Comparing Two Distribution Models of Paul’s Literary Techniques: Poisson Versus Negative Binomial

Abstract

This article explores how literary features are statistically distributed in the Christian apostle Paul’s letters. While several decades of occasional research have applied statistics to Paul’s letters, most if not all previous such approaches have either assumed that Paul’s language follows a normal distribution or ignored the question of statistical distribution entirely. The nature of feature distribution—be the features vocabulary words or second-order features chosen by the analyst—is a crucial component of any statistical analysis, and the dearth of work in this area therefore forms a major hole in mathematical approaches to Paul’s letters. This paper addresses this hole in scholarship by comparing two possible models for Paul’s various literary techniques: the Poisson distribution versus the negative binomial distribution.

1. Introduction

This article explores how literary features are statistically distributed in the Christian apostle Paul’s letters. While several decades of occasional research have applied statistics to Paul’s letters, most if not all previous such approaches have either assumed that Paul’s language follows a normal distribution or ignored the question of statistical distribution entirely.1 The nature of feature distribution—be the features vocabulary words or second-order features chosen by the analyst—is a crucial component of any statistical analysis, and the dearth of work in this area therefore forms a major hole in mathematical approaches to Paul’s letters. This paper addresses this hole in scholarship by comparing two possible models for Paul’s various literary techniques: the Poisson distribution versus the negative binomial distribution.

Recent scholarship has argued that many of Paul’s literary techniques are governed by Poisson distribution, noting theoretical reasons why this should be the case as well as analyzing the validity of this quantitative model.2 Nevertheless, we observed that certain literary techniques are not well modeled by Poisson distribution, in particular exhortations and imperatives. The goal of this paper is to determine whether the negative binomial distribution may serve as a better model for Paul’s use of these techniques. We suggest theoretical reasons why these techniques may follow the negative binomial distribution, and we then fit our observed data to the negative binomial distribution, commenting on the validity of this model.

To produce a fuller picture of Paul’s use of these literary techniques, we also analyzed the occurrence of these same techniques in a sample of works from several diverse authors in the contemporary Greco-Roman world, which were previously identified and assessed as potential comparanda for Paul’s letters.3

• Epictetus’ Discourses.
• 4 Maccabees.
• Epistle to the Hebrews.
• The Damascus Document.
• Aristides’ Panathenaicus.

For each work, we determine whether the negative binomial distribution or the Poisson distribution serves as the better model. By including an analysis of several additional texts written in Greek from the roughly contemporary ancient Mediterranean world, we provide controls against which we may compare Paul’s style. We can thus determine whether and to what extant Paul’s style is typical of this set of Greco-Roman literature and/or if it differs and in which ways.

In Section 2, we explain and explore the two types of distributions. Section 3 surveys data from Paul’s letters and the textual comparanda noted above, focusing on key literary features that we previously found were not well modeled by Poisson distribution. Section 4 examines the implications of this statistical model for stylometry in general and Pauline studies in particular. Section 5 concludes with an experiment that uses insight from these two models to explore whether a passage from Paul’s letters is, in fact, a non-Pauline interpolation.

Throughout, we use the data selected and hand-coded in Robertson’s 2016 monograph (Robertson 2016). That work fully explains the selection of features from both a theoretical and practical standpoint, features which—in contrast to traditional stylometric methods which use vocabulary words and positions—are second-order selections derived from experts in Pauline Studies that reflect Paul’s literary style. The reader is directed to that book for further discussion and data accessibility, but the features are reproduced as follows for ease of reference:

• Universal Claims;
• Appeals to Authority;
• Conversation;
• Prosópopoiia/Éthopoiia;
• Rhetorical Questions;
• Metaphors or Analogies;
• Anecdotes or Examples;
• Imperatives;
• Exhortation;
• Caustic Injunctions;
• Pathos;
• Irony or Satire;
• Hyperbole;
• Oppositions or Choices;
• Figurations of Groupness;
• Plural Inclusive Address;
• Second Person Address;
• First Person Reflection;
• Analysis of Potential Questions/Objections;
• Systematic Argument.

2. Probability Distributions

We begin by reviewing the reasons why the Poisson distribution provides a good model for an author’s use of literary techniques under certain assumptions. Nevertheless, we discuss why these assumptions may not be true for every literary technique and we suggest reasons why the negative binomial distribution may provide a better model in these cases.

2.1. Poisson Distribution

In their study of the Federalist Papers (Mosteller and Wallace 1963), Mosteller and Wallace argued that the occurrences of a given word throughout an author’s corpus follow a Poisson distribution. We outline the mathematical theory of Poisson distributions, following Sullivan’s introductory textbook (Sullivan 2013). A Poisson distribution, or a Poisson random variable, describes the number of “successes” (where the term “success” denotes the occurrence of whatever event is under consideration) that occur in a fixed interval, provided the following conditions are met:

• The probability of two or more successes in any sufficiently small interval is 0.
• The probability of success is the same for any two intervals of equal length. When this assumption is true, we say the population is homogeneous.
• The number of successes in any interval is independent of the number of successes in any other interval, provided the intervals do not overlap.

A Poisson random variable takes values in the non-negative integers, $0,1,2,3,\dots$ and so on. The probability of exactly $k$ successes in a given interval is given by the following:

$$P\left(k\right)={\displaystyle \frac{{\lambda }^k}{k!}}{e}^{-\lambda }$$

where the parameter $\lambda$ is the mean of the random variable. In our context, $\lambda$ represents the average number of occurrences of a given literary technique in a 20-verse block of text. It is important to note that this single parameter determines the Poisson distribution. When we find the best-fitting Poisson distribution for Paul’s literary techniques, we need to find the value of $\lambda$ that yields the closest agreement between our observed data and the values predicted by the Poisson distribution.

We claim that the occurrences of literary techniques in a block of text meet these conditions:

• For the first requirement, if we take a very small section of text, there will be at most a single example of a given literary technique. For example, it is unlikely for a single verse to contain two distinct appeals to authority.
• Regarding the second requirement, we have no reason to suppose that any two randomly chosen blocks of text, as long as they are the same length, should include a different number of instances of a given literary technique. The only exception may be at the beginning and ending of letters, where Paul writes his greetings and farewell, which may make certain literary techniques more or less likely to occur. For example, Paul may be unlikely to make an appeal to authority in the opening or closing of a letter, while he may make more frequent use of second person address. We explore possible exceptions to this requirement in Section 2.3.
• Regarding the third requirement, for many of Paul’s literary techniques, we see no reason that the presence of a given literary technique in one block of text should influence its occurrence in a different block of text, assuming there is no overlap between the two text-blocks. Nevertheless, we note that some literary techniques may not satisfy this requirement. For example, the presence of one exhortation may increase the likelihood of exhortations nearby in the text, and thus they may not be well modeled by a Poisson distribution. We explore this issue below in Section 2.4.

2.2. Negative Binomial Distribution

In this section, we outline the mathematical theory of the negative binomial distribution, following Casella and Berger’s advanced textbook on statistics (Casella and Berger 2002). Much like the Poisson distribution, the negative binomial distribution describes the number of “successes” that occur in a fixed interval, although under a different set of assumptions. A negative binomial random variable takes values in the non-negative integers $0,1,2,3,\dots$ and so on, and the probability of exactly $k$ successes is given by the following:

$$P\left(k\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{k+r-1}{k}}\right){\left(1-p\right)}^k{p}^r$$

where $r$ and $p$ are parameters of the distribution. $r$ is a positive integer, and $0<p<1$. When we find the best-fitting negative binomial distribution for Paul’s literary techniques, we must find the values of $r$ and $p$ that yield the closest agreement between our observed frequencies and the frequencies predicted by the negative binomial distribution.

In their landmark study of accident statistics (Arbous and Kerrich 1951), Arbous and Kerrich investigated the mathematical assumptions that give rise to a negative binomial distribution. They identified two important circumstances which produce a negative binomial distribution. Although their work focused on determining the probability distribution of accidents in the workplace, the underlying principles carry over to our situation in which we seek to determine the probability distribution of literary techniques in an author’s corpus.

2.3. Compound Poisson Distributions

As outlined in Arbous and Kerrich’s study (Ibid, Part II, Section III), the first circumstance that leads to a negative binomial distribution is when the population under study is not homogeneous (as necessary for Poisson distribution). Specifically, the population consists of two or more homogeneous populations. Since each sub-population is itself homogeneous, the occurrence of a given literary technique within that sub-population follows a Poisson distribution. These compound Poisson distributions form a negative binomial distribution in the population.

For stylometry, these sub-populations might consist of subsets of Paul’s corpus. For example, one such subset may be the opening and closing sections of Paul’s letters, where the distribution of literary techniques differs from that of the main body of the letter. Nevertheless, the exact subsets remain unknown without further research.

Another possibility is that the homogeneous sub-populations may consist of different sub-types of a given literary technique. For example, it may be the case that the population of literary techniques we group under the single heading of exhortation in fact consists of several sub-populations marked by specific types of exhortation. That is, we may be able to identify sub-types of exhortations, types $A,B,C,\dots$, and sub-type $A$ is homogeneous and consequently its occurrence follows a Poisson distribution, as do $B$ and $C$, and so on.

2.4. Contagious Distributions

Furthermore, Arbous and Kerrich consider another set of circumstances that give rise to a negative binomial distribution, once again phrased as an alternative to having a population be homogeneous (Ibid, Part II, Section IV). In a homogeneous population, the occurrence of one “success” in an interval of time does not affect the probability of “success” for any other interval of time. In their study of contagious diseases (Greenwood and Yule 1920), Greenwood and Yule considered the possibility that the event that one individual contracts a disease would make it more likely for other individuals in contact with them to contract the disease. Thus, the occurrence of one “success” may increase the likelihood of another “success”. Distributions that obeyed this property were called “contagious distributions” because of their origin in modeling disease spread.

For stylometry, we consider an analogous situation in which the occurrence of a given literary technique may increase the likelihood of another such technique to occur. For example, it may be the case that when Paul uses one exhortation in a verse, the probability that he will use one in a nearby verse becomes greater. Thus, exhortations may exhibit contagious-like behavior, causing a departure from Poisson distribution to negative binomial distribution.

2.5. Comparing These Assumptions

In Section 3, we demonstrate that a subset of literary techniques from Paul’s corpus are better modeled with the negative binomial distribution rather than the Poisson distribution. In these cases, we ought to wonder why these literary techniques follow a different probability distribution than the majority.

It must be that at least one of the requirements for Poisson distribution is not satisfied by these literary techniques, and we must determine whether another set of assumptions can explain it. Above, we introduced two assumptions that would lead to a negative binomial distribution, but the data alone cannot tell us whether it is because the data consist of compound Poisson distribution or they are an example of contagious distributions. Instead, we must employ literary-critical theory to consider which possibility is more plausible. This will be the task of Section 4. Finally, in Section 5, we perform an experiment, based on the hypothesis that our texts consist of compound Poisson distributions, to test whether a given passage is a non-Pauline interpolation.

3. Survey of Greco-Roman Authors

In this section, we analyze the occurrences of four literary techniques: exhortation, imperatives, universal claims, and second person addresses, in the following works:

• The undisputed Pauline letters (omitting Philemon).
• Epictetus’ Discourses.
• 4 Maccabees.
• Epistle to the Hebrews.
• The Damascus Document.
• Aristides’ Panathenaicus.

Within each work, we seek to determine whether the observed occurrences of each literary technique are better modeled by the negative binomial distribution or the Poisson distribution.

We have included these works to sample texts from a variety of socio-literary spheres.4 For example, Aristides represents an author with a high level of formal education. Paul and Epictetus occupy a socio-literary sphere of cultural producers with a somewhat lesser amount of formal education. The remaining documents, Epistle to the Hebrews, 4 Maccabees, and The Damascus Document, are included as representatives of works that have traditionally been compared with Paul’s (Robertson 2016).

In our previous work,5 we found that Paul’s use of universal claims was moderately well modeled by the Poisson distribution, and second person addresses were very well modeled by the Poisson distribution, in contrast to the relatively poor model of exhortations and imperatives. In the sections below, we show that the negative binomial distribution is a better model for Paul’s use of exhortations and imperatives, and possibly universal claims as well. These results for Paul and the other authors under consideration are discussed more extensively in the sections below. In Section 4, we discuss the results of this analysis.

3.1. Undisputed Pauline Letters

In this section, we fit our observed data of Paul’s use of exhortations and imperatives to the negative binomial distribution. We then compare the negative binomial model with the model given by the Poisson distribution, determining which distribution serves as a better model. Throughout this section, we work with the undisputed Pauline letters, Romans, 1 Corinthians, 2 Corinthians, Galatians, Philippians, and 1 Thessalonians. We omit Philemon from this analysis because of its abbreviated length and therefore poor sample size; it also seems to be a strongly different type of letter from the others. To conduct our analysis we divide Paul’s undisputed letters into 73 text-blocks, each consisting of 20 verses.

3.1.1. Summary

For each literary technique, we find the best-fitting distribution under the assumption of a negative binomial model as well as a Poisson model. To find the best-fitting distribution, we determine the value of the parameters for each distribution using the method of Maximum Likelihood Estimation (MLE), a method which determines the values of the parameters that have the highest probability of producing the observed data. For the Poisson distribution, the method of MLE determines that the best-fitting distribution occurs when $\lambda$ is that average value of the observations. For the negative binomial distribution, the value of $r$ cannot be determined by MLE, though for a given value of $r$, the best-fitting distribution occurs when

$$p={\displaystyle \frac{r}{r+\overline{x}}}$$

where $\overline{x}$ denotes the average value of the observations.

We then check the validity of each model by running a ${\chi }^2$-Goodness of Fit Test and reporting the $p$-value. The $p$-value of a ${\chi }^2$-Goodness of Fit Test is the probability that observations drawn randomly from a given distribution, in our case either a Poisson distribution or a negative binomial distribution, would deviate from the expected values as much or more than our actual observations. A low $p$-value implies that it is very unlikely for the difference between our observed values and expected values to arise through mere random variation. Thus, a high $p$-value, close to 100%, indicates that the observed data are consistent with the model, while a small $p$-value, close to 0%, indicates that the observed data are not consistent with the model. The $p$-values of these tests are summarized below in

Table 1. Summary of Goodness of Fit Tests for Paul.

	NB	Poisson
Exhortations	23.80%	12.51%
Imperatives	98.78%	31.65%
Universal claims	76.83%	70.94%
Second person addresses	80.69%	80.52%

.

From this brief summary, we find that the negative binomial distribution is a significantly better model for exhortations and imperatives over the Poisson distribution. On the other hand, the negative binomial distribution has similar $p$-values for universal claims and second person addresses, suggesting that it offers no improvement over the Poisson distribution. The sections below provide more detail on our statistical analyses.

3.1.2. Exhortations

We begin with exhortations, Paul’s literary technique which is the worst fit for the Poisson distribution.

Table 2. NB vs. Poisson models of exhortations (Paul).

Value	Frequency	NB Expected Frequency	Poisson Expected Frequency
0	23	21.99	18.74
1	25	23.46	25.48
2	9	15.01	17.33
3	12	7.48	7.86
4	1	3.19	2.67
5	2	1.23	0.73
6	1	0.65	0.16

below records the observed data for the frequency of Paul’s use of exhortations, as well as the frequencies expected by the negative binomial distribution with $r=5$ and $p=0.787$, as well as those expected by Poisson distribution with $\lambda =1.36$. The reader should understand the first row of this table to indicate that we observed 23 text-blocks containing no occurrences of exhortation, while the predicted number of text-blocks containing no exhortations was 21.99, according to the negative binomial model, and 18.74, according to the Poisson model. The other rows should be interpreted similarly, describing the number of text-blocks containing different numbers of occurrences of exhortation. We graphically compare the two models in Figure 1.

In addition, we calculate the $p$-values for a ${\chi }^2$-Goodness of Fit Test for each model (

Table 3. Goodness of Fit Test for exhortations (Paul).

	NB	Poisson
$p$-value	23.80%	12.51%

). To conduct this test, we combine all text-blocks containing four or more instances of an exhortation into a single category. The $p$-value for this statistical test indicates the probability that a random sample drawn from each theoretical model would produce observed data that deviate this much or more from the expected values.

A comparison of these $p$-values lends credibility to the hypothesis that Paul’s use of exhortations is better modeled by the negative binomial distribution than the Poisson distribution.

3.1.3. Imperatives

Next, we consider imperatives, Paul’s literary technique that was the second-worst fit for the Poisson distribution. Although this literary technique was fairly well modeled by the the Poisson distribution, we seek to determine whether the negative binomial distribution provides an improved model.

Table 4. NB vs. Poisson models of imperatives (Paul).

Value	Frequency	NB Expected Frequency	Poisson Expected Frequency
0	14	15.07	9.06
1	9	8.76	12.50
2	6	5.09	8.62
3	3	2.96	3.97
4	1	1.72	1.37
5	2	1.00	0.38
6	1	0.58	0.09

below records the observed data for the frequency of Paul’s use of imperatives, as well as the frequencies expected by the negative binomial distribution with $r=1$ and $p=0.419$, as well as those expected by the Poisson distribution with $\lambda =1.38$. We graphically compare the two models in Figure 2. For this literary technique, we organized Paul’s letters into 40-verse blocks rather than 20-verse blocks.

Table 5. Goodness of Fit Test for imperatives (Paul).

	NB	Poisson
$p$-value	98.78%	31.65%

below records the $p$-values of a ${\chi }^2$-Goodness of Fit test with the two possible models. As with exhortations, we combine all text-blocks containing four or more instances of an imperative into a single category.

A comparison of these $p$-values in Table 5 lends credibility to the hypothesis that Paul’s use of imperatives is better modeled by the negative binomial distribution than the Poisson distribution.

3.1.4. Universal Claims

We now consider Paul’s use of universal claims. In contrast to the previous two literary techniques, Paul’s use of universal claims is well modeled by the Poisson distribution. For this reason, this literary technique acts as a control.

Specifically, if we find that a negative binomial provides a better model for Paul’s use of universal claims, that may suggest that a negative binomial distribution will often provide a better model for the occurrence of literary techniques regardless of whether the Poisson distribution also provides a good model. A negative binomial distribution has a similar shape to the Poisson distribution, and it involves two parameters, rather than the Poisson distribution’s single parameter, which may provide a mechanism for it to serve as a better model in most situations. On the other hand, if we find that the negative binomial distribution does not provide a better model for Paul’s use of literary techniques, it will suggest that there are some situations in which the Poisson distribution provides a better model and others in which the negative binomial provides a better model.

Table 6. NB vs. Poisson models of universal claims (Paul).

Value	Frequency	NB Expected Frequency	Poisson Expected Frequency
0	9	9.69	8.59
1	18	18.51	18.38
2	19	18.66	19.67
3	16	13.20	14.03
4	4	7.36	7.51
5	6	3.43	3.21
6	1	2.15	1.15

below records the observed data for the frequency of Paul’s use of universal claims, as well as the frequencies expected by the negative binomial distribution with $r=18$ and $p=0.894$, and those expected by the Poisson distribution with $\lambda =2.14$. We graphically compare the two models in Figure 3.

Thus, we see that the Poisson distribution and the negative binomial distribution are comparable models for our observed data. The table (

Table 7. Goodness of Fit Tests for universal claims (Paul).

	NB	Poisson
$p$-value	76.83%	70.94%

) below records the $p$-values of a ${\chi }^2$-Goodness of Fit test with the two possible models. To conduct these tests, we combine all text-blocks containing five or more instances of a universal claims into a single category.

The negative binomial distribution yields a slightly higher $p$-value, though the improvement is small enough that it is not clear whether this higher $p$-value signifies that the negative binomial is the preferred model for universal claims, or if it is merely an artifact of the fact that the negative binomial distribution has two parameters that can be varied to fit observed data, compared to the one parameter of Poisson distribution.

3.1.5. Second Person Addresses

Finally, we consider Paul’s use of second person addresses. In contrast to the exhortations and imperatives, Paul’s use of second person addresses are well modeled by a Poisson distribution. For this reason, this literary technique acts as a control.

Table 8. NB vs. Poisson models for second person addresses (Paul).

Value	Frequency	NB Expected Frequency	Poisson Expected Frequency
0	19	18.47	18.36
1	25	25.21	25.34
2	15	17.37	17.49
3	11	8.06	8.04
4	2	2.83	2.78
5	1	0.80	0.77

below records the observed data for the frequency of Paul’s use of universal claims, as well as the frequencies expected by the negative binomial distribution with $r=100$ and $p=0.986$ and those expected by the Poisson distribution with $\lambda =1.38$. We graphically compare the two models in Figure 4. We note that, in our analysis, the negative binomial distribution became an increasingly good fit for the observed data as $r$ increases without bound, though the improvement becomes smaller and seems to approach some value, and consequently we capped the value of this parameter at $r=100$. We suspect that this behavior reflects the fact that the Poisson distribution can be obtained by the limit of a negative binomial distribution.

Table 9. Goodness of Fit Tests for second person addresses (Paul).

	NB	Poisson
$p$-value	80.69%	80.52%

below records the $p$-values of a ${\chi }^2$-Goodness of Fit test with the two possible models. As with exhortations, we combine all text-blocks containing four or more instances of a second person address into a single category.

As with universal claims, the negative binomial distribution yields a slightly higher $p$-value, though the improvement is small enough that it is not clear whether this higher $p$-value signifies that the negative binomial is the preferred model for universal claims, or if it is merely an artifact of the fact that the negative binomial distribution has two parameters that can be varied to fit observed data, compared to the one parameter of Poisson distribution. This latter possibility seems confirmed by the fact that the negative binomial model only achieves a good fit with the observed data when the parameter $r$ takes high values, as when the negative binomial distribution approaches the Poisson distribution in a limiting sense.

3.2. Epictetus’ Discourses

In this section, we use the Poisson distribution and the negative binomial distribution as competing models of Epictetus’ literary techniques throughout his Discourses. Robertson previously argued that Epictetus’ Discourses serves as a good comparison for Paul’s letters in the sense that both authors and their respective texts occupy what Robertson called similar “socio-literary spheres”, namely a shared intersection of social position and social purpose (Robertson 2016). For this reason, we believe it will be instructive to compare Epictetus’ use of literary techniques with Paul’s, focusing once again on exhortations, imperatives, universal claims, and second person addresses. As with Paul’s corpus, we divide the Discourses into text-blocks of 20-verses, resulting in 131 text-blocks.

3.2.1. Summary

As with Paul’s corpus, we find the best-fitting distribution for each literary technique using the method of Maximum Likelihood Estimation, under the assumption of a negative binomial model and also a Poisson model. We then check the validity of each model by running a ${\chi }^2$-Goodness of Fit Test. The $p$-values of these tests are summarized below in

Table 10. Summary of Goodness of Fit Tests for Epictetus.

	NB	Poisson
Exhortations	93.04%	41.80%
Imperatives	98.64%	27.05%
Universal claims	99.33%	17.19%
Second person addresses	59.70%	62.14%

. A high $p$-value, close to 100%, indicates that the observed data are consistent with the model, while a small $p$-value, close to 0%, indicates that the observed data are not consistent with the model.

We can make several observations from these results in Table 10. First, generally, the negative binomial distribution is the better model for the observed data. This may be expected simply from the fact that the negative binomial distribution involves two parameters, while the Poisson has only one. We caution the reader against concluding that the negative binomial is therefore the better model for this type of data. The fact that the negative binomial distribution involves an additional parameter means there is the risk of over-fitting the data, when compared to a Poisson model.

An analogous situation is the choice between linear-regression versus quadratic-regression to model the correlation between two variables. The quadratic model will necessarily fit the data more closely, since the linear model can be considered a sub-case of the quadratic model (when the degree-two term vanishes). Nevertheless, researchers often prefer to model the relationship with the linear model precisely because it involves fewer parameters.

Additionally, the negative binomial distribution can approach the Poisson distribution in a limiting sense as parameter $r$ grows without bound. For this reason, whenever the Poisson distribution is a good model, there must be a negative binomial distribution which is a similarly good model. Nevertheless, we observe that for Epictetus’ use of second person addresses, the Poisson distribution is the better model. This fact alone indicates that there are some situations where the Poisson distribution is the better model.

Second, and notably, we find much qualitative agreement between Paul and Epictetus in the model for their literary techniques. The negative binomial distribution provides the better model for exhortations and imperatives, while the Poisson distribution is preferred for second person addresses. The only exception is universal claims, though both distributions can be considered good models for Epictetus’ work.

These results further confirm Robertson’s hypothesis that Paul and Epictetus occupied similar socio-literary spheres, a commonality which is reflected in the stylistic similarities of their corpora. The techniques these writers use follow the same model, negative binomial or Poisson, although with different parameters reflecting rates of usage. The fact that they overlap in the distribution of their literary techniques suggests certain similarities in terms of (following Robertson’s proposals) educational backgrounds, social purposes in writing their texts, and social positions, be they aspirational or real.

Third, these qualitative similarities raise another question: When an author occupies a different socio-literary sphere, will their use of certain literary techniques follow different models than those observed in Paul and Epictetus? An investigation into authors from different spheres may provide a control in our study, highlighting the role that socio-literary spheres play in determining the behavior of “top-down” stylometric data. The following sections explain the statistical analyses of each literary technique in more detail.

3.2.2. Exhortations

We begin with Epictetus’ use of exhortations.

Table 11. NB vs. Poisson models of exhortations (Epictetus).

Value	Frequency	NB Expected Frequency	Poisson Expected Frequency
0	35	33.77	27.53
1	36	40.10	42.94
2	29	28.57	33.50
3	18	15.83	17.42
4	9	7.52	6.79
5	4	3.21	2.12

below records the observed data for the frequency of Epictetus’ use of exhortations, as well as the frequencies expected by the negative binomial distribution with $r=5$ and $p=0.763$, as well as those expected by the Poisson distribution with $\lambda =1.56$. We graphically compare the two models in Figure 5.

Table 12. Goodness of Fit Tests for exhortations (Epictetus).

	NB	Poisson
$p$-value	93.04%	41.80%

below records the $p$-values of a ${\chi }^2$-Goodness of Fit test with the two possible models. To conduct these tests, we combine all text-blocks containing five or more instances of an exhortation into a single category.

We see that both distributions serve as good models. The negative binomial distribution has the higher $p$-value, though it may reflect nothing more than the fact that the negative binomial distribution relies on two parameters, compared to the single parameter of the Poisson distribution.

3.2.3. Imperatives

Next, we examine Epictetus’ use of imperatives.

Table 13. NB vs. Poisson models of imperatives (Epictetus).

Value	Frequency	NB Expected Frequency	Poisson Expected Frequency
0	79	78.25	72.62
1	34	35.55	42.84
2	13	12.11	12.64
3	3	3.67	2.49
4	2	1.04	0.37

below records the observed frequency of Epictetus’ use of imperatives, as well as the frequencies expected by the negative binomial distribution with $r=2$ and $p=0.773$, as well as those expected by the Poisson distribution with $\lambda =0.59$. We graphically compare the two models in Figure 6.

Table 14. Goodness of Fit Tests for imperatives (Epictetus).

	NB	Poisson
$p$-value	98.64%	27.05%

below records the $p$-values of a ${\chi }^2$-Goodness of Fit test with the two possible models. To conduct these tests, we combine all text-blocks containing three or more instances of an imperative into a single category.

Both distributions yield high $p$-values, though the negative binomial distribution has a $p$-value almost four times greater than the Poisson distribution. We conclude that the negative binomial distribution is the better model for Epictetus’ use of imperatives, very similar to Paul.

3.2.4. Universal Claims

We continue with an examination of Epictetus’ use of universal claims.

Table 15. NB vs. Poisson models of universal claims (Epictetus).

Value	Frequency	NB Expected Frequency	Poisson Expected Frequency
0	57	58.52	48.68
1	40	38.81	48.19
2	20	19.31	23.85
3	9	8.54	7.87
4	2	3.54	1.95
5	3	1.41	0.39

below records the observed frequency of Epictetus’ use of universal claims, as well as the frequencies expected by the negative binomial distribution with $r=2$ and $p=0.668$, as well as those expected by the Poisson distribution with $\lambda =0.99$. We graphically compare the two models in Figure 7.

Table 16. Goodness of Fit Tests for universal claims (Epictetus).

	NB	Poisson
$p$-value	99.33%	17.19%

below records the $p$-values of a ${\chi }^2$-Goodness of Fit test with the two possible models. To conduct these tests, we combine all text-blocks containing four or more instances of a universal claims into a single category.

Both distributions serve as good models for Epictetus’ use of universal claims, though the negative binomial seems stronger. As with exhortations, the negative binomial’s higher $p$-value may reflect nothing more than the fact that it involves a second parameter.

3.2.5. Second Person Addresses

We end with an examination of Epictetus’ use of second person addresses.

Table 17. NB vs. Poisson models of second person addresses (Epictetus).

Value	Frequency	NB Expected Frequency	Poisson Expected Frequency
0	56	59.86	60.05
1	54	46.70	46.84
2	16	18.40	18.27
3	3	4.88	4.75
4	2	0.98	0.93

below records the observed frequency of Epictetus’ use of second person addresses, as well as the frequencies expected by the negative binomial distribution with $r=100$ and $p=0.992$, as well as those expected by the Poisson distribution with $\lambda =0.78$. We graphically compare the two models in Figure 8. We note that the negative binomial distribution seemed to become an increasingly good fit as $r$ increases without bound. This observation seems to reflect the fact that our data are very well modeled by the Poisson distribution, and a negative binomial distribution becomes a good approximation to the Poisson distribution in a limiting sense.

Table 18. Goodness of Fit Tests for second person addresses (Epictetus).

	NB	Poisson
$p$-value	59.70%	62.14%

below records the $p$-values of a ${\chi }^2$-Goodness of Fit test with the two possible models. To conduct these tests, we combine all text-blocks containing three or more instances of a second person address into a single category.

Here, we conclude that Poisson distribution is the better model for Epictetus’ use of second person addresses. The $p$-value for the Poisson distribution is slightly higher, which is even more striking as the Poisson distribution relies on just one parameter. For this reason, we conclude that the Poisson distribution is the better model.

3.3. 4 Maccabees

Next, we consider 4 Maccabees. Among the four literary techniques we studied in Paul’s corpus and Epictetus’ Discourses, only universal claims occurred with enough frequency to hope to determine the distribution. We note that we grouped the book into 10-verse blocks of text in order to generate more data points. Also, we end our analysis at Chapter 8, verse 8, because the style changes noticeably at this part of the work. We observe no instances of universal claims after Chapter 8, verse 1. This change occurs when the author begins to narrate the torture of the seven brothers, which continues throughout the remainder of the book. The style of the book changes considerably, and for this reason, we feel justified in ending our analysis at that point.

Table 19. NB vs. Poisson models of universal claims (4 Maccabees).

Value	Frequency	NB Expected Frequency	Poisson Expected Frequency
0	8	8.55	7.35
1	6	6.64	7.72
2	5	3.44	4.05
3	2	1.48	1.42

below records the observed frequency of universal claims in 4 Maccabees, as well as the frequencies expected by a negative binomial distribution with $r=3$ and $p=0.741$, as well as those expected by a Poisson distribution with $\lambda =1.05$. We graphically compare the two models in Figure 9.

Table 20. Goodness of Fit Tests for universal claims (4 Maccabees).

	NB	Poisson
$p$-value	84.35%	63.04%

below records the $p$-values of a ${\chi }^2$-Goodness of Fit test with the two possible models. To conduct these tests, we combine all text-blocks containing four or more instances of a universal claims into a single category.

Both distributions provide good models for universal claims in 4 Maccabees. Each distribution yields a $p$-value above 50%, and so we cannot determine one distribution as the better model.

3.4. Epistle to the Hebrews

Next, we consider Epistle to the Hebrews. We analyze the occurrence of three literary techniques: exhortations, universal claims, and second person addresses (

Table 21. Summary of Goodness of Fit Tests for Hebrews.

	NB	Poisson
Exhortations	96.89%	65.36%
Universal claims	43.88%	43.64%
Second person addresses	62.82%	1.48%

). We omit an analysis of imperatives because there are only 12 occurrences in Hebrews, all concentrated in the last four chapters of the letter (and in fact, 8 of those 12 occur in the last 30 verses). For these reasons, we cannot fit the Poisson distribution or the negative binomial distribution to this data.

3.4.1. Exhortations

We begin with an examination of the occurrence of exhortation in Hebrews.

Table 22. NB vs. Poisson models of second person addresses (Hebrews).

Value	Frequency	NB Expected Frequency	Poisson Expected Frequency
0	15	14.60	12.57
1	9	8.83	10.93
2	3	4.00	4.76
3	1	1.61	1.38
4	2	0.61	0.30

below records the observed frequency of second person addresses, as well as the frequencies expected by the negative binomial distribution with $r=2$ and $p=0.698$, as well as those expected by a Poisson distribution with $\lambda =0.87$. We graphically compare the two models in Figure 10.

Table 23. Goodness of Fit Tests for exhortation (Hebrews).

	NB	Poisson
$p$-value	96.89%	65.36%

below records the $p$-values of a ${\chi }^2$-Goodness of Fit test with the two possible models. To conduct these tests, we combine all text-blocks containing two or more instances of an exhortation into a single category.

Each distribution yields a $p$-value above 50%, and so we cannot determine one distribution as the better model.

3.4.2. Universal Claims

Next, we examine the occurrence of universal claims in Hebrews.

Table 24. NB vs. Poisson models of universal claims (Hebrews).

Value	Frequency	NB Expected Frequency	Poisson Expected Frequency
0	5	4.57	4.49
1	9	8.52	8.52
2	4	8.02	8.10
3	8	5.08	5.13
4	4	2.44	2.44

below records the observed frequency of second person addresses, as well as the frequencies expected by the negative binomial distribution with $r=100$ and $r=0.981$, as well as those expected by a Poisson distribution with $\lambda =1.90$. We graphically compare the two models in Figure 11. We note that as $r$ increases to infinity, the negative binomial distribution becomes an increasingly better fit, which may reflect the fact that the negative binomial distribution converges to a Poisson distribution. For this reason, we capped the value of $r$ at $r=100$.

Table 25. Goodness of Fit Tests for universal claims (Hebrews).

	NB	Poisson
$p$-value	43.88%	43.64%

below records the $p$-values of a ${\chi }^2$-Goodness of Fit test with the two possible models. To conduct these tests, we combine all text-blocks containing four or more instances of a universal claims into a single category.

Both models have similar $p$-values, and so we cannot rule either one out, though we tentatively prefer the Poisson distribution because it involves a single parameter.

3.4.3. Second Person Addresses

Finally, we examine the occurrence of second person addresses in Hebrews.

Table 26. NB vs. Poisson models of second person addresses (Hebrews).

Value	Frequency	NB Expected Frequency	Poisson Expected Frequency
0	17	15.25	11.37
1	5	7.50	11.03
2	3	3.69	5.35
3	3	1.81	1.73
4	1	0.89	0.42
5	1	0.44	0.08

below records the observed frequency of second person addresses, as well as the frequencies expected by the negative binomial distribution with $r=1$ and $p=0.508$, as well as those expected by a Poisson distribution with $\lambda =0.97$. We graphically compare the two models in Figure 12.

Table 27. Goodness of Fit Tests for second person addresses (Hebrews).

	NB	Poisson
$p$-value	62.82%	1.48%

below records the $p$-values of a ${\chi }^2$-Goodness of Fit test with the two possible models. To conduct these tests, we combine all text-blocks containing three or more instances of a second person address into a single category.

These $p$-values suggest that the negative binomial distribution is the better model. The negative binomial distribution yields a $p$-value above 5%, suggesting that it is a valid model, while the $p$-value for the Poisson distribution is below 5%, indicating that the data do not follow a Poisson distribution.

3.5. Damascus Document

In this section, we examine the occurrences of universal claims and exhortations in The Damascus Document. We restrict our attention to these two literary techniques because second person addresses and imperatives occur rarely, merely four and five times, respectively. We organize the text into blocks of 20 verses.

3.5.1. Universal Claims

First, we examine the occurrence of universal claims in The Damascus Document.

Table 28. NB vs. Poisson models of universal claims (Damascus Document).

Value	Frequency	NB Expected Frequency	Poisson Expected Frequency
0	3	2.90	2.85
1	4	5.55	5.55
2	6	5.36	5.41
3	5	3.48	3.52
4	2	1.72	1.71

below records the observed frequency of universal claims, as well as the frequencies expected by the negative binomial distribution with $r=100$ and $p=0.981$, as well as those expected by a Poisson distribution with $\lambda =1.95$. We graphically compare the two models in Figure 13. Once again, we capped $r$ at $r=100$ because the negative binomial distribution seems to approach the Poisson distribution as $r$ grows without bound.

Table 29. Goodness of Fit Tests for universal claims (Damascus Document).

	NB	Poisson
$p$-value	89.29%	89.43%

below records the $p$-values of a ${\chi }^2$-Goodness of Fit test with the two possible models. To conduct these tests, we combine all text-blocks containing four or more instances of a universal claims into a single category.

Both models have similar $p$-values, and so we cannot rule either one out, though we tentatively prefer the Poisson distribution because it involves a single parameter.

3.5.2. Exhortations

We end our analysis of The Damascus Document with an examination of exhortations.

Table 30. NB vs. Poisson models of exhortations (Damascus Document).

Value	Frequency	NB Expected Frequency	Poisson Expected Frequency
0	3	5.23	5.18
1	9	6.97	7.00
2	6	4.69	4.72
3	2	2.12	2.13

below records the observed frequency of exhortations, as well as the frequencies expected by a negative binomial distribution with $r=100$ and $p=0.987$, as well as those expected by a Poisson distribution with $\lambda =1.35$. We graphically compare the two models in Figure 14.

Table 31. Goodness of Fit Tests for exhortations (Damascus Document).

	NB	Poisson
$p$-value	46.09%	47.31%

below records the $p$-values of a ${\chi }^2$-Goodness of Fit test with the two possible models. To conduct these tests, we combine all text-blocks containing two or more instances of exhortation into a single category.

Both distributions provide good models for the observed data. Because the $p$-value for the Poisson distribution is larger, despite that the Poisson distribution involves fewer parameters, we conclude that the Poisson distribution is likely the better model for the occurrence of exhortation in The Damascus Document.

3.6. Aristides’ Panathenaicus

We focus on the occurrence of universal claims in Aristides’ Panathenaicus because it is the only one of the four literary techniques under consideration that occurs sufficiently to provide evidence of its distribution. We group the text into text-blocks of two sections.

Table 32. NB vs. Poisson models of universal claims (Aristides).

Value	Frequency	NB Expected Frequency	Poisson Expected Frequency
0	0	3.59	3.52
1	5	6.76	6.76
2	16	6.42	6.48
3	3	4.11	4.15

below records the observed frequency of universal claims, as well as the frequencies expected by a negative binomial distribution with $r=100$ and $p=0.981$, as well as those expected by a Poisson distribution with $\lambda =1.92$. We graphically compare the two models in Figure 15.

Table 33. Goodness of Fit Tests for universal claims (Aristides).

	NB	Poisson
$p$-value	0.01%	0.01%

below records the $p$-values of a ${\chi }^2$-Goodness of Fit test with the two possible models. To conduct these tests, we combine all text-blocks containing three or more instances of universal claims into a single category.

These $p$-values are the smallest seen in this study by several orders of magnitude. We conclude that neither distribution is a good model for the observed occurrences of universal claims in Aristides’ Panathenaicus. It may be worthwhile to remember that Aristides represents the highest socio-literary sphere of Greco-Roman society. The fact that his use of universal claims differs so greatly from both the negative binomial distribution as well as the Poisson distribution may reflect his high level of rhetorical training.

3.7. Summary

In this section, we present another summary of our results, grouping the results from each author by literary technique in

Table 34. Summary of NB vs. Poisson models of exhortations.

Exhortations
	NB	Poisson
Paul	23.80%	12.51%
Epictetus	93.04%	41.80%
Hebrews	96.89%	65.36%
Damascus Document	46.09%	47.31%

,

Table 35. Summary of NB vs. Poisson models of imperatives.

Imperatives
	NB	Poisson
Paul	98.78%	31.65%
Epictetus	98.64%	27.05%

,

Table 36. Summary of NB vs. Poisson models of universal claims.

Universal Claims
	NB	Poisson
Paul	76.83%	70.94%
Epictetus	99.33%	17.19%
Hebrews	43.88%	43.64%
4 Maccabees	84.35%	63.04%
Damascus Document	89.29%	89.43%
Aristides	0.01%	0.01%

and

Table 37. Summary of NB vs. Poisson models of second person addresses.

Second Person Addresses
	NB	Poisson
Paul	80.69%	80.52%
Epictetus	59.70%	62.14%
Hebrews	62.82%	1.48%

.

The only observation we note is that there does not seem to be a convincing pattern for the $p$-values within any single literary technique. Every literary technique that we examined throughout multiple authors has instances in which the negative binomial distribution seems to be the better model, and others for which the Poisson distribution seems better. This lack of pattern may suggest that there is no explanation for why one model is preferable to the other, which would cut across socio-literary spheres. Instead, the preferred model may depend on the socio-literary sphere in which an author is located, influenced perhaps by the level of rhetorical training and literary purpose common to that subset of Greco-Roman culture.

4. Consequences for Stylometry and Pauline Studies

In the previous section, we analyzed the occurrence of exhortations, imperatives, universal claims, and second person addresses in several pieces of Greco-Roman literature. We first note some observations on our results and then consider the following two questions:

• Why is the negative binomial distribution a better model for the occurrence of literary techniques in some cases?
• What insights can be gained by determining the probability model for a particular author’s use of a literary technique?

In Section 1, we outlined two hypotheses that lead to the negative binomial distribution, which we called compound Poisson distributions and contagious distributions. We must now determine whether these hypotheses provide a plausible explanation for the qualitative differences we have observed.

4.1. Observations

First, we note that Paul and Epictetus exhibit similar qualitative behavior for the occurrence of their literary techniques. Both authors use exhortations and imperatives in a way that is better modeled by a negative binomial distribution, while their use of second person addresses is better modeled by a Poisson distribution. We find it significant that among all the authors considered in this study, it is precisely Paul and Epictetus who were hypothesized to belong to the same socio-literary sphere.

Moreover, we note that the Poisson and negative binomial distributions provide good models for all the authors we examined and their use of exhortations, imperatives, universal claims, and second person addresses, with the single exception of Aristides. We found that both the negative binomial distribution and the Poisson distribution are very poor models for Aristides’ use of universal claims, both having a $p$-value below 0.01%. Aristides stands out as the representative of the highly educated, higher status socio-literary sphere. We suspect that his advanced education in rhetoric may have led him to be very conscious of his personal style, which may lead to his atypical results.

4.2. Compound Poisson Distributions

In this section, we consider the possibility that the occurrence of a literary technique follows the negative binomial distribution because it consists of two or more homogeneous sub-populations. There are (at least) three ways Paul’s corpus could be divided into such sub-populations, each with its own implications for stylometry.

First, the text itself could be divided into sub-populations. For example, one could divide each letter into three sub-populations consisting of the introductory section, the main body, and the closing remarks, and each sub-population would be homogeneous. Within each sub-population, the occurrence of exhortations and imperatives could then be analyzed to determine what distribution they follow. If we have correctly identified a homogeneous sub-population, the occurrence of each literary technique would follow a Poisson distribution.

Second, in another approach, the literary techniques themselves could be divided into sub-populations. For example, we could identify different types of exhortations, A-exhortations and B-exhortations, and then determine what distribution each sub-type follows. Again, if we have correctly identified a homogeneous sub-population, it will follow a Poisson distribution. That is, A-exhortations would be well modeled by some Poisson distribution and so would B-exhortations.

A third possibility is that the occurrence of literary techniques may be divided into sub-populations that represent the work of different authors. For example, the occurrence of exhortations in Paul might consist of those written by Paul himself, sub-population A, and those written by a second hand at a later time, sub-population B.

This possibility offers a new avenue for investigating questions of authorship via stylometry. By identifying the literary techniques that follow a negative binomial distribution, we might identify the specific way a later editor has altered a written work. In the example of Paul, we see that universal claims and second person addresses follow a Poisson distribution, but exhortations and imperatives are better modeled by a negative binomial distribution. This may indicate that a later hand inserted several instances of exhortations and imperatives, and their activity created sub-population B, detected by presences of a negative binomial distribution of those literary techniques. On the other hand, it would equally indicate that this later hand did not introduce any universal claims or second person addresses.

4.3. Contagious Distributions

Next, we consider the possibility that the occurrence of a literary technique follows the negative binomial distribution because it has the property of being contagious, in the sense that one occurrence raises the probability of a subsequent occurrence. Two possible conclusions arise.

First, we note that for both Paul and Epictetus, the occurrence of exhortations and imperatives follows the negative binomial distribution, while the occurrence of second person addresses follows the Poisson distribution. If the reason for this qualitative difference is that exhortations and imperatives have the property of being contagious, it is interesting to note that other works do not follow this pattern. Notably, the occurrence of exhortations in The Damascus Document follows the Poisson distribution, in contrast to the negative binomial distribution as in Paul and Epictetus; the occurrence of second person addresses in Hebrews follows a negative binomial distribution, rather than the Poisson distribution as in Paul and Epictetus.

The contagious behavior of exhortations and imperatives may be a common property to writings located in Paul and Epictetus’ socio-literary sphere. Perhaps because of a similar level of education or similar goals in their writing, these authors have developed certain common habits that make it so one instance of an exhortation or an imperative increase the likelihood for another instance to follow. In contrast, the writers located in a different socio-literary sphere—Aristides or the author of The Damascus Document—received different literary training which produced different use of these literary techniques. Our findings on the differences between authors in Paul’s sphere and those outside of it (e.g., Aristides) indicate that features such as exhortation and imperative are not contagious per se across all types of texts and authors, but rather only for certain authors and texts (e.g., Paul and those in his sphere). In this way, our comparative analysis of distributions has shown that certain literary features behave differently across different socio-literary spheres and are not universally contagious in the same way.

Second, in contrast to the above, we might suggest the possibility that the particular distribution is simply a product of the individual writer’s style, rather than a characteristic typical of a whole socio-literary sphere. This possibility would mean that Paul and Epictetus’ use of exhortations, imperatives, universal claims, and second person addresses exhibit the same qualitative behavior by mere coincidence. Such a coincidence is possible, though in the opinion of the authors it is not persuasive. The more likely explanation, we believe, lies in the shared socio-literary conventions of authors within a given sphere, which in turn lends weight to the theorization of the existence of such spheres themselves.

5. Application to Textual Criticism of Paul’s Letters

In this section, we present an experiment which uses an analysis of Paul’s literary techniques via probability distributions to give insight into the textual criticism of Paul’s letters. Whereas the above sections focused on the distributions themselves and what they might reveal about socio-literary groupings at large, here we focus on more applied issues within Pauline Studies to explore how our work on distributions might help illuminate textual and historical questions specific to Paul’s corpus.

5.1. Theory of Compound Poisson Distributions

We begin with the assumption that the reason certain literary techniques in Paul’s letters follow a negative binomial distribution, rather than a Poisson distribution, is because the text consists of two or more homogeneous sub-populations. Earlier we referred to this distribution as a compound Poisson distribution because each homogeneous sub-population follows Poisson distribution.

In the above discussion on compound Poisson distributions, we mentioned three possible ways that homogeneous sub-populations could arise. In this experiment, we assume the third possibility, in which the homogeneous sub-populations are the works of different authors. That is, we assume that Paul’s letters contain a subset of work written by a different author (or several authors), whose personal style would dictate that their use of a given literary technique would follow a Poisson distribution distinct from the one governing Paul’s own style. When this author’s work was incorporated into the final form of Paul’s letters, the compound Poisson distributions from each author’s personal style gave rise to a negative binomial distribution.

5.2. Experiment on Possible Interpolations

This theory of compound Poisson distributions arising from the work of multiple authors in Paul’s letters suggests a method by which one could test whether a given passage is a non-Pauline interpolation. Suppose a collection of verses in Paul’s letters were not written by Paul, but rather by another author whose style determined that their use of a given literary technique followed a Poisson distribution different from Paul’s. By removing this collection of verses, the remaining occurrences of that literary technique should follow a distribution that more closely fits a Poisson distribution rather than a negative binomial distribution, since one of the homogeneous sub-populations has been reduced (if not eliminated). On the other hand, if the remaining occurrences of that literary technique more closely fit a negative binomial distribution, compared to a Poisson distribution, then we find no evidence that the collection of verses came from a different author.

5.2.1. Two Possible Interpolations

In this section, we identify two collections of verses that have been proposed as non-Pauline interpolations. First, we consider 1 Corinthians 11:2–16 and 14:34–35. These passages address the behavior of women in Christian worship and have been given much attention because of their relevance to the question of women’s role in modern Christian churches. It has been suggested that both passages were not written by Paul. For example, Philip B. Payne argued, based on manuscript evidence, that verses 14:34–35 were not in the original text of 1 Corinthians (Payne 1998). Similarly, WM. O. Walker, Jr. argued that verses 11:2–16 are a non-Pauline interpolation (Walker 1975).

Second, we consider 2 Corinthians 6:14–7:1. Hans Dieter Betz argued that these verses are another non-Pauline interpolation (Betz 1973). He proposed that these verses come from a document written by Paul’s opponents in Galatia, which were included in 2 Corinthians by a later redactor.

Without evaluating the arguments in favor of each theory, we will take up these two hypotheses that 1 Corinthians 11:2–16 and 14:34–35 and 2 Corinthians 6:14–7:1 were not written by Paul. We will test these theories by carrying out our experiment described in the previous section. In addition to testing these theories, we show how quantitative work can help illuminate qualitative issues within Pauline Studies, as well as social, cultural, and historical studies more broadly.

5.2.2. Method

To carry out this experiment, we decided to look at individual letters rather than the entire corpus of Paul’s authentic letters. The reason for this decision was that the passages removed from each letter consist of a small number of verses, and consequently we suspect that the effect of their removal would be too small to detect among the entire undisputed Pauline letter collection. For example, the passage we remove from 1 Corinthians consists of a mere 17 verses, and the passage from 2 Corinthians consists of just six verses. Thus, the verses from 1 Corinthians represent about 1% of the 1468 verses from Paul’s undisputed letters, and the verses from 2 Corinthians represent about 0.4% of those 1468 verses.

We then divided the two letters, 1 Corinthians and 2 Corinthians, into text-blocks consisting of 10 verses. We also removed the hypothesized non-Pauline verses from each letter and divided the remaining text into text-blocks consisting of 10 verses.

Next, we counted how many exhortations occurred in each text-block, for both the full letter and the letter with the verses removed. We chose to focus on exhortations because they were one of the few literary techniques that was not well-modeled by a Poisson distribution. Exhortations were modeled much better by a negative binomial distribution, which suggests that this literary technique may be made up of compound Poisson distributions, representing the work of authors in addition to Paul. It may be instructive in the future to carry out this experiment on additional literary techniques.

We then determine the best-fitting Poisson distribution and negative binomial distribution for the data. That is, we find the values of the parameters, $\lambda$ for a Poisson distribution, and $r$ and $p$ for a negative binomial distribution, that best fits the data set, using the method of Maximum Likelihood Estimation. Finally, we calculate several statistics to help us determine whether a Poisson distribution or a negative binomial serves as a better model for the data.

• ${\chi }^2$-statistic. We calculate the ${\chi }^2$-statistic for each model, Poisson and negative binomial, from a Goodness-of-Fit test. In addition, we calculate the $p$-value for this ${\chi }^2$-statistic. The $p$-value represents the probability of observing data that deviates as much or more from the expected values than our observed data. Thus, a larger $p$-value indicates that the data are consistent with a given model.
• Variance to mean ratio (VMR), $s^2/\overline{x}$. We calculate the sample variance divided by the sample mean as a way to estimate ${\sigma }^2/\mu$, the variance divided by the mean. For a Poisson distribution, ${\sigma }^2/\mu =1$, whereas for a negative binomial distribution, ${\sigma }^2/\mu =1/p$, where $p$ is a parameter of the distribution. Since $0<p<1$, this indicates that ${\sigma }^2/\mu >1$ for negative binomial distribution. Therefore, a value of the VMR close to 1 suggests that the data resemble a Poisson distribution, while a value larger than 1 suggests that the data resemble a negative binomial distribution.
• Skewness. The negative binomial distribution is often considered an over-dispersed Poisson distribution. For this reason, the skewness of a distribution is a good way to distinguish a Poisson distribution from a negative binomial distribution. Generally, a negative binomial distribution has greater skewness than a Poisson distribution.

5.3. Results

In the following sub-sections, we provide the results of this experiment, comparing the observed frequencies of exhortations with the frequencies expected by a negative binomial and a Poisson distribution.

5.3.1. Full Text of 1 Corinthians

Table 38. NB vs. Poisson models for full text of 1 Cor.

Value	Frequency	NB Expected Frequency	Poisson Expected Frequency
0	20	20.64	19.52
1	15	14.09	15.42
2	5	5.77	6.09
3	3	1.84	1.60

below records the observed data for the frequencies of Paul’s use of exhortations, as well as the frequencies expected by the negative binomial distribution with $r=5$ and $p=0.863$, as well as those expected by the Poisson distribution with $\lambda =0.79$.

Table 39. Goodness of Fit Tests for full text of 1 Cor.

	NB	Poisson
$p$-value	96.35%	86.22%

below records the $p$-values of a ${\chi }^2$-Goodness of Fit test with the two possible models. The ${\chi }^2$-Goodness of Fit test was conducted by combining all the expected values for three or more into a single category.

In addition, we record the values of the VMR and skewness below.

$$VMR=1.0574$$

$$skewness=1.0270$$

5.3.2. Text of 1 Corinthians with Verses Removed

Table 40. NB vs. Poisson models for 1 Cor with verses removed.

Value	Frequency	NB Expected Frequency	Poisson Expected Frequency
0	19	20.51	18.68
1	16	13.08	15.13
2	3	5.56	6.13
3	4	1.97	1.65

below records the observed data for the frequencies of Paul’s use of exhortations, as well as the frequencies expected by the negative binomial distribution with $r=3$ and $p=0.788$, as well as those expected by the Poisson distribution with $\lambda =0.88$.

Table 41. Goodness of Fit Tests for 1 Cor with verses removed.

	NB	Poisson
$p$-value	49.34%	32.07%

below records the $p$-values of a ${\chi }^2$-Goodness of Fit test with the two possible models. The ${\chi }^2$-Goodness of Fit test was conducted by combining all the expected values for three or more into a single category.

In addition, we record the values of the VMR and skewness below.

$$VMR=1.0990$$

$$skewness=1.1333$$

5.3.3. Full Text of 2 Corinthians

Table 42. NB vs. Poisson models for full text of 2 Cor.

Value	Frequency	NB Expected Frequency	Poisson Expected Frequency
0	14	13.51	13.18
1	8	8.00	8.44
2	1	2.67	2.70
3	2	0.66	0.58

below records the observed data for the frequencies of Paul’s use of exhortations, as well as the frequencies expected by the negative binomial distribution with $r=8$ and $p=0.926$, as well as those expected by the Poisson distribution with $\lambda =0.64$.

Table 43. Goodness of Fit Tests for full text of 2 Cor.

	NB	Poisson
$p$-value	95.76%	94.35%

below records the $p$-values of a ${\chi }^2$-Goodness of Fit test with the two possible models. The ${\chi }^2$-Goodness of Fit test was conducted by combining all the expected values for two or more into a single category.

In addition, we record the values of the VMR and skewness below.

$$VMR=1.2865$$

$$skewness=1.5463$$

5.3.4. Text of 2 Corinthians with Verses Removed

Table 44. NB vs. Poisson models for 2 Cor with verses removed.

Value	Frequency	NB Expected Frequency	Poisson Expected Frequency
0	15	15.63	13.72
1	6	5.86	8.23
2	3	2.20	2.47
3	1	0.82	0.49

below records the observed data for the frequencies of Paul’s use of exhortations, as well as the frequencies expected by the negative binomial distribution with $r=1$ and $p=0.625$, as well as those expected by the Poisson distribution with $\lambda =0.60$.

Table 45. Goodness of Fit Tests for 2 Cor with verses removed.

	NB	Poisson
$p$-value	95.36%	59.98%

below records the $p$-values of a ${\chi }^2$-Goodness of Fit test with the two possible models. The ${\chi }^2$-Goodness of Fit test was conducted by combining all the expected values for two or more into a single category.

In addition, we record the values of the VMR and skewness below.

$$VMR=1.25$$

$$skewness=1.3388$$

5.4. Experimental Control

To interpret the results of removing passages from 1 Corinthians and 2 Corinthians, we must establish a basis for comparison. Above, we described how our results may compare with theoretical Poisson and negative binomial distributions, especially as seen in the statistics of VMR and skewness. In addition, we must compare how our choice of which passage to remove affects these statistics, so that we can ask whether the effect we observe is more than mere chance variation.

To this end, we randomly chose 10 passages of 17 consecutive verses from 1 Corinthians, as well as ten passages of six consecutive verses from 2 Corinthians. We then removed these passages and grouped the remaining text into blocks of 10 verses and calculated the VMR and skewness. The results are recorded in

Table 46. Control statistics for 1 Cor.

1 Corinthians
Passage removed		VMR	Skewness
1:15–1:31		0.8580	0.6443
4:11–5:06		0.9024	0.6907
6:9–7:5		0.9024	0.6907
7:33–8:9		0.9645	0.9079
8:11–9:14		0.9645	0.9079
10:4–10:20		0.9953	1.0339
10:18–11:1		0.9756	1.1212
12:7–12:23		1.0387	0.9961
13:4–14:7		1.1402	1.0756
13:10–14:13		1.1275	1.1660
	Mean	Median	Standard Deviation
VMR	0.9869	0.9701	0.0931
Skew	0.9234	0.9520	0.1901

and

Table 47. Control statistics for 2 Cor.

2 Corinthians
Passage removed		VMR	Skewness
2:11–2:16		0.9461	0.6707
4:4–4:9		0.9461	0.6707
4:6–4:11		0.9461	0.6707
5:1–5:6		1.0260	0.7819
5:4–5:9		1.0260	0.7819
9:7–9:12		1.2865	1.5463
10:13–10:18		1.3137	1.3625
10:14–11:1		1.3137	1.3625
11:19–11:24		1.2865	1.5432
12:14–12:19		1.3137	1.3625
	Mean	Median	Standard Deviation
VMR	1.1404	1.1563	0.1739
Skewness	1.0756	1.0722	0.3879

below, along with summary statistics describing the distribution of VMR and skewness.

5.5. Conclusions

We begin by summarizing, in

Table 48. Results for 1 Cor.

1 Corinthians
	Full Text	Verses removed
$p$-value Poisson	86.22%	32.07%
$p$-value NB	96.35%	49.34%
VMR	1.0574	1.0990
Skewness	1.0270	1.1333

, the statistics we calculated to help us determine whether the Poisson distribution or negative binomial distribution is the better model for Paul’s use of exhortations, both in the full letter and the letter with verses removed.

We see that removing the verses in 1 Corinthians has significantly reduced the $p$-value of the ${\chi }^2$-Goodness of Fit test for both the Poisson model and the negative binomial model. Consequently, these $p$-values will be of little use in determining whether removing the verses makes the occurrence of exhortations more Poisson-like or negative binomial-like.

However, we see a clearer picture when we examine the statistics of VMR and skewness. Both statistics increase when the verses are removed, which indicates that the occurrence of exhortations behaves more like a negative binomial distribution when the verses concerning women are removed from 1 Corinthians. Furthermore, the removal of several randomly selected passages did result in a VMR closer to 1 than the VMR for the full text. Similarly, removing these verses concerning women in 1 Corinthians caused the skew to increase more than the removal of all but one of the randomly selected passages. This outcome is not consistent with the hypothesis that 1 Corinthians 11:2–16 and 14:34–35 are a non-Pauline interpolation.

Our results for 2 Corinthians are summarized above in

Table 49. Results for 2 Cor.

2 Corinthians
	Full text	Verses removed
$p$-value Poisson	94.35%	59.98%
$p$-value NB	95.76%	95.36%
VMR	1.2865	1.25
Skewness	1.5463	1.3388

. Removing the six verses from 2 Corinthians decreased the $p$-value of a ${\chi }^2$-Goodness of Fit test with a Poisson model, while the $p$-value for a negative binomial model increased. Nevertheless, all $p$-values remain quite high, which means, once again, these $p$-values will be of limited use in determining whether the occurrence of exhortations has become more Poisson-like or negative binomial-like.

Both the VMR and skewness statistics decrease when these six verses are removed. Removing several of the randomly selected passages also resulted in a text whose VMR and skew decreased. Nevertheless, these texts had a VMR and skew which do not suggest that the text became more Poisson-like. Of the five randomly selected passages whose VMR and skew decreased, three of them have a VMR below 1, which is not consistent with Poisson distribution, and all five have a skew that falls much below 1, which is also inconsistent with Poisson distribution. For these reasons, those five texts constructed by removing the randomly selected passages did not become clearly Poisson-like. Of the remaining five texts, their VMR and skew increased, indicating that the text became more negative binomial-like. In comparison, removing verses 6:14–7:1 resulted in a text that is distributed slightly more like a Poisson distribution than our original text.

It is striking that none of the texts produced by removing randomly selected passages produced this same change in VMR and skew. Therefore, this change suggests that the occurrence of exhortations becomes more like a Poisson distribution when verses 6:14–7:1 are removed. This finding is consistent with the hypothesis that these verses were not written by Paul.

6. Conclusions

We can highlight two sets of conclusions, one from our broader empirical work across the preceding chapter and one from our specific exploration of 1 Corinthians. On the former, we have provided a novel method to understanding the distribution of stylistic features in Paul’s letters. Both quantitatively and visually, we have thus contributed a new form of descriptive understanding of what we might term the “shape” of Paul’s letters. Because previous quantitative work on Paul’s letters so rarely discusses the notion of distributions at all, this work has highlighted the function of mathematical distributions in understanding this corpus.

Further, we showed that other forms of non-normal distributions are effective models for features of Paul’s letters, namely Poisson distribution and negative binomial distribution. Because the rare work on statistical distributions in biblical literature assumes normal distributions, either explicitly or implicitly, this is an important finding in that it illustrates that we need to explore and analyze other forms of distributions in both traditional stylometry (vocabulary words’ frequency and location) and in our second-order feature selection (stylistic characteristics selected by subject experts). There is more work to be conducted here, and we hope that future work expands and improves upon our findings here.

More narrowly, we found that sometimes a Poisson distribution was a better fit for certain features, while at other times, our features were better modeled by a negative binomial distribution. At its face, this demonstrates that Paul’s letters are complex and not reducible to simple models. Quantitative approaches are sometimes criticized as reductionist, and such findings demonstrate that properly sophisticated empirical approaches illustrate the non-reductive explanatory complexity of statistical modeling.

Concerning the texts and features themselves, furthermore, we found that Paul’s letters and Epictetus tended to overlap in terms of which types of distributions were the best models. This finding supports the hypothesis, seen in previous work, that Epictetus’ Discourses should be considered part of what was termed a shared “socio-literary sphere”. Stated more simply, Epictetus’ Discourses is a better comparandum than other texts sometimes used to try and explain the stylistic structure of Paul’s letter, such as so-called Jewish apocalyptic (Damascus Document) or formal Greco-Roman oratory (Aelius Aristides’ Panathenaicus). Indeed, the latter text was so poorly modeled by both forms of distribution considered here (and accurately applied to Paul’s letters and other comparanda) that we should strongly question the use of such comparanda in anything beyond general comparison studies with Paul’s letters.

We can now move onto our latter set of findings, namely our specific case studies of 1 and 2 Corinthians. There, we applied our distributions to attempt to test an open question in Pauline Studies, namely whether certain passages from the Corinthian correspondences were later interpolations. As these passages pertained to the role of women (1 Cor 11:2–16 and 14:34–35) and idolatry (2 Cor 6:14–7:1), our goal was to provide some quantitative bases for otherwise qualitative discussions on important historical and theological questions. In such applications, the goal of quantitative work is to provide a form of triangulating evidence, which can support or undermine arguments made on the basis of textual, historical, and/or theological grounds. Triangulating evidence such as this is non-definitive, and we make no claims of reducing these complex debates to one or more sets of numbers, but rather that such statistical studies can help advance these debates by providing fresh perspectives and empirical forms of evidence around what can sometimes be conflicting matters of subjective interpretation.

Our approach consisted of removing the verses in question and then comparing various resultant statistical outcomes within our models. In the case of 1 Cor 11:2–16 and 14:34–35, this method did not lend support to the hypothesis that these were non-Pauline interpolations. Such passages were stylistically consistent—in the sense of our distributions analyses—with the rest of the letter. Of course, it is entirely possible that a non-Pauline interpolation was able to sufficiently mimic Paul’s style; other explanatory hypotheses are also possible. But from the perspective of our distributions analyses, our findings lend weight to the position that these passages are authentically Pauline.

In the case of 2 Cor 6:14–7:1, meanwhile, we found the opposite, namely that the removal of these verses improved the accuracy of our distribution modeling elsewhere. In other words, these verses were sufficiently non-Pauline according to their distribution such that their removal was statistically detectable. Again, we should not interpret this as definitive, and multiple explanations are possible such as noting that these verses contain a variety of specific references to the Hebrew Bible. In that sense, such findings merely demonstrate that Paul’s letters appreciably differ from the Hebrew Bible, which is obvious and unsurprising. But Paul’s letters contain a host of such references elsewhere too, so our findings minimally direct us to re-consider the possibility that these verses are non-Pauline. Again, we highlight in closing that such work is intended to triangulate work in other non-quantitative areas: to help validate points of consensus, to help test various hypotheses, to weigh in on active debates and open questions in the field, and to suggest new questions and lines of analysis—all using a different form of evidence than is traditional, namely empirical data.