## Abstract

Sociologists of religion have long been interested in the interaction between religious pluralism and religious vitality. Previous empirical studies approach this theme by drawing on data of denominational participation rates across geographical units, investigating the property of association between the quantity of one minus the Herfindahl–Hirschman Index (religious pluralism), and the total religious participation rate (religious vitality). However, this association could be theoretically spurious. Taking advantage of the median’s statistical property of being less sensitive to the variations of extreme values, this study proposes to apply the median instead of the arithmetic summation of religious participation rates to measure geographical-unit-level religious vitality. This method is illustrated by analyzing the New York State census of religion 1865 and the U.S. county survey 1990.

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## Notes

- 1.
The religious economies theory adopts the market metaphor to describe the dynamics between denominations in a particular area. Each denomination is viewed to be a supplier of religious “goods” on a religious “market,” and their free competition is expected to contribute to the overall “religious” vitality, since the competition encourages the sellers (clergies) to work hard to win more market shares (recruiting more religious followers) for their own “company (church).” The HHI is an instrument used by economists to measure the extent of market concentration, so one minus the HHI reveals the extent of diversity in the religious “supplies” on a particular religious market. In particular, suppose there are three denominations with their participation rate to be respectively x

_{1}, x_{2}, and x_{3}. Then, their market shares are then respectively \(\frac{{x_{1} }}{{x_{1} + x_{2} + x_{3} }}, \frac{{x_{2} }}{{x_{1} + x_{2} + x_{3} }},\) and \(\frac{{x_{3} }}{{x_{1} + x_{2} + x_{3} }}\), and the HHI is \(\left( {\frac{{x_{1} }}{{x_{1} + x_{2} + x_{3} }}} \right)^{2} + \left( {\frac{{x_{2} }}{{x_{1} + x_{2} + x_{3} }}} \right)^{2} + \left( {\frac{{x_{3} }}{{x_{1} + x_{2} + x_{3} }}} \right)^{2}\). Total religious participation rate is the arithmetic summation of individual denomination’s participation rate, that is, \(x_{1} + x_{2} + x_{3}\). - 2.
This study focuses on the area-level investigation that is based on a cross-sectional design. If we switch to the individual-level analysis, some other strategies for handling the non-substantive problem may be available, such as the hierarchical modeling approach.

- 3.
Note that for a given number of denominations, the sum of denominational participation rates has a one-to-one correspondence with the mean of denominational participation rates.

- 4.
Note that most surveys organize data by listing all possible denominations for each area, and imputing zero for the ones with no followers as a convenient data management practice. For instance, the NYS65 that is analyzed below listed 50 denominations for each county in the New York State, but, in reality, most counties have much less than 50 nonzero denominations. In this case, most median values would be zero if we inappropriately base our computation of the median on the unrealistic 50 denominations for each county.

- 5.
However, the extent of unevenness, such as the variance of the denominational sizes, should not be fixed since it is an indispensable component of HHI (e.g., Hannan 1997). If it is controlled for, the extent of religious diversity could become a constant.

- 6.
Suppose the regression model is religious pluralism = β

_{0}+ β_{1}× the number of nonzero denominations +ε, the residual is the difference between the observed value and the model-based predicted value. The residual median is computed in the same manner. - 7.
The point estimate of the correlation coefficient differs a little from that of Voas et al. (2002) because we left out zero-pluralism cases. These cases are mostly driven by the counties with only one non-zero denomination. Since the theoretical argument in the religious economies theory concerns the consequence of the inter-denomination interaction—e.g., market competition – on the vitality of local religious market, these cases should not be considered. However, no material change in the correlation coefficient (always positive) if we preserve these cases.

- 8.
The term partial correlation is used here because both quantities have taken into account the effect of the number of non-zero denominations. Supplementary analysis returns a negative correlation between the median and the number of non-zero denominations, which means that the value of the median is partly influenced by the emergence of small denominations, affirming the necessity of controlling for the number of denominations.

- 9.
This is especially necessary in light of the relatively small effect size of the median-based test.

- 10.
Some cells of the data set matrix could be empty since the number of non-zero denominations are not the same across counties. These empty cells also enter the permutation process. It is necessary to mention that, in some iterations, the summation of denominational participation rates in the simulated data set might be over 100%. However, as discussed by Voas et al. (2002: 222 footnote 13), these cases should be preserved, because “restricting the total participation rate to no more than 100% of the population tends to make the inter-correlations among the sizes of different denominations negative, thus violating the assumptions of our null model in which the sizes of denominations are independent of one another.”.

- 11.
The permutation test differs from the null hypothesis testing because the expected value in the permutation test is not necessarily zero. In other words, if we allow denominations to randomly match with each other, the expected value of the pluralism-vitality link is not always null. In contrast, the routine null hypothesis testing always sets the center of the null distribution to be zero.

- 12.
We also test the significance of the partial correlation using parametric null hypothesis testing and it is marginally significant at the 0.1 level. Again, this test differs from the permutation test.

- 13.
It is worth mentioning that the partial correlation coefficient is statistically significant at the 0.01 level in the routine null hypothesis testing. Also, like the case of the NYS65, there is a negative association between the median and the number of denominations, so controlling for the number of denominations is called for.

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## Acknowledgement

This study was partly supported by the School of Social Development and Public Policy, Fudan University.

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## Appendix

### Appendix

In this appendix, we will be showing the process of computing the threshold denominational size. Denote \(x_{p }\) to be the threshold, we compute the first-order derivative of \(1 - \sum\nolimits_{i = 1}^{n} {\left( {\frac{{x_{i} }}{{\mathop \sum \nolimits_{i = 1}^{n} x_{i} }}} \right)^{2} }\) to find the value of \(x_{p }\). Specifically, we have

To show that religious pluralism reaches its maximum at \(x_{p} ,\) we need to compute the second-order derivative of \(f\). To facilitate computation, denote \({\text{H}} = \sum\nolimits_{i = 1}^{n} {x_{i} }\), \({\text{S}} = \sum\nolimits_{i = 1}^{n} {x_{i}^{2} }\), \({\text{h}} = H - x_{p} = \mathop \sum \limits_{i = 1,i \ne p}^{n} x_{i}\), and \({\text{s}} = S - x_{p}^{2} = \mathop \sum \limits_{i = 1,i \ne p}^{n} x_{i}^{2}\). By definition, both h and s are positive. Then, we have

Since \(x_{p} = \frac{s}{h}, f_{{x_{p} }}^{''} = - \frac{{2s + 2h^{2} }}{{H^{4} }} < 0\), so \(x_{p}\) corresponds to the value of denominational participation rate threshold that maximizes religious pluralism of a given geographical unit.

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Hu, A. Religious pluralism and religious participation: a median-based approach to the non-substantive problem.
*Qual Quant* **52, **969–982 (2018). https://doi.org/10.1007/s11135-017-0498-3

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### Keywords

- Religious pluralism
- Religious prevalence
- Non-substantive connection
- Religious economies theory
- Median-based test