Statistical Appendix

ON CORRELATIONS: If a correlation were perfect—for example, if all port cities had Christian congregations before any inland city did—then the correlation between these two variables would be 1.0. If there were no correlation between the two—that is, if they varied randomly with respect to one another—then the correlation would be 0.0. Hence, the closer to 1.0, the stronger the correlation. However, in the real world, correlations of 1.0 are rare. When working with data such as these, a correlation of 0.4 is quite respectable.

Table 3-1: Travel and Christianization
	Port cities	Inland cities
Had a church by 100 CE	164%	124%
Had a church by 180 CE	122%	141%
No church by 180 CE	114%	135%
	100%	100%
	n =(14)	(17)
r = .430** V = .413* ? = .598**

By comparing across the first row of the table we can see that most port cities (64 percent) had a church by the end of the first century, while few inland cities (24 percent) had a church that soon. Or, reading across the bottom row, we see that only 14 percent of port cities still lacked a church in 180, while many inland cities (35 percent) still were without a church by that year. These are very substantial differences.

Now look at the correlation coefficients, shown at the bottom of the table. There are many ways to calculate correlations, and specialists disagree as to which is best for what sort of data. To forestall such concerns, the three most appropriate measures are reported here. The first, known as Pearson’s product-moment correlation (r), is an extremely robust but conservative measure. The second is Cramer’s V. Many experts recommend V because it does not assume the data are ordinal—in other words, that the cases can be ordered on each variable. The variables used in this book meet minimum standards of ordinality (some cities were Christianized sooner than others, for example, while ports had more travelers than did inland cities). But, rather than endlessly pursue the matter, it is sufficient to provide the coefficients for V, except when the analysis involves 2 × 2 tables, one in which each of two variables has two values. In such cases r and V produce precisely the same value. The third measure is gamma (? ). It is a less conservative measure than either r or V, but it has valuable properties when data have limited variability—that is, when they take only a few values. Gamma tends to yield coefficients that are far higher than either r or V, and on 2 × 2 tables it will go to unity (1.000) whenever there is one empty cell.

As shown below the table, all three of these correlations strongly support the hypothesis. Port cities did tend to be Christianized sooner than inland cities. Notice that the correlation as measured by r (.430) and the correlation as measured by ? (.598) both carry two asterisks, while the correlation as measured by V carries only one. The asterisks indicate statistical significance: the odds that a correlation this large, and based on this number of cases, could not have occurred by random chance. One asterisk means the odds against a chance finding are at least 20 to 1, known as the .05 level of significance. Two asterisks means the odds are at least 100 to 1, or the .01 level of significance. It is not possible, even when analyzing the same set of data, to determine a threshold of significance from one table to the next. Thus, for example, a V of .413 is significant in the table above but would not be significant on some others—since significance has to do with the distribution of cases as well as the value of a correlation. It also is the case that gamma tends to require a much higher coefficient in order to achieve significance.

There is a technical dispute among statisticians as to whether significance is meaningful when the data are not based on random samples. These 31 cities were not selected randomly from among all Greco-Roman cities; rather, the set consists of all cities above 30,000 population. Even so, an excellent case has been made for using significance as a guide to whether or not a correlation is sufficiently large to be meaningful, and that is how significance is to be interpreted here.1 Correlations falling short of the .05 standard (one asterisk) will be dismissed as trivial.

Table 3-2: Distance and Christianization
	Within 1,000 miles of Jerusalem	More than 1,000 miles from Jerusalem
Had a church by 100 CE	71%	7%
Had a church by 180 CE	29%	36%
No church by 180 CE	0%	57%
	100%	100%
	n =(17)	(14)
r = .744** V = .744 ** ? = .950**

Table 3-3: Hellenism and Christianization
	Very Hellenic cities	Less Hellenic cities
Had a church by 100 CE	63%	8%
Had a church by 180 CE	37%	25%
No church by 180 CE	0%	67%
	100%	100%
	n =(19)	(12)
r = .733** V = .767** ? = .928**

Table 3-4: City-Size and Christianization
	Larger cities	Smaller cities
Had a church by 100 CE	75%	30%
Had a church by 180 CE	25%	35%
No church by 180 CE	0%	35%
	100%	100%
	n =(8)	(23)
r = .430* V = .431* ? = .778**

There are only a few numbers of interest in a regression analysis; the others are there for concerned specialists. Among these are the standardized betas. These are somewhat like correlations, except they reflect the effect of each variable independent of the other. That is, .681 represents the impact of Hellenism on Christianization with the effect of city-size removed, and .317 shows the impact of city-size on Christianization with the effect of Hellenism removed. While Hellenism has the greater effect, both variables have a significant independent effect. To see if a particular beta is significant, look to see if the t value is marked by asterisks. The other number of importance is the Multiple R-Square. This is the joint effect of Hellenism and city-size on Christianization: their combined effect. In this example, there is a very large effect (.635), and together these two variables account for 40 percent of the variation in Christianization (.635 × .635 = .403).

Some might suggest the use of dummy variables here on grounds that city-size and Hellenism are not ordinal variables. But surely city-size is ordinal, even if it takes only two values (larger and smaller). Hellenism too is a matter of degree: there was a significant amount of Hellenic influence in all Greco-Roman cities, but some had much more than others. Given the small number of cases (thirty-one) involved here, I usually limit use of regression analysis to three variables; in several instances I use four variables, but only with extreme care and caution. For those readers with specialized statistical concerns, be assured that use of logistic regression did not alter any outcomes.

Table 4-1: Ports and Cybelene Temples
	Port cities	Inland cities
Had a Cybelene temple	57%	12%
No Cybelene temple	43%	88%
	100%	100%
	n =(14)	(17)
r = .483** ? = .818**

Table 4-2: City-Size and Cybelene Temples
	Larger cities	Smaller cities
Had a Cybelene temple	63%	22%
No Cybelene temple	37%	78%
	100%	100%
	n =(8)	(23)
r = .382* ? = .714

Table 4-3: Cybelene Temples and Christianization
	Cybelene temple	No Cybelene temple
Had a church by 100 CE	80%	24%
Had a church by 180 CE	20%	38%
No church by 180 CE	0%	38%
	100%	100%
	n =(10)	(21)
r = .546** V = .556** ? = .875**

Table 4-4: Hellenism and Isiac Temples
	Very Hellenic cities	Less Hellenic cities
Had an Isiac temple by 100 CE	74%	25%
No Isiac temple by 100 CE	26%	75%
	100%	100%
	n =(19)	(12)
r = .477** ? = .787**

Table 4-5 : Ports and Isiac Temples
	Port cities	Inland cities
Had an Isiac temple by 100 CE	93%	24%
No Isiac temple by 100 CE	7%	76%
	100%	100%
	n =(14)	(17)
r = .693** ? = .954**

Table 4-6: City-Size and Isiac Temples
	Larger cities	Smaller cities
Had an Isiac temple by 100 CE	75%	48%
No Isiac temple by 100 CE	25%	52%
	100%	100%
	n =(8)	(23)
r = .239 ? = .532

Table 4-7: Isiac Temples and Christianization
	Had an Isiac temple by 100 CE	No Isiac temple by 100 CE
Had a church by 100 CE	65%	14%
Had a church by 180 CE	29%	36%
No church by 180 CE	6%	50%
	100%	100%
	n =(17)	(14)
r = .583** V = .583** ? = .815**

Table 4-8: Temples to Cybele and Isis
	Had a temple to Cybele	No temple to Cybele
Had an Isiac temple by 100 CE	90%	38%
No Isiac temple by 100 CE	10%	62%
	100%	100%
	n =(21)	(10)
r = .488** ? = .872**

Table 5-1: Distance and the Diaspora
	Within 1,000 miles of Jerusalem	More than 1,000 miles from Jerusalem
Had a significant Jewish community	47%	7%
No significant Jewish community	53%	93%
	100%	100%
	n =(17)	(14)
r = .438** ? = .841**

Table 5-2: Ports and the Diaspora
	Port cities	Inland cities
Had a significant Jewish community	50%	12%
No significant Jewish community	50%	88%
	100%	100%
	n =(14)	(17)
r = .419** ? = .765**

Table 5-3: City-Size and Diasporan Jewish Communities
	Larger cities	Smaller cities
Had a significant Jewish community	50%	22%
No significant Jewish community	50%	78%
	100%	100%
	n =(8)	(23)
r = .272 ? = .565

Table 5-4: Hellenism and Paul’s Visits
	Very Hellenic cities	Less Hellenic cities
Missionized by Paul	42%	0%
Not missionized by Paul	58%	100%
	100%	100%
	n =(19)	(12)
r = .469** ? = 1.000**

Table 5-5: Ports and Paul’s Missions
	Port cities	Inland cities
Missionized by Paul	43%	12%
Not missionized by Paul	57%	88%
	100%	100%
	n =(14)	(17)
r = .354* ? = .678*

Table 5-6: The Diaspora and Paul’s Missions
	Had a significant Jewish community	No significant Jewish community
Missionized by Paul	67%	9%
Not missionized by Paul	33%	91%
	100%	100%
	n = (9)	(22)
r = .597** ? = .905**

Table 5-7: Paul and Christianization
	Missionized by Paul	Not missionized by Paul
Had a church by 100 CE	100%	23%
Had a church by 180 CE	0%	41%
No church by 180 CE	0%	36%
	100%	100%
	n =(9)	(22)
r = 577** V = .614** ? = .933**

Table 5-8: The Diaspora and Christianization
	Had a significant Jewish community	Had no significant Jewish community
Had a church by 100 CE	100%	18%
Had a church by 180 CE	0%	46%
No church by 180 CE	0%	36%
	100%	100%
	n =(9)	(22)
r = .665** V = .753** ? = 1.000**

Table 6-1: Ports and Heretical Schools
	Port cities	Inland cities
Had a heretical school	29%	24%
No heretical school	71%	76%
	100%	100%
	n =(14)	(17)
r = .057 ? = .130

Table 6-2: City-Size and Heretical Schools
	Larger cities	Smaller cities
Had a heretical school	63%	13%
No heretical school	37%	87%
	100%	100%
	n =(8)	(23)
r = .495** ? = .835**

Table 6-3: Diasporan Communities and Heretical Schools
	Significant Jewish community	No significant Jewish community
Had a heretical school	33%	23%
No heretical school	67%	77%
	100%	100%
	n =(9)	(22)
r = .110 ? = .259

Table 6-4: Christianization and Heretical Schools
	Had a church by 100 CE	Had a church by 180 CE	No church by 180 CE
Had a heretical school	23%	50%	0%
No heretical school	77%	50%	100%
	100%	100%	100%
	n =(13)	(10)	(8)
r = .156 V = .436 ? = .225

Table 6-5: Heretical Schools and Marcionite Congregations
	Had a heretical school	No heretical school
Had a Marcionite congregation	62%	38%
No Marcionite congregation	38%	62%
	100%	100%
	n =(8)	(23)
r = .205 ? = .443

Table 6-6: Diasporan Communities and Marcionism
	Had a significant Jewish community	No significant Jewish community
Had a Marcionite congregation	89%	27%
No Marcionite congregation	11%	73%
	100%	100%
	n =(9)	(22)
r = .562** ? = .910**

Table 6-7: Heretical Schools and Valentinian Congregations
	Had a heretical school	No heretical school
Had a Valentinian congregation	88%	9%
No Valentinian congregation	12%	91%
	100%	100%
	n =(8)	(23)
r = .760** ? = .973**

Table 6-8: City-Size and Valentinian Congregations
	Larger cities	Smaller cities
Had a Valentinian congregation	75%	13%
No Valentinian congregation	25%	87%
	100%	100%
	n =(8)	(23)
r = .597** ? = .905**

Table 6-9: Heretical Schools and Montanist Congregations
	Had a heretical school	No heretical school
Had a Montanist congregation	88%	65%
No Montanist congregation	12%	35%
	100%	100%
	n =(8)	(23)
r = .215 ? = .577

Table 6-10: City-Size and Montanist Congregations
	Larger cities	Smaller cities
Had a Montanist congregation	100%	61%
No Montanist congregation	0%	39%
	100%	100%
	n =(8)	(23)
r = .377* ? = 1.000**

Table 6-11: Heretical Schools and Manichaeist Congregations
	Had a heretical school	No heretical school
Had a Manichaeist congregation	88%	13%
No Manichaeist congregation	12%	87%
	100%	100%
	n =(8)	(23)
r = .697** ? = .958**

Table 6-12: City-Size and Manichaeist Congregations
	Larger cities	Smaller cities
Had a Manichaeist congregation	75%	17%
No Manichaeist congregation	25%	83%
	100%	100%
	n =(8)	(23)
r = .539** ? = .869**

Correlations 6-1
	Heretical schools (r)
Valentinians	.760**
Manichaeism	.697**
Montanism	.215
Marcionism	.205

Correlations 6-2
	Demiurgism (r)
Christianization	.186
Diaspora	.127
Hellenism	.134
Ports	.097
City-size	.611**

Table 7-1: Correlations*
	Mithraism (r)
City-size	.019
Ports	.029
Christians	.046
Hellenized	-.158
Diaspora	-.193
Isiac temples	.230
Cybelene temples	.022
Heretical schools	.167
Marcionism	-.100
Valentinianism	.092
Montanism	-.092
Manichaeism	.160
Demiurgism	.156

*The presence of Mithraism in a city is based on the work of Manfred Clauss (2000).

Table 7-2: Ports and Paganism
	Port cities	Inland cities
Substantial paganism	50%	29%
Little paganism	50%	71%
	100%	100%
	n =(14)	(17)
r = .210 ? = .412

Table 7-3: City-Size and Paganism
	Larger cities	Smaller cities
Substantial paganism	75%	26%
Little paganism	25%	74%
	100%	100%
	n =(8)	(23)
r = .439* ? = .789*

Table 7-4: Hellenism and Paganism
	Very Hellenic cities	Less Hellenic cities
Substantial paganism	58%	8%
Little paganism	42%	92%
	100%	100%
	n =(19)	(12)
r = .496** ? = .876**

Table 7-6: Heretical Schools and Paganism
	Had a heretical school	No heretical school
Substantial paganism	88%	22%
Little paganism	12%	78%
	100%	100%
	n =(8)	(23)
r = .591** ? = .924**

Table 7-6: Demiurgism and Paganism
Demiurgism:	High	Medium	None
Substantial paganism	86%	50%	17%
Little paganism	14%	50%	83%
	100%	100%	100%
	n =(7)	(6)	(18)
r = .583** ? = .821**