medoid siglum will be used to label its group. If groups that have the same medoid but are from different partitions need to be distinguished, then the number of groups in the relevant partition will be added to the label in parentheses. If the medoid of a group changes for different partitions of the same data set, then the sequence of medoids will be chained together to form a label. For example, Gr 044 refers to a group whose medoid is 044, Gr Byz (3) to the group with medoid Byz in a three-way partition, Gr Byz (6) to the one with medoid Byz in a six-way partition, and Gr 044/03 to the group whose medoid changes from 044 to 03 in different partitions of the same data set. (Frederik Wisse [1982] introduced group labels comprised of a “Gr” prefix and manuscript siglum.)

The groups that emerge from a three-way partition of the UBS4 data set are in some respects similar to traditional categories: Gr 044 contains a number of “Alexandrian” witnesses, Gr Byz is mainly comprised of “Byzantine” ones, and Gr it-i includes a number of “Western” texts. However, the groups also contain witnesses that are not normally associated with the conventional categories. Some of the witnesses that seem out of place are out of place. In a situation analogous to hammering square pegs into round holes, they do not fit their assigned places. When deciding how to partition a data set, numbers of groups with larger values of the mean silhouette width are preferable. Although the average value of the silhouette widths may be relatively large, individual silhouette widths might be small. Indeed, a case can have a negative value for the statistic, indicating a particularly poor fit to its assigned division. The more negative the silhouette width, the worse the classification. When the UBS4 data set is divided into three groups, witnesses 04, syr-s, arm, and vg are thus identified as not well classified. Nevertheless, assigning them to the indicated divisions still minimizes the sum of distances between medoids and witnesses for the given number of groups.

Dividing the witnesses into six groups gives poorly fitting witnesses the freedom to migrate into new groups where they are more at home. Other witnesses stay in the remnants of groups from the three-way partition, as can be seen in Table 26. In this partition, Gr 03, Gr Byz, and Gr it-ff-2 are reminiscent of traditional “Alexandrian,” “Byzantine,” and “Western” categories. Gr vg is centred on the entity that UBS4 uses to represent the Latin Vulgate. While one might expect Augustine’s quotations and a number of Latin manuscripts to be here, it is surprising to find 038, the Ethiopic (eth), and the Palestinian Syriac (syr-pal) included as well. Two of these, 038 and the Ethiopic, have negative silhouette widths to indicate that they are not a good fit. Nevertheless, this partition suggests that 038, the Ethiopic, and the Palestinian Syriac (actually Aramaic) share something in common with the Latin Vulgate and other members of this group. The UBS4 editors date the Ethiopic and Palestinian Syriac versions after the Latin Vulgate (B. Aland et al. 1983, 26*–28*). It is thus conceivable that these two versions incorporate passages that amount to translations of the Latin Vulgate. This does not mean that the Palestinian Syriac and Ethiopic versions are primarily translations of the Vulgate! According to Metzger (1977, 82), “the text of the Palestinian Syriac version agrees with no one type of text, but embodies elements from quite disparate families and texts.” Rochus Zuurmond (1995, 146) writes, “Whatever the vicissitudes of the Eth may have been, and granted that influences from non-Greek sources may have played their role already at an early stage, the Eth is an immediate descendant of the Greek textual tradition.”

As for 038, also known as Θ or Codex Koridethi, a glance at the corresponding CMDS map shows that it does seem drawn towards the region of textual space associated with the Latin Vulgate. Using the UBS4 distance matrix to rank witnesses by distance from 038 confirms that a number of its closest neighbours are of the Latin Vulgate kind (e.g. Augustine, vg, it-l), as seen in Table 27.

Returning to the six-way partition, all eight members of Gr arm and Gr 205 fall into a category which Streeter (1924, 27) regarded as an “Eastern type,” having sub-varieties associated with the provincial capitals of Syria and Palestine. The witnesses that Streeter regards as primary, secondary, tertiary and supplementary members of each sub-variety are listed in his chart of MSS and local texts (1924, 108), summarized in Table 28.

Larry Hurtado (1981) challenges the view that 032, also known as Codex W or Washingtonensis, has a “Caesarean” text:

If Codex Θ is a good representative of the “Caesarean text,” the poor and unexceptional agreement of Codex W with Θ makes it highly unlikely that W is related in any special way to this text-type. (83)

The textual nature of 032 is thought to change part of the way through the Gospel of Mark, which change Hurtado (1981, 19) locates in the vicinity of Mark 5.6. Streeter (1924, 69) regarded the latter part of 032 as “Caesarean.”

Ranking witnesses by distance from 032 in Mark chapters 6–16 helps to reveal the manuscript’s character in this block (see Table 29). Hurtado is right to say that agreement between 032 and 038 is poor. In fact, the texts of 032 and 038 have an opposite relationship, being further apart than would be expected to happen by chance. Even so, all of the seven closest witnesses to 032 belong to Streeter’s “Eastern” branch, although none is adjacent to 032 in the sense of being closer than expected by chance. However, as mentioned before, lack of statistical significance for a distance does not imply lack of relationship between two witnesses. Instead, an adjacent or opposite relationship may exist but it is not possible to say so confidently without analysis of a more comprehensive data set. In the context of the UBS4 data set for Mark, 032 does not have any close neighbours and might therefore be described as an eccentric text. Nevertheless, it remains true that the closest witnesses to 032 in this data set are members of Streeter’s “Eastern” category.

How are we to explain that 032 is closest to texts of Streeter’s “Eastern” variety yet is unlike 038? The six-way partition is not inconsistent with Streeter’s identification of a distinct textual variety that includes 032, Family 1, 28, the Sinaitic Syriac, Armenian, and Georgian. At the same time, the relevant CMDS map shows that 038 and 565 lie on a trajectory between the Armenian and Georgian versions at one end and “Western” witnesses at the other. Ironically, it seems that the two manuscripts Streeter regarded as primary authorities for the “Caesarean” sub-variety of his “Eastern” branch are mixtures of “Eastern” and “Western” readings in the Gospel of Mark. Others have already noticed the “Western” leanings of 038 and 565. Hurtado (1981, 88) writes, “The quantity of Western readings in Θ and its allies (565, 700) is so great that the present writer would suggest that perhaps the text represented by these MSS is a form of the Western text as it was shaped in the East.” Stephen C. Carlson (2004, 20–21) writes, “The practice of anchoring the ‘Caesarean’ label on the branch containing Θ and 565 now appears unwise, since Θ and 565 come from a family that originated as a late mixture of Branch gamma (to which Origen’s text belongs) and a Western text substantially similar to D” (Carlson includes P45, W, Families 1 and 13, 28, Codex Bobbiensis (i.e. Old Latin k), and Origen’s text in his “Branch gamma”).

The relevant medoids of the six-way partition, namely the Armenian version and minuscule 205, are better representatives of the textual complex that Streeter called the “Eastern type,” and 032 is closer to these than the other medoids. Whether Gr arm and Gr 205 should be associated with Antioch and Caesarea remains an open question. There has always been a suspicion that the Armenian and Georgian versions trace their ancestry back to the Old Syriac. Syrian Antioch or Edessa would be reasonable guesses for the provenance of the Old Syriac. Making that connection would leave Gr 205 (i.e. Family 1) as a candidate for the “Caesarean” branch of Streeter’s “Eastern” text in the Gospel of Mark. (Amy S. Anderson (Anderson 1999, ii) acknowledges the possibility of a connection between Family 1 and Caesarea although in the Gospel of Matthew rather than Mark: “The text and marginal variants of Codex 1582 are shown to be related, though not identical to the text of Matthew used by Origen, raising the possibility of a Caesarean archetype.” Anderson proposes that 1582 should be considered the leading member of Family 1 in Matthew.)

The next preferable number of groups is eleven, represented in Table 30. When a data set is split into two different numbers of groups, first a smaller then a larger number, it sometimes happens that groups present in the first partition remain substantially unchanged in the second. Such groups are more coherent than others, tending not to fragment. Less coherent groups lose members or split into pieces. Cases that have migrated out of one group might combine with others to form a new group in the second partition while other cases form singletons. Examples of these phenomena are seen by comparing the six- and eleven-way partitions. Gr Byz, Gr it-ff-2, Gr 205, and Gr vg are coherent, remaining much the same when the data set is split into a larger number of groups, although they do lose some of their constituents. Gr 03 (6) and Gr arm (6) (i.e. Groups 03 and arm of the six-way partition) fragment in the eleven-way partition. Gr 03 (6) splits into three parts: a smaller Gr 03 (11) with the same core as its parent; Gr 037, which picks up 04 as well; and Gr cop-bo, which gains 892. Gr arm (6) loses 032 and 565, leaving behind syr-s, arm, and geo. More eccentric witnesses such as 032 and it-k are the first to form singletons. Gr 038, comprised of 038, 565, and syr-pal, forms from cases that have migrated out of other groups. There is a good deal of overlap between Streeter’s “Eastern” category and Gr 038, Gr 205, and Gr arm of the eleven-way partition.

Using PAM analysis to divide the UBS4 data set into the next preferable number of twenty-four groups produces the result shown in Table 31. This partition has a claim to being the most “natural” one because the corresponding MSW plot has a large magnitude for this number of groups despite the general tendency for MSW values to decrease as the number of groups increases. If this is the best partition then it is reasonable to describe the UBS4 data set as comprised of many small groups and singletons along with a few larger groups such as Gr Byz, Gr vg, and Gr it-i. Most of the other structures present for smaller numbers of groups have fragmented.

The process of division into ever-larger numbers of groups could be continued although not much would be gained by doing so. Examining partitions with smaller numbers of groups has already revealed the main contours of the data set’s group structure.

Variants in Mark (INTF)

The MSW plot for the INTF data set, seen in Figure 22, displays a series of peaks, each indicating a preferable number of groups for partitioning. The plot for the corresponding control, shown in Figure 23, indicates the noise level for each number of groups. Comparison shows that each peak in the primary example’s MSW plot is worth considering except the one associated with 151 groups. As in the UBS4 data set, division into too many groups produces fairly uninteresting partitions comprised mainly of small groups or singletons. Much of the group structure is revealed by partitions based on the first four peaks in the MSW plot, which occur at two, four, seven, and seventeen groups.

Figure 22

MSW Plot (Mark, INTF, Primary).

Figure 23

MSW Plot (Mark, INTF, Control).

A two-way partition produces Gr A and Gr 1339, represented here in Table 32. The medoid of the first group is the synthetic Ausgangstext (initial text) that is printed in editions such as the ECM, Nestle-Aland’s Novum Testamentum Graece, and the UBS Greek New Testament. It is joined by a number of manuscripts that might be described as some flavour of “Alexandrian,” although not many would place 05 (Codex Bezae) in this traditional category. As it happens, 05 has a negative silhouette width for this partition indicating a poor fit. Gr 1339 is broad, containing many witnesses often characterized as “Byzantine” along with others that are not.

In a four-way partition, shown in Table 33, Gr A remains static while Gr 1339 spawns Gr 209 and Gr 826. These last two largely correspond to Families 1 and 13, respectively. It is interesting to see that Gr 209 contains 032 and that Gr 826 includes 038 and 565. Codex Bezae (05) remains in Gr A but again has a negative silhouette width to indicate a poor fit.

In a seven-way partition, shown in Table 34, Gr A, Gr 209, and Gr 826 recur almost unchanged. Codex Bezae (05) is still located in Gr A, and still has a negative silhouette width to indicate a poor fit. Codex 032 is again included in Gr 209 (i.e. Family 1). Gr 1339 retains a coherent core and continues to produce new groups, namely Gr 041, Gr 517, and Gr 1528.

Gr A, Gr 041, Gr 209, Gr 826, Gr 517, and Gr 1528 remain substantially the same when the data set is divided into the next preferable number of seventeen parts, as can be seen in Table 35. They are not precisely the same, however. For example, manuscripts 33 and 579 drop out of Gr A (7) to form Gr 579 while manuscripts 732 and 863 of Gr 041 (7) combine with 427 of Gr 1339 (7) to form Gr 732. Also, 034 migrates to Gr 07, and 038 combines with 565 from Gr 1528 (7) to form Gr 565. Gr 1339 (7) begets Gr 07, Gr 022, and Gr 1457. Gr 07 is almost as large as what remains of Gr 1339. Gr 022 contains two deluxe manuscripts dated to the sixth century (i.e. 022 and 042). Minuscules 1071, 1273, and 2766 also move into this group, although negative silhouette widths imply a poor fit for 1071 and 1273. Looking at the corresponding CMDS map shows that the singletons for this partition, namely 05, 032, 28, and 792, are somewhat isolated in textual space. Many of the poorly classified manuscripts would move into other groups or form singletons if the data set were divided into yet more pieces. A number of these complexes are already known, as shown in Table 36, that associates groups discovered by this PAM analysis with ones mentioned in Frederik Wisse’s book on manuscript classification. (See Wisse [1982], chapters 5 and 6. Wisse’s groups relate to the Gospel of Luke but evidently carry over to Mark.)

Finally, dividing into a really large number of parts reveals group cores (see Table 37). Singletons are omitted to leave only those sets with more than one member. It is interesting to see that seven of the ten core members of Gr 1339 are from the Saba monastery near Jerusalem. In this group at least, there is a strong correlation between locality and text. (Strutwolf and Wachtel [2011, 5*], suggest that the Saba manuscripts [Gregory-Aland numbers 1328–1348] “may help to answer whether common location results in textual similarity.”)

Fragmentary Witnesses

In order to reduce sampling error to a tolerable level, the analytical procedures used in this article drop a witness if including it would cause any distance to be calculated from less than fifteen places of comparison. Unfortunately, this policy causes certain witnesses whose readings are not defined at every variation unit to be excluded from consideration. In the present context of New Testament data sets, a number of circumstances can cause the reading of a witness to be undefined at a variation site:

• a manuscript may be illegible at the relevant place

• the text preferred by a Church Father may not be discernible due to absence or ambivalence of relevant patristic evidence at the site, or

• the Greek reading supported by a version may not be discernible because a back-translation of the relevant passage is consistent with more than one of the Greek alternatives.

One such witness is P45, a third century papyrus manuscript, which, due to its fragmentary state, has not yet appeared in the analysis results presented in this article. Happily, its textual nature can be explored by restricting analysis to places where its text is legible. Distance matrices, CMDS, DC, and MSW plots for UBS4 and INTF data sets that retain P45 are presented in Table 38, Figures 24, 25, 26, Table 39, Figures 27, 28, and 29 respectively.

Figure 24

CMDS (Mark, UBS4, P45). Animated version available here:

Figure 25

DC (Mark, UBS4, P45). Archival version available here:

Figure 26

MSW (Mark, UBS4, P45).

Figure 27

CMDS (Mark, INTF, P45). Animated version available here:

Figure 28

DC (Mark, INTF, P45). Archival version available here:

Figure 29

MSW (Mark, INTF, P45).

(The distance matrices were calculated by the R script named groups-scripts-dist.R. The witness to be retained is called the reference, in this case P45. Rather than select only those variation units where the reference is defined, the script detects the two least well-defined witnesses on every iteration. It normally drops the least well-defined but will retain it if the reference witness, dropping the other instead. If the witness to be retained is not defined for the minimum number of variation units then the procedure is abandoned at the outset.)

The MSW plots for both the UBS4 and INTF distance matrices that include P45 have prominent peaks corresponding to four groups. Dividing into this many parts produces the partitions in Tables 40 and 41. The four-way partition of the UBS4 data set places P45 in Gr f-1 with 032, 038, f-1, 28, 205, 565, the Sinaitic Syriac, Armenian, and Georgian. Streeter regarded all of these as members of his “Eastern” category. (In the preface to the fifth impression of his Four Gospels, Streeter (1924, vii) quotes Kenyon’s assessment that the character of the text of P45 in Mark is “definitely Caesarean.”)

The four-way partition of the INTF data set places P45 in Gr A, although a negative silhouette width indicates that it is not a good fit there. The CMDS map derived from the INTF distance matrix that includes P45 locates this manuscript approximately the same distance from Gr A, Gr 209, and Gr 1339.

Ranking witnesses by distance from P45 using the INTF distance matrix produces the ordered list in Table 42. All of these distances are larger than the lower critical limit of the confidence interval of distances that can be attributed to chance. That is, P45 is eccentric by the standard of manuscripts included in the INTF data set, being a long way from any of them. Its nearest neighbours are the Ausgangstext (A) followed by 04 from Gr A, 032 from Gr 209, then an array of manuscripts from Gr 1339 of the relevant four-way partition. In the context of the INTF data set, it seems reasonable to describe P45 as an isolated text which is roughly the same distance from Gr A, Gr 209, and Gr 1339.

Origen’s text is another witness that falls victim to the vetting process, and it too can be included by restricting analysis to an appropriate slice of the data set, seen here in Table 43, Figures 30, 31, and 32. (No results are presented for the INTF data set because it does not include patristic citations.) Both the CMDS map and DC dendrogram place Origen with witnesses that Streeter styled as “Eastern.” The MSW plot indicates that a four-way partition is preferable. When the data set is divided into four, shown in Table 44, the group occupied by Origen’s text includes 038, 28, 565, the Sinaitic Syriac, Armenian, and Georgian. Streeter, aware of the tendency for Origen’s citations of the Gospel of Mark to agree with these witnesses, concluded that this kind of text was already established in Caesarea when Origen moved there in 231 and thus provides a “fixed point for the history of the text of the New Testament” (1924, 101–2).

Figure 30

CMDS (Mark, UBS4, Origen). Animated version available here:

Figure 31

DC (Mark, UBS4, Origen). Archival version available here:

Figure 32

MSW (Mark, UBS4, Origen).

Block Mixture

Sometimes the textual affiliation of a witness changes from section to section. One way such block mixture might have occurred is through partial correction of one text to another. A corrector might have begun to “improve” his or her copy by reference to a second text then have lost interest or run out of time before the work was completed. If the corrected text was later copied, then that copy and its descendants would contain as many of the second text’s readings as had been transferred by the corrector then retained by the copyist. Another way block mixture might have occurred is through replacement of damaged leaves by ones copied from a different variety of text.

Shifts in the textual character of a witness can be studied by first dividing a data matrix into consecutive blocks with approximately equal numbers of variation units, producing a distance matrix for each block, then performing multivariate analysis on each block’s distance matrix. There is a trade-off concerning how many blocks to use. On one hand, shifts that occur in a small section of text have a better chance of being detected if the data set is divided into correspondingly small blocks. On the other hand, sampling error increases as the number of blocks increases because of the decreasing number of variation units in each block. Consequently, increasing the number of blocks also increases the number of fragmentary witnesses that must be excluded to maintain the integrity of analysis results.

Every distance in a distance matrix based on incomplete data is a mere estimate of the actual distance that would be obtained if every place of variation between two witnesses were compared. As it happens, the sampling error of the distance estimate is roughly equal to the square root of the number of places sampled. Thus, if two witnesses are compared at one hundred places and disagree at fifty then the estimated simple matching distance is 50/100 (0.5) and the associated sampling error is approximately equal to the square root of one hundred, which is ten. Consequently, in this example it would not be unlikely for the actual distance to be anywhere between (50–10)/100 (i.e. 0.4) and (50 + 10)/100 (i.e. 0.6).

In the CMDS maps, sampling error causes plotted witness locations to be randomly displaced from where they would be given a larger sample. In the DC dendrograms, sampling error can cause witnesses to jump between branches, especially if those witnesses have mixed texts that stand between the textual varieties associated with the branches. One must search for genuine textual shifts against this noisy background, and this dividing of data sets to search for block mixture only makes matters worse.

Dividing data sets into blocks also creates difficulties for PAM analysis. MSW plots are unlikely to suggest the same number of groups for every block. However, it is desirable to use a single number of groups when comparing group membership across blocks. The chosen number of groups should not be so small that actual shifts will be missed or so large that spurious shifts will occur through sampling error. Despite these difficulties, useful analysis results can be obtained provided that each block retains a sufficient number of variation units. Dividing the UBS4 data matrix into four consecutive blocks of approximately equal size results in each one having about thirty-five variables. This is enough to avoid too many witnesses being dropped due to the constraint on the minimum number of variables required to calculate a distance, which in this study is set to fifteen. The sampling error in distances between witnesses for each block is roughly twice that of distances calculated using the undivided data set covering all of Mark. The factor of two occurs because the data set is divided into four, and four divided by its square root is two. The results shown in Table 45, Figures 33, 34, Table 46, Figures 35, 36, Table 47, Figures 37, 38, Table 48, Figures 39 and 40 are obtained when the UBS4 data set for Mark is divided into four blocks and each is separately analyzed.

Figure 33

CMDS (Mk 1.1–4.24, UBS4). Animated version available here:

Figure 34

DC (Mk 1.1–4.24, UBS4). Archival version available here:

Figure 35

CMDS (Mk 4.28–8.15, UBS4). Animated version available here:

Figure 36

DC (Mk 4.28–8.15, UBS4). Archival version available here:

Figure 37

CMDS (Mk 8.26–11.25, UBS4). Animated version available here:

Figure 38

DC (Mk 8.26–11.25, UBS4). Archival version available here:

Figure 39

CMDS (Mk 12.23–16.20, UBS4). Animated version available here:

Figure 40

DC (Mk 12.23–16.20, UBS4). Archival version available here:

(The original data matrix was divided into submatrices using the R script named groups-scripts-from-1-to-N.R. This script produces submatrices from the same set of cases (i.e. witnesses) but a subset of variables (i.e. variation units). The submatrices have approximately equal numbers of variables.)

Comparing the CMDS maps shows that the overall structure of the plots is similar across blocks. However, some texts exhibit substantial shifts in their relative locations. To name a few, Family 1 (f-1), 038, 565, Old Latin Codex Corbeiensis II (it-ff-2), the Ethiopic (eth), and Sahidic Coptic (cop-sa) do not seem to have stable locations across blocks. Each of these shifts is consistent with partial correction of an ancestral text to another standard. For example, the Sahidic is near “Eastern” texts in the first block but near “Alexandrian” ones in the other three blocks. Perhaps the initial text of this Coptic version had an “Alexandrian” flavour throughout but was then partially revised so that it became more “Eastern” in the first few chapters of Mark?

The DC dendrograms also exhibit a similarity of analysis results across blocks with the exception of a few texts that tend to shift from branch to branch. The exceptions are generally the same texts that undergo substantial shifts in the CMDS maps. Codex 032, which is thought to be more “Western” in the initial chapters of Mark, occupies the “Western” branch of the DC dendrogram for the first block.

Partitioning into a single number of groups is desirable for comparison across blocks using PAM analysis. However, the MSW plots for these blocks do not agree on a single preferable number. To choose what seems a reasonable compromise, partitioning into five groups returns a fairly high MSW value for each block while providing enough slots to allow some differentiation but not so many that like texts are liable to fall into different slots through sampling error.

Tables 49, 50, 51, and 52 give five-way partitions of the four blocks. The medoids of some groups change from block to block. Nevertheless, corresponding groups in different blocks can be identified because certain witnesses recur within them. In what follows, groups will be labelled by chaining together the sigla of the associated medoids, using a final ellipsis in cases that have three or more medoids across corresponding groups (e.g. Gr geo/205/…). A few texts such as 04, 044, the Sinaitic Syriac (syr-s), Old Latin k (it-k), and Bohairic Coptic (cop-bo) are not present in every division due to the vetting procedure exercised to reduce sampling error. Nothing can be said about their character in divisions that omit them. (Nevertheless, the script that constructs a distance matrix can be instructed to retain a specific case so it is often possible to obtain analysis results for omitted witnesses if desired.) It is prudent to also treat poorly classified witnesses (i.e. those with a negative silhouette width) as absent. Excluding these from consideration, a comparison of corresponding groups across blocks helps to identify witnesses that exhibit textual shifts, seen here in Table 53.

Some of these shifts may be due to witnesses being located midway between adjacent groups. When a witness is in such a position, slight perturbations of the distance matrix may cause a jump from one group to another. Inspection of the CMDS maps for the four blocks shows that 04, 892, and 1342 are approximately equidistant from the centres between which they shift. That leaves 037, 038, f-1, 28, 205, 565, Latin codices Monacensis (it-q) and Corbeiensis II (it-ff-2), the Sahidic Coptic, and Ethiopic as texts which appear to exhibit block mixture.

Certain texts adhere to one standard in the first and last blocks but another in between. For example, the entity that represents Family 1 in the UBS apparatus (i.e. f-1) starts and ends as Gr Byz but belongs to Gr geo/205/… in the middle; 038 starts and ends as Gr geo/205/… but changes allegiance to Gr it-d/it-ff-2 or Gr vg in the central parts of Mark. Such witnesses may have resulted from partial efforts to make one text conform to another, cosmetic renovations that affected the outer leaves of a book but left the interior untouched.

PAM analysis does not identify 032 (i.e. Codex W) as a shifting text but instead classifies it as a member of Gr geo/205/… in all four blocks. This is surprising because the first chapters of Mark in 032 have long been considered to preserve a “Western” text. Henry A. Sanders (1912, 73) thought the initial part of Mark (up to 5.31) was the Greek equivalent of the Old Latin version, agreeing with Codex Palatinus (it-e) in particular, and noticed an increasing number of agreements with “Syriacising” manuscripts such as Family 1, Family 13, 28, and 565 in the remainder (i.e. after 5.31). Streeter (1924, 598–600) compared the part of 032 following Mark 5.31 with members of his “Caesarean” text (i.e. 038; Families 1 and 13; minuscules 28, 565, and 700) and concluded that it is “a member of the Θ family, the text of which has suffered, but not too greatly, from Byzantine revision.” (Streeter also gives an imaginative account of how 032 might have acquired the mixture of texts it preserves.) The DC dendrogram for Mark 1.1–4.24 places 032 in a branch that is safely described as “Western.” Table 54, which lists the ten nearest neighbours of 032 in each block, also reveals the “Western” tendency of 032 in the first few chapters of Mark. Codices Palatinus (it-e), Veronensis (it-b), and Bezae (05) are among the nearest neighbours of 032 in the first block. In the remainder of Mark, the nearest neighbours of 032 are predominantly members of Gr geo/205/…. This group, which corresponds to Streeter’s “Eastern” text, includes the Armenian, Georgian, and Sinaitic Syriac. It is possible that these versions share the blame for the distinctive text which Greek members of this group, including 032, bear.

Streeter’s statement about Byzantine revision needs to be qualified. While there is Byzantine influence in the second block, as attested by the proximity of 02, 011, and 1243, this component disappears in the third and fourth blocks where members of Gr 03/2427 appear instead. None of the listed texts is unexpectedly close to 032, as indicated by lack of asterisks attached to distances. The text of 032 in Mark seems to be isolated yet cosmopolitan, with varying “Eastern,” “Western,” “Byzantine,” and “Alexandrian” components in consecutive blocks.

Medoids as Representatives

There is such a great cloud of New Testament witnesses that it is often a practical necessity to restrict ones retained in a summary of the evidence to those few that sufficiently represent the many that have to be omitted. The medoids identified by PAM analysis constitute suitable representatives although alternative witnesses must be selected to represent a group when its medoid is absent at some variation sites or is affected by block mixture. The selection procedure begins with a data set that includes as many witnesses as practicable. PAM analysis is performed for every possible number of groups to plot mean silhouette width versus the number of groups. Next, the plot is used to identify a suitable number of groups. PAM analysis is then applied again to partition the data set into the chosen number of groups, identifying the medoids in the process. Just how many groups are selected depends on the purpose. If aiming to produce a compact apparatus then a smaller number would be chosen; a larger number would be appropriate if presenting a comprehensive survey of, say, the Byzantine textual complex. To give an example, the medoids of the eleven-way partition of the UBS4 data set presented above would serve as a useful starting point if seeking representative texts for a compact summary of the textual situation in Mark. As another example, medoids found by a many-way partition of the INTF data set for Mark would be suitable candidates for witnesses that represent the entire spectrum of extant Greek textual varieties of that Gospel.

Multiple Correspondence Analysis

A multivariate analysis technique called correspondence analysis produces a biplot that simultaneously displays both the cases and variables of a data matrix. This allows variables that are useful for differentiating between cases to be identified, thus providing a basis for classification. If the variables are comprised of categorical data, as is the case for the New Testament textual data analysed in this article, then a technique called multiple correspondence analysis (MCA) should be used. (Concerning categorical data, the only meaningful basis for comparison is equivalence. For example, when comparing the encoded readings of two witnesses at a variation site, one can only say whether they are the same or different; it does not make sense to say that one reading is more or less than the other in this context. Even though New Testament textual data is treated as categorical in this article, the readings of a variation unit can have an inherent order. In fact, the INTF’s Coherence-Based Genealogical Method relies on there being a discernible order of development for the readings of a variation unit.) The biplot in Figure 41 is produced by applying MCA to the first block of UBS4 data for Mark.

Figure 41

Multiple Correspondence Analysis Biplot (Mk 1.1–4.24, UBS4).

(The biplot was produced by the R script named groups-scripts-MVA-MCA.R, which uses the “mca” method of the “MASS” package described in (Venables and Ripley 2002). Prior to analysis, the script drops witnesses that have undefined readings in any variation units.)

The plot is cluttered by inclusion of a label for every variant of every variation unit in the data set. Nevertheless, it does illustrate the potential of the technique to identify variants that are useful for classification purposes. The best variants to use are those whose vectors (indicated by red arrows in this diagram) have a large magnitude. Some vectors point directly at known groups, making the associated variants especially suitable for identifying members of that group. To illustrate, the correspondence of the witness label 03 (representing Codex Vaticanus) and the vector pointing to the variant labelled Mk.2.22.2.1 (representing the first variant listed at the second variation unit of Mark 2.22 in the UBS4 apparatus) indicates that this variant is a useful test for the kind of text found in this manuscript. If the analysed data contains so many variants that the biplot is impossible to use, then the coordinates of witnesses and variants can be printed out to identify variation sites useful for classifying texts. The location of an unknown witness with respect to known groups could be quickly established by comparing the variants it supports with those supported by group medoids at variation sites identified by this procedure.