ShareWare OnLine 2/ShareWare OnLine Volume 2 (CMS Software)(1993).iso/infor/timlin3d.zip/SONSGOD.DOS

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                                             Reality Software
                                             P.O. Box 105
                                             Waldoboro, ME 04572
                                             May 4, 1992



                          THE SONS OF GOD


     As briefly discussed in timln3dc.dos, perhaps the most
fascinating and mysterious observation in the timeline data (for
those of us mystically inclined) is the fact that six of seven
Sons of God (Zoroaster, Siddharta Gautama, Vardhamana, Confucius,
Jesus Christ) were born within a seven century span of time from
(approximately) 700 B.C. to 0 A.D.  Only Mohammed's birth lies
outside this time period.  A priori, it is difficult to imagine
why such births would not be spaced throughout the 5,000 years of
recorded history and that the pattern would be essentially
random.  To postulate otherwise, assumes something "special" or
unusual about the period 700 B.C. to 0 A.D. that never ocurred
earlier in history and has never been duplicated since.  In
effect, such a hypothesis would placed the label 'unique' upon
this seven century period of time.
     The clustering of the Sons of God within this time frame has
been noted in many history books, but it is given only passing
mention.  Hypotheses would seem to lurch off into the realm of
the occult and what self respecting historian would wish to enter
that territory?  What circumstances, environmental or otherwise,
might foster the appearance of what I shall term "deity beings"?
My speculation upon such matters will not be presented here.
Rather, I have a more modest goal for this document.

     I shall subject the data to a fairly rigorous statistical
analysis, if only to demonstrate in quantitative terms how
improbable is this clustering of deity beings within the period
700 B.C. to 0 A.D.  The much maligned and often hated statistical
tools have real tactical value.  They serve to confirm (or
refute) our intial intuition and observations that are based upon
a quick visual scan of the pattern in the data.  A quick scan, of
the timeline may be misleading in the sense that subconsciously I
might "wish" to see this clustering, particularly when my study
has primed me to look for it and I "want it to exist". The
serious question is whether or not the apparent clustering meets
scientific criteria of extreme unusualness or improbability.  My
eyes, and yours as well, may be perceiving a trend that is of no
significance.
     I will go slowly with full explanation. In order to provide
you, dear reader, with the motivation to continue, I shall tell
you the conclusion in advance.  From a strict statistical point
of view (which is the only scientific methodology that is
acceptable), the clustering of six Sons of God within the first
seven centuries B.C. is extremely improbable, so much so as to
deserve the label "unique".  It cannot be explained within the
context of the usual (i.e. "normal") distribution of events
presumed to occur at random. This analysis is designed for the
skeptics and is a preliminary exercise that clears the way for an
attempt at a mechanistic explanation for this extraordinary
clustering of deity being births.

     These data are very simple from a mathematical point of view
but the statistical tools that will be employed to analyze them
while not advanced computationally, are complex and subtle
conceptually.  These data lend themselves to cross tabulation in
a simple 2 x 2 table.  In statistics the organization of a body
of data determines the tools that may be appropriately applied in
the analysis.  Cross tablulated data are very common, a battery
of statistical tools is at hand; indeed entire books have been
written on the analysis of data in this form.

     Case 1 will present two closely related tables.  Table 1A
tabulates the presence and absence of Sons of God over the 4,000
years of recorded history documented in the timeline.  Table 1B
presents such observations for the 5,000 years of recorded
history to date; i.e. from c. 3,000 B.C. to c. 2000 A.D.


Case 1: Presence and Absence of Sons of God

     Table 1A: SONS OF GOD: Presence-Absence Analysis

                               SGby                 SGab

    700 BC to 0 AD               6                   694

  c.3000 BC to 900 AD            7                  3993



         Table 1B: SONS OF GOD: Presence-Absence Analysis

                               SGby                 SGab

    700 BC to 0 AD               6                   694

  c.3000BC - c.2000AD            7                  4993

where:

     SGby = Sons of God birth years; and

     SGab = Non-birth years for Sons of God; i.e. years within
     the designated time interval in which births of a Son of God
     did not occur; i.e. were absent.

     Several statistics for these table were computed using
Statgraphics 5.0, a state of the art statistical software
package.  All statistics supported the conclusion of extreme
improbability.  In this and all succeeding tables in this
analysis, there is one degree of freedom, i.e. only one dimension
for data independence.  For Table 1A,the contigency coefficient C
= 0.04619.  This is a measure of association between two
attributes and it is very low here, 4.6% to be precise, and of no
significance in either the strict statistical sense or the
everyday meaning of the word.  The pattern of births for the
Sons of Gods in the time interval 700 B.C. to 0 A.D. is
completely unlike that over the entire 4,000 year span of the
timeline.  For Table 1B where all of recorded history is
examined, C = 0.049 and the conclusion is the same.  Strict
significance levels for cross tabulated data are usually assessed
from the Chi-square statistic (X2).  For Table 1A, X2 = 10.05 and
the significance level (p) is 1.52 x 10-3. A variation in
computation of the X2 known as the Yates correction takes into
account extremely small sample sizes, a situation that we have
here for SGby and SGab. X2y = 7.73 with p = 5,43 x 10-3 for Table
1A.  For Table 1B, X2 = 13.88 with p = 1.95 x 10-3 and X2y = 10.91
with p = 9.59  x 10-3.  Fisher's Exact Test assesses the
probability that the data in the four cells of the table should
be exactly as they are, assuming the total number of observations
for the entire table is fixed (held constant).  Further
assumptions are that the classes for categorizing are mutually
exclusive (they certainly are here - either a deity being is born
in a given year or is not); and that the underlying distribution
of the data while exhibiting a peak and two 'tails' need not be
taken to be 'normal' (i.e. is not the familiar bell shaped
curve).  For our purposes, this a particularly powerful
statistical tool because no assumptions are allowed about the
underlying distribution that characterizes the data.  For
theoretical reasons, it is not applicable to Case 1 but will be
applied to Cases 2 and 3.


Case 2: Sons of God: Model I for Expected Births

     Routine formula based upon theoretical statistical models
allow for the easy computation of expected values of any
observation in data that are cross tabulated.  The expected
values caculated by this procedure are rounded off to the nearest
whole integer; certainly fractional people or deity beings make
no sense and could not exist.

        Table 2A: Model I for Expected Births-Timeline only

                               SGob                 SGex

    700 BC to 0 AD               6                    2

  c.3000BC - c.900AD             7                   10

    Table 2B: Model I for Expected Births-all recorded history

                               SGob                 SGex

    700 BC to 0 AD               6                    2

  c.3000BC - c.2000AD            7                   11

where:

     SGob = observed births of Sons of God (= SGby in Case 1)

     SGex = expected births of Sons of God for time interval as
     computed by the standardized statistical approach.

     In Case 2, we are assessing the degree to which the expected
numbers differ from the observed.  This is a very direct
approach.  Unlike Case 1, the question of interest may be asked
directly.  Are there more Sons of God in the time interval 700
B.C. to 0 A.D. than would be expected assuming their existence to
be spread evenly throughout recorded history?
     For Table 2A, which tabulates the data for the interval
represented by the timeline, C = 0.30. X2 = 2.49 with p = 0.114
and Xy2 = 1.32 with p = 0.250.  For Table 2B, which tabulates the
data for all of recorded history, C = 0.32. X2 = 2.89 with p =
0.089 and Xy2 = 1.63 with p = .202.  While the Chi-square values
are not nearly as high as in Case 1 and the corresponding p
values are greater, there is still a firm case for the
nonidentity of the two rows of data and the extreme improbability
of the observed births of Sons of God.  For Table 2A, Fisher;s
Exact Test gives a p= 0.125 for the one tailed test and p = 0.202
for the two tailed test.  For Table 2B, Fisher's Exact Test gives
a p = 0.100 for the one tailed test and p = 0.202 for the two
tailed test.  The conclusion is that the distribution of Sons of
God birth years as tabulated is very unusual; the probability
that it would fall out as observed is, at most, 20%.


Case 3: Sons of God: Model II for Expected Births

     In the second model for expected births, the expected number
of Sons of God per time interval is computed from an hypothesis
of total randomness; i.e. such births are equally probable in any
year.

         Table 3: Model II of Sons of God: Expected Births

                               SGob                 SGex

   700B.C. to 0 A.D.             6                    1

   c.3000BC - 900AD              7                    7

     The expected numbers calculated under this hypothesis are
the same if the entire span of recorded history c.3000 B.C. to c.
2000 A.D.is considered and therefore this one table encompasses
both scenarios.  For this model, C = 0.33, X2 = 2.53 and p =
0.112 while Xy2 = 1.24 with a p = 0.267.  Clearly, there is a
very significant difference between the observed number of deity
being births in each time interval and the number one might
expect asssuming complete randomness of such events.  Fisher's
Exact Test yields p = 0.133 for the one tailed test and p = 0.174
for the two tailed test thereby confirming the highly improbable
distribution of observations as seen in Table 3.

     Each of the three tabulations of data and their respective
analyses provide the same conclusion.  These statistical tools
confirm in a rigorous manner what our intuition suspected upon
casually glancing at the timeline.  The number of Sons of God
clustered in the interval 700 B.C. to 0 A.D. is truly
extraordinary, exceptional and improbable!


                            References


Siegel, S. 1956. Nonparametric Statistics for the Behavioral
     Sciences. New York: McGraw-Hill.

Simpson, G.G., A. Row and R.C. Lewontin. 1960 rev.ed.
     Quantitative Zoology. New York: Harcourt, Brace & World.

Sokal, R.R. and F.J. Rohlf. 1980 2nd ed. Biometry. San Francisco:
     W.H. Freeman.

SGC. 1991. Statgraphics Version 5.0. Reference Manual. Rockville,
     Md: STSC, Inc.