Subject: Literary statistics for dummies (formerly Replication of strings

From: "Vinton A. Dearing" <[email-masked]>
Date: Wed, 27 Aug 1997 10:37:49 PST

I should like to enter a general plea that discussions of  literary
statistics on the list be more oriented to statistical dummies like 
myself.
     If we are to apply mathematics to language in the general 
sense of both terms we need to be sure that the formulas we choose 
fit the circumstances to which we apply them. With respect to 
language as it is used in literature, while it is true that there are 
many ways to describe an event or express a rule, an idea, or an 
emotion, still there are definite constraints on the details of 
description or expression--constraints of  grammar, vocabulary, 
and style, at a minimum--which allow us, for example, to recognize 
mistakes, plagiarisms, and editorial interventions. Thus, for 
example, to say that verbal parallels between the Gospels should be 
rarer as they are longer if they are purely the result of chance is 
not enough by itself. The shorter parallels are not all the result of 
chance but are conditioned at a minimmum by the limitations of 
language, and this is true for sections within the longer parallels. 
We need to remember also that in a string of words one or more may 
condition the next or some of the next if the words are to convey 
meaning to the average reader/istener, and that (s)he may sooner or 
later be aware of what is to come: in Latin and Milton, if we find no 
verb before what appears to be the object of a verb we expect to find 
it at the end of the clause or sentence. In addition, if precise 
mathematical reasoning is to be introduced, the procedures must not 
only fit the circumstances of language, they must be multilayered. As 
a dummy in these matters, I cannot name the various procedures, but 
there is one, for instance, that has been used to predict how 
sentences can be expected to develop, and there is another combining 
the statistics reached for various language characteristics.
     Statistical dummies like myself need an explanation of any 
statistical term except "average" -- even "median" requires some 
thought on my part. I remember a story told by a History Professor at 
Columbia. He had two teachers of mathematics. One would write a 
problem on the board, make a grinding noise, and write the answer. 
The other proceeded through the solution step by step, each time 
saying, "What does this gimme?" Statistical dummies are always 
asking, "What does this gimme?" and they need answers in 
non-mathematical language. And finally, they need to be shown how 
the procedures employed fit the circumstances analyzed. A procedure 
that will allow you to estimate whether you should stop processing 
hamburger meat, destroy what you have processed and, if you have 
sold any of the batch, spread the alarm is not necessarily a 
procedure that will allow you to estimate whether two witnesses to 
the word of God are not independent, reject one or both and spread 
the alarm. Maybe it is, but the fact that it is needs to be 
explained, not merely asserted. Even a parallel experiment cited as 
proof that the procedure is correct needs to be explained in detail 
to us dummies.
     I say this not only of initial messages to the list but to 
replies. While I was looking for the name of  the procedure that has 
been used to estimate the expectations generated as word follows 
word, which I read in "Literary and Linguistic Computing," I came 
across a review of a publication of "our own" Scholars Press, Bernard 
Frischer's "Shifting Paradigms. New Approaches to Horace's `Ars 
Poetica,'" in Vol. 11, no. 1 (1996), p. 47. The review is by 
G.R.Ledger of the University of Reading in England, an institution 
for which I have a great deal of respect, and this review has given 
me great respect for Ledger. I quote: "The general reader, I suspect, 
would find that Chapter 2 on the stylometric attempt to date the poem 
is, as Huckleberry Finn found parts of the Gospel to be, `interesting 
but tough.' It could be made clearer, for example, that the ratio of 
unique strings to all strings is simply a measure of vocabulary 
richness, otherwise known as the type/token ratio, one among several 
metrics of vocabulary which might have been used (p. 42). In fact it 
is a measure which is dependent partly on text length, and this 
intrusive factor is not accounted for. A simple statement in the body 
of the text could easily have cleared up much avoidable confusion 
about strings and lexemes. Table P, derived from this vocabulary 
measure, should state that the two figures sum to unity, i.e., they 
are not independent variables. This may be obvious to an experienced 
statistician, who would in any case probably not include figures for 
non-unique strings, since they add nothing to the information already 
available from the percentage of unique strings. I should also like 
to have seen at the outset a clear Table displaying the accepted 
dates of each work [analyzed in determining the date of Horace's 
poem], rather than having to grub among the footnotes for this 
information (p. 28). Not everyone has this knowledge in their back 
pocket as it were. However, this is not of major importance. It is 
encouraging to see a distinguished scholar willing to use the 
analytic methods of statistics and stylometry." Bravo, G.R.Lederer, 
though clearly no dummy yourself you speak for those of us who are, 
and with a geniality befitting a lover of mankind.
     By the way, I recommend that all those who wish to undertake or 
have undertaken statistical studies of the Scriptures read through 
back issues of "Literary and Linguistic Computing," and subscribe to 
it. If your institutional library does not have copies, get them on 
interlibrary loan and ask the library to subscribe, or contribute 
your copies to your library. It is expensive, US$145 a year, and I 
sometimes wonder why I spend the money, since much of what it 
contains does not interest me, and sometimes I do not open the 
issues at all, but I save them, because they are full of useful 
information for later perusal. I found Ledger's review before I 
found the article on estimating sentence development and went no 
farther, since it made the point I wished to make. Laus Deo, my 
eye lit on it almost at once.
     Vinton A. Dearing

---

From: "Robert B. Waltz" <[email-masked]>
Date: Wed, 27 Aug 1997 13:29:21 -0500 (CDT)

>statistics on the list be more oriented to statistical dummies like 

[ etc. ]

Tangentially, I would like to point out that the web site below
has an article on mathematics. And whatever certain people out
there say about my skills as a textual critic, I *am* a trained
mathematician.

But I wrote that page somewhat casually. I included most of the
mathematical subjects I regularly work on -- but I don't know
what others need. I can't guarantee I'll be able to supply what
you need; I don't know everything (my geometry, for instance,
is rather pitiful -- not that I can imagine a textual critic
needing to know geometry). And I can't promise to make everything
clear -- as it is, the material already in place walks the
fine line between being too-simplistic-to-be-true and being
too difficult to understand. (Actually, I probably have missed
on both sides of the line.) But if people have questions, I
will try to accommodate them.

Eventually. :-)

-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-

                        Robert B. Waltz
                     [email-masked]

Want more loudmouthed opinions about textual criticism?
Try my web page: http://www.skypoint.com/~waltzmn
(A site inspired by the Encyclopedia of NT Textual Criticism)

---

From: Jim Deardorff <[email-masked]>
Date: Wed, 27 Aug 1997 16:35:05 -0700 (PDT)

On Wed, 27 Aug 1997, Vinton A. Dearing wrote:

> mistakes, plagiarisms, and editorial interventions. 

I hope this discussion doesn't drift too far away from the key point I was
raising, which is that in two independent editings or translations of a
given document, the actual frequency distribution (or histogram if you
like) for numbers of strings of duplicate successive words as a function
of the string's word length, is observed to follow the exponential
distribution. 

Knowing this allows one to judge much better whether a possible plagiarism 
is an actual plagiarism or is just a matter of chance or coincidence.

The jury has been out for a long time on whether the duplication of the
longer word strings found in parallel passages of Mt-Lk (Q verses) is a
result of plagiarism by one or the other evangelist, or not.  Now we have
a procedure for deciding, and if it is not a plagiarism, then some other
mechanism that can mimic it needs to be proposed.  Hence I was interested
in opinions on whether scribal assimilation could likely account for it,
or maybe even the method by which the standard text (N-A) has evolved. 

> Thus, for 
> it at the end of the clause or sentence.

And one must try to include all the other factors that would condition the
independent editor's choice of words -- intended improvements of text and
of theology being extremely important and having a large variety of
different means of implementation.  So it's not surprising if the central
limit theorem takes over, causing the end result to look as if chance had
been at work.  So why not examine the simplest statistical result that can
explain the observations?

Previously I described how the analogy is relevant of how many black balls
in a row one draws out of a mixed bag of 1000 black balls and red balls.
The distribution of numbers of strings of black balls drawn in sequence
turns out to mimic the results of the test case I analyzed in the
distribution of verbal parallels between Ezra and 1 Esdras, if the bag
contains 577 black balls.  An exponential distribution occurs in each
case, with the same fall-off rate and degree of scatter. 

Today a friend of mine ran a computer program for me wherein this analogy
is extended so that the conditional probability in each draw of a ball out
of the bag is not a fixed value (0.577) but itself varies, as I had
suggested yesterday. I let it vary randomly (and uniformly) within the
interval 0.254 and 0.9, the linear average still being 0.577.  The idea is
to simulate the editor who, in deciding, e.g., what his 4th word would be
after a duplicate string of 3, would in one instance have a probability of
0.8 or 0.9 of giving it the same value as was in the text he was editing,
but in another instance only a probability of 0.3 or 0.4 of duplicating
the word.  And same for all the other words he lays down -- different
conditional probabilities depending upon the particular sentence he's in
and depending upon his mood (or the mood he wishes to express it in), etc. 
Again the distribution came out exponential -- on the average, the longer
the string, the fewer of them in the string.  The number of strings of
length I+1 again tended to be 0.58 of the number of strings of length "I".

So I see the exponential distribution as being a very robust solution for
this problem, and fitting the circumstances of any complicated language
having many degrees of expression.  Greek is especially relevant since
word order is exceptionally open to the editor's option.

> In addition, if precise 

I see only three "multilayering" effects that need to be paid attention to
with this exponential frequency distribution: 

1) Don't expect it to necessarily hold well for duplicate word-string
lengths as short as 2 and 3;

2) Expect sampling error (scatter of the data) to become quite evident for
word-string lengths exceeding 10 when the total number of words in one set
of parallel verses is around 9000, or when the total number of words in
duplicated strings of two or longer is around 3400.  For shorter data
bases the sampling error becomes evident for smaller values of word-string
length, "I".  However, the results continue to hold statistically for this
and all higher values of "I", for independent editing, and in no way
become useless from this point (e.g., I=10) on.

3) Allow the exponential fall-off factor b [e.g., Y(I)=A*exp(-b*I)], to
vary a little from one study to another depending upon what language is
involved, who the editor is and what his motives are.  The key point,
however, is that a single value of "b" will prevail in any particular case
of independent editing.  But what we see in the Gospel verbal parallels is
a radical departure (excesses) from exponential for the longer strings,
indicating purposeful duplication or plagiarism somewhere along the line.

> thought on my part. ...

Vinton, please don't keep referring to yourself as a "dummy!"  I'm a
"dummy" when it comes to tc or to the proper choice of words with which to
describe things involved with textual criticism.  E.g., I'm glad to adopt
your phrase, "verbal parallels," for which I've been using something more
cumbersome, to indicate identical (or nearly so) Greek-word parallels as
opposed to just parallels in the text's meaning.  And I'm no statistician
either, having only brushed up against pieces here and there in my former
career as a researcher in the atmospheric sciences.

> I remember a story told by a History Professor at Columbia. ...
> A procedure 
> the alarm.

There's nothing new in the latter, of course.  From the time of Irenaeus,
Origen and Augustine, and no doubt earlier, the tradition was spread of
the Mt-Mk-Lk dependence.  And I'd say that Papias inferred a dependence of
Mark upon Matthew in his statement of how Mark has verses out of order. 
And even the modern consensus allows two gospels to be dependent upon
Mark.  Advocates of these, plus Griesbachians, have long since spread
various alarms of lack of independence. 

> Even a parallel experiment cited as 

I hope I've come closer to doing this now.  Otherwise you'll either need
to wait until my paper comes out (if it does), or I could e-mail you its
Part II, which contains the data analysis and exponential-distribution
fits, upon individual request.  You can understand that it's not easy to
get a paper into print of a new procedure (if it's indeed new) by an
independent scholar whose previous career was not related to biblical
studies.

> .... While I was looking for the name of the procedure that has 
> G.R.Ledger of the University of Reading in England...

If you come across a name for the procedure, please let me know! 
Otherwise, if one doesn't exist, perhaps someone on this list could
suggest one.  So far I'm stuck with "The Expected Frequency Distribution
of Duplicate Strings of Words in Parallel Passages due to Independent
Editing or Translating."

> ... I quote: "The general reader, I suspect, 
> intrusive factor is not accounted for.

I could mention that in this exponential frequency-distribution approach,
the length of text involved in the parallels is more or less proportional
to the constant A in Y(I) = A*exp(-b*I)  .  You want A to be as large as
possible (plentiful data) so that the resulting exponential curve can be
pinned down as well as possible in the 3<I<10 range, and so that its
extension to all larger values of I will also be valid, for indicating the
expectations if the text had been independently edited. 

> it. ...

I am additionally a dummy when it comes to knowing what books or journals
on statistical methods within biblical scholarship might have anything
relevant to say on the exponential distribution for duplicate strings,
which one can observe (out to a certain word length even for the Gospels) 
upon doing the analysis.  Hence I'm testing out this list on this subject. 
If anyone becomes aware of such, I'm the one who most needs to know at the
moment, and then, if it isn't too late, I could perhaps add a reference,
in my paper, as to who was the first to observe this.  As it stands, I'm
not aware of such, and thus have not been able to supply any of you with
any literary reference about the procedure. Perhaps the motivation never
came up before for someone to plow through all duplicate word strings in
relevant parallel passages instead of just noting 10 or 15 of the longer
ones, and then plot their frequency distribution, etc.; it's a good deal
of work, but an ideal project for a graduate student to work on!  I doubt
that it could be safely computerized, as each potential parallel seems to
need careful scrutiny to make sure which words & phrases are really
parallel. 

Jim Deardorff
Research Professor emeritus
Oregon State University